SBIR Phase I Final Progress Report

Grant Number: 1R43NS69310-1A1 Project Title: BOLD-Related EEG Signal Estimation Software

Grantee Organization: Source Signal Imaging, Inc. Project Period: 3/15/2011 – 9/30/2012

PI: Mark E. Pflieger Goal, requirements, and impact. The scientific-technical goal of this SBIR project is to develop a cohesive set of new algorithms for analyzing simultaneous EEG-fMRI experimental data in order to detect and estimate EEG space-time-frequency signals that are coupled specifically with BOLD signals of interest, such as BOLD signals associated with a brain region of interest. Commercially, BOLD-Related EEG (BRE) signal detection and estimation algorithms shall be provided on the EMSE software platform to cognitive, systems, and translational neuroscientists, who shall be supported in their research efforts by Source Signal Imaging (SSI) application engineers (see Attachment [1]). The BRE algorithms shall respect general principles that constrain biophysical models (such as hemodynamic response functions and neuroelectric volume conductors) without, however, depending on specific modeling details. Rather, the computational theory shall serve to promote discoveries of empirical relationships in EEG-fMRI experimental data. We aspire to provide and support data-driven BRE algorithms that will help neuroscientists discover new relationships between the vast realms of neuroelectric and hemodynamic phenomena—new relationships that can promote a deeper understanding of neuroimaging of the adaptive and maladaptive human brain. Assessment of progress and current status. We have made substantial progress with algorithm theory and methods (two major revisions), software technologies for bridging EEG and fMRI preprocessing tools, engagement with EEG-fMRI experimental data, and computational infrastructure. However, algorithmic methods in the third generation (second major revision) are implemented partially, and analysis of the experimental data is incomplete. Therefore, this project needs—and we have been giving it—more effort beyond Phase I in order to demonstrate feasibility as required for Phase II. We are encouraged by incremental progress with implementation and application of the current generation of algorithms. Algorithm theory and methods: First generation. The Phase I application described our first generation of algorithms for BRE (BOLD-related EEG) signal detection and estimation. Key features of our first approach have been maintained throughout this project: (i) the researcher specifies a BOLD signal of interest (SOI) in the hemodynamic domain of phenomena comprising frequencies ~0.01-0.10 Hz; (ii) specificity of the BOLD SOI is determined by its contrast with BOLD signal not of interest (SnOI) in the hemodynamic domain; (iii) the BOLD signal subspace potentially relevant to quasi-statically generated EEG signals is constrained by gray matter voxels located principally in cerebral cortex; (iv) the EEG signal is analyzed spatially one frequency at a time across a spectrum ranging ~0.10-100 Hz; (v) the applied math problem is to find a temporal filter (hemodynamic response function = HRF) and a spatial filter (beamformer-like estimator) such that a flexible nonlinear function of spatially filtered EEG power, convolved with the HRF, is maximally coupled with SOI-specific BOLD signal; (vi) a nonparametric statistical test is needed to detect BRE signal and to assess the strength of EEG-BOLD coupling; (vii) fast BRE signal in the neuroelectric time domain is estimated by synthesizing multi-frequency spatial filters across the EEG spectrum; and (viii) a composite HRF is estimated in the hemodynamic time domain by via its correspondence with fast BRE signal. We learned that our first approach was limited significantly in practice due to two highly general yet “brute force” nonlinear optimization methods: specification of the objective function as conditional mutual information, and numerical optimization by conjugate gradient ascent method. The highly general character of these nonlinear methods comes at a high computational cost; in particular, up-front over-reduction of the BOLD and EEG signal spaces are required in practice. Consequently, too many space-time-frequency degrees of freedom available in each domain (hemodynamic and neuroelectric) are lost before the analysis can proceed. A second related issue is that—although we seek model-independent characterizations of temporal and spatial filters in (v) above—our methods had to be initialized using a canonical hemodynamic response model used in the fMRI field, and a standard volume conductor and source space model used in

