Using Chaos to Control Epilepsy David J. Mogul, Ph.D. Department of Biomedical Engineering Pritzker...

Using Chaos to Control Epilepsy

David J. Mogul, Ph.D.

Department of Biomedical EngineeringPritzker Institute of Biomedical Science & Engineering

Illinois Institute of TechnologyChicago, IL

Illinois Institute of Technology

Epilepsy

• Afflicts over 1% of world population (>60 million people)

• Generalized vs. Partial° Simple partial° Complex partial

– Most common type (~40%)– Usually starts in medial temporal lobe (i.e., hippocampus)– Most likely to be refractory to drugs (>50%)

• Currently, the primary alternative to drugs is surgery


Goals of this Research:

1. Understand the nonlinear dynamics of epilepsy

2. Explore chaos control techniques for manipulating electrical seizures in the brain

3. Ultimately, to create an implantable device that would revert or prevent seizures using low-amplitude, sporadic, precisely-timed electrical impulses


• Nonlinear and aperiodic

• Deterministic, not stochastic

• Unpredictable in the long term: ° Sensitivity to initial conditions

• Contains unstable periodic orbits (UPOs)

• Goal: Use these properties to minimize amount of stimuli needed to effectively control epileptiform bursting

Chaos


Unstable periodic orbits (UPOs)

• The saddle displays in three-dimensions the concept of stable and unstable manifolds


Bursting and chaos control

• Used spontaneous electrical bursting in rat hippocampus as the model of epilepsy° Use of interburst intervals as the encoding parameter

• What is the best way to control bursting?° Simple pacing could kindle more seizures

° Chaos control techniques – potentially a better solution

• Chaos control (or anticontrol)° Perturb a system from a chaotic trajectory to a periodic

one (or vice versa)


Characterization of bursting• Two-dimensional delay-embedding to form

return maps of the system dynamics° Current and previous interburst intervals (IBIs)

• Nature of bursting has been controversial° Stochastic vs. deterministic/chaotic

• Lyapunov exponents° Quantify sensitivity to initial conditions; measure

of global determinism° Initial method used (Kantz, 1994) but it had problems° Short-time expansion rate analysis devised as

alternative• Unstable periodic orbit (UPO) detection

° Sign of local determinism (suggesting chaos) and a key element for control

Part One


Methods of in vitro recording

•Used transverse hippocampal slices from young adult rats•Electrical bursts were recorded extracellularly from the CA3

pyramidal layer•Control stimuli were applied to Schaffer collaterals


Characterization of bursting: Methods

• In vitro bursting is analogous to interictal spikes on an EEG

• In vitro epilepsy models generated spontaneous electrical bursts using three different protocols:° High extracellular potassium (10.5 mM)

° Zero extracellular magnesium

° GABAA antagonists: bicuculline + picrotoxin


Examples of bursts, interburst intervals (IBIs) and a return map

CA

B


Lyapunov exponent calculation• Initial method - measured average expansion rate

of attractor over several time steps• Exponents for experimental data were compared to

the surrogates using paired t-test• Results were positive, and statistically bigger than

those for the surrogates° However, the exponents were too small (~10-3) to

differentiate data from noise

• Problems with calculation: inaccuracies due to extremely fast expansion of initial neighborhoods

• Thus this method was not useful for IBI data


IBI data expanded to over half of the entire attractor within two iterates

1 2 3 4 51

1.5

2

2.5

3

3.5

4

4.5

5

IBIn-1

IBI n

= 1st= 2nd= 3rd

Iterates:

IBIn-1

IBIn


Surrogate data methods• Randomization that provides a null

hypothesis that the data are from a stochastic system° Used to determine significance of chaos measures

• Types used in this work° Gaussian (simple) shuffled (SS)

– Preserve amplitudes but not frequency spectrum– Some consider better for UPOs (Dolan et al., 1999)

– Used for Lyapunov, expansion rate, and UPO analyses

° Amplitude-adjusted Fourier transform (AAFT)– Preserve amplitudes and approximate freq. spectrum

– Preserve short-time correlations

– Used for UPO detection


Short-time expansion rate analysis

• Measured average expansion rate (Lave) of system over one time step° Small clouds of points iterated one time step° Ratio of two major axes of best-fit ellipses was an

estimate of expansion rate

• Lave would be smaller in a deterministic system than in a stochastic system° Also, Lave should be independent of neighborhood

size in chaotic systems ° This provided a way to compare data with

surrogates to assay for determinism


Expansion rate analysis of bursting revealed no differences between data and random surrogates

A Hénon map: plateaus seen in data, not in surrogates

= noiseless= noise, =0.02= noise, =0.2

= surrogates (SS)

Lav

e

# nearest neighbors (% of total points)

Lav

e

# nearest neighbors (% of total points)

= IBI data= IBI surrogates (SS)

B IBI data: no plateaus seen in data or surrogates Simulated System Physiological System


An example of bursting behavior exhibiting signs of chaos

This pattern was similar to a Shil’nikov oscillator - jumping chaotically among a finite number of states


