Bioelectric Source Localization in Peripheral Nerves...Abstract Bioelectric Source Localization in...

Bioelectric Source Localization in PeripheralNerves

by

Jose Zariffa

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Edward S. Rogers Sr. Department of Electrical and Computer Engineering

and

Institute of Biomaterials and Biomedical Engineering

University of Toronto

c©Copyright by Jose Zariffa 2009

Abstract

Bioelectric Source Localization in Peripheral Nerves

Jose Zariffa

Doctor of Philosophy

Graduate Departments of

Edward S. Rogers Sr. Department of Electrical and Computer Engineering

and

Institute of Biomaterials and Biomedical Engineering

University of Toronto

2009

Currently there does not exist a type of peripheral nerve interface that adequately

combines spatial selectivity, spatial coverage and low invasiveness. In order to address this

lack, we investigated the application of bioelectric source localization algorithms, adapted

from electroencephalography/magnetoencephalography, to recordings from a 56-contact “ma-

trix” nerve cuff electrode. If successful, this strategy would enable us to improve current

neuroprostheses and conduct more detailed investigations of neural control systems. Using

forward field similarities, we first developed a method to reduce the number of unnecessary

variables in the inverse problem, and in doing so obtained an upper bound on the spatial

resolution. Next, a simulation study of the peripheral nerve source localization problem

revealed that the method is unlikely to work unless noise is very low and a very accurate

model of the nerve is available. Under more realistic conditions, the method had localization

errors in the 140 µm-180 µm range, high numbers of spurious pathways, and low resolution.

On the other hand, the simulations also showed that imposing physiologically meaningful

constraints on the solution can reduce the number of spurious pathways. Both the influence

of the constraints and the importance of the model accuracy were validated experimentally

using recordings from rat sciatic nerves. Unfortunately, neither idealized models nor models

based on nerve sample cross-sections were sufficiently accurate to allow reliable identification

ii

of the branches stimulated during the experiments. To overcome this problem, an experi-

mental leadfield was constructed using training data, thereby eliminating the dependence on

anatomical models. This new strategy was successful in identifying single-branch cases, but

not multi-branches ones. Lastly, an examination of the information contained in the ma-

trix cuff recordings was performed in comparison to a single-ring configuration of contacts.

The matrix cuff was able to achieve better fascicle discrimination due to its ability to select

among the most informative locations around the nerve. These findings suggest that nerve

cuff-based neuroprosthetic applications would benefit from implanting devices with a large

number of contacts, then performing a contact selection procedure. Conditions that must

be met before source localization approaches can be applied in practice to peripheral nerves

were also discussed.

iii

Acknowledgements

This thesis is the reflection of four years of work, during which I have learned many

valuable lessons about conducting research, lessons that cannot be reduced to any of the

chapters, figures, or references in this document. In this respect, I could not have hoped for

a better supervisor than Dr. Milos Popovic. He gave me tremendous freedom and autonomy

in my research, yet at the same time always came through when I needed help and support.

He taught me to see past my thesis project, to understand the bigger context, and never to

forget the people for whom we carry out biomedical research, and for these things I will be

thanking him for many years to come.

Dr. Popovic may have made the last four years interesting, educational, productive,

and occasionally very funny, but it was all the members of the Rehabilitation Engineering

Laboratory who made it so easy to go to work every day. To have such a large group of

people who are all, without exception, kind, smart, funny, and helpful, is surely some sort

of major statistical anomaly, and I’m immensely grateful to all of them for making my PhD

fun.

This work was not conducted in isolation, but was made possible by the support and

collaboration of a large group of people. First and foremost, Dr. Mary Nagai, whose help

with the experiments was invaluable. I owe many thanks as well to our collaborators at

the University of Freiburg in Germany, Drs. Thomas Stieglitz and Martin Schuettler, who

provided us with the electrodes that made the study possible, and to our collaborators at the

Center for Addiction and Mental Health in Toronto, in particular Dr. Jeff Daskalakis and

Lori Dixon, who provided us with the resources and space necessary for our experiments.

Finally, the members of my thesis committee, Drs. Berj Bardakjian and Adrian Nachman,

gave me a great deal of very sound advice, kept me on the right path, and were always

encouraging.

Academia, of course, is only half the story. My studies, and for that matter everything

iv

else I do, would be impossible without my wife Vanessa, whose love and support have been

unflinching since the moment we set eyes on each other. My parents, likewise, have seemingly

endless confidence in me, and for this selective memory I thank them with all my heart.

And last but, as they say, not least, I wish to thank all of my friends and family who have

supported and encouraged me all this years, even when they had no idea what my research

was about.

v

List of Abbreviations

Action potential AP

Akaike’s Bayesian information criterion ABIC

Boundary element method BEM

Central nervous system CNS

Compound action potential CAP

Cross-validation error CVE

Electrical analysis ELECTRA

Electrical impedance tomography EIT

Electroencephalogram EEG

Finite element FE

Finite element method FEM

Flat interface nerve electrode FINE

Focal underdetermined system solver FOCUSS

Functional magnetic resonance imaging fMRI

Generalized cross-validation GCV

Hematoxylin and eosin H&E

Linear programming LP

Linearly constrained minimum variance LCMV

Local auto-regressive averages LAURA

Longitudinal intrafascicular electrode LIFE

Low resolution brain electromagnetic tomography LORETA

lp norm iterative sparse solution LPISS

Magnetic resonance imaging MRI

Magnetoencephalogram MEG

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Micro-electrode array MEA

Minimum current estimate MCE

Multi-contact cuff MCC

Multiple signal classification MUSIC

Noise-to-signal ratio NSR

Positron emission tomography PET

Peripheral nervous system PNS

Restricted maximum likelihood ReML

Slowly penetrating interfascicular nerve electrode SPINE

Standardized LORETA sLORETA

Truncated singular-value decomposition TSVD

vii

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Modeling phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.2 Source localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.3 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Organization of the Document . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Literature Review 10

2.1 Origin of Bioelectric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Bioelectromagnetic Source Localization . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 The forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Equivalent dipole methods . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3 Distributed linear methods . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4 The dynamic source localization problem . . . . . . . . . . . . . . . . 33

2.2.5 Choice of regularization parameter . . . . . . . . . . . . . . . . . . . 36

2.3 Peripheral Nerve Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.1 Electrode types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.2 Signal processing techniques for extraneural measurements . . . . . . 43

viii

2.3.3 Source localization and pathway discrimination using extraneural mea-

surements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Reduction of the Inverse Problem Solution Space 48

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.1 Finite element model and leadfield construction . . . . . . . . . . . . 52

3.3.2 Element grouping algorithm . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.3 Leadfield comparison metrics . . . . . . . . . . . . . . . . . . . . . . 57

3.3.4 Example and Complexity Analysis . . . . . . . . . . . . . . . . . . . 59

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.1 Size of the reduced leadfield . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.2 Properties of reduced leadfield . . . . . . . . . . . . . . . . . . . . . . 63

3.4.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Simulation of Bioelectric Source Localization in the Rat Sciatic Nerve 70

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.1 Approach to solving the source localization problem . . . . . . . . . . 70

4.2.2 Evaluation of the results . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.1 One-pathway case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.2 Three-pathways case . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Validation of the Source Localization Approach on Physiological Data 91

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

ix

5.2.1 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2.2 Construction of nerve-specific leadfields . . . . . . . . . . . . . . . . . 98

5.2.3 Evaluation of the source localization performance . . . . . . . . . . . 100

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3.1 Using the idealized leadfield without the spatio-temporal constraint . 103

5.3.2 Using the idealized leadfield with the spatio-temporal constraint . . . 104

5.3.3 Using the nerve-specific leadfield without the spatio-temporal constraint107

5.3.4 Using the nerve-specific leadfield with the spatio-temporal constraint 110

5.3.5 Influence of the constraints on the number of peaks in the estimate . 110

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6 Source Localization Using an Experimentally-Derived Leadfield 120

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2.1 Construction of the experimental leadfield . . . . . . . . . . . . . . . 121

6.2.2 Identification of fascicle combinations . . . . . . . . . . . . . . . . . . 123

6.2.3 Evaluation of the results . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7 Influence of the Number and Location of Recording Contacts on the Se-

lectivity of a Nerve Cuff Electrode 133

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.2.1 Evaluation of the classification success rate . . . . . . . . . . . . . . . 134

7.2.2 Evaluation of the influence of the stimulation artefact . . . . . . . . . 137

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.3.1 Comparison of the matrix and single-ring configurations . . . . . . . 139

7.3.2 Influence of the stimulation artefact . . . . . . . . . . . . . . . . . . . 141

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7.3.3 Layout of the most informative contacts . . . . . . . . . . . . . . . . 144

7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8 Discussion 150

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8.2 Comparison of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2.1 Resolution achievable . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2.2 Validation of the simulations . . . . . . . . . . . . . . . . . . . . . . . 156

8.2.3 Implication of the contact configuration study on the source localiza-

tion results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.3 Limitations of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3.1 Experimental issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3.2 Use of CAPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.3.3 Use of simplified FE models . . . . . . . . . . . . . . . . . . . . . . . 159

8.3.4 Use of a low-resolution source localization algorithm . . . . . . . . . . 160

8.3.5 Use of peaks in the estimate as a measure of the number of pathways 160

8.3.6 Focus on spatial over temporal resolution . . . . . . . . . . . . . . . . 161

8.3.7 Focus on a specific electrode and nerve . . . . . . . . . . . . . . . . . 161

8.4 Optimal number of contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.5 Number of pathways to be localized . . . . . . . . . . . . . . . . . . . . . . . 164

8.6 Factors related to the use of a nerve cuff electrode . . . . . . . . . . . . . . . 166

8.7 Implications for future cuff-based peripheral nerve interfaces . . . . . . . . . 168

9 Conclusions 172

Bibliography 175

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List of Figures

1.1 56-contact nerve cuff electrode. . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Schematic overview of bioelectric source localization in peripheral nerves. . . 6

1.3 Anatomy of a peripheral nerve. . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Two implementations of the tripole configuration. . . . . . . . . . . . . . . . 41

2.2 Types of extraneural electrodes . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 Types of intraneural electrodes . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1 Mesh element grouping example. . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Number of columns of the reduced leadfield. . . . . . . . . . . . . . . . . . . 62

3.3 2D projections of the true source distribution and the estimates obtained with

the original and reduced leadfields. . . . . . . . . . . . . . . . . . . . . . . . 65

4.1 FEM geometry cross-sections. . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Source localization metrics for the one-pathway case. . . . . . . . . . . . . . 82

4.3 Localization trial examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Source localization metrics for the three-pathways case. . . . . . . . . . . . . 86

5.1 Placement of the cuff electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 Connector linking the cuff electrode to the amplifier. . . . . . . . . . . . . . 96

5.3 Example of a trial before and after conversion to a common-average reference. 97

5.4 H&E stained sciatic nerve cross-sections. . . . . . . . . . . . . . . . . . . . . 99

5.5 Cross-sections of FE meshes based on nerve samples. . . . . . . . . . . . . . 101

xii

5.6 Summary of the source localization performance (idealized leadfield without

spatio-temporal constraint). . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.7 Summary of the source localization performance (idealized leadfield with spatio-

temporal constraint). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.8 Summary of the source localization performance (nerve-specific leadfield with-

out spatio-temporal constraint). . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.9 Summary of the source localization performance (nerve-specific leadfield with

spatio-temporal constraint). . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.10 Mean number of peaks in the estimates. . . . . . . . . . . . . . . . . . . . . 113

5.11 Spurious and missed pathways in the estimates. . . . . . . . . . . . . . . . . 119

6.1 Means of activity estimates for each branch. . . . . . . . . . . . . . . . . . . 126

6.2 Success rates for identifying the exact combination of active branches (thresh-

old = 0.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3 Success rates for identifying the exact combination of active branches (thresh-

old = 0.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.1 Contact configurations for matrix and single-ring cases. . . . . . . . . . . . . 135

7.2 Example of a trial before and after conversion to a tripole reference. . . . . . 138

7.3 Maximum classification success rate for each configuration. . . . . . . . . . . 140

7.4 Classification success rate using the first 8 contacts of each configuration. . . 141

7.5 Classification success rate as a function of the number of contacts. . . . . . . 142

7.6 Order in which contacts were added. . . . . . . . . . . . . . . . . . . . . . . 143

7.7 Comparison of artefact and success rate variations between contact rings in

Rat 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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List of Tables

2.1 Commonly used weight matrices for weighted minimum norm methods. . . . 26

3.1 Parameters for the finite element model of the rat sciatic nerve. . . . . . . . 53

3.2 Metric Met1 for all test cases. . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 Simulation results using the original and reduced leadfields. . . . . . . . . . . 66

3.4 Computation time comparison for simulations using the original and reduced

leadfields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Parameters for the idealized finite element model of the rat sciatic nerve. . . 74

5.1 Cuff diameters in nerve-specific rat models. . . . . . . . . . . . . . . . . . . . 100

5.2 Influence of constraints on the number of peaks. . . . . . . . . . . . . . . . . 114

7.1 Correlation of the artefact and classification success rate variations between

contact rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

xiv

Chapter 1

Introduction

1.1 Motivation

The human nervous system is responsible for controlling a staggering number of physiological

functions, both autonomic and somatic. Before any medical intervention related to neural

mechanisms can take place, it is crucial to have an understanding of the corresponding

control systems, and this has made the nervous system one of the most active topics of

research in the last century. There are two major obstacles that make deciphering the inner

workings of the nervous system so difficult: its sheer scale, and the technology available to

us. With over 100 billion neurons in the human brain alone [69], identifying the specific

networks corresponding to given tasks is enormously difficult. To compound this problem,

our ability to examine the activity of a specific neuron or group of neurons is limited by

technological considerations.

When monitoring neural activity in the brain, existing methods rely either on non-

invasively detecting changes in blood oxygen levels (usually using functional magnetic res-

onance imaging (fMRI)), or on recording the bioelectric or biomagnetic fields generated

by neural activity. The latter can be recorded non-invasively using electroencephalography

(EEG) or magnetoencephalography (MEG). Alternatively, invasive measurements of bio-

electric fields with different types of electrodes can be used to obtain electrocorticographic

1

Introduction 2

(ECoG) recordings from the surface of the cortex, local-field potentials (LFPs) from small

groups of neurons, or even single-neuron recordings. Methods based on field recordings have

much higher temporal resolution than fMRI, but none of them can equal the latter for high

resolution images of the entire brain. More recently, source localization approaches have been

applied to EEG or MEG recordings in order to try to obtain whole-brain information with

high temporal resolution, but spatial resolution of these methods still remains substantially

inferior to that of fMRI.

In peripheral nerves, monitoring techniques are currently restricted to recording bio-

electric fields, either from within the nerve (intraneural potentials) or right outside of it

(extraneural potentials). These two approaches represent a trade-off between resolution and

invasiveness. Micro-electrode arrays (MEAs) provide a large number of measurements from

specific locations within the nerve, but are more likely to damage the nerve and may be too

big to be used on smaller nerves. Longitudinal intrafascicular electrodes (LIFEs) are equally

selective and less damaging, but can only monitor a few nerve fibers. Nerve cuff electrodes,

on the other hand, use only extraneural measurements, making them less invasive and likely

to damage the nerve, but also less selective.

Although the neural control of physiological systems is carried out by networks of

neurons in the central nervous system (CNS), the difficulty of identifying the neurons that

are involved in a specific task creates the need for additional sources of information. Con-

sequently, using peripheral nerve recordings to monitor the information being conveyed to

and from the CNS is a complementary and attractive approach for helping to decipher

these control systems. Indeed, the “inputs” and “outputs” of the CNS can provide valuable

information about the control mechanisms being used, and task-specific signals are more

easily discriminated because each peripheral nerve innervates a known region of the body.

Nonetheless, the selectivity of peripheral nerve recordings is still too limited for this strategy

to reach its full potential. This is particularly true in the context of applications in humans,

where the higher selectivity of MEAs cannot be fully taken advantage of because of the risk

of nerve damage associated with those devices. Cuff electrodes, on the other hand, have

Introduction 3

been chronically implanted in humans for up to 12 years with no adverse effects [192].

1.2 Objective

In light of the situation described in the previous paragraph, the goal of the study presented

in this thesis was to improve the amount of information that can be obtained from extra-

neural recordings of peripheral nerve activity. More specifically, the aim was to develop a

system that improves the spatial resolution of traditional cuff electrode recordings, such that

bioelectric fields measured can be associated with specific pathways or groups of pathways

within the nerve.

1.3 Applications

The information obtained from higher resolution peripheral nerve recordings could be used

as follows:

1. Attributing recorded afferent and efferent activity to more specific neural pathways,

allowing us to better understand how the CNS controls the activity of a limb or organ

during complex tasks (for example reaching [159]).

2. Using the greater selectivity to improve the performance and capabilities of neuro-

prosthetic systems, which are defined as artificial systems that interact directly with a

damaged nervous system in order to replace or enhance its function.

The second point is broad. Some existing neuroprosthetic systems use nerve cuff

recordings of afferent signals to provide feedback to a stimulation device (for example systems

correcting for foot-drop [55, 75, 51, 52] or assisting grasping [64, 65]; other applications

under investigation in humans include quiet standing [198, 102, 12] and bladder control

[78]). The feedback provided can represent either a discrete event (e.g. heel strike) or a

Introduction 4

continuous quantity (e.g. a joint angle). Discriminating between the activity of different

sensory pathways would allow for more sophisticated control of the stimulation.

A more ambitious application, and one requiring real-time pathway discrimination

with good resolution, would be to convert efferent neural control signals into commands for

a prosthetic limb after amputation, thereby creating a direct neural interface ([191, 27, 26]).

With sufficient resolution, a “natural” interface could be created, in which the signals of

various pathways are used for their original purpose, enabling the user to control the artificial

limb as he or she would a real limb. With lower resolution, “natural” control may not be

possible, but the activation of different fascicles could be used as part of multiple-switch type

of control scheme.

1.4 Approach

The approach proposed to attain our objective consisted of two main components. First,

instead of a traditional nerve cuff electrode (which typically contains three contacts and

produces a single signal), a multi-contact cuff (MCC) was used (here with 56 contacts),

providing a much larger number of measurements obtained from all around the nerve. This

electrode is shown in Figure 1.1. Second, in order to best deal with this larger amount of

data, the task of locating the bioelectric sources within the nerve was treated as an inverse

problem of source localization. This is in contrast with the signal processing techniques

traditionally used to interpret nerve cuff recordings. A schematic overview of the approach is

shown in Figure 1.2. The design and fabrication of the MCC was performed by collaborators

at the Laboratory for Biomedical Microtechnology at the University of Freiburg, Germany.

Therefore, this thesis deals mainly with the mathematical treatment of the inverse problem,

and with the experimental validation of the system.

Introduction 5

Figure 1.1: The 56-contact MCC used in this study.

1.4.1 Modeling phase

The source localization process requires us to have a model describing the relationship be-

tween bioelectric sources at different locations in the nerve and the MCC measurements. In

addition, such a model makes it possible to generate simulated measurements and therefore

to investigate source localization approaches before experimental data is collected. The first

step of this project was therefore to construct a model of a nerve. The anatomy of a typical

nerve is shown in Figure 1.3. Individual fibers are grouped into bundles known as fascicles,

and the membrane delimiting each fascicle is the perineurium. The fascicles are in turn en-

cased in a tissue known as the epineurium to form the nerve trunk. The inside of the fascicles

is known as the endoneurium. Finite-element analysis can be used to model the electric fields

produced by electric sources in regions of complex geometry and inhomogeneous, anisotropic

conductivity. It is used here to create a model that reflects the nerve geometry, the different

Introduction 6

Figure 1.2: Schematic overview of our approach to localizing active pathways in peripheral

nerves: an MCC is used to record the electric fields generated by the neural activity at various

locations around the nerve. These recordings are used as the input to a source localization

algorithm, which produces a three-dimensional estimate of the bioelectric activity in the nerve.

conductivities of the endoneurium, perineurium, and epineurium, the shape and properties

of the MCC placed around the nerve, and those of the tissues and fluids around and between

the nerve and the electrode.

1.4.2 Source localization

Using the nerve model, simulation studies can be conducted to investigate different algo-

rithms for solving the inverse problem of source localization. The mathematical treatment

of this problem can be directly based on another bioelectric source localization problem, elec-

troencephalogram (EEG) source localization [104]. The goal in that context is to determine

Introduction 7

Figure 1.3: Anatomy of a peripheral nerve (from [199])

the distribution of brain activity that is responsible for producing a set of measurements

observed on the scalp. The EEG and peripheral nerve source localization problems are

similar in that they both involve the same underlying biophysics governing the relationship

between the electrical activity of neurons and extracellular potentials. In both cases, any

given set of measurements can be produced by an infinite number of source configurations,

such that the problem lacks a unique solution (this was shown by Helmholtz in 1853 [58]; a

translation of the relevant section of his often-cited paper is available in [59]). Small changes

in the measurements (i.e. noise) can also have a large effect on the solution, making the

problem unstable. A problem with these characteristics fails to meet Hadamard’s definition

of a well-posed problem (namely, that a solution exists, is unique, and depends continuously

on the data [47]). The bioelectric source localization problem therefore belongs to the class

of ill-posed inverse problems.

The most obvious difference between the EEG and peripheral nerve source localiza-

tion problems is the geometry of the region containing the sources, which is why the model

described in the previous section is necessary. In addition, solving this type of ill-posed

problem requires the imposition of constraints onto the solution to overcome the underde-

Introduction 8

termined nature of the problem. These constraints should ideally be tailored to the problem

and accurately reflect available information (e.g. biophysical, anatomical or electrophysio-

logical characteristics). For that reason, it is to be expected that not all the constraints that

are applicable to the EEG source localization problem are applicable to the peripheral nerve

problem, and vice versa. Part of the process of finding an appropriate algorithm for the

peripheral nerve source localization problem is therefore to select useful and well-justified

constraints.

1.4.3 Experimental validation

After having selected a source localization algorithm on the basis of simulated performance,

the last step of the project was to test the complete system (MCC and algorithm) using

physiological measurements. For this purpose, MCC recordings were obtained from rat

sciatic nerves during a series of in vivo experiments. By stimulating specific pathways, the

output of the source localization system can be evaluated by comparing it to the known

positions of the stimulated pathways. The experiments can also be used to validate the

influence of different constraints on the performance, as predicted by the simulation results.

1.5 Organization of the Document

The work presented in this thesis is divided into several individual studies, each of which

stands independently but flows logically from the previous ones. Each of these studies

constitutes a separate chapter, with its own introduction, method, results, and discussion

sections. The remainder of this document is therefore organized as follows. Citations in

parentheses refer to publications by the author that relate to each chapter.

Chapter 2 provides an overview of the relevant literature on bioelectric fields, source

localization, and peripheral nerve interfacing.

Chapter 3 describes a method for reducing the number of variables in the inverse

Introduction 9

problem ([203, 206, 205]).

Chapter 4 provides a simulation study of the peripheral nerve source localization

problem ([204, 207]).

Chapter 5 describes the experimental component of this work, as well as the per-

formance of the source localization approach on experimental data, whereas Chapter 6 is

concerned with an alternative approach to the source localization problem ([200, 201]).

Chapter 7 uses the 56-contact cuff used in this work to investigate the influence of the

number and location of the recording contacts on the spatial selectivity of the cuff ([202]).

Chapter 8 provides a discussion of the results, and the main conclusions are summa-

rized in Chapter 9.

Chapter 2

Literature Review

2.1 Origin of Bioelectric Fields

Peripheral nerves consist of axons linking the body’s sensors and effectors with the more

proximal components of the nervous system. Specifically, these axons can be those of auto-

nomic, dorsal root ganglion, or ventral horn neurons [62]. The axons and their associated

Schwann cells are encased in a collagen matrix, the endoneurium. Nerve fibers are grouped

into fascicles according to a topographical organization, and the endoneurium of each fas-

cicle is surrounded by the perineurium, which acts as a blood-PNS barrier. The fascicles

are then held together by the epineurium, a collageneous tissue that provides most of the

overall nerve’s structure and strength. The axons within a peripheral nerve can be either

myelinated or not, and exhibit a range of diameters. The combination of these characteris-

tics produces conduction velocities ranging from approximately 0.4 to 120 m/s. Myelinated

axons are faster and tend to be used in tasks when speed is important, such as the control

of movement or the perception of rapidly changing stimuli. Conversely, unmyelinated axons

are generally involved in the control of smooth muscle and the perception of slower events

such as pain and temperature changes. The nomenclature of axon fibers is based both on

the size and speed of the fibers and their function. Cutaneous fibers are refered to as, in

10

Literature Review 11

order of decreasing size, Aα, Aβ, Aδ, B, and C. Muscle sensory nerves are classified as Ia,

Ib, II, III, and IV, whereas muscle motor nerves are known as α, β, and γ, again in order of

decreasing size.

In all cases, information travels along nerves in the form of action potentials (APs),

which are temporary changes in a neuron’s transmembrane voltage. When an AP occurs at

a given point along a neuron’s axon, it results in charge redistributions that cause another

AP to be initiated further down the axon. This process repeats itself, allowing the AP to

“travel” down the axon. The frequency of the APs is responsible for encoding information, for

example the intensity of a stimulus [130]. The mechanism underlying the generation of an AP

is the opening of ion channels in the cell membrane, allowing ions to temporarily redistribute

themselves and alter the transmembrane voltage. This movement of ions produces small

electric currents across and along the membrane. Because the extracellular medium is a

volume conductor, these currents generate electric fields in the vicinity of the cell [92].

Therefore, it is possible to obtain information about the activity of the cell by recording

potentials in the extracellular medium, a small distance away from the cell. Because electric

fields add linearly, the activity of a population of cells can also be detected from further

away, yielding for example electroencephalographic (EEG) recordings, which are reflections

of brain activity obtained from the scalp.

The currents arising from ion movements during APs are nonconservative currents

resulting from the conversion of chemical energy to electrical energy [92], and are known

as impressed currents. This electrical activity imposes an electric field in the surrounding

volume conductor, and this field in turn creates a conduction current, known as the return

current, which prevents the impressed current from resulting in a charge buildup. Denoting Ji

the impressed current density, E the electric field, and σ the conductivity of the extracellular

medium, the total current density J can be expressed as shown in Equation 2.1 [92].

J = Ji + σE (2.1)


Note that in practice the conductivity is not constant throughout the medium and

in some regions is a tensor rather than a scalar (i.e. the medium is anisotropic). Because

of the frequency band of the bioelectric signals under consideration and the properties of

the medium, it has been shown that these bioelectric phenomena can be examined under

quasistatic conditions [131]. In other words, at any given instant, the fields can be considered

stationary and any time-dependent effects can be ignored. As a consequence, the electric

field can be expressed as the negative of the gradient of the potential field Φ, such that

Equation 2.1 becomes Equation 2.2.

J = Ji − σ∇Φ (2.2)

According to Maxwell’s equations under quasistatic conditions, the divergence of the

total current density J must be 0 (Ampere’s law, using the fact that the divergence of a curl

is 0). Taking the divergence of Equation 2.2 therefore leads to Equation 2.3.

∇.(σ∇Φ) = ∇.Ji (2.3)

This is known as Poisson’s equation, which governs the potential distribution in the

extracellular medium around active nerves. While analytical solutions to Poisson’s equation

are available and useful for simple cases (for example, sources in an infinite homogenous

medium), they quickly become unwieldy for more complex situations (a more complete

mathematical treatment of this issue can be found in [92]). In more realistic scenarios, the

equation is usually solved using numerical techniques (see Section 2.2.1 for more details).

2.2 Bioelectromagnetic Source Localization

Bioelectromagnetic source localization problems are a type of inverse problems aiming to

determine the location of biological current sources within a volume, based on electric or

magnetic potential measurements obtained from the surface of that volume. Until now, the


main applications of these problems were to determine the location of brain activity using

EEG and/or magnetoencephalographic (MEG) measurements, or to study the distribution

of electrical activity on the heart using electrocardiogram (ECG) measurements. The very

high temporal resolution of the source localization approach makes it an attractive method

in these contexts. Although the problem discussed in this thesis is to localize sources within

a peripheral nerves, the algorithms used for EEG/MEG source localization remain valid.

Before the inverse problem can be solved, it is necessary to be able to solve the

forward problem, that is to say to compute the measurements that would result from a known

source distribution. Once this is achieved, the inverse problems algorithms are divided into

two broad classes: equivalent dipole models and distributed linear models. The underlying

assumption of equivalent dipole models is that the surface measurements can be modeled

using a small number of current sources. The number of sources is chosen in advance,

and an optimization problem is solved to determine which source locations, magnitudes

and orientations will produce measurements most similar to the recorded ones. Distributed

linear models, on the other hand, do not make any assumptions about the number of sources.

Rather, the activity is estimated at a large number of fixed points forming a 3D grid in the

volume. This approach corresponds to estimating the direction and magnitude of a large

number of dipoles, each of which has a fixed location. Because the number of points in

the solution grid is typically much greater than the number of measurements, the problem is

severely underdetermined. In other words, any set of surface measurements can be explained

by infinitely many source configurations. To overcome this problem, various constraints can

be imposed on the solution. These constraints can be purely mathematical in nature, or

can be based on anatomical or physiological information about the problem. Additionally,

the sequential entries in a measurement time series can be viewed as independent, or as

belonging to a system evolving in time. In the latter case, temporal information can be used

to constrain or guide the solution. In all distributed systems, the ill-posed nature of the

problem creates the need for regularization, which can take various forms corresponding to

the different algorithms. In all of those cases, however, the choice of regularization parameter


has a considerable impact on the solution, and therefore a number of methods exist to help

guide this choice.

2.2.1 The forward problem

Both equivalent dipole methods and distributed linear methods rely on the availability of a

model that allows us to compute the measurements resulting from a given source configu-

ration. This computation is known as solving the forward problem [48]. It is a well-defined

problem, and can be solved analytically if the geometry and properties of the volume con-

taining the sources are sufficiently simple [109]. On the other hand, accuracy in the forward

problem is important for the quality of the inverse solution, so models with realistic geome-

tries are often preferable to simplified ones [50, 196, 100, 19]. When the geometry becomes

too complex for analytical solutions to be convenient, the forward problem is instead solved

with the help of numerical methods, such as boundary element or finite element methods

(BEM and FEM, respectively). BEM only discretizes the boundaries between regions of dif-

ferent conductivities, rather than the whole three-dimensional volume [104, 23]. In contrast,

FEM must discretize the whole volume into a three-dimensional mesh. The advantage of

the BEM approach is that possible source locations are not constrained by the shape of a

mesh, as they are in FEM. They can be left unconstrained, which is useful for equivalent

dipole methods, or organized into a regular grid of arbitrary coarseness, which is useful for

distributed linear methods. BEM is the method most commonly used in EEG/MEG source

localization studies (e.g. [109, 163, 194]). On the other hand, the method is not as well suited

as FEM for dealing with volumes that contain regions of anisotropic conductivity, and FEM

is therefore the method of choice in those cases [10, 193, 5, 160]. The operator mapping

electrical activity at any location in the region of interest to the measurements is known as

the leadfield, and it is computed by solving the forward problem. Different quantities can

be chosen to reflect this electrical activity, and it follows that different source models can be

used when solving the forward problem.


The finite element method

The distribution of bioelectric potentials in a conducting volume is governed by Poisson’s

equation, as given by Equation 2.3 of Section 2.1, and subject to Dirichlet or Neumann

boundary conditions, or both. Although this equation has an analytical solution in the case of

a homogeneous region of isotropic conductivity, the analytical approach becomes intractable

when the volume is made up of several regions of different geometries and conductivities,

creating the need for a numerical method such as FEM. The first step in the finite element

(FE) solution is to define an approximation Φ to the potential field Φ in terms of a set of

basis functions, as shown in Equation 2.4 [105].

Φ =n∑

a=1

φaNa (2.4)

The set of piecewise-smooth functions Na spans an n-dimensional space. The criterion used

to select the best such approximation is the minimization of a weighted residual integrated

over the volume Ω, leading to the problem in Equation 2.5.