the EEG field. However, a priori models may introduce biases that can hinder discoveries of empirical neurovascular coupling relationships to be found in concurrent EEG-fMRI data. Finally, a third issue we encountered through engagement with the data is that—in spite of the relative high spatial resolution of fMRI—regional specificity of the ongoing BOLD signal can be difficult to achieve in practice due to high inter-regional correlations with a region of interest (ROI). In fact, whole-brain BOLD signal correlations are the well-known basis for studies of resting and task-related functional connectivity networks using fMRI. However, this presents a problem with the core-shell approach to regional specificity proposed in our Phase I application, because a “core” ROI may be more highly correlated with distant regions than with its immediately surrounding “shell”; yet distant regions also may contribute to the generation of EEG signals in synchrony with ROI generation. Algorithm theory and methods: Second generation. The second generation of algorithms (our first major revision) was based on the following insight: since BOLD SOI is coupled with some (generally nonlinear) function of EEG power, and EEG power is a second order statistic, and second order statistics in EEG are generally characterized by a matrix whose elements are sums of cross-products between pairs of EEG channels, we can initiate the analysis process by maximizing BOLD SOI coupling for each EEG channel pair separately. When the separately optimized couplings are assembled into a full inter-channel coupling matrix, that matrix can be analyzed for its principal spatial components, which are closely related to the desired EEG spatial filter. Thus, the second approach estimated the EEG spatial filter empirically from the start, without model-based initialization or multivariate nonlinear gradient ascent. On the other hand, we still needed a canonical HRF model for the coupling analysis; and we still conceived of the BOLD SOI in terms of an associated ROI. This hybrid approach was an intermediate step that lead to our current third generation of algorithms (second major revision). Algorithm theory and methods: Third generation. Our current algorithms (second major revision) are described in detail in Attachment [2], a manuscript in preparation entitled “Data-driven detection and estimation of BOLD-related EEG signal: computational theory”. The new approach begins (1) by characterizing the total BOLD signal space empirically, based on a temporal component analysis such as ICA, as used typically in fMRI functional connectivity studies (see Attachment [3]). The components define basis functions that span the total BOLD signal space, and the BOLD SOI specified by the researcher must “stand out” against this total background. In particular, any ROI-based SOI must be specified as a linear combination of components so that ROI correlations with distant regions are taken into account and canceled. Thus, although voxel space is still important for SOI specification, the total BOLD signal space as defined by components is primary. Next (2), a linear technique in the hemodynamic frequency domain is used to maximize nonlinear dynamic specificity of the SOI in the total BOLD signal space. This technique works by forming a matrix that characterizes interactions between low frequencies (~0.01-0.10 Hz) in the BOLD signal space. Interactions between different frequencies occur only for nonlinear systems; however, the matrix is analyzed using standard linear methods. Next (3), the insight described above for the second generation approach is extended as follows. Second-order inter-channel statistics in the neuroelectric frequency domain are estimated separately; however, a static nonlinear function (designated by φ) is applied to the matrix as a whole. This is another way in which nonlinearities may be introduced while preserving the computational efficiency and stability of linear methods. Next (4), we depart from the second generation by directly optimizing the EEG-to-BOLD HRF in the hemodynamic frequency domain, without using a canonical HRF model. Once again, linear methods are employed, this time due to the standard assumption that the unknown HRF is convolved with some function of neural activity. This assumption is consistent with our Phase I application, and so is adequate for feasibility assessment. However, from other work, we also know how to extend linear methods to solve nonlinear Volterra models of neurovascular coupling (reference [25] of Attachment [2]). Next (5), we employ phase-randomization in the hemodynamic frequency domain to test nonparametrically for statistically significant neurovascular coupling of the entire channel-by-channel HRF matrix obtained in the previous step. Importantly, this method is used to detect BRE signal and—if so detected—the method determines the “beamspace dimension” of the BRE spatial filter (whereas in our Phase I application, we were resigned to use only a beamspace dimension of 1). Next (6), we synthesize a spatial filter across all (or a range of) neuroelectric frequencies in order to estimate fast BRE in the continuous EEG data. This spatial filter may be used to derive virtual EEG channels of derived data—specifically related to BOLD SOI—which in turn may be analyzed using standard methods of task-

related EEG analysis. Finally (7), we use the fast BRE signal to refocus the HRF analysis across a broader band of EEG frequencies. The ensuing neuroelectric HRF (as opposed to a canonical HRF) may then be incorporated into the design matrix of standard task-related general linear model analyses of the fMRI data. In summary, our revised algorithms employ stable methods of linear matrix algebra to solve nonlinear problems in the hemodynamic frequency domain, where second-order EEG cross-spectra have been doubly transformed (first to the neuroelectric frequency domain, and then to the hemodynamic frequency domain). The resulting algorithms preserve the objectives of our Phase I application while overcoming the computational limitations of our original approach. Software technologies for bridging EEG and fMRI preprocessing tools. In addition to fundamental revisions of our computational approach, we have addressed the important practical matter of how to perform integrated BRE signal detection and estimation using SSI’s Windows-based C++ development platform (the technological basis for EMSE, which contains a complete set of preprocessing tools for EEG) together with commercial-friendly-licensed software for fMRI preprocessing. To this end, we have consulted with Neuroimaging in Python (NiPy) developers giving particular attention to the Nipype pipelines and interfaces (http://nipy.sourceforge.net/nipype/). The NiPy project contains native commercial-friendly Python code for neuroimaging analysis as well as interfaces to other standard fMRI analysis packages, notably, FSL, AFNI, and FreeSurfer. Of these, FreeSurfer has a commercial-friendly license; AFNI does not; and FSL has a mixed license (one for academic use and another for commercial use; we have obtained permission from Oxford University for our current use of FSL as described in Attachment [3]). Python code is portable across computing platforms (Windows, Unix), and we have therefore implemented a Python interface to SSI’s scripting engine technology. That is, native EMSE scripts may incorporate existing Python scripts, or generate and execute Python scripts “on the fly”. Because most available fMRI software runs best on Unix, we have been using VMware (http://www.vmware.com) to run fMRI software on an Ubuntu virtual machine (https://help.ubuntu.com/community/VMware) running on Windows concurrently with EMSE, using a shared data folder to merge EEG and fMRI analysis results. Engagement with EEG-fMRI experimental data. Using standard preprocessing methods, we have engaged with two high-quality EEG-fMRI datasets provided by Prof. G. Ron Mangun, acquired at the UC Davis Imaging Research Center. See reference [1] of Attachment [2] for a description of the experimental paradigm. Eight further datasets are standing by, awaiting completion of our analyses. With respect to the 64-channel EEG data, MRI gradient artifacts and ballistocardiogram artifacts were corrected successfully and consistently using two different commercial software implementations (Scan by Compumedics Neuroscan’s Scan; and EMSE by SSI). Standard analyses of the data produced high-quality ERPs using EMSE. With respect to the fMRI data, we engaged Colleen Chen, a graduate student of the Computational Science Research Program at San Diego State University, who routinely performs fMRI analyses at the nearby Brain Development Imaging Lab (http://www.sci.sdsu.edu/bdil/web/BDIL.html). Preprocessing steps and intermediate results for the two UC Davis datasets are provided in Attachment [3]. In brief, we analyzed these multi-run task-related datasets using standard methods using for functional connectivity analysis—in particular, the MELODIC ICA method provided in the FSL software, which we are using for data-driven estimation of the total BOLD signal space (together with exclusion of artifact components and components related to subcortical networks outside of the EEG source space). Importantly, as shown, we were able to identify standard functional connectivity networks across runs for each individual subject. Computational infrastructure. Finally, a significant proportion of our efforts were devoted to designing and implementing C++ classes for BRE integration EEG and fMRI. Except for fundamental data structures, much of the design parallels the computational theory. Consequently, new C++ classes and modifications to classes have been required for each generation of algorithms. In conclusion, based on the current status, we feel optimistic that continuing efforts beyond Phase I will enable us to prepare a competitive Phase II proposal. Attachments

[1] EMSE as a platform for EEG translational research: from theory to clinical practice. [2] Data-driven detection and estimation of BOLD-related EEG signal: computational theory. [3] fMRI preprocessing of UC Davis datasets.

