Using Chaos to Control Epilepsy

Using Chaos to Control Epilepsy David J. Mogul, Ph.D. Department of Biomedical Engineering Pritzker Institute of Biomedical Science & Engineering Illinois Institute of Technology Chicago, IL

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: • Understand the nonlinear dynamics of epilepsy • Explore chaos control techniques for manipulating electrical seizures in the brain • Ultimately, to create an implantable device that would revert or prevent seizures using low-amplitude, sporadic, precisely-timed electrical impulses

Chaos • 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

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 Part One • 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

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 A C 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 Iterates: = 1st = 2nd = 3rd IBIn IBIn-1

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 B IBI data: no plateaus seen in data or surrogates Lave # nearest neighbors (% of total points) = noiseless = noise, s=0.02 = noise, s=0.2 = IBI data = IBI surrogates (SS) = surrogates (SS) A Hénon map: plateaus seen in data, not in surrogates Physiological System Simulated System Lave # nearest neighbors (% of total points)

An example of bursting behavior exhibiting signs of chaosThis 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 orbit A Raw data B Data after transform C Significance plot

Period-2 and period-3 orbits in two-dimensional histograms A Period-2 orbit B Period-3 orbits

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 Part Two • 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

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 = unstimulated iterates = stimulated iterates = control region A Control without noise B Control with noise, s=.005 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

Effect of Rc on control efficacy: example Demand pacing phenomenon High-[K+] only IBI number (n) = unstimulated IBIs = stimulated IBIs = control region

Adaptive tracking was used to overcome nonstationarity (drift) • Readjusted the fixed point (x*) and stable manifold slope (ls) 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 (DXcm) after next IBI • DXcm 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:DXcm 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 Hénonmap even with added noise (up to s=.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