∫

Ω

waR dΩ = 0, a = 1, 2, ..., n (2.5)

where R = ∇.(σ∇Φ) + Iv is the residual, Iv is the current source density, and wa are the

weighting functions. Using the divergence theorem and substituting the expression for R

into equation 2.5, we obtain Equation 2.6

∫

Ω

∇wa.σ∇Φ dΩ =

∫

Ω

waIv dΩ +

∮

Γ

wa[n.σ∇Φ] dΓ (2.6)

Here, Γ is the boundary of Ω and n is the normal component of the current density.

The weighting functions wa are usually selected to be equal to the functions Na, and

this choice is known as the Galerkin method. By substituting the definition from Equation

2.4 into Equation 2.6, we finally obtain Equation 2.7


n∑

b=1

3∑

i,j=1

[∫

Ω

∂Na

∂xi

σij

∂Nb

∂xj

dΩ

]

φb =

∫

Ω

NaIv dΩ +

∮

Γ

Na[n.σ∇Φ] dΓ (2.7)

x1, x2, and x3 are the three Cartesian coordinates. Note the presence of both Na and Nb.

Equation 2.7 is established for each value of a, ranging from 1 to n. Within each of these

equations, all other weighting functions appear because of Equation 2.4, hence the index b,

which also ranges from 1 to n. Using the boundary conditions and appropriately defined

weighting functions, Equation 2.7 can be written as a linear system where the only unknowns

are the coefficients φ1...n (the vector φ in Equation 2.8, see [105] for details).

Kφ = F (2.8)

The matrix K is know as the stiffness matrix and F is the force vector, these terms

having their origin in structural engineering. K contains the information about the geometry

and conductivities of the volume, whereas F reflects the boundary conditions and sources

within the volume. Once the coefficients have been computed, the final solution can be

recovered using Equation 2.4.

The remaining component of the process is to partition the volume into a mesh

composed of geometrical elements such as tetrahedra or wedges. The key to making the

method efficient is to define each basis function Na so that it is non-zero at one mesh

node and zero at all the other nodes. Between the non-zero node and its neighbours, the

function is generally a polynomial. With a definition of this type, the system in Equation

2.8 become sparse and can be readily solved. By constructing the mesh out of irregularly

shaped geometrical elements, the method is able to deal with complex volume geometries.

Furthermore, the conductivity tensor is defined separately for each mesh element, making it

possible to incorporate regions of different conductivities as well as anisotropic conductivities.


The boundary element method

The starting point of BEM is to segment the inhomogeneous region into sub-regions of

homogeneous conductivity. The effect of sources on the potential at the outermost surface is

then in effect computed by replacing sources within each region by equivalent distributions

of sources on the boundary between that region and the one containing it. The potentials

on all the surfaces are governed by Equation 2.9 [50, 99, 109].

σ0Φ∞(r) =(σ−

j + σ+j )

2Φ(r) +

1

4π

m∑

i=1

(σ−i − σ+

i ).

∫

Γi

Φ(r’)ni(r’).d/d3dr’ (2.9)

where Φ∞ is the potential field that the source distribution would produce in an infinite

homogeneous region of unit conductivity σ0 (this can be computed analytically), d = r - r’

(with magnitude d) is the distance between the observation point r and the source point r’,

Γi is the boundary between the ith and i+ 1th regions, m is the total number of boundaries

between regions, r is a location on Γj, and σ−j and σ+

j are the conductivities inside and

outside the surface Γj, respectively. Equation 2.9 can be written more simply by noting that

the right-hand side of the equation is a linear operator on Φ(r), leading to the formulation

L(Φ(r)) = Φ∞(r) [109]. Like in the FEM, the problem can then be cast as the minimization

of weighted residuals integrated over all the boundaries (Equation 2.10).

∫

(L(Φ(r)) − Φ∞(r))w(r)dr = 0 (2.10)

Then, the solution Φ and the weighting function w(r) are expressed as expansions in terms

of (different) basis functions (Equations 2.11 and 2.12).

Φ(r) ∼=n∑

a=1

φaNa(r) (2.11)

w(r) =n∑

a=1

βaψa(r) (2.12)


The solution basis functions are defined as in FEM, which is to say that each one is equal

to 1 at one node and to 0 at all the others, usually with a polynomial interpolation between

nodes. Using these definitions and Equations 2.10 to 2.12, we can derive the linear system

of equations shown in Equation 2.13 (see [109] for details).

(ψ1(r),Φ∞(r))

. . .

(ψn(r),Φ∞(r))

=

(ψ1(r), L(N1(r))) . . . (ψ1(r), L(Nn(r)))

. . .

(ψn(r), L(N1(r))) . . . (ψn(r), L(Nn(r)))

.

φ1

. . .

φn

(2.13)

The linearly independent basis functions ψa of the residual weighting functions are

often chosen to be equal to the solution basis functions Na (Galerkin form), or alternatively

can be defined as a set of Dirac delta functions each centering on one node (collocation form)

[109]. By carefully choosing the basis and residual weighting functions in this manner, the

first two matrices in Equation 2.13 can be conveniently computed, leaving a linear system

(Equation 2.14) whose only unknowns are the coefficients φa, which in turn yield the solution

Φ.

G = Hφ (2.14)

Like the matrix K in FEM, the matrix H depends only on the geometry and conductivities

of the region, not on the sources present, and therefore can be precomputed. The resulting

BEM system is smaller than in FEM but, unlike in FEM, is dense, not sparse [23].

Source models

Whether equivalent dipole or distributed linear methods are used, bioelectric source local-

ization problems typically seek to describe the distribution of currents in a region. The goal

of the forward problem is therefore to compute the measurements that would result from a

current dipole at a given location. In other words, the sources are modeled as equivalent

dipoles that reflect the current density in the vicinity of their position.


If FEM is being used, the principle of reciprocity can be used to more rapidly compute

the influence of a current dipole in each mesh element on the measurements. The principle

states that the potential measured between two surface points in the presence of a unit cur-

rent dipole is equal to the magnitude of the electric field that would occur at the dipole’s

position and orientation if a unit current was created between the two surface points. As

a consequence, by placing a current source at an active electrode and a current sink at the

reference electrode and computing the resulting electric field throughout the region, the in-

fluence on the active electrode of current dipoles in every mesh element can be obtained. The

number of potential distributions to compute is therefore equal to the number of electrodes,

rather than the number of mesh elements [193]. On the other hand, if BEM is used, the

principle of reciprocity is not as beneficial, but the positions of the dipoles are not restricted

by an FE mesh and can be chosen as the investigator sees fit (e.g. the dipoles can be placed

on a regular grid whose coarseness is appropriate to the problem at hand).

It is also possible to model the sources as something other than current densities.

Grave de Peralta Menendez et al. [42] have proposed a framework, called electrical analysis

(ELECTRA), which establishes relationships between different possible types of sources to

be recovered and the leadfields corresponding to each one. At its core, ELECTRA relies on

the observation that only irrotational sources are capable of producing measured potentials

at the electrodes [39]. Three types of source models are consistent with irrotational current

sources: current density vectors (the current dipoles most often solved for in bioelectric source

localization problems), current source densities (the divergence of the current density vector

field), and electric potentials. By providing a framework relating the different source models,

ELECTRA makes it possible to solve for any one of them. The current source density and

potential are both scalar fields, such that using these source models can reduce the number of

variables in the inverse problem by a factor of three. Nonetheless, the current density vector

source model remains the one used most often in the literature, perhaps because it has the

most direct relationship with the neuronal transmembrane currents. In the remainder of this

thesis, unless otherwise noted, the source model used is the current density vector.


2.2.2 Equivalent dipole methods

Equivalent dipole models rely on scanning the solution space and solving a forward problem

for each location under consideration. The location that results in simulated measurements

that best correspond to the real ones is selected as the solution. The drawback of this

approach is that an exhaustive search of all possible locations can quickly become impractical,

particularly when the method is expanded so that the solution consists of more than one

dipole. Instead of an exhaustive search, therefore, optimization algorithms are used to find

the dipole locations that result in the smallest error [70, 63, 186]. In order to further ease

the process, Mosher et al. proposed uncoupling the unknown variables that have a nonlinear

effect on the measurements (the dipole locations and orientations) from those that have

a linear effect (the dipole magnitudes). The search algorithm is used only to determine

the values of the nonlinear parameters. The linear ones can be factored out of the process

and, once the nonlinear values have been solved for, can be estimated using a pseudo-

inverse approach similar to the one used in the distributed linear methods presented in the

next section [110]. This uncoupling strategy is part of the widely used MUltiple SIgnal

Classification (MUSIC) algorithm. Those same authors later proposed a scheme the avoids

having to search for all the dipoles simultaneously. In the recursive MUSIC algorithm (R-

MUSIC), sources can be identified one by one: the source that comes closest to explaining

the data is selected first and retained, the search is repeated to find the source that best

explains the remaining part of the data, and the process is repeated until all the dipoles have

been localized [108]. R-MUSIC also extended the concept of a source from a single dipole to

a set of several synchronized dipoles.

Scherg et al. [149] have also proposed a forward model that, in addition to the location

and orientation of the dipoles, includes the temporal waveform of activation at each dipole.

The recorded data from a block of time rather than a single instant is then used to evaluate

the ability of a given set of model parameters to explain the data. In other words, temporal

information is incorporated into the problem in order to increase the likelihood of obtaining


a plausible solution.

Regardless of the search method employed, the biggest difficulty inherent in equivalent

dipole methods is that number of dipoles must be chosen a priori. Even if the investigator

has some knowledge about the phenomenon causing the electrical activity, estimating the

number of sources may be quite difficult. Some authors have suggested adding dipoles one at

a time, until the amount of unexplained activity stops decreasing significantly [148]. Another

possibility is to use additional imaging modalities to guide the decision. For EEG/MEG

source localization, fMRI has been used to estimate the number of different active areas,

and that information can be used either to choose the number of dipoles to be localized

[30, 74, 101] or to evaluate the plausibility of sources localized without using the fMRI

information, and to adjust the number of dipole if necessary [1]. The main problem with

this approach is that the relationship between the neural activity underlying EEG/MEG and

the hemodynamic response underlying fMRI is not well understood, thereby making the use

of fMRI information in EEG/MEG source localization potentially problematic [189, 25, 90].

For example, EEG/MEG and fMRI are not always equally sensitive to different source

configurations, meaning that it is possible for an area to be detected as active by one method

but not the other [104]. Furthermore, although EEG/MEG have a temporal sensitivity on

the order of the millisecond, the hemodynamic changes that fMRI responds to are much

slower. In a situation where brain activity associated with a given task is spread across

several areas, each of which may be active at different times, it is impossible to use fMRI

information without losing the fine temporal resolution that is the most attractive feature

of EEG/MEG.

An alternative to using other imaging modalities is to use the mathematical properties

of the recorded signals themselves to choose the number of dipoles. For example, Mosher

et al. use the eigendecomposition of the data recorded over a time interval to estimate the

dimensions of the signal subspace and noise subspace, then use this information to select

the number of dipoles (the relationship between the leadfield of each location and the signal

subspace is then used to estimate the location of the dipoles)[110]. Other spatio-temporal


decompositions that have been proposed include common spatial patterns decomposition,

which relies on comparing the signal of interest with a control signal to identify important

components [72, 73], and independent component analysis [71].

Use of equivalent dipole methods to obtain time series of activity

As mentioned previously, the temporal resolution of EEG and MEG is the most attractive

feature of bioelectromagnetic source localization methods. When using equivalent dipole

methods, the time series of dipole magnitudes can be recovered in a straightforward manner

once the dipoles have been localized. This is made possible by the linear relationship between

those magnitudes and the measurements. In the MUSIC algorithm, the complete relationship

between the measurements and the dipole magnitudes is described by Equation 2.15, using

notation from the original paper [110].

B = H(L,M)S (2.15)

The matrix H represents the influence on the measurements B of a set of dipoles whose

locations and orientations are given by L and M, respectively. S represents the magnitudes

of the dipoles. Computing the entries of H for given values of L and M corresponds to

solving the forward problem. Finding the values of L and M that minimize the difference

between the recorded measurements F and B (the simulated measurements) is a nonlinear

search problem, and is at the heart of equivalent dipole methods, as describe in the previous

section. Once this step has been accomplished, however, the magnitudes S can simply be

obtained as shown in Equation 2.16.

S = H†F (2.16)

H† denotes the pseudo-inverse of H. If the dipole locations are assumed to be constant

over a certain time interval, then Equation 2.16 can be applied at each time instant and

an activation time series obtained. These high-resolution time series are ideally suited to


studying the temporal relationships and frequency components of different brain regions

during a given task [1, 111].

2.2.3 Distributed linear methods

Distributed linear methods represent a different approach to the source localization prob-

lem, and accordingly the set of challenges associated with them is distinct from the issues

encountered with equivalent dipole methods. As discussed above, the equivalent dipole prob-

lem involves a complex but well-determined nonlinear optimization problem. In contrast,

distributed linear methods formulate the problem as a linear system of equations. Indeed,

since the location and orientation of the dipoles are fixed, only their amplitudes remain to

be estimated. Because electric fields add linearly, the relationship between the dipole ampli-

tudes and the measurements is linear. The problem can therefore be expressed as shown in

Equation 2.17.

d = Lj + ǫ (2.17)

d is an Mx1 vector containing the recorded data from the M electrodes contacts, j

is an 3Nx1 vector whose entries represent the magnitudes of the current dipoles, and L is

a Mx3N matrix known as the leadfield matrix whose entry (i,j) represents the influence of

a unit current at dipole j on the potential recorded at electrode i. The leadfield matrix has

three columns for each of N dipole locations, corresponding to the three orthogonal dipole

orientations. ǫ is the additive noise.

The difficulty in distributed linear methods comes from the fact that that system of

equations in Equation 2.17 is severely underdetermined. To obtain a satisfactory solution,

additional information must be incorporated into the problem. This information takes the

form of constraints on the solution, which can be either mathematical in nature or anatom-

ically derived.


Weighted minimum norm methods

The fundamental idea underlying all minimum norm solutions to the bioelectric source local-

ization problem is that, out of the infinite set of solutions to the underdetermined problem,

the chosen solution should be the simplest one. In other words, the goal is to find the sim-

plest configuration of sources that explains the measurements. The notion of “simplest” is

then formalized as the minimization of a given norm.

The use of the solution minimizing the l2 norm was first proposed in the context

of bioelectromagnetic source localization by Hamalainen and Ilmoniemi [49]. Many of the

source localization methods that have been proposed have been shown to be equivalent to

this classical minimum norm approach if no constraints are added [56]. Although this method

could achieve reasonable performance in two dimensions, it was not adequate for realistic

three-dimensional problems [122]. The main drawback of the minimum norm approach is

that sources near the electrode positions have a much greater influence on the measurements

than deeper sources, and the resulting solutions therefore tend to feature mainly superficial

sources. Deep sources are usually severely mislocalized. This problem led to the development

of weighted minimum norm approaches. Instead of minimizing the l2 norm of the solution,

the l2 norm of a weighted version of the solution is minimized instead. A variety of weight

matrices can be chosen, corresponding to different constraints on the solution. The simplest

option is to use a diagonal weight matrix that normalizes each leadfield column by its norm

[122]. Since sources near the electrodes correspond to leadfield columns with larger norms,

this weighting has the effect of compensating for the distance between sources and the

electrodes, thereby improving the localization of deep sources. An alternative basis for

weighting source locations is to use information from other modalities. Specifically, fMRI

images can be used to obtain an estimate of which locations are more likely to be active,

and MRI and PET images provide anatomical information that can be used to restrict the

solution space [22, 87, 127, 86]. By using a non-diagonal weight matrix, more complex spatial

features can be incorporated into the constraints. The most common example of this is the


low resolution brain electromagnetic tomography (LORETA) method, which uses a discrete

Laplacian weight matrix to impose a smoothness constraint [124]. LORETA is better able

to localize sources in three dimensions than the weighted minimum norm solutions described

above. Its disadvantage, however, is that the resulting solutions are blurred, representing a

trade-off between accuracy of peak locations and spatial resolution. The anatomical validity

of the smoothness constraint is also debatable unless the solution grid is very fine [120].

Another method that uses a non-diagonal weight matrix is known as local auto-regressive

averages (LAURA), which couples elements of the solution based on the distance separating

them and the rate of change of electric fields with distance [42].

Having chosen an appropriate norm to evaluate the solution, the problem remains that

the inverse problem is ill-posed and extremely sensitive to noise. Minimum norm solutions

(weighted and non-weighted) fit the data exactly, so when this data is noisy, the attempt to

fit the noise will lead to significant errors in the solution. This problem must be addressed,

through a process known as regularization. In essence, regularization involves accepting

some error in the fit to the data, in exchange for increased stability of the solution [54]. Two

types of regularization methods are commonly used in bioelectromagnetic source localization:

Tikhonov regularization and truncated singular-value decomposition (TSVD). In Tikhonov

regularization, the problem is formulated as shown in Equation 2.18 [179, 54].

j = arg minj‖C−0.5

ǫ (Lj − d)‖2 + λ‖Hj‖2 (2.18)

j is the current source distribution estimate, Cǫ is the noise covariance matrix (the

noise is assumed to be zero-mean Gaussian), and H is the weight matrix. The expressions

for the various weight matrices described in the previous paragraph are given in Table 2.1.

λ is the parameter that balances the minimization of the residual and the minimization of

the weighted solution norm. A small value of λ will give preference to fitting the data. As

λ is increased, more importance is given to the a priori constraints. In other words, the

noisier the data, the larger the value of λ should be, so that more importance is given to the


Table 2.1: Commonly used weight matrices for weighted minimum norm methods.

Method Weight Matrix Expression

Minimum Norm I

Weighted Minimum Norm

Hwmn|Hii = LT1..n,iL1..n,i;Hij = 0, i 6= j

LORETA BHwmn

LAURA HwmnA⊗ I3 where Aii = NNi

∑

k⊂Vi

d−3

ki

Definitions B is the Laplacian operator

⊗ is the Kronecker product

Vi is the set of Ni neighbours of location i, of size at most N

dki is the distance between locations k and i

investigator’s prior knowledge of what characteristics the solution should have. Equation

2.18 has a closed-form solution, which is shown in Equation 2.19 [126].

j = (HtH)−1Lt[L(HtH)−1Lt + λCǫ]−1d (2.19)

As for the TSVD method, it can be derived starting from the expression for the

unregularized, unweighted solution, which is simply the Moore-Penrose pseudo-inverse of

the leadfield, multiplied by the data (Equation 2.20).

j = L†d (2.20)

If the weight matrix is square and non-singular, it can be incorporated in Equation

2.20 by first re-writing Equation 2.17 as Equation 2.21, such that the new solution is as

shown in Equation 2.22.

d = (LH)(H−1j) + ǫ (2.21)

j = H(LH)†d (2.22)


Note that Equation 2.22 is equivalent to Equation 2.19, with λ set to 0 [37]. The

next step involves a singular value decomposition, which decomposes a matrix into two

orthonormal matrices U and V and a diagonal matrix S. U and V can be interpreted as

containing basis vectors describing the structure of the data, whereas the singular values

in S reflect the relative importance of these different components. Now, using the singular

value decomposition LH = USVT, Equation 2.22 can be written as shown in Equation 2.23.

Here, S−1 is defined as the matrix in which each nonzero entry of the diagonal matrix S has

been replaced by its inverse.

j = H(VS−1UT)d (2.23)

The instability of the solution in an ill-posed inverse problem is related to the small

singular values in the matrix S [54]. Based on this insight, the TSVD regularization method

functions by setting to zero the entries of S that are below a certain threshold. The solution

is then obtained by using the modified singular values matrix S in Equation 2.23. Tikhonov

regularization can be understood within the same framework of singular value decomposition,

where it corresponds to adding λ to each singular value in S, with λ being assigned a value

that is much greater than the smallest singular value but much smaller than the largest

singular value. This will reduce the instability associated with taking the inverse of a very

small number, without having a significant impact on the large singular values. Several

methods exist to help with choice of λ; they are described in Section 2.2.5.

Iterative minimum norm methods

Gorodnitsky et al. have proposed an algorithm by which more focal solutions can be achieved

than the ones resulting from the minimization of a weighted l2 norm. The algorithm, known

as the focal underdetermined system solver (FOCUSS), is essentially an iterative application

of a weighted minimum norm method (Equation 2.19), where the weight matrix at each

step is a diagonal matrix whose entries are based on the source estimates from the previous


iteration [37]. In this way, locations that were found to have significant activity in one

iteration are favoured in the next iteration. As the number of iterations increases, the

activity becomes concentrated in a small number of locations, and tends to zero elsewhere.

The process stops when the solution is no longer changing significantly from one iteration to

the next, or if the number of nonzero elements in the solution starts to increase. FOCUSS

can obtain at most as many focal solutions as there are measurements [37, 38].

Before FOCUSS can be applied, an initial estimate must be obtained, which can

be done using any distributed linear method. The original paper proposed a variant of the

depth-weighted minimum norm solution [37]. More recently, good results have been obtained

using sLORETA (see section 2.2.3) as the initial estimate [88]. FOCUSS is very sensitive to

the initial estimate, so having a good estimate is critical to the success of the method. Addi-

tionally, the regularization of the algorithm can be difficult, making it potentially vulnerable

to noise. FOCUSS can be regularized using the Tikhonov or TSVD methods [37], but the

process is made more difficult by the fact that a good choice of regularization parameter

is needed at every iteration. Rao et al. [134] have proposed parameter choice methods to

overcome this problem (see Section 2.2.5).

Standardized minimum norm methods

The classical minimum norm solution given by Equation 2.19 inherently has a magnitude

distortion for each element in the estimated solution. This can be seen using the concept of

the resolution matrix. If the result of the (weighted or unweighted) minimum norm solution

is as given in Equation 2.24, then the relationship between the true source distribution and

the estimated source distribution can be obtained by combining Equations 2.17 and 2.24, as


j = Td (2.24)


j = Td = TLj = Rj (2.25)

For perfect reconstruction, the resolution matrix R should be the identity matrix,

but because of the ill-posed nature of the problem there does not exist a transfer matrix T

that results in R being the identity. Pascual-Marqui proposed using the bias information

contained in R to normalize the results of an unweighted minimum-norm solution, as shown

in Equation 2.26.

jT

l (Rll)−1jl (2.26)

The resulting algorithm is known as standardized LORETA (sLORETA) [123]. Al-

though there is no explicit smoothness constraint like in the original LORETA algorithm (see

Section 2.2.3), sLORETA nonetheless results in a smooth solution with well-localized peaks.

In the noiseless, single-source case, sLORETA has been shown to have zero localization error

[123, 162, 43].

The standardization method used in sLORETA has also been incorporated into the

FOCUSS algorithm by Liu et al [88]. Their algorithm, known as standardized shrinking

LORETA-FOCUSS (SSLOFO), recomputes the resolution matrix at every iteration based

on the updated transfer matrix T, then applies the normalization described by 2.26. Note

that in this case, T results from a weighted minimum norm solution, as per the FOCUSS

algorithm, rather than the unweighted minimum norm solution used in sLORETA. The

resulting algorithm outperforms the original FOCUSS [88].

An alternative method using the resolution matrix was proposed by Grave de Peralta

Menendez et al. [41]. Instead of normalizing the solution elements individually using local

sections of the resolution matrix, they proposed creating an approximate inverse of the

entire resolution matrix. The unmodified resolution matrix is usually not invertible, but the

approximation can be obtained by inverting the sum of the resolution matrix and a diagonal

perturbation. Calling this perturbation D, the approximate inverse is shown in Equation


2.27, and the transfer matrix is updated as shown in Equation 2.28.

M = (D + R)−1 = (D + TL)−1 = D−1 − D−1T[I + LD−1T]−1LD−1 (2.27)

T := MT (2.28)

This method’s aim was to overcome the limitations of partial inversions such as

sLORETA, namely that they work better for single sources than for linear combinations

of sources.

Methods using lp norms

Although the minimum norm methods described so far usually minimize the l2 norm, this

does not have to be the case: a different lp norm can be chosen. In particular, several studies

have investigated the minimization of the l1 norm, which produces sparser solutions than

l2 [94, 95, 187, 165, 194]. Using these methods, solutions are obtained that have at most

as many nonzero elements as there are sensors. The main disadvantage of this approach is

that, unlike the l2 norm minimization problem, the equivalent l1 problem does not have a

direct solution. An optimization problem must be solved instead or, if the dipole orientation

is fixed, a simpler linear programming (LP) approach can be used [32, 187]. The practical

outcome is that the computational demands of l1-based methods are much higher than those

of l2-based methods.

Notable l1-based methods include the approach of Matsuura and Okabe [95], the

Minimum Current Estimate (MCE) introduced by Uutela et al. [187] and the lp Norm

Iterative Sparse Solution (LPISS) by Xu et al. [194]. Matsuura and Okabe minimized the

l1 norm under inequality constraints corresponding to an acceptable data misfit tolerance,

thereby making the procedure more robust in the presence of noise. MCE determines the

dipole orientations using a traditional weighted minimization of the l2 norm, then estimates


the dipole magnitudes by using a LP approach to minimize the l1 norm of the solution. The

minimization is constrained by a regularized equality between the estimated solution and the

data. LPISS is a FOCUSS-like algorithm that replaces the l2 solution norm in Equation 2.18

by an lp (p ≤ 1) norm, as shown in Equation 2.29. By using the l1 norm LPISS reconstructs

sparse distributions more reliably than FOCUSS. Because of the presence of both an l1 and

an l2 norm in Equation 2.29, the problem cannot be solved using LP, and a more complex

optimization problem must be solved. This algorithm is therefore particularly time intensive,

requiring roughly two orders of magnitude more computation time than FOCUSS.

j = arg minj‖Lj − d‖2 + λ‖j‖1 (2.29)

Local linear estimators

Most of the minimum-norm methods discussed so far can be described as global linear

estimators, because they produce an entire source distribution designed to account for the

measurements as well as possible. An alternative approach is to use local linear estimators,

which focus on the recovery of each target location individually, without explicitly minimizing

the overall misfit to the data [43]. The rows of the resolution matrix (Equation 2.25) provide

information about the bias of a given solution method for each individual location, as well

as about the influence of activity at other locations on the target location. Similarly, the

columns of the matrix provide information about the blurring that can be expected when

recovering a single point source. Local linear estimators function by constructing transfer

matrices designed to produce resolution matrices with specific properties, rather than to

minimize the data misfit. Each row of the transfer matrix is treated as an individual problem,

and produces a resolution matrix row (also known as a resolution kernel) with the chosen

properties. Specifically, the problem can be formulated as a minimization of the norm of the

resolution kernel under certain constraints, which vary from one type of linear estimator to

the other.


Examples of constraints that have been proposed include ensuring that the solution

has a gain of one at the target location (distortionless estimators [43]) or making the resolu-

tion kernel as close as possible to a delta function centered around the location of interest and

integrating to one (the oft-cited Backus and Gilbert method first introduced in geophysical

inverse theory [4] and improved in [40]). Note that the sLORETA algorithm is a local lin-

ear estimator, although it was presented separately in Section 2.2.3 because of its frequent

mention in the EEG/MEG source localization literature (it is also worth clarifying that

the original LORETA algorithm is a global linear estimator, and therefore fundamentally

different from sLORETA).

All of the local estimators in the previous paragraph are non-adaptive estimators,

meaning that the solution is derived independently of any data. Adaptive local estimators

have also been described, which assume some knowledge of the measurements. The best-

known example is the linearly constrained minimum variance (LCMV) spatial filter [188],

which is the “beamforming” technique common in radar and sonar technology. As in the

non-adaptive estimators, the idea is to obtain an estimate at each location by constructing a

spatial filter that excludes as much as possible signals originating from outside the location

of interest, while having unit gain at that location. Because of the underdetermined nature

of the problem, these filters do not have enough degrees of freedom to adequately exclude

the signals from all undesired locations. The LCMV approach formulates the problem as

the minimization of the variance of the filter output, constrained by the requirement of unit

gain at the target location. The variance of the filter output (i.e. of the sources) can be

expressed in terms of the signal covariance and the leadfield. Consequently, the beamforming

approach relies on knowledge of the signal covariance matrix and uses this information to

allocate the resources of the filters in a way that blocks the activity at the locations most

likely to interfere with the activity at the target location. The signal covariance matrix is

generally estimated directly from the data. If the underlying signal is stationary and a good

estimate of the signal covariance matrix is available, the beamformer can perform very well

[162]. If the assumption of stationarity cannot be justified, on the other hand, the method


may produce misleading results.

A commonly used variant of the LCMV beamformer is known as synthetic aperture

magnetometry (SAM) [137, 190]. In addition to estimating the source power at each location,

the contribution of the sensor noise to the source power estimate at that location is also

estimated, and the ratio of source power to noise variance is used as the final tomographic

image. The normalization by the noise variance improves the resolution of the algorithm by

emphasizing regions with statistically significant brain activity.

2.2.4 The dynamic source localization problem

In addition to constraints derived from anatomical and mathematical considerations, the

temporal behaviour of the bioelectromagnetic sources can be taken into account in order to

obtain realistic solutions. This is a natural approach considering that the fine temporal res-

olution of bioelectromagnetic measurements is one of their main advantages when compared

to other imaging modalities (for example MRI and PET), and makes them useful in the

examination of temporal characteristics such as frequency, synchronization, and the order of

activation of different regions. Two approaches have been used to impose temporal relation-

ships between the solutions at various time instants: grouping several time instants together

into a single large linear system [24, 154, 153, 155], or using the results of one time instant to

obtain initial conditions for the solution of the next instant [195, 33]. Additionally, methods

have been proposed in which the localization algorithms are applied to a time-frequency

decomposition of the signal, in order to focus on sources producing a certain frequency band

[84, 91, 174, 161, 35, 106, 118].

Coupled spatio-temporal system approach

Ideally, the whole spatio-temporal system could be expressed and solved as a single linear

system, by concatenating the measurement and source vectors for all time instants [8, 24,

154]. This is shown in Equation 2.30, which is very similar to Equation 2.17, with the


exception that the vectors dc, jc, and ǫc include the complete information about the system

at all time instants in the observation time window (i.e. dc = [d(1)Td(2)T...d(n)T]T, with

equivalent definitions for jc, and ǫc). Lc is a block diagonal matrix, where each block is equal

to L.

dc = Lcjc + ǫc (2.30)

The advantage of this approach is that temporal patterns can be included as con-

straints, in the same way that spatial constraints were included in the instantaneous prob-

lem. Since the solutions at different time instants are incorporated into a single vector,

the solution of the inverse problem can include a weight matrix that relates activations at

different times, according to an a priori pattern. One possible constraint is to impose tem-

poral smoothness. The obvious drawback of this approach is that the combined system is

extremely large and quickly becomes intractable, unless the number of variables per time in-

stant is very small. Efficient algorithms have been proposed to try to overcome this problem

[8, 153, 155]. Another notable approach within the framework of coupled spatiotemporal

systems was proposed by Greensite, who described how to explicitly incorporate a non-

informative temporal prior, rather than assume independence or smoothness between time

instants. That is to say, the Greensite prior can be used to eliminate any such assumptions

and reflect the fact that we have no information about the relationship between the time

instants or their independence [44].

State-space representation

An alternative way of incorporating temporal information into the problem is to use the

estimates of past time instants as initial conditions for the current time instant, under the

assumption of temporal smoothness. A weighted difference between the solution and the

initial condition can be incorporated as an additional term in Equation 2.18, as shown in

Equation 2.31, or similarly in Equation 2.32 [195].


jt = arg minjt

‖C−0.5ǫ (Ljt − dt)‖2 + λ1‖Hjt‖2 + λ2‖H(jt − jt−1)‖2 (2.31)

jt = arg minjt

‖C−0.5ǫ (Ljt − dt)‖2 + λ‖H(jt − φjt−1)‖2 (2.32)

Alternative formulations can include using derivative information from previous time

instants to obtain more precise initial conditions [117, 178]. Additionally, Yamashita et

al. point out that Equation 2.32 can be interpreted in terms of a state-space representation,

where the observation and state equations are shown in Equations 2.33 and 2.34, respectively

[195].

dt = Ljt + ǫt ǫt ∝ N(0, σ2Cǫ) (2.33)

jt = φjt−1 + ηt ηt ∝ N(0, τ2(HTH)−1) (2.34)

Equation 2.34 can be modified to incorporate more complex relationships between

time instants, as shown in Equation 2.35, where information from the previous p time points

is used.

jt =

p∑

i=1

Aijt−i + ηt (2.35)

Where the A matrices are state transition matrices. The advantage of this formulation

is that complex relationships between time instants can be expressed without increasing the

size of the system to be solved at each instant (unlike the coupled approach presented in the

previous paragraph). On the other hand, its drawback is that a poor estimate at one time

instant will lead to poor initial conditions in future time instants, potentially causing the

error to propagate.