SSI Vision Statement 130929

EMSE as a Platform for EEG Translational Research: From Theory to Clinical Practice

Translational Research is the engagement of basic research with clinical practice: in particular, the

engagement of cognitive, behavioral, and social electrophysiological research with best clinical practices for preventing, assessing, diagnosing, and treating neurological, psychiatric, and neurodevelopmental disorders.

(a) Clinical practice needs more effective combinations of diagnostic constructs and therapeutic interventions which are based on substantiated theories of adaptive and maladaptive brain-behavioral circuit dynamical mechanisms. These clinical needs motivate the design of EEG experiments that use special behavioral paradigms (including baseline or resting conditions). The EEG experiments are designed to test etiological hypotheses, to devise EEG-based diagnostic biomarkers, to assess the effectiveness of pharmacologic interventions, or to develop therapeutic neurofeedback protocols.

(b) Series of well-designed experiments, incisive data analyses, and sound theoretical interpretations in cognitive electrophysiology and related fields lead systematically to improved clinical practices.

SSI Vision Statement 130929

Technical Support at Source Signal Imaging (SSI) is the engagement of SSI personnel with researchers who use the EMSE software platform to design, conduct, analyze, visualize, and interpret behavioral electrophysiology experiments.

From its classical embodiment as a GUI-intensive suite of software tools for analysis, visualization, and modeling of neurophysiological data and integration with MRI neuroimaging, EMSE has been evolving to serve as a complete software platform for EEG translational research, including: (i) integration of experimental design with nonlinear system modeling (SystemID module); (ii) hardware-neutral, real-time data acquisition integrated with closed-loop behavioral experimentation (rtEMSE ); (iii) fully automated high-throughput analysis (EM|SE|2 = EMSE Scripting Engine); and (iv) multimodal functional integration via hybrid analysis pipelines leveraging other neuroimaging packages (EEG-fMRI).

(c) Researchers have specific use cases that need to be supported on the EMSE platform. SSI technical support personnel are available to help.

(d) The EMSE platform enables various research scenarios, both classical and new. Software Development leverages and organizes SSI software technologies (algorithm libraries and

programming frameworks) in order to create, maintain, and evolve the EMSE platform. (e) SSI software developers translate functional requirements of EMSE into detailed designs that

call upon programming frameworks and algorithm libraries. (f) SSI software developers implement the detailed designs and SSI quality engineers test against

the functional requirements. Algorithms Research at SSI engages applied mathematics to formulate theories that solve advanced

computational problems of scientific interest. Solutions are implemented as algorithms within suitable programming frameworks. Computational performance characteristics are studied and optimized.

(g) A scientific computational problem is formulated and solved in mathematical terms. (h) A theoretical solution to the computational problem is embodied as a testable algorithm.

In conclusion: Sound mathematical theory is the foundation for computational algorithms which, when provided as accessible content on the EMSE platform, can facilitate basic research that leads systematically to improved clinical practices.

Interaction between translational research labs and SSI technical support occasionally can lead to

formal research collaborations. Interaction between SSI technical support and SSI software development can generate requests for

new features to be implemented and reports of bugs to be fixed. Interaction between SSI software development and SSI algorithms research can give rise to a

creative and productive research & development cycle. Workshops sponsored by SSI & partners are fruitful occasions that can engender collaborations and

generate feature requests. Product planning is a formal lifecycle process at SSI that also engages marketing partners to

prioritize implementation of routine feature requests and commercialization of advanced R&D projects.

Abstract—Given a concurrent EEG-fMRI experiment with multiple runs, we describe a data-driven computational procedure for detecting neuroelectric-hemodynamic couplings associated with specific EEG frequencies and fMRI brain signals of interest (SOIs). For each SOI and frequency, a hemodynamic response function (HRF) is estimated for every pair of EEG channels with respect to a second-order quantity derived from spectral cross-products. A resulting HRF matrix is submitted to a nonparametric randomization procedure to assess the statistical significance of its principal EEG spatial components. SOI-specific BOLD-related EEG (sBRE) signal is detected when one or more components exceed a statistical threshold. From the above-threshold components, spatial filters are constructed which may be used to estimate fast sBRE signal from the EEG data over a range of frequencies. In addition, a reduced SOI-specific HRF is obtained which may be used to improve general linear modeling of the fMRI data. Thus, EEG-related HRFs (temporal convolutions) and BOLD-related neuroelectric source estimators (spatial matrices) are jointly derived from the concurrent data without biophysical modeling.

I. INTRODUCTION

Computational integration of experimental data from electrodynamic neurophysiology and hemodynamic neuroimaging modalities is needed for noninvasive estimation of neurovascular coupling dynamics [26], mutual improvement of spatial-temporal resolution [30], and a deeper theoretical understanding of each modality [29]. Joint EEG-fMRI has contributed to advances in basic cognitive neuroscience [7], epilepsy {Lemieux}, and schizophrenia [5].

Here we describe a new approach for computational integration of simultaneously recorded EEG and fMRI.

