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Brain Dynamics and Control with Applications in Epilepsy

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Brain Dynamics and Control with Applications in Epilepsy
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NAIR, SANDEEP PARAMESWARAN ( Author, Primary )
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2008

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Animal models ( jstor )
Chaos ( jstor )
Electrodes ( jstor )
Epilepsy ( jstor )
Humans ( jstor )
Modeling ( jstor )
Rats ( jstor )
Seizures ( jstor )
Signals ( jstor )
Warnings ( jstor )

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University of Florida
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University of Florida
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Copyright Sandeep Parameswaran Nair. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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6/30/2007
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659813697 ( OCLC )

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BRAIN DYNAMICS AND CONTROL WITH APPLICATIONS IN EPILEPSY By SANDEEP PARAMESWARAN NAIR A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Sandeep Parameswaran Nair

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This dissertation is dedicated to my pa rents, M.P. Nair and Rajalakshmy Nair.

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iv ACKNOWLEDGMENTS Firstly, I would like to thank Dr. J. Chri s Sackellares for his guidance and support throughout my research tenure in the Brain Dynamics Laborat ory at the University of Florida. His passion, drive, and intelligence have motivated me to become a better engineer and researcher. I extend sincere gratitu de to my advisor, Dr. Panos Pardalos, for his support and assistance and for taking me under his wing during a difficult phase of my graduate school life. I can never thank him enough for his generous advice, guidance and kind encouragement. I would also like to thank Dr. Paul Carney, Dr. William Ditto and Dr. Steven Roper for serving on my a dvisory committee and for their insightful comments and valuable suggestions. I am forever indebted to my colleagues at the Brain Dynamics Lab, the Animal Neurophysiology Lab and the Center for App lied Optimization, and especially Dr. DengShan Shiau, for all his help and insight rende red over the years. He has always taken the time to review my work and provide vital fe edback on my progress or lack thereof. I thank Dr. Wendy Norman for sharing her wea lth of knowledge in animal care and techniques with me. I would also like to thank my fellow graduate students, Michael Bewernitz, Jeff Liu, Anant Hegde and Wichai Suharitdamrong. I am fortunate to have them not only as colleagues but also as my good friends. I woul d like to especially thank my friend and colleague, Linda Dance, for her help and advice at va rious stages of my research. She has been a constant source of in spiration and I thank her for always lending

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v a helping hand with a smile. I thank Dr. Jose Principe, Dr. Leon Iasemidis and Dr. Vitaliy Yatsenko for their significant contribu tions to my research efforts. I also wish to extend my acknowledgements to the administrative staff in the J. Crayton Pruitt family Department of Biom edical Engineering a nd the Department of Industrial and Systems Engineering, especial ly April and Cindy, for their assistance. I thank past and present staff and my coll eagues at the Malcolm Randall VA Medical Center, including Sun, Stephanie, Carol, Th eresa, and David. I would also like to acknowledge financial support provided by NIH-NIBIB, Alpha-One Foundation and the Children’s Miracle Network. I wish to thank my family, especially my parents, my sister, Suchitra, and brother, Suresh for their financial and emotional suppor t. They have been my greatest source of inspiration. I would also like to thank my fiance, Keerthi, for her love and patience. I thank God Almighty for the blessings.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...............................................................................................................x LIST OF FIGURES...........................................................................................................xi ABSTRACT.......................................................................................................................xv CHAPTER 1 INTRODUCTION........................................................................................................1 Epilepsy as a Dynamic Brain Disorder.........................................................................3 Dynamical Nature of Temporal Lobe Epilepsy.....................................................4 Dynamic View of Neuronal Activity....................................................................4 Dynamic Aspect to Controlling Seizures..............................................................6 Chaos Theory and Epilepsy..........................................................................................7 Scope of Research in Epilepsy.....................................................................................8 Benchmark 1: Validation of Animal Models......................................................10 Benchmark 2: Finding Mark ers of Epileptogenesis............................................11 Benchmark 3: Improve and Validat e Seizure Forecasting Techniques...............12 Benchmark 4: Design New Treatment Strategies...............................................14 Contributions of this Dissertation...............................................................................15 Organization of Chapters............................................................................................19 2 A MODEL OF MESIAL TEMPORAL LOBE EPILEPSY.......................................21 Role of Animal Models in Epilepsy Research............................................................21 What is a Good Animal Model?.................................................................................22 Models of Temporal Lobe Epilepsy...........................................................................24 The Kindling Phenomenon..................................................................................24 Self Sustaining Status Epilepticus.......................................................................25 The Continuous Hippocampal Stim ulation Model of Human MTLE........................27 Mechanisms.........................................................................................................27 Model Preparation...............................................................................................28 Electroencephalogram and Seizure Description..................................................34