Taking the implications of the state-space approach further, the same group suggested

using a Kalman filtering approach to solve the source localization problem [33]. In order

for the problem to remain tractable, the high-dimensional problem was decomposed into


a set of uncoupled low-dimensional problems. This requires a number of assumptions, for

example concerning the connectedness of different brain regions and the dynamical models

underlying the activity. The method has the potential to be powerful, on the condition that

the assumptions can be kept both realistic and informative.

Time-frequency approach

Another way to incorporate temporal information into the problem is to focus on limited

frequency bands. Sources corresponding to each frequency band can be localized separately,

which can be of considerable interest for studying issues such as oscillatory activity in a

specific region of the brain, or the synchronization of activity among different regions. Early

methods of this type used the Fourier transform (both amplitude and phase) of the recorded

signal to focus on specific frequencies and explain them using dipole fitting methods [84, 91,

174]. Because the Fourier transform is not appropriate for the analysis of non-stationary

signals, these methods led to ones based on various types of time-frequency decompositions,

for example wavelets. Different decompositions have been combined with dipole fitting

approaches [118], the MUSIC algorithm [161], beamforming approaches [20], and distributed

linear methods [35, 106]. In this last category, the study by Gonzalez Andino et al. suggested

using the time-frequency decomposition to identify times at which the frequency spectrum is

simplest and therefore likely to correspond to a small number of generators, then performing

source localization at those times.

2.2.5 Choice of regularization parameter

Both Tikhonov and TSVD regularization involve a parameter that has a significant impact on

the solution and must be carefully selected. The parameter is λ in Tikhonov regularization,

and the number of singular values to be truncated in TSVD. There is no simple way to find

the best values for these parameters, but several methods exist to guide the choice. We

restrict ourselves here to methods that have been used in the EEG/MEG source localization


literature; a more general review can be found in [54].

L-curve

The most commonly used parameter-choice method in the EEG/MEG source localization

literature is the L-curve method [53, 54]. It creates a plot of the norm of the residuals versus

the norm of the solution, as the regularization parameter is varied. In log-log scale, this plot

is typically shaped like an “L”, whose corner corresponds to the best tradeoff between the

simplicity of the solution and the data misfit.

In some cases, the log-log curve may not be a perfect “L”, in which case choosing the

point of maximum curvature may not yield the best regularization parameter. Rao et al.

observed this phenomenon in relation to the FOCUSS algorithm, and proposed a modified

L-curve method in which the parameter is restricted to values producing a residual norm

within a given interval [134]. This approach ensures that the parameter choice is always

reasonable and results in a more robust algorithm.

Cross-validation

Cross-validation is a common method in model selection problems. One entry (or a subset)

of the data is left out, the model is trained on the remaining data, and the predicted value

of the data that was left out is compared to the correct values. This is repeated for each

data point or subset, and the mean predicted error is used as a metric of the quality of

the model fitting. In a regularization context, the chosen regularization parameter is the

one that minimizes the cross-validation error (CVE). Pascual-Marqui suggested using CVE

minimization as the regularization method of choice for his sLORETA algorithm [121].

A variant of the cross-validation method, generalized cross-validation (GCV), is dis-

cussed in the regularization literature [54], but rarely mentioned in the context of bioelectric

source localization. One inverse electrocardiography study found the performance of GCV

to be similar to that of the L-curve [164].


AIBC

Many source locations algorithms have been derived using a Bayesian approach [152, 21, 31,

128, 147, 126, 180, 68]. It can be shown that under certain common assumptions (e.g. Gaus-

sian priors), the resulting algorithms are equivalent to the ones derived using the minimum-

norm least-squares approach described above. If more complex prior information needs to

be incorporated, however, or if we want to incorporate information from several models si-

multaneously [181], the Baysian approach is attractive because of its versatility. From a

regularization point of view, the Baysian formalism allows us to treat the regularization

parameters as model hyperparameters that need to be determined and who are associated

with their own prior probability distributions. One example of this approach is the use of

an approximation of the Akaike’s Bayesian Information Criterion (ABIC) by Yamashita et

al. [195, 33]. The ABIC is a metric for the evaluation of a statistical model, which includes

both a measure of the log-likelihood of the parameters and of the number of parameters. By

minimizing the value of the ABIC, a model is found that fits the data as well as possible

while still keeping the number of parameters as small as possible. Aside from the obvious

application as a regularization tool, Yamashita et al. point out that the ABIC can be used

as a way to compare several models (i.e. different source localization algorithms or differ-

ent priors). The actual minimization of the ABIC is performed by solving an optimization

problem, making this method potentially more time-consuming than the L-curve or cross-

validation. Furthermore, if probability distributions are not assumed to be Gaussian, the

use of the ABIC becomes analytically very complex.

ReML

An alternative statistical approach to the regularization problem has been suggested by

Phillips et al. [128, 126]. They advocate computing the restricted maximum likelihood

(ReML) solution, which in this case is equivalent to the more general method of using an

expectation-maximization (EM) algorithm [31]. This approach is well established in the


statistical literature, in situations where both parameters and hyperparameters of a model

must be estimated from the data. Here, the model parameters are the dipole magnitudes

describing the source distribution, and the hyperparameters are the variances of the source

and sensor signals (the ratio of which is the regularization parameter used in the previous

techniques). The algorithm consists of an iterative process in which the dipole magnitudes

and regularization parameters are each held fixed in turn, while the value of the other

component is optimized to best account for the data. This guarantees that the resulting

dipole and regularization parameters will account for the data in an optimal way, in the

absence of any additional information.

2.3 Peripheral Nerve Monitoring

As the previous section demonstrates, inverse problems of source localization are complex and

difficult to solve. They are therefore not a natural choice if simpler alternatives are available.

In the case of monitoring the activity of peripheral nerves, there are a variety of other

methods available [142, 115], but none of them are entirely satisfactory. Any given system

for extracting information from recordings of the electrical activity of a nerve can be divided

into two components: a recording device, and a set of mathematical tools for interpreting

the data obtained. Until recently, the majority of research in peripheral nerve recordings

has been focused on the development of better electrodes, while relying on traditional signal

processing techniques. Indeed, if the electrode signals do not contain the desired information,

no amount of processing will be able to compensate. The fundamental trade-off in the

design of peripheral nerve electrodes is to balance the amount of information obtained with

the increased complexity and invasiveness of the device. A survey of the available types of

electrodes reveals that they typically represent various attempts to deal with this trade-off

in a way that is satisfactory for a specific application.


2.3.1 Electrode types

Extraneural Electrodes

Extraneural electrodes do not penetrate into the nerve, and therefore have less chance of

damaging it than intraneural electrodes. On the other hand, because of the greater distance

between the nerve fibers and the recording site, the information that these electrodes provide

is less spatially specific.

The most common type of extraneural electrode is the nerve cuff electrode [166, 89,

112, 113, 167]. A nerve cuff is an insulating sheath that encircles the nerve, with electrical

contacts placed on the inner surface of the tube (Figure 2.2(a)). This configuration isolates

the electrode contacts from signals originating outside the nerve segment contained by the

cuff. In addition, cuff electrodes are easier to implant and less invasive than intraneural

electrodes. Because of these advantages, much attention has been paid to the potential of

nerve cuffs to serve as chronically implanted neural interfaces. Studies have examined the

effects of the cuffs on the underlying nerves after implantation periods of several months

[76, 80, 171, 46, 138, 11, 96, 139], and although minor nerve damage was observed in certain

studies, it was usually not severe enough to cause any functional deterioration. In particular,

a design known as the spiral cuff has been used, which consists of a “self-sizing” cuff that

adjusts to the diameter of the nerve and has been shown to cause less damage in chronic

implantations than cuffs with fixed sizes [112, 46, 138, 139]. In humans, nerve cuff electrodes

have been used for periods of up to 12 years [192].

The traditional contact configuration for a cuff electrode is the tripole configuration,

in which the potential at a ring contact at the centre of the cuff is measured using as reference

the average potentials from two other ring contacts, one at each extremity of the cuff. In

practice this derivation is implemented using either the quasi-tripole or the true tripole

configurations, which are shown in Figure 2.1. Sources outside a cuff produce a signal that

varies linearly along the length of the device, but this is not true of sources located inside the

cuff [170, 132, 133, 2]. The tripole configuration therefore attenuates signal produced outside


Figure 2.1: Two common implementations of the tripole configuration: the quasi-tripole

(a) and the true tripole (b). From [133].

the cuff, but not those from inside. A significant drawback of the device is that a cuff with a

single tripole configuration produces a single signal, which reflects all the activity anywhere

in the nerve, and makes the traditional cuff a device with very poor spatial selectivity. To

remedy this problem, alternative cuff designs can be used that include a higher number

of contacts. Depending on the number of contacts and their configuration, a certain level

of selectivity can be achieved (see Section 2.3.3 for more details). Whenever using cuff

electrodes, it is important to realize that the device acts as a filter on the recorded signal,

and that the geometry of the cuff and contact configuration have a significant impact on the

shape of the recorded signal [168, 173, 199].

A modified version of the cuff electrodes is the Flat Interface Nerve Electrode (FINE)

[183, 60, 150], which aims to improve selectivity by slowly reshaping the nerve into a flatter

configuration, such that every fascicle is close to the surface (Figure 2.2(b)). It has been

demonstrated that if the forces applied are small enough, the nerve can be successfully

reshaped without damaging it [184]. Although it has not yet been used in a clinical setting,

the FINE is a promising technology for selective recording and stimulation of peripheral

nerve signals, because it addresses the key issue of deeper sources having less influence on

the measurements than surface sources.

Other types of extraneural electrodes exist, although they are usually used in more


specific contexts than cuff electrodes. For example, epineurial electrodes (Figure 2.2(c))

are sutured to the surface of the nerve [115], and their stable positioning gives them good

selectivity for certain fascicles close to their location. On the other hand, they provide no

information about nerve fibers further away. Helicoidal electrodes (Figure 2.2(d)) are helix-

shaped electrodes [115] that are easy to place and remove, but whose selectivity is limited by

their open shape. Epineurial and helicoidal electrodes are usually used for stimulation rather

than recording, most notably for phrenic nerve stimulation [18] and vagus nerve stimulation

[97], respectively.

Intraneural Electrodes

Intraneural electrodes penetrate the nerve, making them much more spatially selective than

extraneural electrodes, but also more delicate to implant and more likely to damage the

nerve. Intrafascicular electrodes provide the best spatial selectivity available, by recording

directly from a small area inside a nerve fascicle. For example, the longitudinally implanted

intrafascicular electrode (LIFE) [83, 81, 82, 27, 103] consists of a thin insulated wire im-

planted directly into the fascicle. The recording site is a small length of wire where the

insulation is removed (Figure 2.3(a)). Although providing excellent functionality if only a

small region is of interest, these electrodes are difficult to implant, and implanting more than

a small number of them quickly becomes impractical. They are therefore not appropriate

for systems in which a large number of nerve fibers need to be monitored.

Other common types of intraneural electrodes are the penetrating microelectrodes and

microelectrode arrays [142, 115]. These exist in a variety of configurations, including 1D, 2D,

and 3D arrays. Single microelectrodes can be inserted percutaneously, while larger arrays

may require more invasive implantation procedures. These multielectrode arrays (MEA)

can contain more than 100 needle or wedge-shaped electrodes recording from within fascicles

(Figure 2.3(b)), and therefore currently represent the best option for monitoring a large

number of locations within a peripheral nerve [143, 144, 7, 191]. The major disadvantages


of MEAs are that they are very invasive, modifying the shape of the nerve and possibly its

conduction properties, they are difficult to implant, and they are difficult to manufacture,

requiring sophisticated microfabrication techniques. In addition, chronic experiments with

these devices have revealed poor stability of the recording over time, although stimulation

could be maintained with much higher stability [6].

The slowly penetrating interfascicular nerve electrode (SPINE) [182] is a different

electrode design, which is placed around the nerve like a cuff electrode but features inner

plates that penetrate the nerve (Figure 2.3(c)). The contacts are thus positioned between

the fascicles, rather than inside them. The penetrating elements are inserted into the nerve

slowly, minimizing damage. The electrode is therefore designed to combine the simplicity

and safety of a cuff electrode with intraneural recording capabilities. The trade-off is that

the number of contacts and the specificity of the information obtained is lower than in an

MEA.

Another nerve interfacing technology is the regeneration sieve electrode [29, 114, 79],

which is placed between the two stumps of a transected nerve. The device has numerous small

holes surrounded by electrode contacts. When nerve fibers regenerate, they grow through

the holes in the electrode, such that the contacts will be specific to individual axons or small

groups of axons (Figure 2.3(d)). While an interesting concept and useful for some studies in

animal models, there are few situations in which regenerative electrodes are appropriate for

use in humans, because of the heavy nerve damage involved and the time required for fiber

regeneration. Currently the only target application for these electrodes in amputees, where

the nerve has already been severed.

2.3.2 Signal processing techniques for extraneural measurements

In order to successfully control an assistive device, the signals recorded from a cuff electrode

must be processed and used to generate appropriate commands. The simplest way to ac-

complish this is to filter the signal so as to retain only the frequencies of interest, group


(a) (b)

(c) (d)

Figure 2.2: Types of extraneural electrodes: a) traditional cuff electrodes, b) FINE, c)

epineurial electrode and d) helicoidal electrode.

the measured samples into time bins of a few milliseconds, perform bin-by-bin integration of

the rectified signal, and finally compare the value obtained for each bin to a threshold [55].

The rectified and bin-integrated signal can also be used in conjunction with more sophis-

ticated techniques, for example by being used as the input to an artificial neural network

[102, 12, 16]. Alternatively, the variance of the contents of each bin can be computed (instead

of integrating), and Student’s t-tests performed between bins to detect statistically signifi-

cant increases that mark the presence of a signal [78]. Other approaches include subspace

decomposition methods, which use statistical information about a segment of the signal

to determine whether it should be characterized as noise or as a mix of signal and noise.

Specifically, this information can be obtained by examining the singular values of matrices

containing information either about the autocorrelation or the third-order cumulant (under

the rationale that noise can be modeled as a Gaussian process, and therefore has no non-

zero cumulants of order greater than two) [185]. Lastly, if a model of the expected signal is

available, then optimal filtering techniques such a Wiener filter or a matched filter can be


(a) (b)

(c) (d)

Figure 2.3: Types of intraneural electrodes: a) LIFE, b) MEA, c) SPINE, and d) sieve

electrode.

used to separate the signal from the noise [67].

2.3.3 Source localization and pathway discrimination using extra-

neural measurements

Simple nerve cuffs with a single channel reveal only one aspect of the nerve’s activity, and

provide little opportunity to go beyond traditional signal processing techniques such as the

ones described in the previous paragraph. As the electrodes become more sophisticated and

start providing several recording channels, more options become available for analyzing the


data obtained. Nerve cuffs are then no longer used simply as a binary tool to detect whether

or not a signal is present, but can be used to provide more complex information.

The first pathway discrimination method to be proposed was based on the conduc-

tion velocity of the nerve fibers. Because different types of fibers have different conduction

velocities [69], identifying the velocity provides one way of discriminating the activity of

various pathways. Before sophisticated nerve cuffs were available, this type of method was

explored using extraneural measurements at two different sites (algorithms explored included

correlation-based methods [57] and matched filters [3]). Developments in nerve cuff manu-

facturing technology led to the introduction of a velocity-selective system that uses several

measurements sites within a single cuff [173, 136, 135]. Using measurements at more than

two sites provides increased robustness.

Alternatively, methods have been proposed that attempt to discriminate pathways

not by using velocity information, but by taking advantage of the fact that an AP traveling

along a fiber at a given location in the nerve will produce different measurements at dif-

ferent contacts of a multi-contact cuff. Early attempts examined the recording differences

in traditional cuffs (with eight to twelve contacts) when action potentials traveled in vari-

ous fascicles or pathways [85, 169, 145, 17]. Developing this approach further, later studies

were able to quantify to what extent sources between and within fascicles produced different

measurement patterns, using FINEs with thirteen channels or less [125, 197]. However, iden-

tification of the active fascicle was not discussed in detail, beyond matching measurements

with known patterns. Extending this methodology to finer resolutions would soon become

infeasible, because the number of possible patterns would grow very large, all the more so

if any number of locations were allowed to be active at the same time. Other attempts to

separate the activity of different fascicles in a nerve based on extraneural recordings have

included the use of blind source separation (again with a FINE) [177, 175, 176] and linear

regression (using a four-channel round cuff) [14]. In both of those cases, the resolution was

limited to discriminating the activity of two fascicles. An earlier study attempted to localize

activity at an arbitrary location in the nerve with an eight-contact cuff, but used a very


simplified model of the nerve’s electrical properties and therefore obtained only coarse map-

pings [85]. Simple selectivity studies with multi-contact spiral cuffs have also been reported

that focused on discriminating activity related to specific innervated muscles, rather than to

specific fascicles [140, 141].

The method proposed in the rest of this thesis, based on an inverse problem of source

localization, in effect generalizes the ideas in the previous paragraph and aims to obtain a

more flexible framework, capable of differentiating activity not only between fascicles but also

within a single fascicle (albeit with limited resolution). This approach has previously been

suggested for the purposes of electrode targeting in the spinal cord [107]. That study focused

on localizing a single locus of activity in a two-dimensional cross-section of the spinal cord,

and found that both distributed linear and equivalent dipole methods were able to obtain

localization errors of less then 300µm. They determined that the process was reasonably

robust to errors in conductivity values and electrode position, but less so to high levels of

noise and to errors in the partitioning of the region into white matter and gray matter areas.

Preliminary research into the peripheral nerve source localization problem has also recently

started to emerge from other groups [28].

Chapter 3

Reduction of the Inverse Problem

Solution Space

3.1 Introduction

The aim of the first step in our study was to reduce the number of variables to be estimated

in the source localization problem by taking into account the inherent limitations in the

measurement setup. The guiding principle for the reduction was that if the properties of

the nerve, the electrode, and the discretization of the solution space are such that sources

at two adjacent locations will produce indistinguishable measurements, then the activity at

those two locations can be represented by a single variable. The main benefits of eliminating

unnecessary variables would be improvements in computation time and storage requirements,

as well as improved insight into what resolution limitations are intrinsic to a given nerve

anatomy and electrode geometry.

FEM is a more appropriate method than BEM for constructing a leadfield for the

peripheral nerve source localization problem, due to the anisotropic conductivity of the

region. The accuracy of the FE model both in terms of anatomy and tissue conductivities

will influence the quality of the solution when the leadfield is applied to measurements

48

Reduction of the Inverse Problem Solution Space 49

obtained in vivo. When a FE model is used to construct the leadfield, the fineness of the

mesh will have a significant impact on the accuracy of the matrix entries. Unfortunately,

each element in the mesh corresponds to a variable to be solved for in the inverse problem,

creating a conflict between the needs to make the leadfield as accurate as possible and to keep

the number of variables as small as possible. A simple way to reduce the number of variables

without coarsening the original FE mesh would be to decimate the solution space for the

purposes of the inverse problem. Here, we propose an alternative method to reduce the

number of variables, where the decisions are based on quantitative information contained in

the leadfield. Each column of the leadfield matrix describes the set of measurements obtained

when a unit source is placed at a given location (the forward field of that source). Therefore,

if the difference between two columns corresponding to adjacent locations is extremely small,

then the two corresponding sources can be said to be indistinguishable by the measurement

setup. This implies that two variables are being used when only one is useful. We investigated

how much reduction in the number of variables can be achieved by grouping together leadfield

columns that are indistinguishable, and what, if any, detrimental impact this transformation

may have on the process of solving the inverse problem.

The notion of varying the mesh coarseness in accordance with the capabilities of

the measurement setup has been previously proposed in the context of a different inverse

problem, electrical impedance tomography (EIT) [13, 66]. By using the differences between

leadfield columns as a quantitative grouping criterion, we are adapting this idea to a bio-

electric source localization problem, and developing a method by which the accuracy of the

forward problem does not need to be compromised.

3.2 Theory

Determining whether two leadfield columns correspond to indistinguishable elements is equiv-

alent to determining whether all the entries in the two vectors are indistinguishable down to

a given precision. The precision of the sensor and the amount of noise present will determine


the minimum difference that must exist between two values for them to be reliably distin-

guishable. Calling this minimum distance δ, two vectors v1 and v2 are deemed distinguishable

if they meet the condition in Equation 3.1.

‖v1 − v2‖∞ > δ (3.1)

The remaining necessary step before being able to apply this criterion is to determine

δ. We can consider sensor precision and measurement noise within a single framework by

interpreting sensor error as noise. To relate δ to the noise level, two values will be considered

distinguishable if they are separated by at least twice the standard deviation of the noise. By

using the definition of noise-to-signal ratio (NSR) in Equation 3.2 (the standard deviation

of the elements of the noise vector over the standard deviation of the elements of the signal

vector), the condition stated above can be reformulated as shown in Equation 3.3.

NSR =std(noise)

std(signal)(3.2)

‖v1 − v2‖∞ > 2 ∗NSR ∗ std(signal) (3.3)

Using this criterion, groups of indistinguishable adjacent elements can be formed. The

leadfield columns corresponding to all the elements in a group are then replaced by a single

vector equal to the average of all the selected original columns, thereby reducing the total

number of columns in the leadfield.

It is typical for the leadfield in a bioelectric source localization problem to contain

three columns for each location to be solved for, corresponding to the three orthogonal

components of the dipole at that location. In that case, two elements can truly be said to

be indistinguishable only if all three pairs of leadfield columns prove indistinguishable. In

addition, in order to guarantee that the elements are indistinguishable for a unit source of

any orientation, the threshold in Equation 3.3 must be divided by√

3. To see this, consider

the following argument. If the elements are distinguishable for one of the three orthogonal


orientations corresponding to the leadfield columns, then they are not grouped and no further

examination is necessary. Therefore, for the rest of this discussion, the following inequalities

are assumed to be true:

‖v11 − v21‖∞ ≤ δ (3.4)

‖v12 − v22‖∞ ≤ δ (3.5)

‖v13 − v23‖∞ ≤ δ (3.6)

where v11, v12, and v13 are the three leadfield columns corresponding to the first mesh element,

and v21, v22, and v23 are those corresponding to the second mesh element. If the orientation

of the source is described by the triplet (a,b,c), then the measurements produced when that

source is placed in the first of the two mesh elements will be given by a ∗ v11 + b ∗ v12 + c ∗ v13,

with an analogous expression for the second mesh element. The difference between the

measurements produced by that source in each of the two mesh elements will then satisfy

the following inequality, which follows from Equations 3.4 to 3.6 and the properties of the

infinity norm:

‖a(v11 − v21) + b(v12 − v22) + c(v13 − v23)‖∞ ≤ (a+ b+ c)δ (3.7)

Because the source under consideration has unit magnitude, the values (a,b,c) must

satisfy the constraint:

√a2 + b2 + c2 = 1 (3.8)

It is easily verified that the highest value of (a+ b+ c) attainable under constraint

3.8 is√

3. Combining this information with Equation 3.7, we can conclude that in order to

guarantee that the measurement difference will remain under a certain threshold ∆ regardless

of the orientation of the unit source, then the threshold used in Equations 3.4 to 3.6 should

be equal to δ = ∆/√

3.


In peripheral nerves, however, the current sources generating the electric potentials

outside of the axon can be modeled as dipoles oriented along the axis of the nerve [92]. We

will therefore restrict the solution space to dipoles oriented in that direction only, since the

components in the other two directions should be significantly smaller. This choice reduces

the size of the leadfield from Nx3M to NxM. Furthermore, the grouping of elements is based

on the comparison between a single pair of vectors, instead of three pairs, and using the

threshold in Equation 3.3 without adjustment.

It should be noted that the criterion outlined here assumes that there is no significant

difference in the noise level at the various electrode contacts. This is a simplification, because

potentials originating from sources outside the cuff vary linearly with longitudinal position

along the cuff [2], such that noise levels will be different for contacts located at different points

along the cuff. If sufficient information about the noise distribution at the various contacts

was available, it could be incorporated into the distinguishability criterion by making the

threshold different for each entry in the vector difference.

3.3 Methods

3.3.1 Finite element model and leadfield construction

An FE model of a unifascicular section of the rat sciatic nerve was constructed for this portion

of the study. The model consisted of a cylindrical nerve surrounded by a cuff electrode. The

nerve consisted of a single fascicle and was modeled as three concentric cylinders representing

the endoneurium, perineurium and epineurium layers. The nerve and the cuff were separated

by an encapsulation tissue layer and a saline layer. The whole structure was placed in a saline

bath. The dimensions and conductivities of the various part of the nerve model are given

in Table 3.1 and based on similar models and anatomical studies described in the literature

(relevant references for each parameter are given in the table).

From this model, five meshes of varying coarseness were generated. The solution space


Table 3.1: Parameters for the finite element model of the rat sciatic nerve.

Parameter Values References

Nerve length 5 cm

Endoneurium radius 415 µm [151]

Perineurium width 35 µm [36, 119, 125]

Epineurium width 35 µm [36, 119]

Encapsulation tissue layer width 7.5 µm

Saline layer width 7.5 µm

Cuff length 2.3 cm [157]

Cuff width 30 µm [157]

Cuff radius 500 µm [157]

Cuff starting height 1.35 cm

Saline bath length 5 cm

Saline bath radius 0.485 cm [125, 197]

Endoneurium conductivity 8.26 × 10−2 S/m [36, 98, 197]

(radial)

Endoneurium conductivity 0.571 S/m [36, 98, 15, 197]

(longitudinal)

Perineurium conductivity 2.1 × 10−3 S/m [36, 15, 197]

(all directions)

Epineurium conductivity 8.26 × 10−2 S/m [15, 197]

(all directions)

Encapsulation tissue conductivity 6.59 × 10−2 S/m [119]

(all directions)

Saline conductivity (all directions) 2 S/m [36, 119, 15, 197]

Cuff conductivity (all directions) 1 × 10−7 S/m [197]


for the source localization problem was restricted to the endoneurium. The number of mesh

elements for the endoneurium region only was 75600, 38400, 19200, 12000, and 8400 for the

five meshes, respectively. The other variable of interest is the number of contacts in the cuff

electrode. Four values were considered: 104, 56, 24, and 12. The 56 contacts were chosen

from the original 104, the 24 from the 56, and the 12 from the 24, such that each set was a

subset of the previous case (specifically, the contact configurations consisted of 13 rings of 8

contacts, 7 rings of 8 contacts, 3 rings of 8 contacts, and 3 rings of 4 contacts, with the rings

in all cases being positioned symmetrically with respect to the middle of the cuff). The 56-

contact electrode was modeled on the MCC used in the rest of this thesis [157], and the other

cases are merely extensions or subsets of this pattern of contacts. The electrode dimensions

in the model are also taken from this design. The 1 mm diameter of this cuff is very close to

the approximate diameter of the rat sciatic nerve, hence the tight fit of the cuff in the model.

In total, the five coarseness levels and 4 electrode patterns resulted in 20 leadfield matrices.

The finite element analysis was conducted using the SCIRun computing environment (an

open source platform for the modeling, simulation and visualization of scientific problems

[158]), and the rest of the leadfield computations were performed using the Matlab software

(The Mathworks Inc., Natick, MA). The procedure for obtaining the leadfield from the finite

element model is described by Weinstein et al. [193].

3.3.2 Element grouping algorithm

The algorithm described in the Theory section is designed to reduce a leadfield matrix based

solely on the entries of that matrix, rather than using information contained in a specific set

of measurements. If we want to incorporate information about the noise level into the choice

of threshold, however, the standard deviation of the signal must be known (see Equation 3.3).

In order to circumvent the problem, we base the threshold calculations on a collection of

sample signals obtained from the leadfield columns themselves. Each column corresponds to

the measurements obtained when a unit source is placed at a given location, and therefore


can be seen as a simple sample signal. We choose a subset of all these possible sample

signals. For each of the column vectors in this set, the standard deviation of the entries of

the vector is computed. The average of these standard deviation values is then used as the

signal standard deviation in Equation 3.3. The columns included in the subset are those

corresponding to the mesh elements lying in the endoneurium between the heights of 2 and

3 cm along the 5 cm nerve model, under the assumption that the signals originating close

to the midpoint of the cuff are most representative of the signals that we are interested in

localizing.

Having now obtained all the information necessary for determining whether two el-

ements are distinguishable, we can proceed to use this criterion to form groups of indistin-

guishable elements. Although theoretically groups could consist of non-adjacent elements

spread throughout the solution space, the leadfield reduction process could then no longer be

interpreted as a search for the inherent resolution achievable by the measurement setup. In

this study, the additional restriction is therefore imposed that groups should be formed from

connected mesh elements. This has the additional advantage of considerably reducing the

time required to test every eligible pair of elements for distinguishability. To further simplify

the grouping process, we use the fact that the mesh was constructed using an extrusion pro-

cess that results in distinct layers of elements, and only compare elements to others within

the same layer. Within each layer of the mesh, the algorithm used is as follows. First, a

chain of connected elements is formed, going along the outer edge of the layer and spiraling

inwards until the center is reached. If a point is reached where all the elements connected to

the current element are already part of the chain, then the chain continues at the next free

element. Once all the elements have been added, comparisons are performed on successive

elements of the chain. For instance, the first and second elements are compared, and if they

are indistinguishable they are grouped together. If the third element is also indistinguish-

able from both of the first elements, it is added to the group. This process is continued

until an element is found that cannot be added to the group. A new group is then begun,

and so on until all the end of the chain is reached. A sample grouping is shown in Figure


Figure 3.1: Example of the results obtained when the grouping algorithm is applied to

a layer of the FE model at z = 24.94mm, under the assumption of 0.1% NSR. Elements

surrounded by thick lines belong to the same group.

3.1, corresponding to a layer of the FE model midway up the cuff electrode (this particular

grouping is based on the assumption of a 0.1% NSR).

The algorithm just described clearly does not capture all possible groupings. The

chaining process essentially restricts the comparisons to a one-dimensional geometry instead

of a two-dimensional one, and some elements that are indistinguishable will therefore not be

grouped. Additionally, restricting the comparisons to elements within the same layer will also

omit some possible groupings. The spiral shape is dependent on the geometry of the mesh,

which is irregular and so can cause some distortion and influence which elements are grouped

(this effect is visible in Figure 3.1 toward the centre of the cross-section). Unfortunately,

developing an algorithm able to maximize the number of three-dimensional groupings is far

from a trivial task, and is not the main concern of this study. The chain algorithm used

is capable of significantly reducing the size of a leadfield, as will be shown in the Results

section, and therefore is deemed sufficient for the purposes of this study.


The following NSR values were used to choose a grouping criterion: 0.1%, 5%, 10%,

15%, and 20%. Having an error equal to 0% is unrealistic, if only because of numerical

precision issues. For the purpose of this study, a value of 0.1% was therefore used instead of

0% to capture very small errors, such that elements that are extremely similar will be still

be grouped together.

3.3.3 Leadfield comparison metrics

The success of the leadfield reduction process will be evaluated primarily in terms of the

number of columns of the reduced leadfield, compared to the original leadfield. Nonetheless,

the advantages derived from a smaller matrix may not be worthwhile if the new matrix has

a detrimental impact on the quality of the inverse problem solution. For this reason, it is

important to have a set of metrics that reflect the difficulty of solving the ill-posed problem

using a given matrix.