II. COMPUTATIONAL THEORY

Motivation of the computational approach begins with multi-run concurrent EEG-fMRI data. The researcher specifies one or more BOLD signals of interest (SOIs) which may include brain regions of interest or large-scale networks of interest. Each BOLD SOI is enhanced in order to maximize its specificity with respect to other BOLD signals throughout the brain. During each fMRI TR interval (i.e., repetition time period), second-order neuroelectric quantities are derived for all EEG channel pairs and for each neuroelectric frequency. A static nonlinear function (such as a logarithm) may be applied to the entire matrix of second-

*Research reported in this manuscript was supported by the National

Institute of Neurological Disorders and Stroke of the National Institutes of Health under Award Number R43NS69310. The content is solely the responsibility of the author(s) and does not necessarily represent the official views of the National Institutes of Health.

M. E. Pflieger is with Source Signal Imaging Inc., La Mesa, CA 91942 USA (phone: 619-464-0003; e-mail: mep@sourcesignal.com).

order quantities. Each resulting time series is input to a linear time-invariant (LTI) system which has BOLD SOI as output. For each neuroelectric (high) frequency, the LTI system is characterized via a channel-by-channel matrix of hemodynamic response functions (HRFs) which are estimated in the hemodynamic (low) frequency domain. A nonparametric randomization test is used to identify how many (if any) EEG spatial components are coupled with the BOLD SOI, and the results are used to construct a space-frequency filter bank for estimating SOI-specific BOLD-related EEG (sBRE) signal and its hemodynamic response.

A. Given experimental data We begin with data from a multi-run concurrent EEG-fMRI experiment which may (but need not) involve a task to be performed by the participant. We suppose that MRI scanner and ballistocardiogram artifacts have been removed from the EEG data, and that a suitable preprocessing pipeline has been applied to the fMRI BOLD data.

Symbols related to the data are tabulated as follows: TABLE 1. PARAMETER INDICES AND SIZES

Symbol Table for Discrete Parameters: Indices and Sizes Parameter Description Index Size

Neuroelectric (fast) time (~ms) t

T

Hemodynamic (slow) time (~s ) t

T

Neuroelectric (high) frequency (>~0.1Hz) f

F

Hemodynamic (low) frequency (<~0.1 Hz) f

F

fMRI gray matter voxel v V BOLD spatial basis function (component) b B

BOLD SOI subspace dimension s S EEG subinterval of fMRI scan interval i I

EEG channel j J

Experimental run k K BOLD data phase randomization r R

BOLD-related EEG beamspace dimension d D

TABLE 2. SYMBOLS FOR EXPERIMENTAL DATA

Symbol Table for Time and Frequency Domain Data Description Symbol

EEG potential on run k for channel j at fast

time t

within subinterval i of scan interval t

[ ]( )kijX t t

Fourier transform of [ ]( )kijX t t

at frequency f

[ ]( )kijX t f

BOLD signal on run k at time t

for voxel v ,

component b , or SOI dimension s

( ), ( ), ( )kv kb ksY t Y t Y t

Fourier transform of ( )kY t

at frequency f

( )kY f

Note that time and frequency domain interpretations are resolved via the parameter [] and variable () names. Parameters and variables may be swapped as needed.

Data-Driven Detection and Estimation of BOLD-Related EEG Signal: Computational Theory*

Mark E. Pflieger et al. (in preparation)

B. BOLD signal of interest (SOI) BOLD signals measured at gray matter voxels may be coupled with neuroelectric generators of scalp-recorded EEG. However, due to both local and distant voxel-voxel correlations of BOLD signals over the entire brain, the effective dimension of the BOLD signal space B is much smaller than the total number of gray matter voxels V . In practice, the BOLD signal space dimension B may be obtained by temporal component analysis such as temporal principal component analysis (PCA) or independent component analysis (ICA). Temporal component analysis produces B components or basis functions that collectively span the entire BOLD signal space. Each component b has an associated time series ( )kbY t

(for each experimental run k ) and an associated spatial pattern ( )bP v distributed over all gray matter voxels v V∈ . The spatial pattern of a component may coincide with a brain region. More generally, however, it corresponds to a distributed network of brain regions which have correlated (or anti-correlated) BOLD signals. If the BOLD signal of interest (SOI) does not coincide with one particular component, then the SOI’s desired spatial pattern ( )sP v (e.g., a region of interest) may be reconstructed approximately as a linear combination of component spatial patterns with β coefficients obtained via linear least squares

1

( ) ( )B

s sb bb

P v P vβ=

≈ ∑ (1)

so that the time series associated with the BOLD SOI is

1

( ) ( )B

ks sb kbb

Y t Y tβ=

= ∑

. (2)

In sum, the BOLD SOI may correspond to a localized brain region or a distributed brain network, as specified by the researcher. As a special case, the researcher’s BOLD SOI s may coincide with a single component b . More generally, the BOLD SOI may be a projection from the total BOLD signal space to a subspace comprising a small number S of dimensions.

C. Maximally specific BOLD SOI We aim to derive a bank of EEG spatial filters that maximize neuroelectric-hemodynamic coupling with the BOLD SOI specifically. Therefore, because EEG is sensitive to whole-cortex activity, we must consider and factor out BOLD signal not of interest (SnOI). This subsection describes a hemodynamic frequency domain filter that maximizes specificity of BOLD SOI versus SnOI.

Consider a thought experiment in which BOLD signal is perfectly correlated in time across all voxels. Because no temporal information is available for making discriminations, it would be impossible to identify SOI-specific BOLD-related EEG signal using such data. Although this case is extreme, large-scale correlations are well known in fMRI data and, in fact, are the basis for identifying resting state network phenomena [3]. Hence, only the distinctive SOI features that remain after removing correlations with SnOI

may be used to detect SOI-specific neuroelectric-hemodynamic coupling. Our filter for extracting the distinctive SOI features is designed as follows.