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vii 3 TIME SERIES ANALYSIS AND FEATURE EXTRACTION................................38 Time Domain Analysis...............................................................................................39 Cross-Correlation................................................................................................39 Autoregressive Moving Average Model.............................................................40 Autoregressive Integrated Moving Average Models..........................................41 Autoregressive—Autoregressi ve Moving Average Models...............................42 Frequency Doma in Analysis.......................................................................................42 The Fourier Transform........................................................................................42 The Wavelet Transform.......................................................................................46 Coherence............................................................................................................47 Nonlinear Information Theory Based Analysis..........................................................49 Entropy................................................................................................................49 Approximate Entropy..........................................................................................50 Mutual Information.............................................................................................51 Nonlinear Chaos Theory Based Analysis...................................................................53 Glossary of Terms...............................................................................................54 Fractal Dimension...............................................................................................55 Box Counting Dimension.............................................................................57 Information Dimension................................................................................57 Correlation Dimension.................................................................................57 State Space Reconstruc tion—Method of Delays................................................58 Lyapunov Exponents...........................................................................................60 iEEG Dynamics in a Model of MTLE........................................................................63 Data Description and Selection...........................................................................63 Nonlinear Energy Operator—Teager Energy......................................................64 Test for Nonlinearity...........................................................................................67 Estimation of Short-Term Maximum Lyapunov Exponent.................................69 Results from Short Term Maximum Lyapunov Exponent Analysis...................79 Estimation of Phase Space Average Angular Frequency....................................82 Results from Average Angular Frequency Analysis...........................................84 Discussion............................................................................................................87 Nonlinearity and Temporal Dynamics.........................................................87 Potential Use of Nonlinear Signal Energy for Seizure Detection................88 Similarities with Human MTLE...................................................................88 4 EPILEPTOGENESIS AND SE IZURE PREDICTABILITY.....................................89 Statistical Test for Spatiotemporal Dynamical Analysis............................................92 Seizure Predictability in an Animal Model of MTLE................................................99 Description of Data............................................................................................100 Automated Seizure Warning Algorithm............................................................101 Statistical Evaluation of Seizure Predictability.................................................104 Results of Predictability Study..........................................................................106 Discussion..........................................................................................................110 Spatiotemporal Transitions during Epileptogenesis.................................................112 Description of Data............................................................................................114