Rank and condition number

The rank of the leadfield matrix should be equal to the number of electrode contacts. Oth-

erwise, some of the measurements would be linearly dependent on others and not providing

any additional information. It is therefore important to check that a reduced leadfield still

has full row rank, in order to confirm that the reduction process has not led to a loss of

information. The other matrix property that will be examined is the condition number,

which is a measure of the sensitivity of the solution of a linear system of equations to errors

in the right hand side [34, 54]. The condition number can be computed as the ratio of the

largest and smallest singular values of the matrix. Matrices with relatively small condition

numbers are said to be well-conditioned, otherwise they are ill-conditioned. The leadfield

matrix in the source localization problem is typically ill-conditioned, and a variety of regu-

larization techniques exist to deal with this problem (usually by finding ways to minimize

the destabilizing effect of very small singular values, for example by adding a constant to all


singular values or by truncating the smallest ones) [54]. Our concern here is therefore not

the absolute value of the condition number, but we need to check that the condition num-

ber of the reduced leadfield is not significantly greater than that of the original leadfield.

Otherwise, the reduction process may have made the problem more difficult to solve. The

metric used is therefore the ratio of the condition number of the reduced matrix to that of

the original matrix.

Geometry noise

The two main sources of error in the source localization problem are measurement noise

and geometry noise. Geometry noise is the error introduced by inaccuracies in the leadfield

matrix, which occur because the models used to solve the forward problem are not com-

pletely accurate representations of the true anatomy. During the leadfield reduction process,

the columns corresponding to elements deemed indistinguishable are averaged together. De-

pending on how rigorous a threshold is used, this averaging process could introduce some

additional geometry noise. For each column in the original leadfield, a geometry noise vector

is defined as the difference between the original leadfield column and the corresponding col-

umn in the reduced leadfield. By analogy with the measurement noise, the geometry noise

level is then defined by using this geometry noise vector and the original leadfield column

in Equation 3.2. The average of this value for all the columns in the original leadfield is

then used as a metric quantifying the amount of additional geometry noise introduced by

the reduction process. Defining Met1 to be this metric, signali to be the ith column of the

original leadfield, and gei to be the geometry error vector corresponding to that column, the

metric is computed as shown in Equation 3.9 (where M is once again the number of leadfield

columns).

Met1 =

(

M∑

i=1

std(gei)

std(signali)

)

/M (3.9)


3.3.4 Example and Complexity Analysis

Although the rank, condition number and geometry noise metrics have the advantage of not

being dependent on a specific source configuration, it is nonetheless beneficial to consider spe-

cific examples. These examples can provide some confirmation that the leadfield reduction

process does not significantly increase the localization error, as well as illustrate the reduc-

tions in computation time that can be achieved. We focus here on two cases: the 56-contact,

75600-element model, with 0.1% noise and 10% noise. The reduced leadfields constructed for

NSR = 0.1% and NSR = 10% are used for the first and second of those cases, respectively,

and the localizations are compared with those obtained using the original leadfield.

For each case, we generate simulated measurements corresponding to an action po-

tential traveling along a single myelinated fiber. A myelinated mammalian nerve fiber action

potential is first simulated using the model described by Sweeney et al. [172]. In order to re-

main consistent with the EEG/MEG source localization literature, equivalent current dipoles

are used to approximate the electrical activity of the nerve fibers. The magnitude waveform

of the current dipole is therefore obtained from the first derivative of the transmembrane

potential during the action potential [129]. The waveform is then propagated from one node

of Ranvier to the next at a speed of approximately 50 m/s [172]. The nodes of Ranvier are

placed 1 mm apart, which is consistent with a 10 µm-diameter fiber. For the purposes of

this example we restrict the localization to a single time instant, when the action potential’s

peak is in the nodes of Ranvier near the middle of the cuff. Once the current dipole locations

and magnitudes have been obtained in this way, the measurements are computed using the

original leadfield, and the appropriate amount of noise is added.

The source localization is performed using the sLORETA algorithm [123]. For each

of the two example cases, 100 trials are conducted, with the position of the active fiber in

the nerve cross-section generated randomly at every trial. To evaluate the results, the three-

dimensional solution is projected onto a two-dimensional cross-section, because our primary

interest is to determine which pathway is active. From the two-dimensional projection,


localization error is obtained as the distance between the true pathway location and the

location of the peak of the estimate. The error is averaged over 100 trials for each of the two

example cases. Note that in order to compare the results, the reduced estimate is mapped

back to the full solution space by assigning to each variable in the original space the value

of the corresponding variable in the reduced space.

The computation time gains that can be achieved using the leadfield reduction de-

pends on the algorithm used to solve the inverse problem. In the case of sLORETA, recall

from Section 2.2.3 that the first step of the algorithm is to solve a minimum-norm least-

squares problem, shown in Equation 3.10, with the closed-form solution in 3.11.

j = arg minj‖(Lj − d)‖2 + λ‖j‖2 (3.10)

j = Td = LT[LLT + λI]−1d (3.11)

Here, the value of λ is chosen using the cross-validation functional [121]. The second

step of sLORETA is to normalize the solution from 3.11 using the resolution matrix, as


jT

l (Rll)−1jl (3.12)

In the case of constrained orientations, where there is a single variable per dipole

location, jl is simply the magnitude of the lth dipole, and Rll is the lth diagonal entry of the

resolution matrix.

Recall that L is of size NxM, with M much bigger than N. As a result, examination

of Equation 3.11 reveals it to be of complexity O(MN2). Each diagonal entry of R can be

computed in O(N), such that the total normalization process can be performed in O(NM).

The choice of regularization parameter relies on an eigenvalue decomposition that is limited

by the rank N of the leadfield [121], and therefore independent of M. We expect to see these


complexities reflected in the computational times of the examples when the original and

reduced values of M are used.

3.4 Results

3.4.1 Size of the reduced leadfield

The number of columns in the original and reduced leadfield are compared in Figure 3.2, for

the five meshes, four electrode contact sets, and five NSR values. In all cases, the original

leadfield could be reduced to half its original size or less. As more measurement uncertainty

was incorporated into the choice of threshold, the size of the reduced leadfield decreased.

This result is in accordance with expectations, since it implies that noise deteriorates the

achievable resolution.

When the grouping thresholds were based on the assumption of noiseless measure-

ments (NSR = 0.1% in Figure 3.2), the number of electrode contacts had very little influence

on the number of columns in the reduced leadfield, suggesting that in ideal conditions the

geometries and conductivities of the nerve and cuff may be the key factors influencing the

reduction achievable, rather than the number of electrode contacts used. On the other hand,

when measurement uncertainty was incorporated into the choice of threshold, the number

of electrode contacts started to be more important. Figure 3.2 shows that in those cases

the reduced leadfields had progressively more columns as the number of electrode contacts

increased, meaning that the increased number of measurements slightly improved the res-

olution in the presence of noise. The sizes of the reduced leadfields corresponding to the

different contact configurations did not, however, vary linearly with the number of contacts.

This suggests that an optimal number of contacts could be found, representing a good trade-

off between achieving better resolution and increasing the number of contacts necessary. In

the present study, the difference between 12 and 24 contacts was very small, implying that

very little is to be gained from using the larger of those two sets. On the other hand, the


0 2 4 6 8

x 104

0

1

2

3

4x 10

4 NSR = 0.1%

Original size

Red

uced

siz

e

104 contacts56 contacts24 contacts12 contacts

0 2 4 6 8

x 104

0

1

2

3

4x 10

4 NSR = 5%

Original size

Red

uced

siz

e

0 2 4 6 8

x 104

0

1

2

3

4x 10

4 NSR = 10%

Original size

Red

uced

siz

e

0 2 4 6 8

x 104

0

1

2

3

4x 10

4 NSR = 15%

Original size

Red

uced

siz

e

0 2 4 6 8

x 104

0

1

2

3

4x 10

4 NSR = 20%

Original size

Red

uced

siz

e

Figure 3.2: Number of columns of the reduced leadfield as a function of the number of

columns of the original leadfield and of the number of electrode contacts, when the grouping

criterion is based on NSR = 0.1%, 5%, 10%, 15%, or 20%. The curves for 104 and 56

electrode contacts nearly overlap, as do the ones for 24 and 12 electrode contacts.


difference between 24 and 56 contacts was appreciable. The difference between 56 and 104

contacts was also noticeable but somewhat smaller, which suggests that out of the four con-

figurations examined, the 56-contact configuration is the best choice for this problem. As

the noise level continued to increase and the size of the all the reduced leadfields decreased,

however, the size differences due to the number of contacts became less significant (compare

for example the NSR = 5% and NSR = 20% cases in Figure 3.2). In other words, there

was a limit to the amount of uncertainty that the number of contacts could compensate for.

Lastly, it is important to note that all of these observations are valid for all five of the meshes

investigated.

3.4.2 Properties of reduced leadfield

The reduced matrices had full row rank for every combination of mesh, number of contacts,

and noise level, meaning that the reduction process never led to any measurements becoming

linearly dependent on others.

The ratio of condition numbers between reduced and original leadfields was in all

cases smaller than 1, indicating that none of the leadfields became more ill-conditioned than

they already were as a result of the reduction process. The mean of these ratios for all the

reduced leadfields constructed was 0.905 ± 0.043, although no significant relationship to the

mesh coarseness, number or contacts, or noise level was observed.

Table 3.2 shows the values of the Met1 metric for all of the reduced leadfields. As

expected, the amount of geometry error increases as the criterion for indistinguishability

is relaxed (a clear example is provided by the row in bold). For all cases, the geometry

error remained relatively small, with Met1 remaining under 2.5% when the least stringent

grouping criterion is used, and under 1% when the most stringent criterion is used. These

values suggest that the increased geometry error is a small enough price to pay for the

significant reduction that was achieved in the number of variables.


Table 3.2: Metric Met1 for all test cases.

NSR = 0.1% NSR = 5% NSR = 10% NSR = 15% NSR = 20%

104 contacts

75600 elements 0.6 +/- 1.4% 1.1 +/- 1.7% 1.3 +/- 1.7% 1.6 +/- 1.8% 1.8 +/- 1.9%

38400 elements 0.6 +/- 1.5% 1.1 +/- 1.9% 1.4 +/- 1.9% 1.6 +/- 1.9% 1.9 +/- 2.0%

19200 elements 0.7 +/- 1.8% 1.3 +/- 2.3% 1.6 +/- 2.3% 2.0 +/- 2.3% 2.1 +/- 2.3%

12000 elements 0.7 +/- 2.3% 1.3 +/- 2.7% 1.7 +/- 2.7% 1.9 +/- 2.7% 2.0 +/- 2.7%

8400 elements 0.4 +/- 1.6% 0.7 +/- 2.1% 0.9 +/- 2.2% 1.2 +/- 2.2% 1.3 +/- 2.3%

56 contacts

75600 elements 0.6 +/- 1.4% 1.1 +/- 1.7% 1.4 +/- 1.7% 1.7 +/- 1.8% 2.0 +/- 2.0%

38400 elements 0.6 +/- 1.5% 1.1 +/- 1.9% 1.4 +/- 1.9% 1.7 +/- 2.0% 2.0 +/- 2.1%

19200 elements 0.7 +/- 1.8% 1.3 +/- 2.3% 1.7 +/- 2.3% 2.0 +/- 2.3% 2.2 +/- 2.4%

12000 elements 0.7 +/- 2.3% 1.4 +/- 2.7% 1.7 +/- 2.7% 1.9 +/- 2.7% 2.1 +/- 2.7%

8400 elements 0.4 +/- 1.6% 0.7 +/- 2.1% 1.0 +/- 2.2% 1.2 +/- 2.2% 1.4 +/- 2.3%

24 contacts

75600 elements 0.6 +/- 1.4% 1.3 +/- 1.7% 1.6 +/- 1.8% 1.9 +/- 2.1% 2.1 +/- 2.5%

38400 elements 0.6 +/- 1.6% 1.3 +/- 1.8% 1.6 +/- 1.9% 1.9 +/- 2.2% 2.1 +/- 2.5%

19200 elements 0.7 +/- 1.8% 1.5 +/- 2.2% 2.0 +/- 2.3% 2.2 +/- 2.5% 2.4 +/- 2.7%

12000 elements 0.8 +/- 2.4% 1.6 +/- 2.7% 1.9 +/- 2.7% 2.1 +/- 2.7% 2.2 +/- 2.8%

8400 elements 0.4 +/- 1.6% 0.9 +/- 2.1% 1.1 +/- 2.2% 1.3 +/- 2.4% 1.4 +/- 2.6%

12 contacts

75600 elements 0.6 +/- 1.4% 1.3 +/- 1.7% 1.7 +/- 1.9% 2.0 +/- 2.2% 2.2 +/- 2.6%

38400 elements 0.6 +/- 1.6% 1.4 +/- 1.9% 1.7 +/- 2.0% 2.0 +/- 2.3% 2.2 +/- 2.7%

19200 elements 0.7 +/- 1.8% 1.6 +/- 2.2% 2.1 +/- 2.4% 2.3 +/- 2.6% 2.4 +/- 2.7%

12000 elements 0.8 +/- 2.4% 1.7 +/- 2.7% 2.0 +/- 2.7% 2.1 +/- 2.8% 2.3 +/- 3.0%

8400 elements 0.4 +/- 1.6% 0.9 +/- 2.1% 1.2 +/- 2.2% 1.4 +/- 2.4% 1.5 +/- 2.6%


Figure 3.3: 2D projections of the true source distribution, the estimate obtained with the

original leadfield, and the estimate obtained with the reduced leadfield, for one trial in the

NSR = 0.1% example.

3.4.3 Simulation results

Figure 3.3 shows the 2D-projections of the results of one trial in the 0.1% noise case, using

both the original and reduced leadfields. It is clear from the figure that the localization

performance for this trial was virtually identical regardless of whether the original or the

reduced leadfield was used. Table 3.3 displays the average over 100 trials of the localization

error for both examples, with the original and reduced leadfields in each case. The results

show that the mesh reduction process resulted in a negligible localization error increase in the

first example and a small decrease in the second example. These results support our claim

that the reduction process does not significantly reduce the quality of the inverse problem

solution.

Table 3.4 displays the total computation times for the two examples, as well as the

breakdown into the different components of the algorithm. The table also shows the ratios

of these different values for the simulations conducted with the original and reduced lead-

fields. The computation times were obtained using Matlab running on a desktop PC with a

3.0 GHz Pentium IV processor. As expected, the reduction had no effect on the speed of the

regularization process, and the ratio of computation times for the normalization step was

roughly equal to the ratio of the number variables in the two leadfields. The ratio for the


Table 3.3: Simulation results using the original and reduced leadfields. All simulations were

conducted with 56 contacts, and all means correspond to 100 trials.

NSR = 0.1% NSR = 10%

Size of the reduced leadfield 30252 21472

(original leadfield size is 75600)

Mean localization error using 0.0801 +/- 0.0505 0.1359 +/- 0.1108

the original leadfield (mm)

Mean localization error using 0.0807 +/- 0.0505 0.1333 +/- 0.0940

the reduced leadfield (mm)

solution of the minimum-norm least-squares problem was slightly larger than expected, but

a substantial reduction was nonetheless achieved. The discrepancy between the expected

and observed reduction may be due to the fact that our analysis was based on a “naıve”

approach to matrix multiplications and did not take into account any of the optimizations

that may be present in the Matlab software.

3.5 Discussion

By using the similarity of leadfield columns as a criteria for grouping several mesh ele-

ments into a single variable, we were able to achieve substantial reductions in the number

of variables that the inverse problem aims to estimate. This solution space reduction was

investigated in the context of the peripheral nerve source localization problem, although it

could also be applied to other bioelectric source localization tasks. The reduction is valu-

able for several reasons. First, the inverse problem is made less ambiguous by the clearer

distinctions between the leadfield columns. The smaller discrepancy between the number of

measurements and the number of variables also means that the problem will be somewhat

better conditioned. Most importantly, the smaller leadfield matrix will translate into faster

computations. This can be particularly advantageous when iterative algorithms are used


Table 3.4: Computation time comparison for simulations using the original and reduced

leadfields (all values are means over 100 trials.)

NSR = 0.1% (Mreduced/Moriginal = 0.4)

Computation step Computation time Computation time Ratio

(original) (reduced) (reduced/original)

Regularization 0.3136 s 0.3046 s 0.9918

Minimum Norm Least-Squares 0.3926 s 0.2467 s 0.4865

Normalization 0.3532 s 0.1397 s 0.3959

Total 1.0594 s 0.6910 s 0.6123

NSR = 10% (Mreduced/Moriginal = 0.284)

Computation step Computation time Computation time Ratio

(original) (reduced) (reduced/original)

Regularization 0.2611 s 0.2615 s 1.0022

Minimum Norm Least-Squares 0.3290 s 0.1176 s 0.3576

Normalization 0.3427 s 0.0953 s 0.2781

Total 0.9328 s 0.4743 s 0.5087

(e.g. FOCUSS [37]), and would become crucial if real-time implementation were attempted

as part of a control system for a neural prosthesis. The reduction in required storage space

would also be valuable for implanted systems. The reduction process did not have a negative

impact on any of the metrics used to assess the difficulty of the inverse problem.

Despite these advantages, however, it should be kept in mind that the smaller number

of variables is a mixed blessing, since it implies lower resolution. The purpose of the technique

presented here is not to make the number of variables arbitrarily small, but rather to try to

approach the number that best represents the inherent resolution that is achievable for the

given nerve properties and measurement setup, without having superfluous variables that

simply make the problem more difficult. In that sense, the proposed technique provides


us with information that could not be obtained using a simple decimation approach, and

is therefore preferable even if both methods provide the same amount of solution space

reduction.

The incorporation of information about the noise level in the criterion to determine the

distinguishability of two mesh elements was also explored. Not surprisingly, increasing the

uncertainty increased the number of element groupings, because pairs that were only barely

distinguishable using perfect measurements could no longer be reliably separated. This is

simply another way of saying that uncertainty in the measurements will negatively impact the

achievable resolution. Additionally, it was found that a large number of electrode contacts

had more impact on grouping decisions when uncertainty was present. Indeed, having more

contacts would provide better spatial sampling of the electric fields, and therefore provide

more opportunities to detect differences between the fields generated by sources in adjacent

locations when very small differences are obscured by noise. This finding argues in favor of

using a large number of contacts in practice, where noise cannot be completely avoided. On

the other hand, it was found that increasing the number of contacts past a certain number

started yielding diminishing returns, suggesting that an optimal number could be found. The

impact of the number of contacts on the selectivity of cuff electrodes has been previously

studied by Yoo and Durand [197], who also noted the existence of a plateau. The optimal

number in that study proved to be 7, however there was a single ring of contacts. The

higher number found here suggests that it is possible to take better advantage of having

more contacts by distributing them in several rings along the length of the cuff, rather than

a single ring. It is important to clarify that our conclusion about the existence of an optimal

number of contacts applies only to improvements in resolution, and says nothing about

the impact that a larger number of measurements will have on the accuracy of the inverse

problem solution.

Note also that in practice, precise information on the noise level may in many cases

be unavailable, in which case grouping decisions will have to be based on the NSR = 0.1%

criterion, or on a conservative noise estimate. Otherwise, some resolution might be needlessly


lost.

It should be kept in mind when analyzing the results of this study that the specific

size reductions achieved are dependent on the grouping algorithm, which is not optimized.

Small changes in the algorithm will affect both the groupings that are formed among indis-

tinguishable elements and the overall number of groupings. It would therefore be worthwhile

to continue improving the grouping algorithm in order to maximize the achievable reduction.

Nonetheless, the present study demonstrated that even with a simple algorithm a substan-

tial decrease in the number of variables could be obtained, highlighting the usefulness of this

solution space reduction technique.

Another factor that was fixed in this study was the geometry of the nerve. Although

the geometry was simplified here to a single cylindrical fascicle, the solution space reduction

technique is in no way dependent on this geometry. In can be applied to any mesh geom-

etry for which a leadfield has been computed. Certain situations, such as a mesh that is

not comprised of well-defined layers or whose cross-section is not roughly compatible with a

spiral shape, may require modifications of the grouping algorithm, but the basic reduction

strategy would remain valid. Changing the geometry would of course alter the exact amount

of reduction achieved, but since the technique proposed here is simply exploiting the limited

resolution inherent in cuff electrode measurements, substantial reductions should be achiev-

able regardless of the details of the geometry. Likewise, the exact impact of the geometry on

the relationship between the number of contacts and the amount of reduction has not yet

been established in the general case, but the underlying insight that having more contacts

can compensate up to a point for the loss of information due to noise is independent of

geometry.

Chapter 4

Simulation of Bioelectric Source

Localization in the Rat Sciatic Nerve

4.1 Introduction

Before attempting to apply bioelectric source localization to experimental peripheral nerve

recordings, we conducted a simulation study of the process. The goal of the simulations was

to obtain an estimate of the expected performance under different conditions. Specifically,

we are interested in how effectively the performance can be improved by constraining the

problem in physiologically meaningful ways, and what amount of performance degradation

can be expected as the noise level increases.

4.2 Methods

4.2.1 Approach to solving the source localization problem

In order to obtain an estimate of the source distribution from the cuff electrode measure-

ments, there are two problems that need to be addressed: the forward and the inverse

problem. As described earlier, the goal of the forward problem is to compute the measured70

Simulation of Bioelectric Source Localization in the Rat Sciatic Nerve 71

potentials that would result at each contact from a source at a given location. This informa-

tion can be represented in matrix form, and is known as the leadfield matrix. The leadfield

is needed to solve the inverse problem, whose goal is to estimate the source distribution from

the measurements.

Due to the ill-posed and underdetermined nature of the inverse problem of bioelectric

source localization, in order to arrive at a unique solution it is necessary to impose con-

straints on the solution [104]. These constraints should be determined by the anatomical

and physiological features of the problem at hand. In our simulations, we investigated the

performance of the source localization approach when varying levels of information about the

nerve were incorporated into the problem. An overview of the process is given in this section,

and details about each aspect of the simulations are presented in the following sections.

1. Create two numerical models of the nerve: a) one with an idealized geometry and b)

the other with a more realistic geometry. A leadfield was generated from each model,

and the more realistic model was used to generate simulated measurements, which were

used in all simulations.

2. Develop a spatio-temporal constraint based on the electrophysiological behaviour of

myelinated axons.

3. Evaluate the localization performance on four cases, which were: (1) using the idealized

leadfield and no spatio-temporal constraint (case IL), (2) using the idealized leadfield

and the spatio-temporal constraint (IL-C), (3) using the correct leadfield (i.e. the one

obtained from the more realistic geometry and used to generate the measurements) and

no spatio-temporal constraint (CL), and (4) using the correct leadfield and the spatio-

temporal constraint (CL-C). The first of theses cases incorporates the least information

about the problem into the inverse problem solution, whereas the last case incorporates

the most information.


The forward problem

The forward problem is well-defined and can be solved analytically for simple geometries

and conductivity distributions. However, the anisotropic conductivity of nerves and the

potentially irregular shape of the fascicles mean that we must resort to numerical techniques.

Specifically, the forward problem is solved using FEM. In EEG/MEG source localization, the

method of choice is often BEM (e.g. [109, 163, 194]), but FEM is better suited to anisotropic

conductivities [193, 160] and for that reason is used here.

The first FEM model was built based on the idealized geometry of a unifascicular

section of the rat sciatic nerve. This extended unifascicular geometry is a simplification

compared to the real anatomy, which would branch progressively into several fascicles. The

main components of the model were a cylindrical nerve surrounded by a cuff electrode and

placed in a saline bath. The nerve was modeled as three concentric cylinders representing

the endoneurium, perineurium and epineurium layers. The nerve and the cuff were separated

by an encapsulation tissue layer and a saline layer (left panel of Figure 4.1). The dimensions

and conductivities of the various part of the nerve model are the same as those given in

Table 3.1, with the exception of the parameters that are shown in Table 4.1 along with their

new values. 56 electrode contacts were placed on the inside of the nerve cuff, organized in 7

rings of 8 electrodes each. The dimensions of the cuff electrode and the layout of its contacts

are based on the “matrix” MCC used throughout this project [157]. The ratio of the bath

and nerve diameters is large enough to avoid boundary effects [197], and the nerve segment

is long enough that dipoles placed at its ends have a negligible impact on the measurements.

The number of mesh elements in the model’s endoneurium was 56,400.

The second FEM model was based on a trace of a cross-section of a rat sciatic nerve,

at the level where the nerve begins to divide into its tibial and peroneal branches. The image

that formed the basis of the trace was obtained from the literature (Figure 1C in [151]). The

dimensions of the cuff and the bath were the same as in the previous model, as were the

conductivities. In order to fit the nerve into the cuff, it was scaled to 90% of its original size.


Figure 4.1: Cross-sections of the idealized (left) and more realistic (right) FEM geometries

used to construct the leadfields.

This is a considerably simpler process than morphing the mesh to conform to the shape of

the cuff, and was deemed acceptable for the purposes of this study. It is important to clarify

that this geometry is still simplified, in that it is uniform in the longitudinal direction, and

as such does not represent the progressive branching of the nerve. The cross-section that

was selected as the basis for the trace was chosen because it corresponds approximately to

where the half-point of the cuff would be located on the sciatic nerve, assuming the cuff was

implanted just proximal to the division of the nerve into its main branches. It has been shown

that the shape of the nerve will conform to that of the cuff during chronic implantation [139].

Therefore, the geometry used here corresponds to a situation in which the cuff has only been

implanted for a short time, which is interesting for the purposes of this study, because the

difference between the “realistic” geometry and the idealized one will be greatest at that

time. The cross-sections of both FEM models are shown in Figure 4.1. The number of mesh

elements in the model’s endoneurium was 218,400 (the more complex geometry required a

finer mesh).


Table 4.1: Parameters for the idealized finite element model of the rat sciatic nerve.

Parameter Values References

Endoneurium radius 360 µm [151]

Epineurium width 35 µm [36, 119]

Encapsulation tissue layer width 40 µm [125]

Saline layer width 40 µm [125, 119]

Saline bath radius 0.48 cm [125, 197]

The finite element analysis was conducted using the SCIRun computing environment

[158], and the rest of the leadfield computations were performed using the Matlab software.

The procedure for obtaining the leadfield from the finite element model is described by

Weinstein et al. [193].

Simulated measurements

To generate simulated measurements, a myelinated mammalian nerve fiber action potential

was first simulated using the model described by Sweeney et al. [172]. In order to remain

consistent with the EEG/MEG source localization literature, equivalent current dipoles were

used to model the electrical activity of the nerve fibers. The magnitude waveform of the

current dipole was therefore obtained from the first derivative of the transmembrane potential

during the action potential [129]. The waveform was then propagated from one node of

Ranvier to the next at a speed of approximately 50 m/s [172]. The nodes of Ranvier were

placed 1 mm apart, which is consistent with a 10 µm-diameter fiber. The length of the

simulation was 2 ms.

Once the locations and time courses of the current sources had been determined,

the simulated measurements at the electrode contacts were obtained using the second FEM

model described in the previous section. The reference for the measurements was the average

of the two reference contacts present in the “matrix” cuff design (see [157]). Noise was then


added to each set of measurements, with the noise standard deviation set to 0%, 10%,

20%, 30%, or 40% of the signal standard deviation. The signal standard deviation for the

purposes of generating the noise was estimated by computing the standard deviations of the

measurements at each of the 8 contacts in the middle ring of the cuff, then averaging those

values. The resulting noise standard deviation was used to generate Gaussian white noise

time series for each of the 56 contacts. This noise represents the remaining contamination

after appropriate filtering and noise reduction measures have been used.

Measurements corresponding to two situations were simulated: a single active fiber,

and three active fibers. Each of the two cases was repeated 100 times, with the positions

of the active fibers within the endoneurium generated randomly every time. For the case of

three active fibers, the waveforms for all three fibers were identical but a random time shift

was applied. The maximum allowable time shift was a quarter of the length of the simulation.

Given the distance between the nodes of Ranvier and the length of the nerve model, each

fiber was composed of 50 dipoles, each with its own time course. The source localization

task described in the following sections was therefore dealing with regions featuring either

50 or 150 dipoles with varying magnitudes.

The inverse problem

There are two broad categories of EEG/MEG source localization methods: equivalent dipole

methods and distributed linear methods. Equivalent dipole methods assume that the po-

tential measurements can be explained using a small number of equivalent dipoles, whose

number is set a priori or estimated using the data and whose location, orientation, and

magnitude must be determined using search algorithms to fit the measurements (see Section

2.2.2). In the peripheral nerve problem, the combination of multiple active fibers and the

distribution of activity along the length of the fibers make the assumption of a small number

of dipoles very hard to justify. The distributed linear methods, on the other hand, formulate

the problem as the estimation of the magnitudes of a large number of dipoles whose loca-


tions and orientations are fixed. Electric fields add linearly, so the relationship between the

measurements and the dipole magnitudes can be expressed as the linear system in Equation

4.1.

d(n) = Lj(n) + ǫ(n) (4.1)

Here, d(n) is the Mx1 vector of measurements at instant n, j(n) is the 3Nx1 vector

of current dipole magnitudes at instant n, and ǫ(n) is additive noise (assumed Gaussian).

The Mx3N matrix L is the leadfield matrix, which relates the potential measurement at each

electrode contact to the existence of a unit source at each possible location. It is constructed

as described in the section on the forward problem above. There are three leadfield columns

for each of N mesh elements, corresponding to the three orthogonal dipole orientations.

Note that the system at each time instant can be considered as independent of the other

time instants because quasistatic conditions hold [92].

The difficulty of recovering j(n) from d(n) and L comes from the fact that the problem

is underdetermined and ill-posed. In order to obtain a stable solution, additional constraints

must be added to the problem. An overview of the most commonly used types of constraints

was given in the Literature Review section, and can also be obtained in several reviews

[122, 104]. Here, the method chosen is the standardized low resolution brain electromagnetic

tomography (sLORETA) method [123]. In the absence of additional information about the

solution, sLORETA usually outperforms other instantaneous distributed linear methods, but

does this at the cost of decreased spatial resolution. In other words, it produces images that

are blurred but have well-localized peaks; this smoothness is imposed on the solution as a

constraint to deal with the ill-posed nature of the problem. The algorithm has been shown

to localize single sources with zero error in the noiseless case [123]. sLORETA consists of two

steps: first, a regularized minimum-norm least-squares solution of Equation 4.1 is obtained;

second, that solution is normalized using the diagonal of the resolution matrix, which pro-

vides information about the bias of the solution for each entry in j(n). The regularization


parameter is chosen here by means of the cross-validation error function, as suggested by

Pascual-Marqui [121, 123].

Our choice of algorithm is based on three criteria. First, it should not make the

assumption that there are only a small number of active sources, as was already discussed

above. Second, it should be reasonably fast, in view of possible application in a neuro-

prosthetic system. Lastly, it should not assume that the source distribution generating the

measurements is stationary over a certain time interval, because outside of a controlled ex-

periment there would be no way to ensure that the set of active pathways is not changing

over the observation period. sLORETA satisfies all three of these requirements.

In order to reduce the number of variables that need to be solved for in the inverse

problem, we can further restrict the location and orientation of the dipoles. Active fibers can

only be located in the endoneurium, so the perineurium and epineurium regions are removed

from the solution space. The current dipoles that are used to model the extracellular field of

an action potential are oriented axially along the fiber [92], so that dipoles in the other two

orthogonal directions can be eliminated from consideration, thereby reducing the number of

variables by two thirds. In a real nerve, the fibers are not completely straight, but rather

exhibit a slight wave pattern, meaning that our decision to use a single dipole direction is

an approximation. Nonetheless, dipoles along the axis of the nerve will capture the largest

component of the activity. When we proceed to using this methodology on experimental

data in the next chapter, the inaccuracy introduced by neglecting the radial components of

the current dipoles should lead to a considerably smaller degradation in performance than

that which would result from tripling the number of variables.