UPO detection: Methods

• UPOs allow us to look for determinism on a local scale

• Applied transform method (So et al., 1997)

° Compared with 50 surrogates for significance

° Used windows of 256 IBIs to overcome nonstationarity

° Searched for period-1, 2, and 3 UPOs

• Tested transform on surrogates themselves


Detection of a period-1 orbitA Raw data

B Data after transform

C Significance plot


% S

igni

fica

nce

2

3

4

5

5

4

3

2

% S

igni

fica

nce

10

20

30

40

50

60

70

80

90

100

Period-2 and period-3 orbits in two-dimensional histograms

A Period-2 orbit B Period-3 orbits

% S

ign

ific

ance

IB I (n-1 )

5

4

3

2IB I (n )

23

4

50

20

40

60

80

100


UPO detection: Results

• 73% of all experiments contained at least one statistically significant period-1 or period-2 orbit

• Period-3 orbits were found in all three epilepsy models

• UPOs were found to be valid° Probability of finding significant peaks in data was

significantly higher than for surrogates– 0.28 (data) vs. 0.06 (surrogate), P<0.004 for SS

– 0.22 (data) vs. 0.07 (surrogate), P<0.004 for AAFT


Characterization of bursting: Summary

• High prevalence of UPOs provided evidence of local determinism° UPOs were significant and valid

• Bursting may be globally stochastic with local areas of determinism

• Chaos control might be possible but made more difficult where there are high noise levels


Control of bursting: Methods• Technical issues

° Burst detection: hardware, not software

° Real-time data acquisition and processing– Problems with Windows OS - unreliable

– Real-time data acquisition board

° On-board microprocessor: data input and control

° Host computer: fixed point detection, display, data storage, adaptive techniques

• Control algorithms

• Factors affecting control° Control parameters, e.g. control radius (Rc)

° Noise and nonstationarity

Part Two


Control of bursting: algorithms

• Stable manifold placement (SMP)° Used for experiments varying Rc

• Adaptive techniques° Used in addition to SMP° Adaptive tracking - re-estimated fixed point and

stable manifold


The goal of SMP control is to perturb the state point onto the stable manifold

• With SMP, only the fixed point (z*) and slope of stable manifold are needed


Chaos control was first successfully tested on the Hénon map

A Control without noise B Control with noise, =.005

= unstimulated iterates= stimulated iterates= control region

The Henon map is a well-known mathematical system that exhibits chaotic behavior.


Control of bursting with basic SMP was somewhat successful

= unstimulated IBIs= stimulated IBIs= control region


High-[K+] only

1300 1350 1400 1450 1500 1550 1600

1.8

1.9

2

2.1

2.2

IBI number (n)

IBI l

ength

(s)

IBI number (n)

Demand pacing phenomenon

= unstimulated IBIs= stimulated IBIs= control region

Effect of Rc on control efficacy: example


Adaptive tracking was used to overcome nonstationarity (drift)

• Readjusted the fixed point (x*) and stable manifold slope (s) estimate after each natural burst


Adaptive tracking improved control quality over longer periods


Short, close encounters with period-1, 2, and 3 orbits were occasionally seen

Two possible period-2 orbits in the same experiment


Forcing protocol• Rationale

° To help validate our fixed point estimates

° To help assess the feasibility of control

• Procedure° Forced points onto arbitrary points instead of the

stable manifold

° Measured change in center of mass (Xcm) after next IBI

Xcm should be smaller when forced to fixed points (on stable manifold) than arbitrary points

° Compared for fixed points found both with transform and with adaptive tracking


Forcing protocol measures change in distribution over time w.r.t. placement


Example of a forcing experiment


Results of forcing experiments

• For all fixed points: Xcm significantly smaller for fixed points than for arbitrary points (P<0.004, Wilcoxon signed rank test)

• Analyzed by fixed point type & direction of shift and found no difference in results

• This suggested that fixed point detection was valid


Control of bursting: Summary• Good control was obtained for the Hénon map even

with added noise (up to =.2) and added drift• Some control of bursting was achieved using SMP

alone• As control radius decreased, control variance

decreased, but % of stimulated IBIs increased° At extremely small control radii, demand pacing-like

phenomenon resulted

• Adaptive tracking improved control efficacy ° Seemed to counter the effects of nonstationarity

• Forcing experiments suggested that fixed points were indeed valid


Summary and Conclusions

• Nonlinear dynamical analysis and chaos control techniques were applied to spontaneous epileptiform bursting in the rat hippocampal slice

• Bursting was found to be globally stochastic with local regions of determinism (UPOs)

• Control of bursting was successful but greater control needs to be explored


Future directions

• Effect of control of bursts/spikes on seizure activity with different protocols

• Anticontrol of in vitro bursting

• Characterization and control of in vivo interictal spikes° Hippocampal slice preparation severs many

connections (intrinsic and extrinsic)

° In vivo spiking may actually contain less noise than in vitro bursting

° Chaos control of spiking could conceivably be easier than control of bursting

Using Chaos to Control Epilepsy David J. Mogul, Ph.D. Department of Biomedical Engineering Pritzker...

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