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viii Study Methods...................................................................................................115 Results and Discussion......................................................................................115 5 CONTROLLING CHAOS: THEORY, MODELS AND APPLICATIONS TO BRAIN DISORDERS...............................................................................................119 Motivation for Controlling Spatiotemporal Chaos...................................................119 Goals of Chaos Control............................................................................................120 Computational Models in EEG Analysis..................................................................121 Need for Computational Models.......................................................................121 Coupled Map Lattice Models............................................................................123 Calculation of Lyapunov exponent............................................................125 Spatiotemporal dynamics in coupled map lattices.....................................126 Summary of Chaos Control Techniques...................................................................130 Adaptive Feedback Control......................................................................................133 Additive Control................................................................................................134 Multiplicative Control.......................................................................................135 Additive with Multiplicative Control................................................................138 Optimization of Feedback Parameters......................................................................139 Application to Brain Disorders.................................................................................141 6 A STATE DEPENDENT BRAIN STIMULATION PARADIGM FOR SEIZURE CONTROL...............................................................................................................147 Background and Significance...................................................................................147 Neuronal Activity Modulation via Electrical Stimulation........................................152 Uniform DC Electric Fields...............................................................................152 Localized DC Electric Fields.............................................................................153 Low Frequency Stimulation..............................................................................154 High Frequency Stimulation..............................................................................154 An Emerging View of Seizure Control....................................................................155 Acute Hippocampal Stimulation...............................................................................157 State Dependent Brain Electrical Stimulation..........................................................159 System Description............................................................................................161 Stimulation Methodology and Control..............................................................165 AHS Effects on iEEG Dynamics.......................................................................167 AHS Effects on Seizure Frequency...................................................................170 Post hoc Analysis..............................................................................................172 Discussion..........................................................................................................174 7 CONCLUDING REMARKS AND FUTURE DIRECTIONS.................................178 Summary...................................................................................................................178 Brain Dynamics in the MTLE Model................................................................178 Nonlinearity and temporal dynamics.........................................................179 Spatiotemporal dynamics...........................................................................180 Similarities with human MTLE..................................................................181

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ix Seizure Predictability in the MTLE Model.......................................................182 Markers of Epileptogenesis...............................................................................183 Optimization Based Control Strate gies and Modeling EEG Dynamics............183 State Dependent Therapeutic Inte rvention for Seizure Control........................ 184 Future Directions......................................................................................................185 APPENDIX PROOF OF THEOREM 5.1...................................................................188 LIST OF REFERENCES.................................................................................................191 BIOGRAPHICAL SKETCH...........................................................................................212

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x LIST OF TABLES Table page 3-1 Summary of iEEG data used for dynamical analysis...............................................64 3-2 Summary of Teager Energy based seiz ure detection results for four rats................66 4-1 Summary of test results obtained from spatiotemporal analysis in four animals.....96 4-2 Summary of STLmax analysis on interictal segments from four rats.........................99 4-3 Summary of iEEG data obtained from fi ve animals used for seizure predictability study.......................................................................................................................101 4-4 Evaluation of overall warning perfor mance of the three test algorithms...............108 4-5 Summary of data and resu lts from epileptogenesis study......................................114 6-1 Summary of stimulation results..............................................................................167

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xi LIST OF FIGURES Figure page 1-1 Development of a state-depe ndent seizure control device.......................................10 2-1 Animal headstage.....................................................................................................30 2-2 Approximate relative locations of electrodes on the animal’s skull........................30 2-3 Structural damage due to electr ode placement and induction of SE via hippocampal stimulation..........................................................................................31 2-4 Experimental timeline..............................................................................................33 2-5 Sample epochs of iEEG recorded during interictal, ictal and postictal states from four different brain site s of an MTLE model...........................................................35 2-6 Three minutes of iEEG.............................................................................................35 3-1 Power spectrum (Spectrogram) of iEEG recorded from the hippocampus of an animal model of MTLE............................................................................................45 3-2 Coherence measure at different freq uencies over time between the hippocampus and frontal cortex of an animal model of MTLE.....................................................48 3-3 Ictal segment of filtere d iEEG recorded from the hippocampus of an animal model of MTLE, and the reconstruc ted iEEG segment in 3-D space......................59 3-4 iEEG signal energy...................................................................................................65 3-5 Correlation integral and statis tical significance of nonlinearity...............................68 3-6 The derivative of the correlation f unction calculated for an ictal segment..............73 3-7 The variation of STLmax with the length T of the data segment for data in the preictal and ictal state of a human epileptic seizure.................................................77 3-8 The variation of STLmax with the IDIST2 parameter for data in the preictal and ictal state of a human epileptic seizure.....................................................................78 3-9 STLmax profiles derived from an iEEG signal recorded from the hippocampus of an MTLE model.......................................................................................................79