Spatio-temporal constraint

Keeping in mind that the basic approach to solving underdetermined inverse problems is

to constrain the solution based on our knowledge of the problem, we investigate a spatio-

temporal constraint based on the electrophysiology of nerve fibers. Assuming that we are


applying what follows to myelinated fibres, a spatio-temporal constraint can be implemented

based on the formula in Equation 4.2.

V (x, t) = V (x− ds, t− ds/v) (4.2)

V(x,t) is the transmembrane voltage at position x along the fibre and at time t, ds

is the spatial distance separating two consecutive nodes of Ranvier, and v is the conduction

velocity. ds and v can be estimated from the diameter and type of the fibre. The formula is

simply saying that the activity at a node is ideally identical to the activity at the previous

node ds/v seconds in the past.

Putting this equation to use assumes that the type (myelinated vs. unmyelinated)

and diameter of the fibres at a given location in the nerve are known. Alternatively, even if

no information about the location of different fibre types is available, the constraint can be

applied if a single type of fiber is active at a given time, by assuming that the whole nerve

is composed of fibers of that type. In that situation, before performing the localization, one

can identify the nerve conduction velocity (and thus fiber type) from the MCC electrode

using a method such as the one proposed by Rieger et al. [136].

In order to incorporate this information as a constraint, temporal coupling must be in-

troduced into the problem. Ideally, the whole spatio-temporal system could be expressed and

solved as a single linear system, by concatenating the measurement and source vectors for all

time instants [24, 154]. This is shown in Equation 4.3, which is very similar to Equation 4.1,

with the exception that the vectors dc, jc, and ǫc include the complete information about the

system at all time instants in the observation time window (i.e. dc = [d(1)Td(2)T...d(n)T]T,

with equivalent definitions for jc and ǫc, and n being the total number of time samples). Lc

is a block diagonal matrix, where each block is equal to L.

dc = Lcjc + ǫc (4.3)

The problem with this approach is that the number of variables is multiplied by


the number of time instants, making the system intractable for any realistic application.

Therefore, for computational reasons, we restrict ourselves to coupling two time instants at

a time. Specifically, in accordance with Equation 4.2, we solve the systems corresponding

to the pairs of time instants (t(1), t(1)+ds/v), (t(2), t(2)+ds/v), etc. ds/v is rounded to

the nearest integer. To couple the solutions of the two time instants together in the desired

manner, a coupled leadfield and a non-diagonal weight matrix are constructed as shown in

Equation 4.4. In general, the weight matrix in a distributed linear method for this type of

problem is applied to the norm of the solution in the minimum-norm least-squares problem

and is used to apply a priori constraints to the solution. The incorporation of a weight

matrix into the sLORETA algorithm is described in the original paper by Pascual-Marqui

[123].

Lc =

L 0

0 L

Hc =

I −A0 I

(4.4)

I is the identity matrix corresponding to the number of variables in a single time

instant, and A contains the spatial element of the constraint. If elements i and j of the

solution vector correspond to locations along the same pathway and separated by a distance

ds, then Ai,j is set to 1. In this way, each row of A corresponding to a location in a

constrained pathway contains a single entry of value 1. Hc therefore constrains those values

in the solution at time t+ds/v to have as close a value as possible to the element situated

ds lower in the solution at time t by penalizing differences in these values, as desired.

In the case of our simulations, the constraint assumed that the whole nerve was

composed of myelinated fibres of diameter 10 µm with nodes of Ranvier spaced 1 mm apart

and a conduction velocity of 50 m/s. These values are the same as the ones used to generate

the simulated measurements.


4.2.2 Evaluation of the results

The characteristic of the solution that is of most interest to us is the position of the active

pathways. For that reason, we are primarily interested in the position of the sources in a

cross-section of the nerve, rather than in their longitudinal position. The three-dimensional

solution can therefore be projected onto a two-dimensional cross-section for the purposes

of computing an error metric. This is done by summing the absolute values of the activity

of all the mesh elements with the same cross-sectional position. The value associated with

each location in the resulting two-dimensional projection is therefore a representation of how

much activity was spread along the length of the nerve at that position. This process implies

that a nerve fibre has a straight path with very little radial deviation along the length of

the nerve segment; because there may in fact be small deviations in vivo, we stress that the

actual source localization in no way depends on this assumption. It is simply used to obtain

useful metrics to evaluate the performance of the method in the context of the simulations,

and is justified here because the simulated nerve fibres were in fact straight.

To further simplify the evaluation of the localization process and obtain concise met-

rics, we sum the estimates from all the time instants of a given trial before performing the

two-dimensional projection. The three-dimensional activity over the time interval is there-

fore summarized as a single two-dimensional source distribution. It is important to clarify

that summing the estimates of all the time instants does not presuppose stationarity of the

underlying sources; it simply provides an indication of what sources were active at some

point in the time interval.

Lastly, the estimated source locations are obtained by finding the local maxima in the

final two-dimensional projection. These local maxima are used to estimate the localization

error, as well as the number of missed and spurious pathways. The metrics are computed as

follows:

1. The mesh is interpolated onto a regular grid. The local maxima are detected on this

grid by comparing the activity at each location with the activity at all locations within


50 µm. The peak coordinates are obtained, and the results mapped back onto the

FEM mesh cross-section.

2. Each peak in the estimate is associated with the true pathway closest to it. A local-

ization error is obtained for each of the true pathways by computing the peak-to-peak

distance between that pathway and the closest of the estimated peaks assigned to it.

The other peaks associated with that true pathway, if any, are counted as spurious

peaks.

3. True pathways that do not have any peaks associated with them in Step 2 are counted

as missed pathways.

4.3 Results

4.3.1 One-pathway case

The mean values over 100 trials of the localization error, number of spurious pathways, and

number of missed pathways when a single true pathway is present are shown in Figure 4.2.

When the idealized leadfield was used with no constraints, the localization error was relatively

independent of the noise level, with non-monotonic variations between a minimum of 0.137

mm and a maximum of 0.166 mm. When the spatio-temporal constraint was added, similar

non-monotonic variations were observed between a minimum of 0.134 mm and a maximum

of 0.182 mm. When the correct leadfield was used, a clear relationship with the noise was

observed: the error increase monotonically from 0.078 mm to 0.166 mm without the spatio-

temporal constraint, and from 0.081 mm to 0.175 mm with the constraint. The statistical

significance of this finding was confirmed by the fact that, when testing the relationship

between error and noise against a null hypothesis of no correlation, p was less than 0.05

only in the cases where the correct leadfield was used. The difference in results between

the simulations with the idealized leadfield and those with the correct one is due to the


−5 0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

Noise Level (%)

Loca

lizat

ion

Err

or (

mm

) Average Localization Error, 1−Pathway Case

ILIL−CCLCL−C

−5 0 5 10 15 20 25 30 35 40 45 50

0

2

4

Noise Level (%)

Spu

rious

Pat

hway

s

Average Spurious Pathways, 1−Pathway Case

ILIL−CCLCL−C

−5 0 5 10 15 20 25 30 35 40 45 50−1

0

1

2

3

Noise Level (%)

Mis

sed

Pat

hway

s

Average Missed Pathways, 1−Pathway Case

ILIL−CCLCL−C

Figure 4.2: Localization errors, spurious pathways, and missed pathways for the one-

pathway case (n = 100). IL: idealized leadfield; IL-C: idealized leadfield with spatio-temporal

constraint; CL: correct leadfield; CL-C: correct leadfield with spatio-temporal constraint.

geometry error. In other words, even when the amount of measurement noise is low, the

localization algorithm must still deal with a large amount of error if the leadfield is based

on an inaccurate model of the region. The relatively constant error in the idealized leadfield

cases additionally seems to suggest that there is a plateau in the amount of localization error

as the noise increases, at least for the range of values examined.

Spurious pathways were seen to constitute the biggest obstacle to the applicability of

the source localization approach. The number of spurious pathways increased monotonically

with the noise in all cases (this relationship was statistically significant in all cases except

IL, possibly because the number of spurious pathways seems to reach a plateau early in that


situation). When the idealized leadfield was used, the metric varied from 1.05 to 3.24. When

the spatio-temporal constraint was added, the range of values was from 1.14 to 2.51. When

the correct leadfield was used without the constraint, the range was 0.02 to 2.62, and with

the constraint it was 0.06 to 1.74. As clearly visible in Figure 4.2, this data reveals that

the amount of information incorporated into the source localization problem has a direct

impact on the number of spurious pathways: the spatio-temporal constraint led to a marked

decrease regardless of the leadfield used, and the combination of the correct mesh with the

constraint noticeably outperformed all of the other cases. One-way ANOVA followed by

a multiple comparison test between the four cases, performed at every noise level, showed

statistically significant (p < 0.05) differences between all pairs except the pair (IL-C, CL).

This confirmed the differences qualitatively visible in Figure 4.2. The exception was the

0% noise case, in which the choice of leadfield led to significant differences but the spatio-

temporal constraint did not. Overall, the presence of the spatio-temporal constraint led

to minor increases in localization error at high noise levels, but marked reductions in the

number of spurious pathways, which seems to be a worthwhile trade-off. The error increase

was not statistically significant (p > 0.05), and may be due to the larger number of variables

in the coupled problem. No missed pathways were observed, which is not surprising given

that a single pathway was present in the region.

Figure 4.3 (panels a), c) and e)) shows an example of a localization trial. The smooth-

ing effect of sLORETA is clearly visible, as is the presence of spurious pathways. Note that

the trials shown in this figure (for both the one- and three-pathways cases) are selected to

help the reader visualize the concepts being discussed; they are not necessarily the most

representative of the method’s performance. For that information, the reader should rely

rather on the metrics discussed in the text and shown in Figures 4.2 and 4.4.


Contact column1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

Contact ring

1−Pathway Measurements

(a)

Contact column

Contact ring

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

3−Pathways Measurements

(b)

(c) (d)

(e) (f)

Figure 4.3: Left side: a) Normalized simulated measurements obtained at all 56 contacts

for one of the one-pathway trials with a noise level of 20%. c) Estimated pathways obtained

by applying the source localization procedures to the measurements in a), using the idealized

leadfield combined with the spatio-temporal constraint. e) True location of the pathway for

this trial. Right side: Equivalent data for one of the three-pathways trials. The sharp peaks

seen in the estimates are the output of the peak detection algorithm, superimposed on the

source localization solution. Likewise, the nerve outlines on the floors of figures c)-f) have

been added to help the reader visualize the location of the pathways within the nerve.


4.3.2 Three-pathways case

Panels b), d), and f) of Figure 4.3 show an example trial for the three-pathways case. The

mean values over 100 trials of the localization error, number of missed pathways, and number

of spurious pathways when three true pathways are present are shown in Figure 4.4. The lo-

calization error when the idealized leadfield was used with no constraints was again relatively

independent of the noise level (p > 0.05 under a null hypothesis of no correlation), varying

non-monotonically between 0.152 mm to 0.181 mm. With the spatio-temporal constraint,

the range was 0.155 mm to 0.179 mm. The errors when the correct leadfield was used were

once again an increasing function of the noise (p < 0.05), ranging from 0.083 mm to 0.182

mm without the constraint, and from 0.087 mm to 0.180 mm with the constraint. These

results are very similar to those seen in the one-pathway case.

The general trends for spurious pathways were also similar to those in the one-pathway

case, including the effects of the constraints. Although the increase with noise was not

strictly monotonic in all cases, that was nonetheless the trend (p < 0.05 in all four cases).

The number of spurious pathways when the idealized leadfield was used varied from 0.47 to

1.72 without the constraint, and from 0.64 to 1.15 with the constraint. When the correct

leadfield was used, the range was from 0.02 to 1.24 without the constraint, and from 0.02 to

0.88 with the constraint.

When multiple true pathways were present, the number of missed pathways increased

dramatically. The general trend was an overall decrease as the amount of noise increased,

although this proved statistically significant only in the cases with the correct leadfield.

When the idealized leadfield was used, the number of missed pathways ranged from 0.84 to

0.31 without the constraint, and from 0.84 to 0.58 with the constraint. When the correct

leadfield was used, the range was from 1.44 to 0.38 without the constraint, and from 1.57 to

0.62 with the constraint.


−5 0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

Noise Level (%)

Loca

lizat

ion

Err

or (

mm

) Average Localization Error, 3−Pathways Case

ILIL−CCLCL−C

−5 0 5 10 15 20 25 30 35 40 45 50

0

2

4

Noise Level (%)

Spu

rious

Pat

hway

s

Average Spurious Pathways, 3−Pathways Case

ILIL−CCLCL−C

−5 0 5 10 15 20 25 30 35 40 45 50−1

0

1

2

3

Noise Level (%)

Mis

sed

Pat

hway

s

Average Missed Pathways, 3−Pathways Case

ILIL−CCLCL−C

Figure 4.4: Localization errors, spurious pathways, and missed pathways for the three-

pathways case (n = 100). IL: idealized leadfield; IL-C: idealized leadfield with spatio-temporal

constraint; CL: correct leadfield; CL-C: correct leadfield with spatio-temporal constraint.

4.4 Discussion

The localization of active pathways in a peripheral nerve was approached as an inverse

problem of bioelectric source localization, using simulated measurements from a 56-contact

nerve cuff electrode. This is an ill-posed inverse problem, and problems of this class can

only be solved satisfactorily if sufficient appropriate constraints are imposed on the solution.

We therefore compared the performance of the approach using idealized and correct models

of the nerve geometry, and in the presence of a spatio-temporal constraint based on the

electrophysiology of myelinated nerve fibres. As expected, the overall performance improved

as more information was incorporated. Nonetheless, most of the simulated cases had mean


localization errors in the 140 µm to 180 µm range (in a 720 µm-diameter endoneurium, with

10 µm-diameter individual fibres), and unacceptably high numbers of spurious pathways.

Only at very low noise levels and with accurate constraints did the performance reach levels

that would make the approach reliably usable in practice. Even in those situations, however,

the number of missed pathways was high, due partly to the low resolution of the sLORETA

method. Note that the decrease in the number of missed pathways as noise increased in

the three-pathways case is related to the way that the metrics were computed. Recall that

each true pathway is associated with the closest estimated pathway. Therefore, if there are

several spurious estimated pathways distributed across the region, then there is a greater

chance that some of them will be associated with a true pathway and therefore reduce the

number of missed pathways. This explains why the number of missed pathways decreases

as the number of spurious pathways increases, and furthermore why cases that have the

best performance in the other metrics (e.g. the combination of the correct leadfield and

the spatio-temporal constraint) have more missed pathways. In light of this limitation of

our metrics, the most revealing values for the number of missed pathways are the ones

corresponding to cases with very few spurious pathways. For instance, the simulations using

the correct leadfield with the constraint at 0% and 10% noise have close to 0 spurious

pathways. In those cases, the number of missed pathways is approximately 1.5, or half the

number of true pathways. This is a very high proportion, but it is not entirely unexpected.

Recall that sLORETA imposes smoothness on the solution, and therefore is not capable of

reliably distinguishing closely spaced sources. Therefore, if two pathways are close to one

another, they will be lumped together in the estimate, thereby producing missed pathways.

In addition, our determination of the location of the estimated pathways is based on local

maxima. Upon visual inspection, however, it becomes clear that pathways do not always

produce local maxima, but may significantly affect the shape of the solution in other ways

(for example by producing elongated ridges). There is therefore a need for a better method

of identifying the estimated pathways, but given the variability of shapes and magnitudes

that may occur this is not a trivial problem.


Some remarks are in order regarding the way that information was incorporated into

the problem in this study. The simulations that use the correct leadfield and no noise should

be considered a mostly theoretical situation useful for examining the effects of adding in-

formation, because such an exact correspondence will never be found in practice, even if

detailed anatomical information is available. On the other hand, when noise is incorporated

into the simulations, it can be interpreted as a combination of measurement noise and geom-

etry noise (i.e. mismatch between the leadfield and reality), such that the simulations with

high noise may be more indicative of the expected performance in practical situations where

anatomical information is available. As for the spatio-temporal constraint, it has already

been mentioned that, in the absence of anatomical information about the layout of fibre

types in the nerve, it is useful only if a single type of fibre dominates the nerve’s activity at

a given time. Although it is feasible to ascertain in practice whether or not this assumption

is reasonable [136], it does nonetheless restrict the range of practical situations in which the

constraint would be usable. It is therefore also used in this study as much as a theoretical

tool to examine the effects of constraints as a suggestion for a practical technique. Overall,

there is no doubt that more work is required to develop constraints that are both useful from

the point of view of the inverse problem and whose practical implementation is realistic.

More generally, these simulations indicate that in order for the peripheral nerve source

localization approach to ever be usable as part of a neuroprosthetic system, several advance-

ments are needed. First, noise reduction is essential, and could take the form of improved

instrumentation, better isolation of external signals at the cuff level, or neuromodulation

techniques to boost the amplitude of the recorded nerve signal [146]. Second, methods

should be investigated to obtain precise images of a nerve’s anatomy in vivo. For exam-

ple, an adaptation of electrical impedance tomography techniques [9] to peripheral nerves

could be considered, and studies have already shown that fascicles within a nerve can be

imaged using ultrasound [77]. It is interesting to note, however, that in our simulations

the performance using only the spatio-temporal constraint with the idealized leadfield (IL-

C) was not very far from that using only the correct leadfield without the spatio-temporal


constraint (CL), at least for cases with non-zero noise. This raises the possibility that if

sufficient physiological and contextual information could be provided as constraints, and the

noise brought down to a manageable level, then an idealized geometry could perhaps still

be used. This would be analogous to using a three-sphere head model in the EEG/MEG

source localization problem, as opposed to a patient-specific MRI-based anatomical model.

Another cause for cautious optimism is that the performance for the very idealized case

of noiseless measurements and full constraints approached acceptable levels, which implies

that the very large number of dipoles in the region (50 per pathway in this study) may

not be an insurmountable obstacle in itself to pathway localization. Importantly, although

this study was conducted with an eye to applying the method to neuroprosthetic systems,

the potential for neural system identification (in this case identifying the peripheral control

signals involved in specific tasks) is also very interesting, and has a different set of restric-

tions. In particular, in a context where repeated trials and off-line processing options are

available, the noise could be significantly reduced by trial averaging and a wider range of

algorithms explored, since computation time would no longer be as much of an issue. It may

therefore be worthwhile to keep exploring the source localization approach in this context.

For neuroprosthetic applications, sub-fascicular resolutions do not appear realistic on the

short term, and the usefulness of the method as a framework for determining the activation

of combinations of several fascicles will be contingent on future developments like the ones

discussed above (particularly if numerous small fascicles are involved, as is more likely to be

the case in humans).

The present study aimed to estimate the overall viability of the source localization

approach. There are various topics that were not discussed in detail here but would become

very relevant if the performance was improved enough to make the approach usable. First,

it is important to keep in mind the very high temporal resolution of the method, which is

not reflected in the metrics used in this study. For each dipole location, the algorithm pro-

duces a complete activation time series, making it possible to study transmembrane current

waveforms and firing frequencies for precise locations in the nerve. Second, the influence of


the number of contacts on the performance would need to be explored. We limited ourselves

here to using a model of an existing electrode, but as manufacturing techniques improve so

will the number of contacts that can be placed on a cuff, and this will undoubtedly have an

influence on the source localization performance. There may also be a plateau to the benefits

of increasing the number of contacts, such that the optimal number should be sought. As-

pects of this question will be explored further in Chapter 7. Third, the number of pathways

that the method can accurately localize in a given situation should be determined. The

use of three pathways in this study was motivated by the need to study a simple multi-

pathway case, rather than by physiological considerations. That said, the term “pathway”

should not necessarily be equated with a single nerve fibre, particularly given the limited

resolution and the coarseness of the mesh. Multiple closely spaced nerve fibres that have

a functional relationship and a roughly synchronous firing pattern (e.g. compound action

potential (CAP)) may therefore be considered a single pathway, as could a very small fas-

cicle. Similarly, the currents generated by a few isolated nerve fibres that fire at the same

time as a larger coordinated group of fibres elsewhere in the nerve may be drowned out

and confused with noise; in that case, they may be missed by the algorithm, but may not

significantly hamper the localization of the “main” pathway. The last issue that deserves

to be discussed is that of computation time. Our choice of algorithm was partially based

on its speed. On a 2.33 GHz dual-processor workstation, localization for a single time in-

stant using the idealized leadfield and no constraints took approximately 1 second. When

using the correct leadfield or the spatio-temporal coupling, the computation time increased

roughly proportionally to the number of variables to be solved for. While these computation

times are not yet suitable for real-time implementation, it is nonetheless a realistic target as

computational speed continues to increase and more efficient implementations of the algo-

rithm are explored. As mentioned earlier, in situations where speed is not an issue, slower

algorithms could be explored, for example ones based on lp norms (e.g. [194]).

Chapter 5

Validation of the Source Localization

Approach on Physiological Data

5.1 Introduction

The next step in the study is to collect experimental data under controlled conditions and

use it to test the source localization approach in practice. Despite the fact that the sim-

ulation results in the previous chapter suggest that the performance may be poor due to

high sensitivity to noise and model inaccuracies, experimental data is needed to validate the

qualitative observations regarding the influence of the spatio-temporal constraint and more

accurate leadfield. In addition, even if the bioelectric source localization approach presented

thus far proves to be of limited utility, we will still seek to determine what useful information

can be extracted from the MCC recordings obtained under realistic conditions, which can

only be done accurately using experimental data. For convenience, from this point on, we

will refer to both the spatio-temporal constraint and the use of a nerve-specific leadfield (in-

stead of one based on an idealized model) as “constraints”. Although the two strategies are

of a different mathematical nature, they both represent attempts to improve performance

by incorporating additional information into the problem.

91

Validation of the Source Localization Approach on Physiological Data 92

5.2 Methods

5.2.1 Data collection

Animals

Seven old male Long-Evans breeders (640 g to 850 g) (Charles River Laboratories Inc.,

Wilmington, MA, USA) were used. All rats were acclimatized for one week prior to use in

the experiment. Food and water were provided ad libitum. A 12 hour lights on/off cycle

was used. All animal care and use procedures conformed to those outlined by the Canadian

Council on Animal Care (CCAC). Experiments were performed on seven rats but, due to

technical difficulties with the first animal, that data had to be discarded. Hence, data from

only six animals was analyzed in this study.

Anaesthesia

All animals were anesthetized with a single bolus injection of pentobarbital (60 mg/kg,

intraperitoneal), and their lower backs and legs were shaved and treated with povidone-

iodine. When an adequate depth of anesthesia was attained (loss of corneal reflex and loss

of sharp pain sensation), the animals were positioned prone on the operating table.

Surgical exposure

An oblique incision was centered over the posterior (dorsal) aspect of the hip. The incision

was extended proximally to the midline and distally parallel with the fibers of the gluteus

maximus to the posterior margin of the greater trochanter. The incision was then directed

distally, parallel with the femoral shaft to the posterior fossa of the knee.

The deep fascia was exposed and divided in line with the skin incision. By blunt

dissection, the gluteus maximus was split in line with its fibers and retracted to expose the

sciatic nerve and short external rotator muscles. Care was taken not to disturb the superior

gluteal vessels in the proximal part of the exposure.


The sciatic nerve was exposed as far proximally as possible to allow adequate exposure

for application of the recording cuff. The recording cuff was applied to the sciatic nerve

following application of the three stimulating cuffs (see details in the next section).

The sciatic nerve was then followed distally and three branches were identified: the

sural nerve, peroneal nerve, and tibial nerve. The soft tissue surrounding each of these nerves

was carefully blunt dissected to allow a stimulating cuff to be applied to each nerve.

Electrode placement and recording parameters

A matrix design polyimide spiral nerve cuff electrode [157] (Figure 1.1) was placed on the

sciatic nerve, just proximal to its division into its peroneal and tibial branches. This cuff

was 23 mm long, 1 mm in diameter and contained 56 contacts, arranged in 7 rings of 8

contacts. This electrode was used to record the nerve activity during the experiments. In

addition, three tripolar stimulating polyimide spiral nerve cuffs (8 mm long and 1 mm in

diameter) were placed around the tibial, sural, and common peroneal nerves. The center

ring of the stimulating electrodes contained 8 contacts that were shorted together, resulting

in traditional tripole cuffs. The stimulating cuffs were placed first (Figure 5.1(a)), followed

by the recording cuff (Figure 5.1(b)).

The measurements from the cuff on the sciatic nerve were acquired using a SynAmps2

64-channel amplifier (Neuroscan Inc., Herndon, VA, USA), with a sampling rate of 20 kHz

and a gain of x2010. The signals were band-pass filtered between 300 Hz and 3 kHz. The

reference for the recordings was a contact included in the matrix cuff design and located just

outside the cuff. A needle electrode in the calf was used as the ground.

Direct fascicular stimulation using nerve cuff electrodes

The tibial, peroneal, and sural nerves were stimulated using the 8 mm cuff electrodes, first in-

dividually, then in every possible combination. The stimulation pulses were generated using

Compex Motion stimulators (Compex SA, Switzerland). Although the intended stimulation


(a)

(b)

Figure 5.1: a) The tibial, peroneal, and sural nerves are exposed. Each has a stimulating

cuff wrapped around it. The sciatic nerve has been exposed but the recording cuff has not yet

been placed. b) The exposed sciatic nerve with the recording cuff wrapped around it.

parameters consisted of 10 µs 2 mA pulses (2 mA being comfortably higher than the thresh-

olds reported in the literature for pulses of this duration [145, 116, 45, 183]), in practice the

pulses were shorter and had more variable amplitudes, due to technical difficulties noticed

only after the fact and explained in more detail below. 100 trials were conducted for each

fascicle, at a frequency of 2 Hz.

Nerve samples

A section of the sciatic nerve of each rat was removed and preserved in a formaldehyde

solution. These samples were later used to obtain images of the nerve cross-sections (see


Section 5.2.2), thereby providing anatomical information that can be incorporated into the

source localization process.

Issues encountered

Three difficulties were encountered during the experiments. Their impact on our study is

described below.

1. Recording cuff connector: The current design of the 56-channel recording cuff was

created with in vitro experimentation in mind. Using it in vivo presented a challenge

because the connector linking the electrode to the amplifier was very bulky (Figure

5.2). As a result, it had to be held in place manually, which implies that there was

some variation in the cuff position over the course of an experiment. This will have to

be taken into account when interpreting the results of the source localization using the

nerve cuff data. During one of the experiments, we were able to clamp the connector in

place, and the data from that rat will therefore provide an indication of how much of a

detrimental impact positioning the connector by hand might have had. Clamping was

unfortunately not possible in the other cases, because of the placement angle and the

very limited tension that the cuff could bear without coming off the nerve. The only

satisfactory solution to this problem will be a redesign of the cuff and its connector by

our collaborators at the University of Freiburg, however this was not possible within

the time frame of the current study. To our knowledge, there currently exists no other

nerve cuff design with a sufficient number of contacts. The limitations imposed by the

connector were therefore in our opinion unavoidable in the context of this project.

2. Stimulation artefact: Although we stimulated using very short pulses in an attempt

to limit the stimulation artefacts, capacitive effects nonetheless resulted in substantial

artefacts that overlapped with the action potentials. This issue arose because the

amplifiers were not blanked during the stimulation (our recordings were performed

using AC coupling in order to achieve the necessary gain, and the amplifier’s blanking


Figure 5.2: Connector linking the 56-contact cuff electrode to the amplifier and data-

acquisition system. During the recordings, the connector was held up either manually or

using a clamp.

feature was not available in this mode). The amplifiers did not saturate, but they were

susceptible to an impulse artefact with a time constant of approximately 0.5ms and thus

overlapping with the signal of interest. The simplest way to avoid this problem would be

to use equipment with which amplifier blanking is possible, but such resources were not

available in the context of this project. On a more positive note, we can take advantage

of the large number of contacts in the cuff to implement artefact reduction techniques,

such as using a common average reference. Nonetheless, some contamination of the

signal is unavoidable. An example trial is shown in Figure 5.3.

3. Stimulation pulse characteristics: It was discovered after the experiments that

the stimulators used were not able to adequately control the pulse characteristics,

due to the fact that our desired parameters were at the very limit of the stimulators’

specifications. Therefore, instead of 10 µs 2 mA pulses, the stimulators produced pulses

2-4 µs long and with amplitudes in the 0.7 to 3.8 mA range approximately. Fortunately,

these pulses were still able to reliably produce action potentials in the nerve. This was

established both from the muscle twitches observed during the stimulation and from

the spatio-temporal distribution of activity in the cuff, which is consistent with action


0 0.5 1 1.5−2

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V (

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)

Rat 1, Tibial Nerve, Trial 1

Figure 5.3: Example of the recordings of one tibial branch trial. The upper left-hand plot

shows the raw recordings for all 56 channels. The upper right-hand plot shows those same

recordings after conversion to a common-average reference. The lower plots show the same

data for one contact only, taken from the middle ring of the cuff.

potentials rather than with signals originating outside the cuff. For the purposes of this

study, the presence of action potentials in a given pathway is more important than the

exact amplitude of those action potentials, such that this problem with our stimulation

does not constitute a significant limitation of the study. In addition, another technical

issue with the stimulator resulted in four successive action potentials being generated

at every trial. For simplicity, all of the analysis that follows was conducted using only

the first action potential from each trial.


5.2.2 Construction of nerve-specific leadfields

Samples of the sciatic nerves of the rats used in each experiment were collected. A cross-

section was cut 1 cm proximal to the point where the nerve branched into its tibial and

peroneal components, which corresponds approximately to the mid-point of the recording

cuff. The cross-sections were stained with hematoxylin and eosin (H&E staining) in order

for us to obtain an image detailing the position and shape of the fascicles within the nerve.

These images were then traced and converted into finite-element models. Note that the three-

dimensional models were created by extruding the cross-section in both direction, resulting in

models with no longitudinal variation in the anatomy. Because of the progressive branching

of the fascicles along the length of the nerve, these models are still simplifications compared to

the real anatomy, but they are more detailed than the idealized model presented in Chapter

4. The cross sections obtained from the samples are shown in Figure 5.4.

Recall that the MCC is a spiral cuff designed to be implanted on nerves with a

diameter of 1 mm, which corresponds approximately to the diameter of a rat sciatic nerve.

As visible in Figure 5.4, however, the nerves used in practice were slightly larger. This is

likely a consequence of the fact that we deliberately used the largest rats available to us in

order to have space to implant all the electrodes needed for the experiments (the recording

cuff as well as the three stimulating cuffs). Because of the spiral cuff design, which is intended

to allow the cuff to adjust to the size of the nerve, it was still possible to wrap it around the

nerves without excessive compression or deformation. The spread of contacts around the

cuff wall, however, is designed for a 1 mm diameter, such that having a larger diameter will

result in an certain section of the cuff’s inner wall being devoid of contacts. Furthermore,

we do not know where this section is located. In order to deal with this situation in the

FE models, the inter-contact distance was held constant and the number of contacts was

determined by the diameter of the cuff. For instance, if the cuff had a diameter of 1.5 mm

instead of 1 mm, then the number of contacts was 12 contacts per ring instead of 8, for a

total of 84 contacts instead of 56. The leadfield constructed from this model would therefore


(a) Rat 1. (b) Rat 2. (c) Rat 3.

(d) Rat 4. (e) Rat 5. (f) Rat 6.

Figure 5.4: Sciatic nerve cross-section of each rat after hematoxylin and eosin (H&E)

staining. The sections were cut 1cm proximal to the point where the nerve branches into the

tibial and peroneal nerves.


Table 5.1: Cuff diameters in nerve-specific rat models.

Rat Cuff diameter Number of

(mm) contacts

1 1.5 12

2 1.5 12

3 1.5 12

4 1.4 11

5 1.4 11

6 1.4 11

have 84 rows. Of course, only 56 channels were actually recorded during the experiments,

and so for the purposes of source localization only 56 of these rows were used at a time,

corresponding to 8 adjacent columns of contacts. More detail on the process of determining

which 8 columns to use is provided in Section 5.2.3. Table 5.1 gives the cuff diameter and

corresponding number of columns of contacts for each of the rats. The diameters were chosen

based on the images in Figure 5.4. Cross-sections of the resulting meshes are shown in Figure

5.5.