Let ,SOIkY be the F S×

SOI matrix where each element ( )ksY f

is the Fourier transform of (2), and form the F F×

average SOI inter-frequency cross-spectral matrix

SOI ,SOI ,SOI1

1 K

k kkK

∗

=

≡ ∑G Y Y . Similarly, let kY be the F B×

total signal (SOI + SnOI) matrix for run k where each element is ( )kbY f

; then form the average total signal inter-

frequency cross-spectral matrix 1

1 K

k kkK

∗

=

≡ ∑G Y Y . Next,

perform complex singular value decomposition (SVD) [13] ( )0.5 0.5

SOI 1 1 1svd − − ∗=G G G U W U (3)

of the SOI matrix normalized (i.e., sphered) with respect to the total signal matrix via the inverse symmetric square root

0.5−G . Examine the largest singular values from the diagonal of 1W and retain S S′ ≤ of these. The Appendix provides a statistical specificity test for setting a threshold on the singular values. If 0S ′ = , then the SOI is not sufficiently distinctive to be discriminated from SnOI, and thus there is no basis for further analysis with respect to the SOI. Otherwise, form the matrix S ′U comprising the first

S ′ columns of 1U . The F F×

SOI projection matrix is

0.5SOI S S

∗ −′ ′≡P U U G . (4)

Finally, obtain the F S×

SOI-specific BOLD signal matrix in the hemodynamic frequency domain:

,SOI SOI ,SOIk k=Y P Y . (5)

The filtered hemodynamic time domain signal ( )ksY t

is obtained by inverse transforming the matrix elements

( )ksY f

. However, we note that (16) below uses the frequency domain representation. In addition, we note that cross-frequency interactions reflect nonlinear dynamics of the BOLD signal. Thus, this section uses linear methods (SVD) in the frequency domain to extract distinctive nonlinear features of BOLD SOI, i.e., SOI patterns of cross-frequency interactions that stand out against average patterns found in the total BOLD signal.

D. Second-order EEG-derived quantities A fundamental second-order quantity derived from the

EEG data is the inter-channel cross-spectrum for a scan interval

1 2 1 2 1 2

1

1[ ]( ) [ ]( ) [ ]( ) [ ]( )I

kj j kj j kij kiji

C f t C t f X t f X t fI

∗

=

= ≡ ∑

(6)

which is the subinterval-averaged Fourier transform of the cross-correlation function of EEG channels 1j and 2j for fMRI scan interval t

of run k . For example, if the fMRI scan interval (TR) is 2 s and the EEG subinterval is 250 ms,

then there are 15I = 50%-overlapping subintervals, and the neuroelectric frequency bin width is 4 Hz (= 1/0.25s). Let [ ]( )k f tC

be the J J× matrix whose element at row

1j and column 2j is 1 2

[ ]( )kj jC f t

, and estimate the experimental average inter-channel cross-spectral matrix

1 1

1[ ]( ) [ ]( ) [ ]T K

kt k

f f t fTK•

= =

= ≡∑∑C C C

. (7)

Then the spatially normalized (or sphered) inter-channel cross-spectral matrix is 0.5 0.5[ ]( ) [ ] [ ]( ) [ ]k kf t f f t f− −≡C C C C

(8)

where 0.5[ ]f−C

is the inverse symmetric square root of

[ ]fC

stabilized via complex SVD. The element at row 1j

and column 2j is denoted by 1 2

[ ]( )kj jC f t

. Next perform SVD on (8)

( ) 2 2 2svd [ ]( )k f t ∗=C U W U

(9)

where 2 21 22 23diag{ , , , }w w w=W , a diagonal matrix of rank-ordered non-negative real singular values. Indices k , f

, and t

have been suppressed on the right-hand side. To the extent that EEG spatial patterns and their amplitudes on scan interval t

of run k resemble the experimental average patterns and amplitudes estimated by (7), we expect typical singular values around 1.0 after spatial normalization. A singular value greater than 1.0 indicates a larger-than-average amplitude of the corresponding pattern; whereas a fractional singular value less than 1.0 indicates a smaller-than-average amplitude of the corresponding pattern. Because both decreases and increases relative to the experimental average may be relevant to BOLD-related coupling, a log (e.g., base 2) transform (2.01.0, 1.00.0, 0.5-1.0) may be applied to attenuate average values while accentuating both larger- and smaller-than-average values. More generally, consider any invertible scalar function

: (0, ] [ , ]φ ∞ → −∞ ∞ such as the identity function

1( ) ( )x x xφ φ −= = (10) or a power function

1 1/

( )

( )

x xy y

β

β

φ

φ −

=

= (11)

or a logarithmic function

1

( ) log( )

( ) exp( / )

x xy y

φ β

φ β−

=

= . (12)

Any such function may be applied to the whole matrix

( )2 21 22 2

[ ]( ) [ ]( )

diag{ ( ), ( ), }

k kf t f t

w w

φ φ

φ φ ∗

≡

=

C C

U U

(13)

that is, by applying φ to each singular value of equation (9). The resulting element at row 1j and column 2j is denoted

by 1 2

[ ]( )kj jC f tφ

.

Empirically, our motivation is to discover forms of φ that optimize coupling with BOLD signal.

E. HRF estimation in EEG sensor-sensor space In the hemodynamic time domain, for each pair of EEG sensors, a φ-transformed EEG inter-channel cross-spectral time series is convolved as input with an unknown hemodynamic response function (HRF) H in order to approximate the SOI-specific BOLD time series as output:

1 2 1 2

( ) [ ]( ) [ ]( )ks sj j kj jY t H f C f t dτ φ τ τ≈ −∫

. (14) By the convolution theorem, this transforms to a linear relationship in the hemodynamic frequency domain

1 2 1 2

( ) [ ]( ) [ ]( )ks sj j kj jY f H f f C f fφ≈

(15) so that a best-fitting HRF can be estimated across runs as

1 2

1 2

1 2

2

( ) [ ]( )[ ]( )

[ ]( )

ks kj jksj j

kj jk

Y f C f fH f f

C f f

φ

φ≈

∑∑

. (16)

Thus, the time domain HRF is 1 2

[ ]( )sj jH f t

. Let 1 2

maxsj jt

be the time at which the HRF peaks in a specified time interval, and define the J J× peak HRF matrix for SOI component s :1

1 2 1 2

max[ ] [ ]( )s sj j sj jf H f t ≡ H

. (17)

As a step toward identifying EEG spatial filters (i.e., channel vectors) which are most strongly coupled with BOLD SOI component s , we perform complex singular value decomposition [13] on [ ]s fH

:

( ) 3 3 3svd [ ]s f ∗=H U W U

(18)

where 3 31 32 33diag{ , , , }w w w=W are rank-ordered singular

values. Indices s and f

have been suppressed on the right-hand side.

F. BOLD signal phase randomization procedure Our next step is to assess the statistical significance of the singular values 3W nonparametrically by generating an empirical distribution of nonsignificant singular values obtained from surrogate data sets constructed by destroying the possibility of BOLD-EEG coupling. This is achieved by phase-randomizing the BOLD data across runs and hemodynamic frequencies as follows:

( ) ( ) exp{ }rks ks rkfY f Y f iθ=

(19)

where r indexes one of R randomizations, and 0 2rkfθ π≤ <

is a random phase angle. Note that each

randomization remains fixed for all BOLD SOI components and EEG frequencies. For each randomization r , substitute ( )rksY f

for ( )ksY f

in equation (16) and proceed to compute

3 31 32 33diag{ , , , }r r r rw w w=W as in equation (18). By

1 Alternatively, the HRF matrix may be constructed (a) at a specified

hemodynamic latency; (b) by stacking matrices obtained in (a) for a range of latencies; or (c) by stacking matrices across hemodynamic frequencies.

construction, these singular values are sampled from the null distribution that obtains when BOLD-EEG coupling is zero.

G. Detection of SOI-specific BOLD-related EEG signal To assess the statistical significance of coupling, let d be an integer less than the number J of EEG channels, and let

( )dN w be the number of randomizations for which

3r dw w≥ , where 3r dw is the rank d singular value obtained from one of the randomizations. In addition, let 3d dw w≡ be the corresponding the rank d singular value from the original matrix 3W . Then the estimated probability that a the rank d singular value in the null distribution exceeds or equals dw is ( ) ( ( ) 1) / ( 1)d d dp w N w R= + + . Thus, we detect SOI-specific BOLD-related EEG (BRE) signal when

( )dp w α< , where α is a probability threshold criterion such as 0.05α = . We proceed sequentially

1, 2, ,d d d D= = = until the criterion fails to be satisfied at 1d D= + . The beamspace dimension D is the largest integer for which the criterion is satisfied consecutively.

H. Space-frequency filter bank for sBRE signal estimation We proceed to construct a bank of EEG spatial filters—one set of filters for each neuroelectric frequency f

—that estimate SOI-specific BOLD-related EEG (sBRE) signal, i.e., EEG signal optimally and significantly coupled with the BOLD SOI. The channel-by-channel peak HRF matrix constructed in (17) contains EEG spatial patterns that optimize peak neuroelectric-hemodynamic coupling with the BOLD SOI. The principal patterns are the columns of matrix 3 [ ]s fU

of (18), and the phase randomization

procedure is applied to detect [ ] 0sD f ≥

EEG patterns that are statistically coupled with BOLD SOI component s . Thus, we construct the [ ]sD f J×

sBRE estimation matrix as

[ ] [ ]s Dsf f∗≡E U

(20)

where [ ]Ds fU

comprises [ ]sD f

columns of 3 [ ]s fU

.

I. Estimation of fast sBRE signal from EEG data The sBRE estimation matrix [ ]s fE

is a spatial filter that may be applied in the neuroelectric frequency domain to EEG subintervals ( , , )k t i

of spatially normalized multi-

channel data 0.5[ ] [ ]( )kif t f−C X

, where [ ]( )ki t fX

is a

column vector having J elements [ ]( )kijX t f

. The result

0.5[ ]( ) [ ] [ ] [ ]( )ski s kit f f f t f−=Z E C X

(21)

estimates the sBRE signal (a [ ]sD f

-dimensional vector) in

the neuroelectric frequency domain at frequency f

on experimental run k , scan interval t

, and subinterval i .

It may appear that [ ]( )ski t fZ

can be converted directly to the neuroelectric time domain via the inverse Fourier transform. However, direct conversion to the neuroelectric

time domain is not meaningful because the number of beamspace dimensions [ ]sD f

and the EEG spatial patterns

associated with beamspace dimensions vary with f

. One solution is to obtain a set of common spatial patterns across frequencies by SVD of the stacked estimator matrix

0.5

4 4 4

0.5

[ 1] [ 1]svd

[ ] [ ]

s

s

f f

f F f F

−

∗

−

= = =

= =

E CU W V

E C

(22)

so that sBRE signal is estimated in the frequency domain as [ ]( ) [ ]( )ski s kit f t f′ ′=Z E X

(23)

where s′E ( D J′× ) restricts the rows of 4∗V to D′ common

beamspace dimensions. Now the fast sBRE signal can be obtained via the inverse Fourier transform as [ ]( )ski t t′Z

. We note that finding a frequency-independent sBRE estimator matrix s′E is consistent with the more restrictive quasi-static approximation that is assumed almost always when volume conductor models are constructed to solve EEG and MEG forward and inverse problems. In addition, because the Fourier transform is linear, we can almost write a direct and much simplified time domain estimator as

( ) ( )s st t′ ′=Z E X

(24) where indices for the experimental run, scan interval, and EEG subinterval are no longer needed: “almost” because s′E is in general a complex matrix whereas ( )tX

is recorded as a real vector time series. Thus, (24) obtains with the further assumption—also consistent with the quasi-static approximation—that the imaginary part of s′E is negligible.