5.2.3 Evaluation of the source localization performance

The source localization was applied to the experimental data using the same approach as in

the simulation study. Specifically, the sLORETA algorithm was applied in four cases, namely

using the idealized and nerve-specific leadfields, each with or without the spatio-temporal

constraint. We chose not to use the leadfield reduction method described in Chapter 3

because it is not compatible with the spatio-temporal constraint and using it only on the

unconstrained cases would complicate the comparisons. However, a comparison of the results

using the idealized leadfield and no constraint, with and without the leadfield reduction

technique applied, found that the estimates in both cases were nearly identical (results not


Figure 5.5: Cross-sections of the finite-element meshes produced for each of the rats using

the cross-sections in Figure 5.4. The colour-coding of the different types of tissue or material

is the same as in Figure 4.1

shown). This reinforces the validity of the technique, and the fact that it is not used here is

simply due to the need to limit the number of factors that vary between the different cases

studied, not to any limitations of the technique.

Before computing the source localization estimate, the data was converted to a

common-average reference, and the corresponding modification was made to the leadfield

as well. In each trial, channels with excessive variance or very small amplitude compared to

the other channels were marked as bad channels, and were not used in the localization. In

other words, the rows of the leadfield corresponding to those channels were removed. The

removal was performed before conversion to the common average reference, in both the data


and the leadfield.

For each trial, the source localization was applied to a small time interval delimited

by the peaks of the action potential recordings at the first and last contacts (excluding bad

channels), plus 0.1 ms before and after this interval. The result of the source localization

consists of a three-dimensional estimate for each time instant in the time interval, for each

trial. As a consequence, the amount of data generated by this process is considerable,

and must be summarized in a convenient form in order to gauge the success of the source

localization. The following method was therefore used. First, an estimate of the activity

of each “pathway” (longitudinal column in the FE model) was obtained by summing the

absolute values of the estimated activities in all the elements of that column over the time

interval. Pathways with levels of activity that were equal to at least 50% of the maximum

level of activity in that trial were considered active pathways. Lastly, for each case (i.e.,

combination of branches in a given rat under given constraints), a two-dimensional mesh

cross-section was generated, where the intensity of each mesh element was determined by

the number of trials in which that pathway was judged to be active. This representation

gives a concise view of where the bioelectric activity was estimated to have originated in the

nerve.

Evaluation of the results is somewhat complicated by the fact that the position of the

contacts with respect to the nerve varied from one animal to the next. Attempts to record

the positions of the contacts were hampered by the bulky connector. The implications of

this are different depending on which leadfield is being used.

In the case of the idealized leadfield, the geometry is symmetric, and therefore rotating

the contacts will rotate the estimate but not otherwise alter it. As a result, the images should

be interpreted not by using the absolute location of the activity, but rather by examining

whether stimulating the different nerve branches generated activation at distinct locations

and whether the number and combination of active branches could be identified from the

results.

In the case of the nerve-specific leadfield, the geometry is not symmetric, such that


rotating the contacts will result in an altogether different estimate, not just a rotated one.

We therefore rotate the contacts and perform the source localization for every possible ori-

entation. If, as described earlier, the diameter of the nerve was such that a section of the

cuff was devoid of contacts, then this rotation process will also take into account all possible

positions of the blank section. Because estimates are obtained for each possible rotation of

the contacts, the question remains of which set of estimates should be chosen. In the absence

of better information, we assume that the rotation that is closest to the actual position of

the contacts during the experiments will yield the best performance, and therefore simply

choose the best set of estimates when evaluating the performance. For each rat, the rotation

is chosen based on the overall performance for all fascicle combinations, and is fixed for that

animal (i.e., all estimates presented in the Results section use the same rotation for a given

rat). More details regarding the performance metrics that are used to choose the rotation

are given in Section 5.3.3.

5.3 Results

5.3.1 Using the idealized leadfield without the spatio-temporal

constraint

The first case tested was that of the idealized leadfield, without using the spatio-temporal

constraint. The summarized results of the source localization process are shown in Figure

5.6, for each rat and combination of branches. The figure shows that although some dif-

ferences can be observed between the three single-branch cases, these differences are minor.

Furthermore, the results for branch combinations are not recognizable as combinations of

the single-branch cases, nor can the correct number of pathways be readily obtained from

the source localization outcome. Indeed, in all animals, several of the possible combinations

yielded patterns of activity that were effectively indistinguishable from one another. It is

therefore clear that the source localization approach did not perform adequately, and in the


form used in this section is not a viable method for identifying the active pathways. The

most likely causes for this failure are a too-large difference between the actual anatomy and

the model used to construct the leadfield, as well as the noise and stimulation artefact in-

terference. These combined factors led to a level of error that was too high for the source

localization to be successful.

5.3.2 Using the idealized leadfield with the spatio-temporal con-

straint

The second localization attempt consisted of once again using the idealized leadfield, this

time in combination with the spatio-temporal constraint. The application of the constraint

requires an estimate of the conduction velocity in the active pathway. This information was

estimated for each trial by computing the time delay between the signal’s arrival at the first

and last rings, given that the distance between these rings is known. The time at which

the signal arrives at a given ring was taken to be the mean of the times at which peaks

were detected for all the contacts in that ring. For the purposes of computing the spatial

distance between successive nodes of Ranvier, the fiber diameter was estimated from the

conduction velocity using a proportionality factor of 6 [69]. Note that we are treating the

CAP as having a single conduction velocity, which is a simplification. In other words, we are

using the dominant velocity in the CAP for the purposes of the spatio-temporal constraint

and ignoring the others. The summarized results of the source localization are shown in

Figure 5.7.

The figure shows results that are very close to the unconstrained case (at least in terms

of the spread of the activity, even if the intensities in the figures are sometimes different), and

therefore did not lead to a noticeable improvement in performance. This is not surprising,

because if the leadfield diverges too much from the real anatomy (as is suspected here), then

the relationships between variables on which the constraint relies will also not be sufficiently

close to reality to improve performance. It was also the case in the simulations that the


Figure 5.6: Summary of the source localization performance for each combination of

branches in each rat, when the idealized leadfield is used without the spatio-temporal con-

straint. Refer to the text for the meaning of the intensities in the figures.



branches in each rat, when the idealized leadfield is used and the spatio-temporal constraint

is applied.


constraint reduced spurious pathways mostly by eliminating small erroneous fluctuations in

the estimate; it generally did not lead to very large qualitative changes in the estimate, and

therefore it is not surprising that it did not have such an effect on the experimental data

either. An analysis of the number of peaks in the estimate is conducted in Section 5.3.5.

5.3.3 Using the nerve-specific leadfield without the spatio-temporal

constraint

The next step was to use the leadfields constructed from the nerve sample cross-sections,

without the spatio-temporal constraint. The summarized results are shown in Figure 5.8.

As discussed above, the performance shown for each rat corresponds to only one possible

rotation of the position of the contacts with respect to the nerve anatomy. The rotation

shown was chosen by first performing the source localization for every possible rotation,

and selecting visually the one that yielded the best performance. The first criterion for

performance was to be closest to the ideal of having clearly different patterns for each single-

branch case, with the activity being strongest in a different fascicle each time in the cases

where the separation of the fascicles is visible in the anatomy. The second criterion was to

be closest to the ideal of having multi-branch cases that are recognizable combinations of

the single-branch cases.

When comparing Figure 5.6 to Figure 5.8, we can make the qualitative observation

that the nerve-specific leadfield led to somewhat better separation of the single-fascicle cases.

Nonetheless, it is in most cases difficult to clearly identify a single active fascicle and, worse,

even when there is a clear difference between the cases, the fascicle that appears most active

in the estimate is not always the correct one. For example, in Figure 5.8, the Rat 1 estimate

when the peroneal branch was being stimulated seems to suggest that the sural branch is

dominant, and vice versa (the fascicles in the figure corresponding, in order of decreasing size,

to the tibial, peroneal, and sural branches). In addition, in all cases where the division of

the fascicles was visible in the cross-section, the single-fascicle estimates should show a clear



branches in each rat, when the nerve-specific leadfield is used without the spatio-temporal

constraint


focus of activity in one fascicle, and this was usually not the case. In the case of multiple

fascicles, the use of the nerve-specific leadfield did not make an appreciable difference. In

both Figures 5.6 and 5.8, multiple-fascicle estimates generally either produced a pattern

that was reflective of only one of the active fascicles, or produced a pattern that was not a

recognizable combination. The former case is likely due to higher amplitude of the CAPs for

that fascicle (e.g. due to irregularities in the stimulation parameters, as explained above, and

the sizes of the branches), although a viable source localization approach should of course

be able to take into account sources of different intensities.

In the numerous cases where unexpected patterns were produced, for both single-

fascicle and multi-fascicle cases, there was nonetheless a significant amount of reliability

between trials (recall that red mesh elements in the figures represent pathways that were

active in almost all trials). This suggests that performance degradations were not due to

random types of noise, but to more consistent sources of error. The chief candidates here

are the stimulation artefact and the modeling errors. Because substantial reduction of the

artefact was achieved (Figure 5.3), it seems unlikely that it alone could account for errors

of this magnitude. The most likely conclusion is therefore that the nerve-specific leadfields

were still too coarse an approximation to achieve successful localization. The main culprits

are likely to be the fact that the model does not take into account longitudinal variations in

the anatomy, the fact that the nerve may have been pushed into a slightly different shape

when the cuff was placed on it, and the uncertainty in the positions of the contacts around

the nerve (within each one of the rotations that we took into account, there is room for some

shift, because the contacts were modeled as punctual locations but in fact extend over a

small rectangular area).

The conclusion to be drawn from these results is therefore that although using the

nerve-specific leadfields led to a slight improvement over the idealized leadfield, in the form

of increased distinguishability between single-fascicle cases, this improvement was still too

small to make the source localization useful in practice. Indeed, taking any of the images in

Figure 5.8 in isolation (i.e., without having the other estimates as a point of comparison),


we cannot correctly infer the number and identity of the active fascicles, and this is the only

test that is relevant in a real-life neuroprosthetic application.

5.3.4 Using the nerve-specific leadfield with the spatio-temporal

constraint

The last case examined was that of the nerve-specific leadfields in combination with the

spatio-temporal constraint. These results are shown in Figure 5.9.

The application of the spatio-temporal constraint had similar effect when the nerve-

specific leadfield was used as when the idealized constraint was used. That is to say, it had

very little effect on the outline of the active regions within the cross-section. Once again

this may be due to the coarseness of the approximation, as well as to fact that the expected

effects of the spatio-temporal constraint (as per the simulation study) may be too subtle to

be easily visible in the type of display used in Figures 5.6 to 5.9. In the next section, we

attempt to compare the performances of the four cases in a more quantitative manner.

5.3.5 Influence of the constraints on the number of peaks in the

estimate

Recall that the simulation study in Chapter 4 evaluated the performance by examining the

localization error, the number of spurious pathways, and the number of missed pathways.

Computing the localization error requires knowledge of the source’s true location, which

is not available here. A qualitative evaluation of the localization accuracy can be made

in Figures 5.8 and 5.9 for rats in which the fascicles were separated, because the correct

fascicle is then known. As discussed above, this criterion indicates that the localization

errors in most cases were large. This observation is, however, difficult to translate into a

rigorous quantitative assessment, because of the spread of activity among several fascicles

in most estimates and the fact that we can only use this approach for four of the rats when



branches in each rat, when the nerve-specific leadfield is used and the spatio-temporal con-

straint is applied.


using the nerve-specific leadfields, and none of them when using the idealized leadfield. For

these reasons, we turn rather to the number of peaks in the estimates in order to perform a

quantitative validation of the conclusions of the simulation study.

For each combination of rat, branch combination and constraints, the number of peaks

in the estimate for each trial was computed in the same way as in the simulation study. The

means were computed and are reported in Figure 5.10. Before discussing the data in this

figure, it is necessary to point out that the recordings from Rat 3 were less reliable than

the others. Anticipating the results in Chapter 7, the data suggests that these recordings

were obtained with an improperly closed cuff, and therefore are largely contaminated by

the stimulation artefact (this will be more rigorously demonstrated in Section 7.3.2). As an

additional consequence of this problem, delimiting the time intervals on which to perform the

localization in each trial proved less reliable in the recordings from this rat. Overall, then, the

results from Rat 3 cannot be considered reliable, but for completeness they were nonetheless

reported above and in what follows. In the case of metrics that aggregate information from

several animals, we include the results both with and without Rat 3 included.

The data in Figure 5.10 allows us to determine the effect of each combination of

constraints on the number of peaks in the estimate. There are 5 rats with 7 branch com-

binations, for a total of 35 cases to examine (42 if we include Rat 3). To compare two

combinations of constraint, we perform a one-way ANOVA test comparing the two sets of

trials for each of the 35 cases. Out of these 35 results, we count how many are significant (p

< 0.05) and, of those, the number of cases in which each combination of constraints had a

higher mean number of peaks. The results are summarized in Table 5.2, in which the terms

“idealized” and “nerve-specific” refer to the leadfield that was used and “constrained” and

“unconstrained” refer to whether or not the spatio-temporal constraint was applied. As per

the discussion above, the entries in the table include a second number in parentheses that

corresponds to the results when Rat 3 is included.

The results in lines 1 and 2 of the table allow us to gauge the effects of adding the

spatio-temporal constraint, and show that, among the cases that were statistically significant,


T P S TP TS PS TPS0

1

2

3

4

5

6

7Rat 1

Pea

ks

T P S TP TS PS TPS0

1

2

3

4

5

6

7Rat 2

Pea

ks

T P S TP TS PS TPS0

1

2

3

4

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6

7Rat 3

Pea

ks

T P S TP TS PS TPS0

1

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6

7Rat 4

Pea

ks

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1

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7Rat 5

Pea

ks

T P S TP TS PS TPS0

1

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7Rat 6

Pea

ks

Idealized/Unconstrained Idealized/Constrained Nerve−specific/Unconstrained Nerve−specific/Constrained

Figure 5.10: Mean number of peaks detected in the estimates, for all combinations of

branches and constraints, in each rat. The abbreviations are as follows: Tibial (T), Peroneal

(P), Sural (S), Tibial and Peroneal (TP), Tibial and Sural (TS), Peroneal and Sural (PS),

and Tibial, Peroneal, and Sural (TPS).


Table 5.2: Influence of constraints on the number of peaks. The column “A > B” provides

the number of cases in which the combination of constraints in column A had a higher mean

number of peaks than the combination in column B, and analogously for “A < B”.

Line Combination Combination A > B A < B No significant

A B difference

1 Idealized/ Idealized/ 21(26) 0 14(16)

Unconstrained Constrained

2 Nerve-specific/ Nerve-specific/ 12 1(8) 22


3 Idealized/ Nerve-specific/ 16 14(18) 5(8)

Unconstrained Unconstrained

4 Idealized/ Nerve-specific/ 9 13(20) 13

Constrained Constrained

5 Idealized/ Nerve-specific/ 18 13(20) 4


6 Idealized/ Nerve-specific/ 7 16(22) 12(13)

Constrained Unconstrained

the constraint had a clear tendency to reduce the number of peaks. This is consistent with the

results of the simulation study. In contrast, the effect of using the nerve-specific rather than

idealized leadfield (lines 3 and 4 of the table) was much smaller, and no clear tendency could

be identified one way or the other. This result differs from the finding in the simulation

study, but is not surprising given that the nerve-specific leadfield is likely to still be no

more than a coarse approximation of the true anatomy. In the simulation study, we had

access to an exact model. When comparing cases “Idealized/Unconstrained” and “Nerve-

specific/Constrained” (line 5), between which the simulations had predicted the difference to

be greatest, we find only a slight tendency towards a reduction of the number of peaks by the


constraints (and only if Rat 3 is excluded). This discrepancy is not surprising given the results

for the individual effects of the constraints: the brunt of the reduction is produced by the

spatio-temporal constraint, whereas the use of the nerve-specific leadfield is just as likely to

produce an increase as a decrease. Line 6 shows that there were fewer peaks when using only

the spatio-temporal constraint than when using only the nerve-specific leadfield. Although

the simulations did not make any strong predictions in this respect, these observations are

consistent with the other results in Table 5.2.

In summary, then, an examination of the total number of peaks confirmed the pre-

dictions of the simulations regarding the influence of the spatio-temporal constraint, namely

that it can reduce the number of spurious pathways. The prediction that the use of the

nerve-specific leadfield would have a similar effect was not confirmed, but this is easily ex-

plained by the different accuracies of the models used in the simulations and experimental

localizations.

The remaining quantitative metrics that are available to use are the number of spuri-

ous and missed pathways. These cannot be computed in the same way as in the simulations,

where we had access to all the true pathway locations, so we must use a simplified definition

instead. The only information available to us in the experimental recordings is the num-

ber of branches that were stimulated in a given trial. Therefore, we define the number of

spurious pathways as the number of detected peaks in excess of the number of stimulated

branches. The number of missed pathways is defined analogously in cases where there are

fewer detected pathways than stimulated branches. The results are shown in Figure 5.11,

but keep in mind that this is simply a different way of visualizing the data already presented

in Figure 5.10 (i.e., Figure 5.11 is Figure 5.10 with the appropriate constant subtracted from

each column in the bar plot).

Figure 5.11 suggests that there were generally fewer spurious pathways in cases where

several branches were stimulated. Viewed another way, the total number of pathways was not

very different regardless of the number of stimulated branches, which is a further indication

that the source localization process was not a reliable reflection of the underlying activity.


It may also be related to the fact, observed in Figure 5.8, that the estimates for combined

cases were sometimes very similar to the estimates for one of the underlying cases. The other

observation to emerge from Figure 5.11 is that there were many more spurious pathways than

missed pathways. This result is also consistent with our previous observations, because the

algorithm’s difficulty in identifying a clear dominant source of activity is reflective of the

several sources of error present in the recordings.

5.4 Discussion

The results of this chapter show that our attempts to apply a source localization algorithm to

MCC recordings did not succeed in producing a reliable and selective neural interface, which

is to say that the estimates obtained could not be used to correctly ascertain the number and

identity of the branches being stimulated. Difficulties encountered during the experiments

included slight movement of the cuff and a stimulation artifact. While these factors mean

that we are not testing the source localization under ideal conditions, they are representative

of realistic conditions (i.e. a moving nerve, and interference from bioelectric sources outside

the cuff), such that our results are indicative of the expected performance in practice. The

performance observed here was not unexpected given the results of the simulations in the

previous chapter. In light of those conclusions, the source localization experiments that were

described in this chapter were mainly intended, first, to validate the results of the simulations

in the previous chapter and, second, to provide insight into other practical factors that may

affect source localization in real-life situations but were not revealed by the simulations.

The experimental data validated the finding that the application of a constraint can

have a beneficial impact on performance; in the case of the spatio-temporal constraint this

change takes the form of a reduction in the number of spurious pathways. The use of a

nerve-specific leadfield was not observed to have a similar benefit. Although this is contrary

to the results of the simulations, it is easily explained by the fact that the anatomical model

constructed from the nerve sample cross-sections is not a perfect representation of the true


nerve anatomy. In situations where the more complex model increased the number of peaks,

we may also attribute this effect in part to the more irregular anatomy and, in several cases,

clear separation between fascicles, which may have resulted in a more fractured estimate

and therefore more peaks. The “realistic” model used in the simulations still had a fairly

regular shape, and only a small separation between the fascicles, and may therefore have

avoided this problem. It is also possible that using an incorrect nerve-specific model does

more harm than good when compared to a more idealized model: a higher level of detail is

only beneficial if the detail is reasonably accurate. This hypothesis may explain why Rat

6, which has the simplest anatomy, produced the results that were the most in accordance

with the simulations (Figure 5.10). Even though the nerve-specific leadfield was as likely to

increase as to decrease the number of peaks in the estimate, it did have some minor benefit in

the form of decreased overlap between the estimates for single-fascicle cases. Unfortunately,

there was no indication that this decrease in overlap was combined with an increase in

localization accuracy, such that this cannot be considered a major improvement due to the

nerve-specific leadfield. In addition, this result is partly biased by the fact that the rotation

of contacts that was chosen for each rat was based in large part on its ability to minimize

overlap between single-fascicle estimates.

The process of applying the source localization approach to real data recorded from

a peripheral nerve did not identify any new factors, but it did demonstrate the large im-

pact of several issues on the performance. Chief among these was, as expected, the high

sensitivity of the performance on the accuracy of the measurements and the model. The

combination of electrode motion, stimulation artefact and model inaccuracy prevented the

proposed approach from providing an accurate representation of the neural activity. Another

issue emphasized by the experiments is that of estimating the position of the cuff contacts

with respect to the nerve. If their position is misjudged by more than a very small amount,

then this will constitute another source of error that will significantly degrade performance.

For the purposes of the experiments in this chapter, we chose to deal with this problem by

rotating the position of the contacts in an effort to find the best performance. This approach


is sufficient for our current purposes, namely to evaluate the impact of using a nerve-specific

leadfield, as long as we accept the hypothesis that the most accurate rotation will yield the

best performance. Nonetheless, the problem remains that the position of the contacts can

be varied in a continuous manner, and so trying a finite set of rotations does not capture all

possibilities and may not produce the optimal performance. In addition, the method is too

time-consuming to be used in practice. Before source localization techniques can be applied

in the context of a neuroprosthetic system, a method of correctly estimating the relative

position of the cuff and the nerve will be needed. This could, for example, take the form

of techniques to image the area inside the cuff (e.g. using a variant of electrical impedance

tomography, as mentioned in Chapter 4). Using cuffs with non-symmetrical shapes and fixed

sizes may also be beneficial in this respect, unlike the round spiral cuff used here.

An additional consideration when applying the source localization to experimental

data is to what extent the representation of bioelectric activity as a group of dipolar sources

is appropriate. The simulations did not shed light on this issue because the distributed

dipolar model was used to generate the measurements in addition to being used in the

inverse problem. With experimental data, however, if the source model does not accurately

reflect the complexity of the bioelectric activity distribution in the nerve, performance may

well suffer.


T P S TP TS PS TPS2

1

0

1

2

3

4

5

6Rat 1

Mis

sed

S

purio

us

T P S TP TS PS TPS2

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0

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6Rat 2

Mis

sed

S

purio

us

T P S TP TS PS TPS2

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0

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6Rat 3

Mis

sed

S

purio

us

T P S TP TS PS TPS2

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0

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6Rat 4

Mis

sed

S

purio

us

T P S TP TS PS TPS2

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6Rat 5

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sed

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purio

us

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6Rat 6

Mis

sed

S

purio

us

Idealized/Unconstrained Idealized/Constrained Nerve−specific/Unconstrained Nerve−specific/Constrained

Figure 5.11: Spurious or missed pathways as measured by the number of peaks in the

estimates compared to the number of stimulated branches, for all combinations of branches

and constraints, in each rat. The abbreviations are the same as in Figure 5.10.

Chapter 6

Source Localization Using an

Experimentally-Derived Leadfield

6.1 Introduction

The results of the previous section show, first, that the source localization approach as applied

so far results in poor performance and, second, that one of the major factors accounting for

this limitation is the discrepancy between the actual anatomy of the nerve and the model used

to construct the leadfield. In this chapter, we investigate an alternative formulation of the

source localization process that does not rely on an a priori anatomical model. Instead, we

use recordings of a few trials of each single-branch case as a training set to build a collection

of known vectors, each vector consisting of the 56 values recorded at all the contacts at a

given time instant. A new “leadfield” is constructed from these vectors. The problem is then

to identify which combinations of these vectors can best account for new recordings, when

either single or multiple branches are active.

Constructing a leadfield out of experimentally observed patterns has both advantages

and disadvantages. On one hand, it does not depend on having an anatomical model of the

nerve, and therefore eliminates a potential source of errors. On the other hand, because the

120

Source Localization Using an Experimentally-Derived Leadfield 121

observed patterns are matched only to a fascicle rather than a more specific location within

the nerve, the solution to this new problem can identify no more than the combination of

active fascicles; sub-fascicle resolutions are not possible. Of course, reliable identification of

fascicle combinations would already be a significant step forward from the performance in

Chapter 5, and so is very much worth investigating.

6.2 Methods

6.2.1 Construction of the experimental leadfield

For the remainder of this chapter, we will use the term “experimental leadfield” to refer to

a collection of observed measurement vectors that is built using a training set and will be

used to classify future observations. This leadfield is a matrix in which each column is a

56-element vector corresponding to the measurements recorded from the MCC at a given

instant. The goal is to construct a set of such vectors that are sufficiently representative of

the activity of each fascicle to be able to correctly identify future recordings. The leadfield

constructed in this way is thus analogous to the leadfield in previous chapters with the

exceptions that:

• Each column corresponds to an instantaneous spatial pattern of activity produced by

an entire fascicle, instead of by a dipolar bioelectric source in a very specific position.

Note that each fascicle will be associated with several vectors, because different patterns

of activity will be produced as a CAP travels in that fascicle along the length of the

cuff.

• The leadfield is constructed using a training set of experimental observations, rather

than using an anatomical model of the nerve. It is therefore not sensitive to modeling

errors.


The experimental leadfield is constructed using a training set that includes only ob-

servations of single-fascicle activity. Although multi-fascicle cases are of interest and are

included in the testing set, they are not included in the training set because, in order to

be useful, the system should be able to identify combinations of fascicles based only on its

knowledge of the single-fascicle base cases. This principle is analogous to including in a

traditional leadfield only vectors corresponding to single-dipole cases, rather than explicitly

including in the leadfield all possible combinations of dipoles.

The process of constructing the experimental leadfield is as follows:

1. In each trial, identify bad channels in the same way as in Chapter 5 and set the data

for those channels to 0. Afterwards, convert the data to a common-average reference.

2. Divide the observed trials into a training set and a testing set, for each of the seven

combinations of fascicles (see Chapter 5 for details on the data collection procedure).

In the case of the multi-fascicle combinations, all trials belong to the testing set, as

explained above. In each single-fascicle case, the trials are divided into 5 groups, and

the performance will be measured using 5-fold cross-validation. Accordingly, the multi-

fascicle performance will be evaluated 5 times, using a different training set each time

but always the same testing set (i.e., all multi-fascicle trials).

3. Examine the training set of each single-fascicle case and record the measurement vectors

that occur within it. The goal is to build a set of distinct vectors. Therefore, each new

vector must be examined against all the previously seen vectors in order to determine

whether it is a new pattern or one that has already been recorded. Ideally, each

trial should produce exactly the same activity, so that a small number of patterns

(corresponding to the different positions of the CAP along the cuff) should occur many

times. In practice, this may not be the case, due to noise and the slight movement of

the cuff.

For each trial, the time instants examined are those in the same interval used for the


source localization in the previous chapter. A collection of vectors is built using these

time instants from all the trials corresponding to a given fascicle. From this collection

of vectors we identify recurring patterns as follows.

(a) Normalize each vector with respect to its entry with the largest absolute value.

(b) Compare each new vector to all previously observed ones. Two vectors are deemed

to be the same if the l2 norm of their difference is less than 30% of the norm of

the first vector.

(c) If a vector has not been observed before, record it and move on to the next one.

Otherwise, the current vector must be incorporated into our collection of known

vectors. For the purpose of future comparisons, each vector is represented by the

mean of all its occurrences. For example, the second time a vector is seen, both

occurrences are replaced by a single entry that is the mean of the two. When a

third vector is found that resembles that mean, the entry is replaced by the mean

of all three vectors, etc... The number of times that a vector has been observed

is also recorded.

4. Once all vectors have been examined, any vector that occurred only once is deemed

not to be useful for identifying future observations and therefore is removed.

5. The remaining vectors are gathered to form the columns of the experimental leadfield.

The leadfield contains the vectors identified for all of the single-fascicle cases, but we

keep track of which columns correspond to which fascicle.

6.2.2 Identification of fascicle combinations

Once the leadfield has been constructed according to the procedure in the previous section,

two situations can arise: the problem can be either underdetermined or overdetermined. The

far more likely of the two, and the one dealt with here, is that of an underdetermined problem:


the number of measurements is much smaller than the number of different patterns observed,

so that the leadfield has more columns than rows. This is the same type of situation that

we faced when using the model-based leadfield. Many of the principles mentioned in that

context for solving an ill-posed inverse problem can therefore be applied here as well. The

sLORETA algorithm, however, is less meaningful here because the variables now represent

more abstract entities than FE mesh elements, and lack clear spatial relationships between

them. The concept of “smoothness” that is central to sLORETA is therefore no longer

relevant. We turn instead to the simpler weighted minimum-norm approach (Section 2.2.3).

This method does not have a geometrical assumption underlying it and simply solves the

inverse problem in the simplest way (finding the minimum-norm least-square solution), with

only the added complexity of compensating for the different norms of the leadfield columns.

Examining the problem more closely, it becomes apparent that a typical solution should

consist of only a small number of non-zero variables. Indeed, a CAP in a given fascicle

should ideally correspond to a single column of the experimental leadfield. Even considering

a complex case of several active fascicles, some of which may contain two or three CAPs at

different positions, the total number of variable will still be fairly small. Accordingly, it is

reasonable to choose an inverse problem method that produces sparse solutions. We choose

for this purpose to use FOCUSS (Section 2.2.3), using as the initial estimate the weighted

minimum-norm estimate. Both methods are regularized using Tikhonov regularization, with

the regularization parameter chosen using the L-curve method (Section 2.2.5).

6.2.3 Evaluation of the results

Each trial was evaluated as follows. First, the estimate for each time instant was normalized

with respect to its entry with the largest absolute value. Second, for each variable in the

experimental leadfield, the mean estimated activity over the time interval was computed.

Next, for each of the three branches, the mean of these results over all the variables corre-

sponding to that branch were computed. This resulted in a set of three numbers for each


trial. The absolute values were taken and the three numbers normalized with respect to the

highest one. The final product was a set of three values between 0 and 1 describing the

relative estimated activations of the three branches during this trial.

6.3 Results

Excluding Rat 3, the number of columns in the 25 experimental leadfields (5-fold cross-

validation in each of 5 rats) ranged from 176 to 384. This shows that there was a high

level of variability in the measurements: there are three fascicles being used to construct the

leadfield, and given the sampling rate and approximate conduction velocity, it would take on

the order of 10 time samples for a CAP to propagate through the cuff. In the ideal situation,

therefore, there would be only about 30 columns in the experimental leadfield (creating

a overdetermined problem and calling for different methods to solve it). Furthermore, as

described above, these columns are those that remain after patterns that were observed only

once are eliminated, such that the total number of observed vectors was even higher. On the

other hand, the highest number of repetitions for a given pattern in each rat varied from 31

to 127, showing that there was still a measure of regularity to be found in the recordings.

Figure 6.1 shows the mean of the three activity estimates for each branch combination

and each rat. These means are taken on the agglomeration of the results in all 5 testing

sets. Examination of the figure reveals that in the single-branch cases the algorithm was

successful in identifying the stimulated branch as by far the most active. Activity estimates

of the other branches were in most cases small, but not insignificant. As for the multi-branch

cases, the algorithm was less successful in identifying the active branches. Although a few

cases were close to being accurate (e.g. Rat 1, tibial + peroneal and tibial + sural), inactive

branch activities estimates were still high, and on the whole the method was not reliable.

Figure 6.1 gives a useful overview of the algorithm’s ability to assess the activity of

the different branches. From the point of view of a neuroprosthetic system, however, we are

interested in knowing how often the algorithm can correctly identify the exact combination of


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Figure 6.1: Means of the activity estimates for the three branches, for each rat and branch

combination. The abbreviations are the same as in Figure 5.10.

active branches. In order to measure this aspect of the performance, the activity estimates

computed above were thresholded at 0.2. A branch is deemed active if it is above this

threshold, and inactive otherwise. We then computed the percentage of trials in which the

combination of active branches is exactly accurate, for each branch combination and rat. The

success rates were averaged across the 5 testing sets, and the results are shown in Figure 6.2.