J. Hemodynamic response function of the sBRE signal

Let [ ]( )skidZ t f

be beamspace component d of the

estimated BRE signal vector [ ]( )ski t fZ

of (21); then form

[ ] 2

1 1

1[ ]( ) [ ]( )sD fI

sk skidi d

C f t Z t fI

φ= =

′ ≡

∑ ∑

(25)

which is function φ applied to the subinterval averaged sum

of squared magnitudes of the [ ]sD f

orthogonal beamspace dimensions of the sBRE estimate. After Fourier transformation, substitution of [ ]( )skC f f′

for 1 2

[ ]( )kj jC f fφ

in (16) produces the desired hemodynamic response function

[ ]( )sH f f

(time domain equivalent = [ ]( )sH f t

) as the

counterpart of 1 2

[ ]( )sj jH f f

.

III. DISCUSSION

We have detailed a computational theory that aims to bridge a considerable gap between EEG and fMRI BOLD phenomena, which are measured on temporal frequency scales separated by about three orders of magnitude: the neuroelectric high frequency range (~0.1-100 Hz) and the

hemodynamic low frequency range (<~0.01-0.1 Hz). Our approach employs the Fourier transform twice. The first transform converts continuous EEG (~ms time scale) to spectra in subintervals (~250 ms) which are averaged to estimate inter-channel cross-spectra for each fMRI TR (~2 s). On this ~0.5 Hz scale, the second transform is used to solve a convolution equation in order to estimate a neuroelectric-hemodynamic response function for each pair of EEG channels. The resulting channel-by-channel matrix is associated with a particular EEG frequency and is analyzed for EEG spatial patterns which are most strongly coupled with BOLD signal of interest.

Thus, taking advantage of concurrent EEG-fMRI measurements, we seek to identify space-frequency properties of EEG generated by cortical neuroelectric sources that are coupled specifically with BOLD hemodynamic signals of interest. However, functional connectivity analyses with fMRI reliably disclose hemodynamic correlations between distant brain regions that may, in turn, be coupled with concurrent generators of scalp EEG. Consequently, it appears vital to take into account BOLD signal correlations throughout the brain, and especially correlations with the BOLD signal of interest (SOI).

{Discussion to be continued…}

ACKNOWLEDGMENT Research reported in this manuscript was supported by the

National Institute of Neurological Disorders and Stroke of the National Institutes of Health under Award Number R43NS69310. The content is solely the responsibility of the author(s) and does not necessarily represent the official views of the National Institutes of Health.

APPENDIX We describe a statistical test of SOI specificity for

deciding how many singular values to retain from the rank-ordered singular value matrix 1 11 12 13diag{ , , , }w w w=W of equation (3) in section II-C. Perform K random permutations ( K is the number of summands in the formation of SOIG and G ) over all B basis functions (components) R times. Note that each permutation has no effect on the total signal matrix G , because the total signal uses all basis functions, and their order of indexing is immaterial. However, the SOI matrices ,SOIrkY that compose

,SOIrG will be changed for each K -fold randomization r , and thus the rank-ordered singular value matrix

1 11 12 13diag{ , , , }r r r rw w w=W will be altered. Note that each randomization r constructs K different

signal subspaces, each of which has dimensional characteristics similar to the SOI subspace. Hence, the singular values obtained after each randomization are not specific to one particular signal subspace. Consequently, the randomization procedure destroys SOI subspace specificity, and may be used to construct a distribution for the null

hypothesis that the observed singular values were obtained in the absence of SOI subspace specificity.

For each integer s S′ ≤ , let ( )sN w′ be the number of randomizations for which rsw w′ ≥ , where 1rs r sw w′ ′≡ is the rank s′ singular value from the diagonal of 1rW for some randomization r ; and let 1s sw w′ ′≡ be the corresponding rank s′ singular value from the original matrix 1W . Then the estimated probability that a nonsignificant rank s′ singular value (obtained from nonspecific signal subspaces) exceeds or equals sw ′ is ( ) ( ( ) 1) / ( 1)s sp w N w R′ ′= + + . Thus, an acceptance criterion for SOI specificity is

( )sp w α′ < , where α is a probability threshold (e.g., 0.05α = ). We choose the largest s S′ ′= (number of

retained singular values) for which the criterion is satisfied consecutively. If 0S ′ = , then the test indicates that SOI cannot be differentiated adequately from SnOI.

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1

1

fMRI Preprocessing of UC Davis Datasets Colleen Chen, graduate student, SDSU Computational Science Research Center; prepared for Source Signal Imaging, Inc. Introduction The functional neuroimaging data was collected from two participants across multiple scans of task stimulus. The goal is to identify functional connectivity networks in individual participants by combining within-participant scans of task data. The networks are parcellated into spatial maps for feature extraction. We analyzed the task data by applying “resting state” methods whereby ignoring the events. We then applied subject-level ICA by extracting components across runs. Multi-sessions temporal concatenation (MSTC), a procedure based on independent component analysis, extracts common spatial maps across different scans within the same subject. We want to see if it is possible to identify canonical brain networks in individual subjects who are engaged in task. Methods Data were processed using the Analysis of Functional NeuroImages software [11] (afni.nimh.nih.gov) and FSL 5.0 [12] (www.fmrib.ox.ac.uk/fsl). Functional images were slice-time corrected, motion corrected (3dvolreg) to align to the middle time point, and aligned to the anatomical image using FLIRT [13][14] with six degrees of freedom. FSL’s nonlinear registration tool (FNIRT) was then used to standardize images to the MNI152 standard image (3mm isotropic) using sine interpolation, and the outputs were blurred to a global full-width-at-half-maximum of 10mm. Given recent concerns that traditional filtering approaches can cause rippling of motion confounds to neighboring time points [15], we used a second-order band-pass Butterworth filter [16][17] to isolate low-frequency BOLD fluctuations (.008 < f < .1 Hz) [18].

Regression of a total of 16 nuisance variables was performed to improve data quality [17]. Nuisance regressors included six rigid-body motion parameters derived from motion correction and the derivatives. White matter (WM) and ventricular masks were created at the participant level using FSL’s FAST image segmentation [19] and trimmed to avoid partial-volume effects. An average time series was extracted from each mask and was removed using regression, along with its corresponding derivative. All nuisance regressors were band-pass filtered using the second-order Butterworth filter (.008 < f < .08 Hz) [16][17]. We did not regress out the global signal.

Motion was quantified as the Euclidean distance between the six rigid-body

motion parameters for two consecutive time points. For any instance greater than

2

2

1mm, considered excessive motion, the time point as well as the preceding and following time points were censored, or “scrubbed” [16]. If two censored time points occurred within ten time points of each other, all time points between them were also censored. Both subjects had more than 90% of time points and retained more than 150 total time points remaining after censoring. Runs were then truncated at the point where 150 usable time points was reached. Motion over the truncated run was summarized for each participant as the average Euclidean distance moved between time points (including areas that were censored). Analysis-ICA Independent component analysis (ICA) is a data-driven approach to defining functional networks that does not require a priori information. Spatial components extracted using ICA have been shown to correspond to known functional networks and to be consistent within and across individuals. Temporally concatenated ICA will be run by concatenating the multiple scans of each subject and using FSL MELODIC to extract a set of subject-level components. Dimensionality will be automatically estimated using Laplace approximation to the Bayesian evidence.

Once data were preprocessed to remove noise artifacts from in-scanner head

motion and those of physiological origin, we applied ICA to the datasets from the two participants across multiple scans. We concatenated the time courses across scans for each participant and the normalized the data by subtracting the mean and dividing by the standard deviation. This standardized data produces cleaner looking components. We also ran ICA without this normalization, in order to compare results. Results: Subject #1- normalized

Visual- medial

Frontoparietal- left lateralized

3

3

Executive control- medial frontal areas (anterior cingulate and paracingulate)

Visual- occipital pole

Sensorimotor

Default Mode Network- medial parietal (Precuneous and posterior cinculate), bilateral inferior, ventromedial frontal cortex

4

4

Frontoparietal- right lateralized

Auditory- superior temporal gyrus and posterior insular

5

5

Executive control- medial frontal areas

6

6

Subject #2- normalized

Visual- medial

Frontoparietal- right lateralized

Default Mode Network

Visual- occipital pole

7

7

Auditory

Executive Control

8

8

Auditory

9

9

Discussion Using automatic dimensionality estimation, MELODIC detected a total of 35 independent components (ICs) for subject #1, and 23 ICs for subject #2. The difference in the number of components detected reflects the variability across subjects and that ICA is susceptible to this cross-subject variability. Further, ICA may be useful in detecting noisy components due to head-motion, as seen in figures below:

It may be useful to defect noise using this analysis and remove the noisy components from further statistical analysis. Overall, ICA seems a promising tool for parcellations of the brain into networks that can later be used as feature components in combining datasets from different modalities, such as EEG. With the addition of more participants and data, it would be possible to conduct group ICA for extraction of most robust canonical networks (refer to figure below from Smith et al.).

10

10

The 10 maps correspond to interpretable functional categories and can be considered the ‘‘major representative’’ functional networks as derived independently from both activation meta-analysis and resting data (Smith et al.). The maps’ functional networks will be listed:

1. Visual 2. Visual 3. Visual 4. Default Mode Network 5. Cerebellum 6. Sensorimotor 7. Auditory 8. Executive control 9. Frontoparietal 10. Frontoparietal

11

11

References [11] Cox, R. W. (1996). "AFNI: Software for analysis and visualization of functional magnetic resonance neuroimages." Computers and Biomedical Research 29: 162-173. [12] Smith, S. M., M. Jenkinson, et al. (2004). "Advances in functional and structural MR image analysis and implementation as FSL." Neuroimage 23 Suppl 1: S208-219. [13] Jenkinson, M., P. Bannister, et al. (2002). "Improved optimization for the robust and accurate linear registration and motion correction of brain images." Neuroimage 17(2): 825-841. [14] Jenkinson, M. and S. Smith (2001). "A global optimisation method for robust affine registration of brain images." Med Image Anal 5(2): 143-156. [15] Carp, J. (2011). "Optimizing the order of operations for movement scrubbing: Comment on Power et al." Neuroimage. [16] Power, J. D., K. A. Barnes, et al. (2012). "Steps toward optimizing motion artifact removal in functional connectivity MRI; a reply to Carp." Neuroimage. [17] Satterthwaite, T. D., M. A. Elliott, et al. (2013). "An improved framework for confound regression and filtering for control of motion artifact in the preprocessing of resting-state functional connectivity data." Neuroimage 64: 240-256. [18] Cordes, D., V. M. Haughton, et al. (2001). "Frequencies contributing to functional connectivity in the cerebral cortex in "resting-state" data." AJNR Am J Neuroradiol 22(7): 1326-1333. [19] Zhang, Y., M. Brady, et al. (2001). "Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm." IEEE Trans Med Imaging 20(1): 45-57.