Figure 6.2 is in accordance with Figure 6.1 in showing that the algorithm was more

successful at correctly identifying single-branch cases than multiple-branch ones. Excluding

Rat 3, the mean success rate over the 15 single-branch cases was 68.5%, with a minimum


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Figure 6.2: Success rate for identifying the exact combination of active branches, for each

rat and branch combination. The standard deviation is based on the 5 repetitions of the

cross-validation process. The abbreviations are the same as in Figure 5.10. A branch is

considered active if its activity estimate is greater than 0.2.


of 19.4% and a maximum of 95%. On the other hand, the mean success rate over the 20

multiple-branch cases was 25.3%, with a minimum of 1.3% and a maximum of 54.6%. Note

that by consulting Figure 6.1, we can see that single-fascicle cases with low performance

in Figure 6.2 were mostly due to inactive branches erroneously being identified as active,

rather than the correct active branch being missed. The exact proportions of false positives

and false negatives of course depends on the threshold that we use. The use of 0.2 was an

attempt to balance the need of the single- and multi-fascicle cases. If the performance of

the multi-fascicles cases was of no interest, it is clear from Figure 6.1 that the number of

false positives in the single-fascicle cases could be reduced by raising the threshold. This is

illustrated in Figure 6.3, which is obtained in the same way as Figure 6.2 except with the

threshold set to 0.6 instead of 0.2. This change of threshold raises the mean success rate

of the single-branch cases to 89.8%, and lowers that of the multiple-branch cases to 11.0%.

Rat 5, which had the poorest performance, was also the rat in which the least repetition

was found among the observed vectors in the training set, illustrating the importance of a

reliable training set when interpreting cases in the testing set.

6.4 Discussion

In this chapter we investigated a potential solution to the issue of anatomical model depen-

dency in the bioelectric source localization process. By constructing a collection of measure-

ment patterns across the 56 contacts, observed during a few single-fascicle training trials,

we were able to create a leadfield derived entirely from experimental data, rather than an

a priori anatomical model. In doing so, we eliminated one source of error (model inaccu-

racies), at the expense of abandoning the possibility of sub-fascicular resolution. Given the

performance described in the previous chapter, this seemed like a very worthwhile trade-off.

The results using the experimental leadfield were significantly improved in the single-

fascicle cases. On the other hand, success rates for correctly identifying multi-fascicle cases

were still very low. Discriminating the activity of three fascicles when only one is active can-


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Figure 6.3: Success rate for identifying the exact combination of active branches, for each

rat and branch combination, when the threshold for considering a branch active is raised to

0.6.


not be considered a novel accomplishment considering the results reported in the literature

(as reviewed in Section 2.3.3), but the method proposed here is a novel way of addressing

this task. This line of enquiry is therefore worth pursuing further: casting the issue as an

inverse problem of source localization provides a rich mathematical framework and opens

the door to a large number of techniques. This large amount of existing theory was one of

the reasons behind our initial decision to apply source localization techniques to peripheral

nerves, and it remains just as valid for the experimental leadfield as for the model-based

leadfield.

Of course, one of the benefits of using sophisticated source localization algorithms is

the possibility of dealing with multiple simultaneous sources. This begs the question of why

the experimental leadfield approach was unable to satisfactorily deal with the multi-fascicle

cases. Despite having eliminated the issue of model inaccuracies, there remain other sources

of error, such as interference from the artefact. The movement of the cuff may have also

had led to a situation where performance degraded more with longer periods of time: in

the single-fascicle cases, the training and testing cases were obtained in short succession in

a single series of trials. Multi-fascicle cases were usually obtained a few minutes later, such

that more cuff movement may have occurred, making it more difficult for the algorithm to

“recognize” the base cases that it had been trained on.

In addition to practical issues of this sort, we must remember that the experimental

leadfield does not change the fact that we are dealing with an ill-posed inverse problem with

an infinite number of solutions and poor stability in the face of measurement errors. From

that point of view, the results of this chapter are indicative of the issues that may arise when

several sources are present, even if simple localization is possible. The previous chapter had

poor performance even in the single-fascicle cases, and therefore was less informative in terms

of the differences between single- and multi-fascicle situations. Interference from the artefact

may account for much of the performance gap between these two situations when using the

experimental leadfield, because measurement error will make successful reconstruction of

complex source configurations more difficult than reconstruction of simpler ones. Nonethe-


less, since the performance of single-fascicle cases was fairly high despite the artefact, we

can conjecture that the poor performance of the experimental-leadfield approach for multi-

fascicle cases is partly the result of fundamental issues in this type of problem, such as the

fact that different combinations of sources can produce the same measurements. It may be

that the constraints and regularization techniques that can best improve the performance in

this problem are different from the ones that are most effective with model-based leadfields.

It is also possible that interactions within the nerve, in particular electric field in-

teractions between fibers, may invalidate the assumptions that the recordings from branch

combinations will be strictly linear summations of single-branch recordings. In that case, it

could be helpful to incorporate system identification methods producing input-output maps

that account for nonlinear behaviour [93].

A disadvantage of the experimental leadfield compare to the model-based leadfield

is the necessity of obtaining training data. We sought to limit this disadvantage by ob-

taining training data only from the single-fascicle cases. It is likely that the necessity of

obtaining this data would not be an insurmountable obstacle to applying this technique in

practice: the training data could be obtained fairly rapidly intra-operatively when the cuff

is implanted, assuming that it is possible to individually stimulate the fascicles of interest,

either directly or through indirect methods such as cutaneous stimulation or passive limb

movements. Additionally, the model-based leadfield’s advantage in this respect is only signif-

icant if no calibration of the anatomical model is required. If it becomes necessary to collect

various pieces of data to refine the model (e.g. ultrasound imaging of the nerve), then the

practicality of both methods becomes comparable. A related issue is that it may be possible

to obtain training data for only a few of the fascicles in a nerve, a situation that would

make the remaining fascicles unidentifiable and in effect impose a different type of resolution

limit on this approach to source localization. The elements of the experimental leadfield

also do not need to correspond to specific nerve branches, but may instead be associated

with more “high-level” functional concepts (e.g. a particular movement in a particular limb,

which may produce activity in several pathways). This alternative approach would modify


the stimulation protocol needed to construct the experimental leadfield.

Chapter 7

Influence of the Number and Location

of Recording Contacts on the

Selectivity of a Nerve Cuff Electrode

7.1 Introduction

In the face of the poor performance of the source localization approach, using both model-

based and, to a lesser extent, experimental leadfields, we take a step back and try to deter-

mine whether the MCC genuinely contains more information than simpler cuff configurations

used in the literature previously. To this end, in this chapter we use the simpler problem

of discriminating the three possible types of single-fascicle activity, when a training set is

available. Our goal is to evaluate the improvement in fascicle discrimination that can be

achieved with the matrix cuff compared to a set of contacts laid out in a single ring, as in

previously employed multi-contact tripole configurations [197]. To investigate this issue, we

use the matrix cuff and compare the performance of the full grid-like contact configuration

(the 56-contact “matrix” configuration) to the performance when using only the 8 contacts

in the middle ring of the cuff (the “single-ring” configuration). If the matrix performance is

133

Influence of the Number and Location of Recording Contacts on the Selectivity of a Nerve

Cuff Electrode 134

found to be superior, we will seek to determine in addition whether the improvement is due

to the number of contacts used or to their position (or both). The large number of contacts

and grid layout of the matrix cuff makes it ideal to study these issues. In addition to the

source localization considerations, information about the optimal placement of contacts in

a nerve cuff to maximize selectivity would have direct applications in the design and use of

this type of electrode in neuroprosthetic systems.

7.2 Methods

The analysis in this chapter was performed on the data collected as described in Chapter 5,

with the exception that only the single-fascicle cases were used. The methodology described

in the remainder of this section therefore pertains to the processing of the signals.

7.2.1 Evaluation of the classification success rate

We sought to determine whether or not the recordings from the 56-contact matrix cuff gen-

uinely contained more useful information than measurements from a simpler configuration.

We considered the case of a simple feature-based classifier for differentiating the activity of

the three different fascicles, when only one of them is active at a time. In the context of our

experiments this means that our goal is to determine which fascicle was being stimulated in a

given randomly chosen trial, using the measurements from the recording cuff. We compared

the performance of this classifier when using data from all 56 contacts to the performance

when using only data from the 8 contacts in the middle ring of the cuff (ring 4 of 7). The

two configurations are illustrated in Figure 7.1.

The classification process was conducted as follows for each animal:

1. For each trial, the data was converted to a “tripole” reference, which is to say that

the average of all the contacts in the first and last rings was used as the reference (the

term tripole is used loosely here, since there are more than three rings in the cuff).


Cuff Electrode 135

Figure 7.1: Contact configurations for the matrix (a) and single-ring (b) cases. The con-

tacts in dark gray are the ones that are available for use in the classification process. The

contacts of the first and last ring are averaged to produce the reference, for both configuration.

Once this was done, the data was normalized using the largest absolute value in this

trial over all contacts. Because of this normalization, the classification is based on

the distribution of activity among the contacts, and therefore on spatial information,

rather than on the magnitude of the activity.

2. A set S of contacts to be included in the feature vector was defined.

3. For each trial, the peak of the action potential recorded at each of the contacts in S

was found (the peaks may not all occur at the same time, since the contacts can be at

different longitudinal positions along the cuff). The feature vector was then defined as

the potential of each contact at its peak, resulting in a vector with one entry for each

element in S.

4. The trials from each nerve were partitioned into a training set and a testing set. The

feature vectors from the training set were averaged for each fascicle, resulting in one


Cuff Electrode 136

mean feature vector for each of the tibial, peroneal, and sural branches. Each of these

mean feature vectors was normalized. The three vectors were then collected into a

matrix L.

5. For each trial in the combined testing sets, the normalized feature vector F was classi-

fied by finding the least-squares solution to the overdetermined system LX = F . The

fascicle corresponding to the largest value in X was chosen as the one responsible for

the observed activity in this trial.

6. The classification success rate is the percentage of trials in the testing set that are

assigned to the correct fascicle.

In order to ensure that the results were not biased by the choice of trials included in

the training set, the evaluation of the classifier was performed using 10-fold cross-validation.

In each trial, channels with excessive variance or very small amplitude compared to

the other channels were marked as bad channels and set to 0 before computing the feature

vector. Trials were discarded when more than a quarter of the channels in S were bad or

when the temporal spread of the peaks across all contacts was greater than 1 ms.

Our main concern is whether or not the matrix cuff allows for more accurate classi-

fication than the single-ring configuration. In addition, we would like to establish whether

or not all 56 contacts are needed for an improvement (if any is found). In other words, we

would like to know if the benefit of the matrix cuff stems from having more channels of

information, or if a small subset of contacts could also lead to better performance simply by

virtue of having 56 possible contacts to choose from instead of 8. To answer these questions,

both configurations were investigated by adding one contact at a time and tracking the per-

formance as more contacts were added. The set of available contacts during this process

was in one case all 56 contacts, and in the other case the 8 contacts in the middle ring

(refer once again to Figure 7.1). At each step, the contact added was the one that improved

the performance the most. In other words, we first computed the performance using each


Cuff Electrode 137

contact individually (i.e. S had a single element, and the full cross-validation procedure

was performed) and retained the best one. Next, we investigated each remaining contact

in combination with the first contact selected, and again retained the best one. The third

contact was then combined with the first two, and so on, until all the contacts from the set

of interest had been added.

7.2.2 Evaluation of the influence of the stimulation artefact

The interpretation of the results will be complicated by the presence of the large stimulation

artefact in the recordings, as described in Section 5.2.1. Figure 7.2 illustrates this with an

example of one trial, showing both the raw data and the data after conversion to the tripole

reference. This figure is very similar to Figure 5.3, with the exception that the latter used a

common-average reference rather than a tripole reference. The tripole reference was used in

this part of the study for consistency with the single-ring configuration. We must consider the

possibility that stimulation at different sites produces slightly different stimulation artefacts,

and that the classifier is partly taking advantage of this information. If this were the case,

we would expect that the classification success rate would be superior when large artefacts

are present. In order to investigate this possibility, we use the fact that the magnitude of

the artefact is expected to vary between rings of contacts. Indeed, theoretically, the electric

field produced by sources outside the cuff should vary linearly along the length of the cuff

[170, 132, 133, 2]. By examining the magnitude of the signals recorded at each contact before

converting the data to the tripole reference, we can estimate how the size of the artefact varies

between rings. This information can then be converted to an estimate of how the artefact

will vary between rings after the tripole reference is applied. Lastly, to determine whether

the classifier is relying heavily on information in the artefacts, we compute the classification

success rate using each ring in turn as the set S described above. If the classification uses

the artefact, we expect that the performance using the different rings will be correlated with

the estimated size of the artefact at those rings.


Cuff Electrode 138

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Figure 7.2: Example of the recordings of one tibial branch trial in Rat 1. The upper left-

hand plot shows the raw recordings for all 56 channels. The upper right-hand plot shows

those same recordings after conversion to a tripole reference. The lower plots show the same

data for one contact only, taken from the middle ring of the cuff.

To estimate the artefact variations, the recordings of each contact are first averaged

over all trials of all three fascicles combined, then rectified and integrated. The signals used

in this step are the raw measurements, recorded with respect to the outside contact rather

than using the tripole reference. The size of the artefact at each ring is estimated using the

average of the obtained values of each contact in the ring. The resulting set of seven values

(one per ring) is normalized using the largest value. By subtracting the mean of the first

and last values and taking the absolute values of the results (to take into account the tripole

reference in the classification), an estimate is obtained of how the classification performance

would be expected to vary from ring to ring if the size of the artefact was the determining

factor. Lastly, the correlation between this series and the performance actually obtained is


Cuff Electrode 139

computed.

7.3 Results

7.3.1 Comparison of the matrix and single-ring configurations

Figure 7.3 shows the maximum success rate achieved for each animal using each method.

These results demonstrate that in all cases a better classification success rate was achieved

using the matrix configuration. The question now is whether the improvement is due sim-

ply to the sheer number of contacts. To resolve this issue, the success rate was computed

using the first eight selected contacts of the matrix configuration, versus the eight contacts

of the single-ring configuration. The results are shown in Figure 7.4, and once again the

matrix configuration results in clear improvements. For each comparison, an ANOVA test

was conducted using the 10 results of the cross-validation procedure for each contact config-

uration. The differences in Figures 7.3 and 7.4 were shown to all be significant (p < 0.05),

with the exception of the rat 4 comparison in Figure 7.4, although the matrix configuration’s

performance was still higher in that case.

Figure 7.5 plots the classification success rate of the matrix configuration as a function

of the number of contacts for each rat. Markers on each plot indicate the point at which

maximum performance is achieved, and the point at which the success rate exceeds the

maximum success rate achieved with the single-ring configuration. As an example of the

contact selection process, Figure 7.6 shows the order in which the contacts were selected in

the case of Rat 1 for each of the two configurations, up to the number of contacts at which

the maximum success rate is reached (refer to Figure 7.5).

Several conclusions can be drawn from this data. First, discrimination of the activity

of different fascicles is feasible, which confirms the information found in the literature [197,

14, 176] and the results of Chapter 6. Second, the use of the matrix cuff can significantly

improve the classification success rate. Lastly, optimal or near-optimal performance can be


Cuff Electrode 140

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*

*

*

*

*

*

Figure 7.3: Maximum classification success rate achieved with the matrix and single-ring

configurations, for each rat. The standard deviations shown are based on the set of 10 results

obtained for each case during the 10-fold cross-validation process. The asterisk denotes a

statistically significant difference (p < 0.05).

achieved with fewer than 10 contacts. This implies that the superior performance of the

matrix cuff is not due to the absolute number of contacts, but rather to the possibility of

sampling the extracellular fields in locations that contain the most useful information. These

results were consistent across all of the animals, but the maximum classification success rate

varied widely, with a range of 83.9% to 100%. In addition, the contacts selected as providing

the most information were not consistent between animals. These variations could be due

to a number of factors, mainly related to the position of the cuff on the nerve, the quality

of the electrical connection established at each contact, and noise issues.


Cuff Electrode 141

Rat 1 Rat 2 Rat 3 Rat 4 Rat 5 Rat 60

20

40

60

80

100

120

Classification Success Rates Using the First 8 Contacts ofMatrix vs. Single−Ring Configurations

Suc

cess

Rat

e (%

)

Matrix (first 8 contacts)Single−ring (all 8 contacts)

*

*

*

*

*

Figure 7.4: Classification success rate achieved with the first 8 selected contacts of the

matrix and of the single-ring configurations, for each rat. The standard deviations shown

are based on the set of 10 results obtained for each case during the 10-fold cross-validation

process. The asterisk denotes a statistically significant difference (p < 0.05).

7.3.2 Influence of the stimulation artefact

An example of the comparison described in Section 7.2.2 is shown in Figure 7.7. The esti-

mated normalized artefact distribution is shown, as well as the expected performance vari-

ations if the classification was based on the artefact, and the actual performance variations

observed. The correlation between the expected and observed variations was 0.07, which

corresponds to a p-value of 0.88 when considering a null hypothesis of no correlation. Table

7.1 shows the correlations and p-values for all six animals. The p-value was considerably

larger than 0.05 in five of the six cases, such that we cannot conclude that the stimulation

artefact plays a significant role in the classification success rate. Although these results

do not allow us to state that the stimulation artefacts have absolutely no influence on the


Cuff Electrode 142

0 10 20 30 40 50 6030

40

50

60

70

80

90

100

Classification Success Rate as a Function of the Number of Contacts Added

Number of Contacts

Suc

cess

Rat

e (%

)

Rat 1Rat 2Rat 3Rat 4Rat 5Rat 6

Figure 7.5: Classification success rate achieved using the matrix configuration, as a func-

tion of the number of contacts used. The first markers (o) indicate the point at which the ma-

trix configuration starts outperforming the maximum success rate achievable with the single-

ring configuration. The second markers (X) indicate the maximum success rate achieved with

the matrix configuration.

performance, they do show that the artefacts are not the dominant factor, and that the

comparisons between the different contact configurations are based on neural activity. The

case of Rat 3, in which p ≤ 0.05, suggests that there may have been an incomplete closure of

the cuff in that experiment, leading to a much more predominant stimulation artefact in the

recordings. All results for this rat should therefore be treated with caution, as mentioned

earlier in Chapter 5.

It should be mentioned that although Figure 7.7 shows a roughly linear variation of

the artefact, as expected, this pattern was not observed in all of the animals. The lack

of linearity in the other animals can be attributed to variations in the impedances of the


Cuff Electrode 143

Figure 7.6: a) Order in which contacts were added in Rat 1 when using the matrix con-

figuration. Only the first 7 contacts are shown because that is the number required to reach

the maximum success rate for this animal (see Figure 7.5). b) Corresponding results when

using the single-ring configuration. In the case the maximum success rate was reached with

6 contacts.


Cuff Electrode 144

Table 7.1: Correlation of the artefact and classification success rate variations between

contact rings.

Animal Correlation P-value

Rat1 0.07 0.880

Rat2 0.08 0.861

Rat3 0.85 0.015

Rat4 -0.57 0.177

Rat5 0.37 0.420

Rat6 0.33 0.475

contacts, as well as to small shifts in the cuff position during the experiments (recall that

the artefacts are estimated using an average of all the trials for a given animal). It is

for this reason that we examined the correlation between the variations in artefact and in

classification success rate, rather than checking for a pre-determined pattern in the success

rate variations.

Computing the classification success rates for every ring of contacts also allowed us

to confirm that none of them outperformed the success rate obtained using the matrix

configuration (results not shown).

7.3.3 Layout of the most informative contacts

Having established that a small number of contacts can be used to achieve high performance,

the question arises of whether it is possible to identify the optimal contacts, and potentially

incorporate this information in future cuff designs. We therefore examined the order in which

the contacts of the matrix configuration were selected, in other words which contacts proved

most informative for the purposes of fascicle classification accuracy.

Establishing a common set of useful contacts among all rats proved difficult, which is

not surprising given that the alignment of the contacts with the fascicles was not the same


Cuff Electrode 145

from one experiment to the next. Nonetheless, it was observed that in all cases the selection

algorithm started by choosing contacts from several different radial positions along the nerve

(not necessarily aligned at the same longitudinal position). The number of contacts that were

added before any repetition of the radial position occurred varied between 4 and 6, with an

average of 5.33 over the six animals. This result indicates that there is value in sampling

different radial positions around the nerve, which of course is to be expected because it

allows different contacts to be close to different fascicles. The fact that the different radial

positions selected were not necessarily aligned longitudinally is also very important, because

it illustrates the value of having more than one contact to choose from when attempting

to record from a given fascicle. These observations are well illustrated by the Rat 1 results

shown in Figure 7.6.

7.4 Discussion

We demonstrated that by using a matrix cuff electrode it was possible to obtain a better

fascicle classification success rate than when using signals only from contacts in the middle

ring of the cuff. We further showed that the difference was not due to the sheer number of

contacts, since the matrix cuff could outperform the single-ring configuration even with a

small number of contacts. These results are in accordance with expectations, because they

support the idea that classification success rate can be improved by selecting the locations

around the nerve that contain the most useful information.

The locations of the most useful contacts cannot necessarily be determined a priori,

because they will not depend only on the locations of the fascicles. Rather, variations in

the impedances of the cuff contacts, the details of the interface of each one with the nerve

(i.e. distance, amount of interfering tissue, etc.) and the noise level are likely to play a large

role. In addition, even if the approximate placement of the fascicles can be estimated, their

relative positions will not be completely constant along the length of the cuff, particularly

if the device is long. As a result, how the selectivity will vary with the longitudinal position


Cuff Electrode 146

is not known in advance. Furthermore, the optimal number of contacts will depend on the

number of fascicles that we are attempting to discriminate in a given nerve. Because of

these issues, the results presented here cannot be used to design a cuff with a small number

of optimally-placed contacts. Rather, they argue in favor of implanting a device with a

large number of contacts, then conducting an optimization procedure that will indicate

which subset of the contacts should actually be used. Having a large initial set of contacts

available is all the more beneficial when one considers the issue of chronic implantation.

With time, morphological changes will occur, in the form of connective tissue accumulation

and reshaping of the nerve itself [112, 46, 138, 139]. The optimal subset of contacts may

therefore not be constant. If the contact selection procedure could be conducted not only

during the initial implantation but on a regular basis, the nerve cuff’s performance could be

maintained at a higher level over time. Another aspect to this issue is illustrated by Figure

7.5, which shows that the accuracy not only can be maximized with a modest number of

contacts, but can actually decrease when too many contacts are added. We can hypothesize

that certain contacts contain very little classification information, either because of their

position or because of high impedance or noise. Including such contacts in the classification

procedure could therefore cause more confusion than improvement. This phenomenon argues

in favour of having a contact selection procedure regardless of the amount of information

bandwidth that can be accommodated.

The main limitation of our study is the presence of the stimulation artefact, which

casts doubt on the exact classification success rate that could be achieved in its absence.

Nonetheless, we have shown that its impact was limited. Similarly, the unintended variations

in the stimulus pulses (as described in the Data Collection section of Chapter 5) likewise

raise the possibility that the classification was partially based not on the spatial position

of the fascicles, but on differences in the neural activity generated in each one. The fact

that the data in each trial is normalized (as described in the Methods section) helps to

compensate for possible differences of this kind. Furthermore, the doubt created by this

issue pertains more to the actual values of the success rate achieved than to the difference


Cuff Electrode 147

between the matrix and single-ring configurations, and as such has little bearing on the main

conclusion of our study (i.e. the benefit of choosing among numerous contact locations).

Another small but important limitation is that the algorithm that we used to select the

best contacts had the benefit of simplicity but was not necessarily optimal. Indeed, the

contacts were selected one at a time, rather than exploring the entire space of possible

contact configurations, which would have been computationally prohibitive. If different

contact selection algorithms were explored, they would most likely have some impact on

the classification success rates. Nonetheless, the simple algorithm was quite sufficient for

demonstrating that the matrix configuration was beneficial and that only a small number

of contacts was needed. A more significant drawback is that the results in this study are

based on recordings of compound action potentials, rather than spontaneous activity. The

larger amplitudes of these signals were helpful in establishing clear measurement patterns

corresponding to each nerve, achieving successful classification, and evaluating with greater

certainty the influence of the number and location of the contacts. The smaller signal-

to-noise ratio that can be expected in certain types of natural activity (e.g. [140]) would

likely result in poorer classification performance. Once again, however, this limitation does

not invalidate our conclusions regarding the varying usefulness of different contacts and the

benefits of carrying out a selection procedure.

Lastly, it is important to keep in mind that the findings described in this study deal

with a reasonably simple case, specifically the identification of the active fascicle when only

one fascicle is active and a training set is available. The more complex case of identifying

combinations of fascicles without a training set cannot be adequately handled with such

simple techniques, evoking the need for more complex methods such as source localization

algorithms. Nonetheless, the comparison of the matrix configuration with the single-ring

one has important practical applications. By using a matrix-type cuff and performing some

preliminary training recordings, it should be possible to improve the performance over current

devices while still using a small number of contacts, thereby avoiding wiring and power

consumption issues stemming from using large numbers of contacts (the combination of


Cuff Electrode 148

nerve cuff electrodes with multiplexer circuits to access different contacts has previously been

explored in the literature [156]). Even when multiple fascicles are simultaneously active (as

will likely be the case in practice), the optimal number of contacts may not be the same for

all situations, but the contact selection method proposed here will still be useful by helping

to identify which contacts are most useful, by virtue of having a good interface with the

nerve and positions that allow them to discriminate among different fascicles.


Cuff Electrode 149

1 2 3 4 5 6 7

0.7

0.8

0.9

1Normalized Artefact

1 2 3 4 5 6 70

0.2

0.4Normalized Artefact (Tripole Reference)

1 2 3 4 5 6 760

70

80

90Classification Success Rate (%)

Figure 7.7: Comparison of artefact variations and classification success rate variations

between contact rings in Rat 1, as described in Section 7.2.2. The top plot shows the normal-

ized estimated variations of the stimulation artefact in the raw recordings, as a function of

the contact ring. The middle plot shows the variations once the use of the tripole reference

has been taken into account. The bottom plot shows the variations in the success rate as a

function of the contact ring, when each ring is used in turn to classify the recorded activity.

The middle and bottom plots are poorly correlated, arguing against the hypothesis that the

stimulation artefact plays an important role in the classification success rate.

Chapter 8

Discussion

8.1 Summary

This thesis was motivated by the lack of existing technology to monitor the electrical activity

of pathways within a peripheral nerve with a satisfactory combination of spatial specificity,

spatial coverage, minimal damage to the nerve, and applicability to different nerve sizes. We

sought to determine whether applying a bioelectric source localization approach to MCC

recordings could provide a useful trade-off between these different issues. Any attempt to

relate extraneural field measurements to the activity of specific locations in the nerve is

fundamentally a source localization problem. Applying to peripheral nerves the theory that

has been developed to address similar problems, most notably EEG source localization,

is therefore a logical step that had not yet been taken in the literature, partly because

cuff electrodes have only recently begun to incorporate sufficient recording contacts for the

approach to make sense. On the other hand, there are substantial obstacles to effectively

solving the source localization problem in practice. In particular, it is an ill-posed inverse

problem, a notoriously difficult type of situation to deal with. The problem requires an

accurate model of the region containing the sources, is sensitive to noise, and does not

have a unique solution, which creates the need for sensitive regularization techniques and

150

Discussion 151

hampers our ability to achieve very high spatial resolution. While these issues, particularly

the sensitivity to noise, may give the impression that this type of approach is poorly suited

to nerve cuff recordings, we must keep in mind that these obstacles do not stem from a

particular choice among several possible methods. Instead, they arise directly from the

nature of the task. In other words, we are not using a source localization approach because

of the characteristics of this method; we are using it because the task that we are addressing

is a source localization problem. On a more positive note, the approach can be applied to

instantaneous electric field recordings, and as a result has extremely high temporal resolution,

which is essential for correctly decoding neural activity.

The first issue we addressed was how to minimize the huge discrepancy between the

number of measurements and the number of unknowns in the source localization problem.

Our strategy was to reduce the redundancy present in the variables. Although a relatively

fine FEM model was used to solve the forward problem as accurately as possible, the mesh

elements were not all distinguishable from the point of view of the inverse problem. The

distinguishability of adjacent elements was quantified using comparisons of the corresponding

forward fields (leadfield columns), and elements that could not be discriminated from one

another were fused together, thereby reducing redundancy. In the idealized geometry used

for this part of the study, the number of variables was reduced by more than half through

this process. Although there was a minor improvement in numerical conditioning, the major

benefit was a reduction of computation times and storage requirements, important factors if

an on-line implementation is ever to be achieved. Furthermore, since the reduced leadfield

better reflects the inherent resolution of the problem, the proposed method has a secondary

application as a technique to analyze the difficulty of successful localization and establish an

upper bound on resolution for a given combination of nerve and electrode.

The next step was to conduct a simulation study to assess the potential of bioelectric

source localization in peripheral nerves. The influences of two factors were investigated:

using a leadfield based on an accurate versus an idealized model of the nerve anatomy, and

applying a spatio-temporal constraint based on the electrophysiology of myelinated fibers.

Discussion 152

Simulated measurements corresponding to either one or three pathways were generated using

the realistic nerve model, and the source localization performance was evaluated in terms

of localization error, spurious pathways, and missed pathways. The approach in its present

form was not found to be sufficiently reliable for sub-fascicular localization in practice, due

to mean localization errors in the 140 µm-180 µm range, high numbers of spurious pathways,

and low resolution. Nonetheless, the improved anatomical model and the spatio-temporal

constraint were shown to produce a marked reduction in the number of spurious pathways.

We concluded that methods by which the noise in nerve cuff recordings could be reduced

and anatomical information obtained in vivo should be investigated if the source localiza-

tion approach is to become viable. In the short term, it is more realistic to focus on reliably

identifying combinations of whole fascicles, rather than to seek to obtain sub-fascicular in-

formation.

Having analyzed the problem through simulations, we sought to validate our findings

through the collection of experimental data. The MCC was placed on the sciatic nerves of

rats, and recordings obtained while the tibial, peroneal, and sural nerves were stimulated

in every possible combination. Once again, localization was performed both with an ide-

alized leadfield and a more realistic one, as well as with and without the spatio-temporal

constraint. In this case, the more realistic leadfields were based on cross-sections of sciatic

nerve samples collected after each experiment. In contrast to the exact anatomical model

used in the simulations, these models are still only approximations. None of the four con-

straint combinations tried led to satisfactory performance, in the sense of allowing us to

infer the number and identity of the active fascicles from one of the estimates. Given the

high levels of error created by the stimulation artefact and inaccurate model, these results

are qualitatively in accordance with the predictions of the simulations. More specifically,

the predicted effect of the spatio-temporal constraint was validated, as it was able to reduce

spurious pathways in practice as well as in theory. The effects of having a more realistic

model were not consistent with the predictions of the simulations, but this is easily explained

by the fact that the “realistic” model used for the experimental data was a much coarser

Discussion 153

approximation than the one used in the simulations. However, although the nerve-specific

model used on the experimental data did not reduce the number of peaks, it did somewhat

decrease the overlap between the estimates for different single-fascicle cases.

In the face of the poor performance of the traditional source localization approach,

we investigated a novel alternative designed to reduce the dependency on anatomical model

accuracy. Instead of constructing the leadfield by solving the forward problem using a model

of the nerve, we constructed an “experimental” leadfield using a collection of patterns ac-

tually observed during a set of training trials. Each pattern was associated with a fascicle,

such that this method is inherently designed to discriminate between fascicles and cannot

provide sub-fascicle resolution. The results for single-fascicle cases were reasonably good:

the correct branch was reliably identified, although this was sometime accompanied by false

positives for the inactive branches. The success of the single-fascicle cases using this tech-

nique proves that the failure of the model-based source localization to reliably identify these

same cases was due to model inaccuracies (either in the anatomy or because of the use of a

distributed dipolar source model), rather than to a fundamental lack of information in the

measurements. On the other hand, results for multi-fascicle cases were not reliable. This can

be attributed to remaining sources of error, such as interference from the artefact, as well

as to the usual difficulty of recovering complex source configurations in the context of an

ill-posed inverse problem. Nonlinear interactions when multiple pathways are present may

also play a role.

In light of the source localization results using both the model-based and experimental

leadfields, the last part of this thesis was concerned with determining how much benefit the

matrix cuff electrode really provides compared to more traditional single-ring multi-contact

nerve cuffs. We focused on the simpler problem of discriminating single fascicles in the

presence of training data and using data from several time instants (i.e., the method is

not strictly instantaneous). We found that very high classification success rates could be

achieved, and that the matrix cuff significantly outperformed the single-ring configuration.

Additionally, we found that an improvement could be achieved even when using the same

Discussion 154

number of contacts in both configurations, simply by virtue of having more flexibility to

choose contact locations in the matrix cuff. The implications for the design and use of

cuff electrodes are that it is beneficial to create devices with high number of contacts, then

perform a calibration procedure after implantation to select a smaller set of contacts that

will actually be used. From the point of view of source localization, there are both pros and

cons to these results. On one hand, we showed that the matrix cuff really does provide more

information that previous designs, and thus has an advantage for source localization. On the

other hand, the results indicate clearly that some contacts are more useful than others, which

implies that the source localization may have less than 56 measurements to take advantage

of or, worse, that some contacts may actually degrade performance.

8.2 Comparison of the results

8.2.1 Resolution achievable

The basic question in this thesis is to determine the spatial resolution that can be achieved

by applying a source localization approach to MCC measurements. The studies presented

in the preceding chapters offer insight into different aspects of this question. The solution

space reduction study of Chapter 3 provided a method to quantify the number of different

locations in the nerve that are theoretically distinguishable. This is useful in that it provides

an upper bound on the resolution that could be achieved. It was found that while single-axon

resolution is not feasible, small groups of axons could be distinguished, meaning that sub-

fascicular resolution is theoretically possible for large fascicles, and that small fascicles can

be discriminated. Again, however, these results deal with the theoretical distinguishability

of regions of the nerve under simple conditions (e.g., only one source is active at a time).

In order to estimate the actual localization performance that could be achieved in more

realistic cases, we must use a different approach. This was the role of Chapter 4, which

contained a simulation study of the source localization, in both single- and multi-pathway

Discussion 155

cases. Spurious and missed pathways proved to be significant obstacles but, for the purposes

of the current discussion, the point of interest is that the localization error was in the

140 µm-180 µm range. This level of uncertainty corresponds to a resolution lower than the

one suggested by Chapter 3, confirming that the source localization resolution in a more

realistic case can be expected to be far from the theoretical upper bound previously derived.

The resolution predicted by the simulations suggests that while discriminating large fascicles

should be possible, smaller fascicles may be confused, and sub-fascicular resolution is unlikely.

The results of Chapter 5 provide a further level of complexity to the issue, by examining

the performance on experimental data. Although the design of the experiment was such

that only whole-fascicle resolution was investigated, even this proved challenging given the

multiple sources of error in the source localization: discriminating single fascicle cases proved

unsuccessful. This part of the study is therefore more informative about practical challenges

than it is about resolution. Chapters 6 and 7, taking into account all the results so far,

used methods which were explicitly limited to whole-fascicle discrimination and excluded

any possibility of sub-fascicular resolution. In both those studies, single-fascicle cases could

be successfully identified. This confirms that the recordings contain enough information at

least for fascicle-level resolution, including a small fascicle (the sural branch), but we must

keep in mind that many human nerves in which these techniques might be applied contain

a much larger number of fascicles, on the order of several dozen. Additionally, multi-fascicle

cases could not be successfully identified, which brings up a central issue in the resolution

question: the amount of localization error is dependent on the number and configuration

of the bioelectric sources. Therefore, it is not possible to obtain a precise estimate of the

resolution given only the shapes of the electrode and nerve. Any discussion of this issue

will inevitably be confined to qualitative observations, unless discussing a specific source

configuration (which would be of little interest). Nonetheless, the multiple studies in this

thesis provide, in the opinion of the author, some useful insight into what is achievable and

what is not.

Discussion 156

8.2.2 Validation of the simulations

As already discussed, the localization results on the experimental data validated the predicted

impact of the spatio-temporal constraint, but not of the nerve-specific leadfield, although

this latter discrepancy is easily explained by differences in the model accuracy. Another issue

related to the validation is that the localization error was predicted by the simulations to be

acceptable in the idealized case of low noise and useful constraints, but as noise increased,

the error increased and the constraints ceased to be beneficial. In the face of the inaccurate

experimental results, this raises the question of how much noise must be present before the

constraints stop being useful. Consulting Figures 4.2 and 4.4, we see that starting at 20%

noise, there is no longer any observable difference in the localization errors of the different

constraint combinations. Given the multiple sources of error during the source localization

on experimental data, it is very likely that the noise was above this threshold (recall that

“noise” in this context can include factors such as model inaccuracies). The simulations

therefore did not predict useful performance under the conditions that were present during

the experiments, and the localization results in Chapter 5 are not inconsistent with the

predictions.

8.2.3 Implication of the contact configuration study on the source

localization results

The results of Chapter 7 demonstrated that good single-fascicle discrimination can be achieved

with a small number of contacts, and also that some contacts may actually be detrimental to

the performance. We therefore need to consider the implications of this information on the

results of the other chapters. On one hand, it raises the possibility that performance could be

slightly increased by excluding some contacts, but at the same time, the numbers of contacts

identified as optimal in the context of the simple problem of Chapter 7 are fairly low (Figure

7.5), and cannot be expected to be sufficient for precise source localization. Initial attempts

were made to re-apply the source localization (using both model-based and experimental

Discussion 157

leadfields) using only the set of contacts selected in Chapter 7, but this decreased rather

than increased performance (results not shown). Although these attempts were made only

on a small portion of the data, they are in accordance with expectations: the methods used

in Chapters 5 and 6 are much more complex than those in Chapter 7, so it is not surprising

that the set of contacts that proved optimal in the latter was not appropriate in the former.

The results of Chapter 7 are still relevant in that they show that not all contacts are equally

useful, but it is clear that the best set of contacts is not the same depending on the algo-

rithm being applied, and that a contact selection procedure must be tailored to the specific

technique being used. Conducting analogous selection procedures for the source localization

methods is, however, much more demanding in terms of time and computational resources,

and was prohibitive in the context of this thesis. In order to overcome these problems, it

may be necessary to develop new contact selection procedures tailored to these techniques,

rather than to use “naıve” approaches like the one in Chapter 7.

8.3 Limitations of the study

8.3.1 Experimental issues

As described in Chapter 5, we encountered three experimental difficulties: movement of

the recording cuff electrodes due to a bulky connector, stimulation artefacts, and poorly

controlled stimulation pulses. The latter issue was relatively minor because the pulses were

still able to reliably produce CAPs, the exact magnitude of which was not crucial to the

study. The other two difficulties were more serious because they significantly increased the

amount of error in the measurements. This is particularly problematic given that the task at

hand is so sensitive to noise. We were therefore not able to test the source localization under

ideal conditions. On one hand, this is counter to our stated goal of evaluating the potential

of the method, but on the other hand, our experimental difficulties were closely related to

the type of difficulties that can be expected in clinical applications. Slight cuff movements

Discussion 158

can occur as a result of body motion, and interference can be expected from other source

of biopotentials (e.g. electromyogram signals). The source localization performance in our

experiments is therefore likely more reflective of the performance that could be expected in

practice. An additional clue as to the impact of these factors on the study is that the cuff

connector was held in place manually for Rats 2 to 6, but clamped in place for Rat 1. If the

motion of the cuff was the main reason for the poor performance, we would expect Rat 1 to

have significantly better performance than the others. Although the recordings for that rat

were somewhat cleaner visually, the source localization was not more successful, so we can

conclude that the slight motion of the cuff was not the dominant factor.

8.3.2 Use of CAPs

The recordings used for this study originate from CAPs elicited via direct electrical stimu-

lation of the nerve branches, and are therefore different from what could be expected during

natural activity. A CAP is produced by a large number of axons firing synchronously. In

contrast, natural activity is generally sparse and asynchronous, such that the total number

of APs occurring within the cuff at any given instant is small. This implies that a signal

recorded when a CAP traverses the cuff will be much larger than signals obtained during

natural activity. The larger amplitudes of our signals were helpful because they increased the

signal-to-noise ratio and clarified the differences between measurement patterns produced by

different branches. How these results translate to the smaller spontaneous neural activity is

again a noise issue and therefore the discussion provided for other sources of error in this

study applies to it as well. The notion that the neural signals will be smaller in practice that

those used here makes it all the more apparent that noise reduction strategies will be vital if

the source localization approach is ever to be applicable. In the mean time, our decision to

deal with the easier case of the CAPs was logical from an experimental point of view, in that

it allowed us to more easily and precisely control which combination of fascicles contained

bioelectric activity in a given trial. Furthermore, given the poor performance obtained it is

Discussion 159

obvious that it would be premature at the moment to apply the source localization approach

to natural neural activity. On the other hand, if the performance on CAPs were to reach

acceptable levels, applying the source localization approach to natural activity would be an

essential next step in validating the technique.

8.3.3 Use of simplified FE models

Although we investigated the localization performance using both idealized and more realistic

FE models, neither type was a completely accurate reflection of the nerve’s true anatomy.

Even in the models based on the nerve sample cross-sections, the trace was not perfectly

accurate and included some approximations (relating for example to the thickness of the

perineurium and the encapsulation tissue layer, as well as to the assumption that the nerve

had the same shape with and without the cuff). Even more significant was the fact that

the cross-sections were simply extruded in both directions to obtain the 3D FE model.

These longitudinally uniform models therefore do not take into consideration the cross-

sectional variations due to the progressive branching of the fascicles. This simplification

is detrimental to the accuracy of the leadfield, and therefore to the success of the source

localization. Constructing a completely accurate full 3D FE model is, however, a daunting

task that would require very sophisticated imaging techniques. At present no such methods

are available clinically, such that a source localization approach that is useable in practice

cannot rely on them. The use of simplified FE models in this thesis is therefore reflective of

the level of information that might be available in clinical application, and therefore more

appropriate for predicting the method’s performance. Similarly, the diameter of the nerve

in the idealized model proved to be too small compared to the measurements from the nerve

samples. As mentioned in Chapter 5, this is likely due to the fact that we deliberately used

large rats, whereas the idealized model was based on information in the literature that was

likely obtained from rats with more average sizes. Once again, however, the goal of having

an idealized model is to dispense with the need to tailor the method to individual nerves,

Discussion 160

and so having a discrepancy in the diameter would not be unexpected in practice.

8.3.4 Use of a low-resolution source localization algorithm

We chose to solve the inverse problem using the sLORETA algorithm. This choice was based

on a careful consideration of the alternatives and the specific characteristics of the peripheral

nerve problem (which differ from those of the EEG source localization problem). Nonetheless,

the fact remains that sLORETA is explicitly designed to produce low-resolution solutions,

which is the price it pays for higher accuracy. In a situation such as the peripheral nerve

problem, where extremely high spatial resolution is desirable, it is obvious that sLORETA

is not an ideal algorithm; it is simply better than the alternatives. There is a need for higher

resolution algorithms tailored to the needs of peripheral nerve source localization, but until

the accuracy of the method can be improved significantly over what was obtained in the

present study, improving the resolution will remain a much lower priority.

8.3.5 Use of peaks in the estimate as a measure of the number of

pathways

The procedure to determine the number of active pathways from the source localization es-

timate, for both simulated and experimental data, is described in Chapter 4. As mentioned

there, this method is not entirely without its flaws. In particular, the presence of an active

pathway may modify the shape of the estimate without producing a distinct peak, for exam-

ple by producing an elongated ridge. Several factors can influence the shape of the estimate

in multi-pathway cases, including the distance between the sources and their relative mag-

nitudes. Finding a method that can reliably determine the number of pathways in such a

widely varying range of situations is a very difficult task. Therefore, for the purposes of this

thesis, the number of peaks was judged to be an acceptable metric for gauging the success of

the localization and the influence of the constraints, even though it is not a perfect method.

The main impact of this limitation will be on the number of missed pathways: if a pathway

Discussion 161

is not detected because it did not create a clear peak, then the number of missed pathways

may be artificially high. The impact of this consideration is, however, minor enough that

our conclusions about the number of missed pathways (e.g. role of sLORETA blurring) still

hold true.

8.3.6 Focus on spatial over temporal resolution

A major reason why bioelectric source localization techniques have been drawing attention

is their high temporal resolution, which gives them access to crucial features of neural ac-

tivity that are unavailable to other slower modalities, such as fMRI or PET. The question

of temporal resolution therefore deserves a mention here, even though little attention was

devoted to it in this thesis. This is not strictly speaking a limitation of the study, because

this data is available: time series were collected from all recording contacts with a sampling

rate of 20 kHz. Until the neural activity can be associated with specific pathways, however,

the characteristics of the signal in time are of little interest because we do not know what

they refer to. If, on the other hand, only temporal information is of interest (as a reflection

of the amalgamated activity in the entire nerve), then it can be obtained with a considerably

simpler electrode and recording setup.

8.3.7 Focus on a specific electrode and nerve

All of the results in this thesis are concerned with performing source localization using a

1 mm-diameter matrix MCC on a rat sciatic nerve. The focus is therefore on a small nerve.

It is possible that in a larger nerve, or with a different electrode shape (e.g. a FINE), the

results may be different. Even if the localization error did not decrease, a similar error would

seem less significant if the overall region was larger, and may allow for better discrimination of

fascicles if they were spaced further apart. For a truly thorough assessment of the potential of

peripheral nerve source localization, different nerve sizes and electrode configurations should

therefore be investigated. In this thesis, however, we focused on the situations that we were

Discussion 162

in a position to test experimentally, in order to be able to validate our results. In addition,

small nerves are of interest because they are more difficult to access when using intraneural

peripheral nerve interfaces such as MEAs, and therefore are a natural target for methods

based on extraneural recordings.

8.4 Optimal number of contacts

The general task of obtaining spatial information about nerve activity from nerve cuff mea-

surements is very difficult. It stands to reason that as we seek to achieve better resolutions,

the amount of information needed will increase. A natural question throughout this study is

therefore how many contacts are needed to achieve the performance goals of a given applica-

tion. In the context of the source localization problem, the number of contacts that will yield

good performance depends on a variety of factors, including the complexity of the region in

the cuff and the source configuration. It is therefore a very thorny issue to try to determine

an optimal number of contacts, and if one were to do so for a very specific situation, it would

almost certainly not generalize to other similar situations. For these reasons, in this thesis

we did not directly investigate the question of which number of contacts is best. Many of

our observations, however, shed some light on different aspects of this issue.

First, our study of the solution space reduction method showed that the amount of

noise had a direct impact on how many contacts were useful for that application. When

there was very little noise the extra contacts were not very beneficial, and when there was

too much noise the benefit again decreased, but between those two extremes the number of

contacts could improve the resolution achievable. It is crucial to keep in mind, however, that

the measure of performance in that study was the number of different regions that could

theoretically be distinguished from one another. It therefore represents an upper bound on

the localization performance, but by no means suggest that this performance will be close to

that bound. There is therefore a distinction between the influence of the number of contacts

on the possible resolution, and on the accuracy of the localization.

Discussion 163

The second main indicator of the influence of the number of contacts was our compar-

ison of the matrix and single-ring cuff configurations. Based this time on experimental data,

and tackling a simplified problem of fascicle identification, we arrived at three conclusions.

First, the way to obtain the best performance was to start with a large number of contacts

and narrow that set down to a smaller subset of the most useful ones. The subset ultimately

selected varied from rat to rat, which further argues against the possibility of designing a cuff

with a set of contacts whose number and locations are optimal. Rather, this result suggests

that calibration procedures may be preferable, although of course the feasibility of this may

become more of an issue in problems more difficult than our simple fascicle discrimination

task. Second, we determined that high performance could be achieved with a small number

of contacts, although this is expected to be highly application-dependent. Third, we high-

lighted the fact that the location of the contacts is crucial, not just their number. Therefore,

varying for example the number of contacts in a single ring and examining the results is

not an appropriate way to investigate this issue. The importance of the locations further

complicates the goal of trying to identify an optimal number of contacts: the issue is not

only how many contacts do we need, but also where should we put them.

Although interesting in their own right and having concrete practical application,

neither of the two studies above directly addresses the fundamental question of this thesis,

which is whether or not bioelectric source localization can be applied successfully to periph-

eral nerves. The performance in that respect was poor, despite using a number of contacts

larger than what was shown to be useful in both the studies just discussed. Even taking into

account the fact, illustrated in Chapter 7, that too many contacts can in fact decrease perfor-

mance, there is clearly a gap between the performance of the source localization and that of

the two simpler problems. More than anything, this illustrates the difficulty of generalizing

any conclusions about the best number of contacts.

Taking all of these factors into consideration, it is the opinion of the authors that

the only way to satisfactorily answer the question “How many contacts do I need?” is by

conducting application-specific pilot studies that use a large number of contacts and inves-

Discussion 164

tigate the performance of various subsets. The general results should then be adapted to

each subject using a calibration procedure. Ultimately, higher number of contacts will not

necessarily always be better, but it is extremely difficult to arrive at specific conclusions a

priori for a given application.

8.5 Number of pathways to be localized

The simulations investigated the case of three simultaneous active pathways and, likewise,

the experiments involved three fascicles. In practice, the number of pathways simultaneously

active may be much higher. For example, the human femoral nerve contains upward of 20

fascicles [150]. The question is then how many pathways can be simultaneously localized

with the type of approach proposed in this thesis. Of course, at the current low level of

performance, the question is premature. Nonetheless, it will become important in the future

if the current obstacles are overcome.

Although the results presented in this thesis do not directly address the issue of a

maximum number of pathways, some of our observations can nevertheless allow us to make

some predictions. First, we have highlighted the fact that the accuracy of the localization

is dependent on the complexity of the source configuration. Second, we found that both

spurious and missed pathways are obstacles to the localization. Missed pathways will become

even more of an issue when there are more pathways, if algorithms are not developed that

avoid smoothing the solution while still retaining accuracy and robustness. Spurious peaks

in the solution may also be even more misleading if the true number of pathways is high,

such that small features in the estimate are actually reflective of activity of interest. Taking

into consideration all of these factors, it seems unlikely that situations with more than 5 or

6 simultaneously active pathways could be accurately and completely reconstructed using

this type of measurement setup. This limit will therefore have to be taken into account in

the development of neuroprosthetic applications. If there are more than 5 or 6 pathways

that are of interest (i.e., not including small pathways whose activity may be interpreted as

Discussion 165

noise without loss of functionality), then alternative strategies may become necessary. This

may include using more than one electrode and placing them more distally, after the nerve

has split into more branches, each of which has fewer fascicles. Alternatively, it may be that

intraneural recording devices such as MEAs will be found to be preferable for large nerves

with many fascicles, whereas MCCs will be preferable for smaller and more delicate nerves

with fewer fascicles.

A closely related issue, which is not specific to the method used to localize the bio-

electric source, arises once we have identified the approximate location of the activity. The

question is then to determine the functional relevance of this information in cases where

fibers with different roles are situated very close together. Identifying the conduction veloc-

ity (Section 2.3.3) can help alleviate this problem if the fibers of interest are of different sizes.

Nonetheless, cases will arise in which fibers have similar conduction velocities and locations,

but different functional implications for the control of a neuroprosthesis (as an example of

this situation, consider the case of a muscle spindle afferent and a Golgi tendon afferent). In

those situations, it may be feasible to classify the nature of the activity based on temporal

patterns. In other words, signal processing methods of the type that have been proposed

for nerve cuff recordings of whole-nerve activity (e.g. artificial neural networks [16]) could

be applied to the signals corresponding to only to a particular region in the nerve. We can

reasonably expect that the performance of these methods would only increase as the number

of different pathways affecting the measurements is reduced. In the case of efferent activity, if

the goal is to control an assistive device through a direct neural interface, then an additional

option would be to treat each of the pathways that can be reliably monitored as a binary

switch. In this way, a multi-switch control scheme could be established that would allow the

user to achieve sophisticated control, without any need to interpret the original functional

relevance of the signals in the pathways.

Discussion 166

8.6 Factors related to the use of a nerve cuff electrode

An important consideration when evaluating the results of this study is that a nerve cuff

electrode increases the uniformity of the electric fields inside of it. In other words, the spatial

variations in the potentials around the nerve are smaller than they would be if there was no

cuff [168, 98]. This is a consequence of the more restricted extracellular environment created

by the cuff, and suggests that were will be an upper limit to the number of recording con-

tacts that are useful. Sampling several point that are too close together to have appreciable

potential differences will not increase the amount of meaningful information obtained. Re-

calling the results of our contact selection study (Chapter 7), however, we know that a limit

on the number of useful contacts may not necessarily translate into a limit on the number of

contacts that are placed on the cuff initially. The increased field uniformity inside a cuff will

also impose a limit on the source localization resolution achievable from these recordings.

The results of the leadfield reduction technique presented in Chapter 3 are useful in this

respect because they provide a tool to help gauge what the attainable resolution might be.

Further, an inherent resolution limit bolsters our decision to use sLORETA, by making its

smoothing of the solution less of an issue. It is also essential to clarify that the increased

field uniformity by no means makes attempts to identify the locations of bioelectric sources

within the nerve hopeless. Among the results of this study, the most informative ones with

respect of this issue are the simulation results presented in Chapter 4. There, we saw that

under favorable conditions (low noise, accurate leadfield, and useful constraints), good per-

formance could be achieved. The FE model used in these simulations accurately reflects the

restricted extracellular medium in which the fields are propagating and therefore the results

are indicative of how big an obstacle the field uniformity really is. We can use the simulations

to draw conclusions about these theoretical limits without being hampered by the numerous

sources of error that were present in the experimental data. In addition, several published

studies (reviewed in Chapter 2) have shown that discrimination of the activity of different

pathways in a nerve is possible, which provides clear evidence that the use of a nerve cuff

Discussion 167

does not completely abolish the presence of useful spatial variations in the fields.

Although the restricted extracellular environment has the disadvantage of increasing

field uniformity, it also has the benefit of increasing the magnitude of the recorded signal.

This was in fact the core motivation behind the introduction of the first nerve cuff electrodes.

From the point of view of bioelectric source localization, the increase in signal-to-noise ratio

afforded by a cuff is very valuable. There is therefore a trade-off between spatial variability

of the potentials in the cuff and the signal-to-noise ratio, a trade-off which seems worthwhile

based on the simulation results presented in this study, and other published studies of nerve

cuff selectivity. Nerve cuff designs have also been introduced that are divided into several

chambers [61]. The selectivity of electrodes in different chambers is thus increased, while at

the same time retaining the benefit of larger signals. While a cuff with numerous contacts

may require numerous chambers and therefore pose a greater challenge in terms of manufac-

turing, the concept is sound and likely to improve the trade-off mentioned above. The nerve

cuff variant know as the FINE, which flattens the nerve to get better access to all fascicles,

has also been shown to improve selectivity, and therefore is a promising technology to com-

bine with the source localization approach. At the time of writing, however, no FINE had

been developed that had the grid-like configuration of contacts that was shown in Chapter

7 to be useful. When discussing the pros and cons of nerve cuffs we may be reminded of the

benefits of intraneural electrode designs, which can combine both high selectivity and high

signal-to-noise ratios, but of course those technologies have their own significant drawbacks

(e.g. damage to the nerve and unsuitability for smaller nerves in the case of MEAs, or lack

of spatial coverage in the case of LIFEs).

A final interesting issue is specific to the use of the MCC. In this study we used data

from all 56 contacts simultaneously, in such a way that a common reference was needed for

all of them. This highlighted a new difficulty that was not present in other nerve cuff studies

in the literature. In the traditional nerve cuff design, the tripole configuration is designed

to minimize the interference from sources outside the cuff, but only if the recording contact

is placed half-way between the two reference contacts (see Section 2.3.1 of the Literature

Discussion 168

Review). When there is an entire grid of contacts, like in the MCC, it is no longer possible

to reduce interference equally at all contacts while still maintaining a single reference for

all of them. In this thesis, we explore two reference options: a common-average reference

(as often used in EEG) was used for the source localization analysis, whereas a tripole

reference was used in Chapter 7 in order to maintain consistency between the different contact

configurations studied. We observed little difference in the noise reduction capabilities of

these two references. In both cases, the noise reduction was most effective in the centre ring

of contacts, as expected. The use of the MCC therefore intrinsically creates a situation in

which the signal-to-noise ratio at different contacts may be significantly different, which in

turn may affect the results of the source localization. Note that if an estimate of the noise

distribution is available, it can be taken into account in the inverse problem using the matrix

Cǫ of Equation 2.18, but a precise estimate may be difficult to obtain.

8.7 Implications for future cuff-based peripheral nerve

interfaces

A general remark regarding the use of a source localization approach in a peripheral nerve is

in order. In the EEG/MEG context, source localization techniques have both a much higher

temporal resolution and a lower spatial resolution than other modalities, such as fMRI. In the

peripheral nerve case, the high temporal resolution remains quite attractive, because of the

short time spans involved in trains of action potentials. The low spatial resolution remains

an area in obvious need of improvement but, in contrast with the EEG/MEG context, alter-

native methods for achieving spatial discrimination of the electrical activity within a nerve

without risking tissue damage are very limited. EEG/MEG source localization studies are

also hindered by the difficulty of assessing the correctness of the results, which is particularly

important in light of the ill-posed nature of the problem, but in peripheral nerves the tech-

nique can be validated in a more definite manner, either by stimulating known pathways (as

Discussion 169

done here) or by using simultaneous MEA recordings. In fact, applying source localization

techniques to MCC recordings is, from a mathematical point of view, the most complete and

flexible framework available to us for creating a peripheral nerve interface that can discrim-

inate between spatial locations in the nerve. Given the poor performance obtained in this

thesis, it is natural to ask what the implications are for future cuff-based peripheral nerve

interfaces, and whether there is a limit to the potential of this type of technology. Indeed,

although cuff-based neuroprostheses have been demonstrated to have some selectivity both

in recording and stimulation, there is no reason to assume that it will ever be possible to

achieve arbitrarily fine resolution. The study in Chapter 3 illustrates this fact quite clearly.

The first question to ask is therefore what neuroprosthetic applications are realistic

for multi-contact nerve cuff electrodes. The main factor will be the number of distinct

regions in the nerve that need to be distinguished. These regions may correspond to different

fascicles, or simply to different pathways in a large unifascicular nerve, but it seems unlikely

that their number will be far above single-digit quantities. However, this estimate only

takes into account discrimination based on spatial variations in the extraneural fields. This

method can be combined with other approaches that rely on temporal information. One

realistic technique would be to use the MCC to detect conduction velocity, then translate

this information into the type of fiber that is active in each of the spatially distinguishable

regions. In this manner, the amount of functional information that may be extracted from

an MCC-based neural interface may be significantly increased. Nevertheless, as long as the

spatial information remains coarse, there will be applications that will not be achievable.

At present, it seems that these types of interfaces can have a role to play for example in

monitoring bladder activity or leg position, but creating an interface with a complex artificial

hand, which involves many more degrees of freedom, is unlikely. As mentioned above, the

size of the nerve may also have an impact on the efficacy of the method and therefore on the

applications for which it is appropriate.

The second question is how the performance of MCC-based interfaces can be brought

from the current low level described in this thesis to the level where the applications dis-

Discussion 170

cussed in the previous paragraph can actually be implemented. Our results highlight several

possible avenues for improvement. Primarily, the results of Chapter 6 suggest that further

research into the experimental leadfield approach would be beneficial. In an inverse problem

context, this could take the form of appropriate constraints and regularization techniques

to improve the performance in multi-fascicular situations. More generally, a promising ap-

proach for identifying combinations of branches may be to establish nonlinear input-output

maps between the active pathways and the cuff measurements [93]. In this way, the com-

plex spatio-temporal interactions between bioelectric sources in the nerve may be captured

more accurately than what is possible within the linear framework of the distributed dipoles

approach used here. As for improving model-based source localization in peripheral nerves,

research should be conducted into how to obtain more accurate anatomical models of the

nerve. The nerve’s shape changes during chronic cuff implantation, and encapsulation tissue

forms, meaning that what is really required is a diagnostic method that can be applied in vivo

to obtain an image of the region inside the cuff, as often as required. This information can

then be used to update the model used in the source localization and ensure that it remains

accurate. One example of a method that could be investigated for this purpose is to adapt

the electrical impedance imaging technique [9] to peripheral nerves. Alternatively, ultra-

sound has been shown to be able to image fascicles within a nerve [77], so that if transducer

technology advanced to the point of being implantable around a nerve, this avenue could

also be explored. Our results have also confirmed that applying physiologically meaningful

constraints can have a positive impact on performance. Investigating new constraints for

this purpose will remain premature until a reasonably accurate model of the nerve is avail-

able, but may be warranted in the future. In this respect, using several implanted electrodes

may open new possibilities for constraining the solution. For instance, having simple cuff

electrodes placed distally on the branches of a nerve could indicate which branch is active.

This information could then be used as a constraint when source localization is applied to

measurements from a larger MCC placed more proximally on the main nerve trunk, in order

to achieve finer resolution within that branch. Lastly, higher-order multipole expansions

Discussion 171

could be used as a source model, in order to determine if the dipole model is too simple and

therefore a source of error.

Chapter 9

Conclusions

Having completed our study of bioelectric source localization in peripheral nerves, our main

conclusions can be summarized as follows:

• The theoretical resolution that can be obtained using an MCC, when a single source

is present, is on the order of a small group of fibers or a small fascicle. The number of

variables in the inverse problem of source localization can be matched to this resolution,

eliminating unnecessary computation time and storage requirements.

• Approximate localization of active pathways in peripheral nerves by applying tradi-

tional source localization algorithms to MCC recordings is theoretically possible for a

small number of pathways, under near-ideal conditions of very low noise, informative

constraints about the solution, and accurate nerve and source models. Even under

those conditions, distinguishing closely spaced pathways may not be possible.

• As soon as the noise exceeds very modest levels (i.e., an NSR of 10%), simulations

showed that performance will be hampered by localization errors in the 140 µm-180 µm

range and high numbers of spurious pathways, in addition to the low resolution. There-

fore, under realistic conditions, the proposed approach is not expected to be reliable.

This was confirmed by applying source localization to experimental recordings from172

Conclusions 173

rat sciatic nerves, and noting that the results could not be used to accurately identify

the branches that were stimulated.

• The application of a physiologically meaningful constraint can reduce the number of

spurious pathways both in theory and practice, but had little effect on the localization

error. The use of an accurate model of the nerve anatomy was similarly useful in theory,

but in practice models custom-made based on cross-sections of the nerves used in the

experiments were not successful in improving performance. This suggests that minor

improvements in model accuracy are not sufficient and that very accurate models are

required. As a result, model inaccuracies such as incorrect or insufficient anatomical

detail as well as the use of the distributed dipoles source model are very likely to

prevent our approach, as initially proposed, from being clinically practical.

• An alternative and novel formulation of the source localization problem, in which the

leadfield is constructed using training data instead of a model, outperforms the model-

based approach and is able to correctly identify single-branch cases, confirming that

MCC recordings can be used for this task. The failure of this method to correctly

identify multi-branch cases can be ascribed to a combination of experimental difficul-

ties, the inherent difficulties of ill-posed inverse problems, and nonlinear interactions

within the nerve. More work is warranted to characterize the influences of these dif-

ferent factors and compensate for them. Based on the results presented in this thesis,

methods that rely on experimentally derived relationships between the active branches

and the measurements are a more promising approach to peripheral nerve interfacing

than model-based alternatives.

• A “matrix” type cuff, in which contacts are laid out in a grid, contains more useful

information than a cuff with a single ring of contacts. This can be attributed to differ-

ences between the interface of each contact with the nerve, and their position relative

to the fascicles. The matrix cuff can take advantage of its better spatial sampling and

Conclusions 174

use only the most useful contacts, therefore achieving superior performance without

necessarily needing to use a larger number of contacts. Consequently, applications

that use MCCs would benefit from implanting devices with large numbers of contacts

spread over the inside of the cuff and then performing a contact selection procedure.

The best number of contacts to use depends on the application.

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