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xii 3-10 Mean STLmax value comparison between channels..................................................80 3-11 Comparison of mean values of STLmax among states...............................................80 3-12 STLmax values during two epochs of alert and rest state in an animal model of MTLE.......................................................................................................................81 3-13 Calculation of averag e angular frequency and Lmax.................................................83 3-14 Sample average angular frequency profiles.............................................................84 3-15 Mean average angular frequency value comparison between channels...................85 3-16 Comparison of mean average a ngular frequency between states.............................85 3-17 Average angular frequency values during two epochs of alert and rest state in an animal model of MTLE............................................................................................86 4-1 T-index calculated from STLmax values of groups of electrodes..............................94 4-2 T-index calculated from average an gular frequency values of groups of electrodes..................................................................................................................95 4-3 Comparison of mean T-i ndex values calculated from STLmax between states.........97 4-4 Comparison of mean T-index values calculated from average angular frequency between states...........................................................................................................98 4-5 Automated seizure warning algorithm...................................................................103 4-6 Estimated SWROC curves derived from recordings in the first rat for ASWA and two null seizure warning schemes...................................................................108 4-7 Overall performance indices from ASWA and two null warning schemes with respect to different seizure warning horizons........................................................109 4-8 Predictability power of ASWA derived from AAC and from FTF........................109 4-9 Transitions in Rat A* during the latent period.......................................................115 4-10 Transitions in Rat B* during the latent period.......................................................116 5-1 A simple classification of mode ls used in EEG dynamical analysis......................123 5-2 Change in amplitude and spatiotempor al behavior in a five cell coupled map lattice with varying coupling..................................................................................128 5-3 Change in amplitude and Lyapunov e xponents in a coupled map lattice system with varying coupling.............................................................................................129

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xiii 5-4 Change in amplitude and Lyapunov e xponents in the iEEG from an animal model of MTLE......................................................................................................129 5-5 Probability distributions of finite step Lyapunov e xponents of a logistic map......135 5-6 Multiplicative Control............................................................................................137 5-7 Probability distributi ons of finite step mean Lyapunov exponents of a CML model controlled via multiplicative control strategy..............................................138 5-8 Probability distributi ons of finite step mean Lyapunov exponents of a CML model controlled via a combination of multiplicative and additive control strategy...................................................................................................................139 5-9 Error calculated as a difference between mean global Lyapunov exponents and target Lyapunov exponents of a two cell CML......................................................141 5-10 Schematic diagram for seizure control...................................................................142 5-11 Phase portrait of STLmax of a rodent seizure...........................................................144 5-12 Proposed learning algorithm for emulating evolution of Lyapunov exponent......145 6-1 A desired effect of a controller based on resetting theory......................................157 6-2 An automated seizure warning based st ate dependent closed-loop seizure control system.....................................................................................................................160 6-3 Block schematic of components and de sign of an automated seizure warning based electrical stimulation para digm in the animal model...................................163 6-4 Long -term iEEG/Video Monitoring and Stimulation Station...............................164 6-5 The CK1601 PC parallel port re lay board with 8 terminals...................................164 6-6 Flow chart of the software control for automated stimulation...............................165 6-7 iEEG, STLmax and T-index plots before and after a stimulus.................................168 6-8 Automated seizure warning system show ing a warning prior to the onset of a seizure.....................................................................................................................169 6-9 Automated seizure warning system showing an example of stimulation of the hippocampus after a warning.................................................................................169 6-10 Sustained reversal of dynamical convergence among hippocampal and frontal cortical brain regions by acu te hippocampal stimulation.......................................170 6-11 Automated seizure warning system show ing a seizure warning and stimulation..170

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xiv 6-12 Seizure distribution before and after a stimulus block...........................................171 6-13 Comparison of mutual information va lues before divergent and non-divergent stimulations............................................................................................................173 6-14 Comparison of STLmax values before divergent an d non-divergent stimulations...173 6-15 Comparison of approximate entropy valu es before divergent and non-divergent stimulations............................................................................................................174

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xv Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BRAIN DYNAMICS AND CONTROL WITH APPLICATIONS IN EPILEPSY By Sandeep Parameswaran Nair December 2006 Chair: Panagote M. Pardalos Cochair: William L. Ditto Major Department: Biomedical Engineering Developing novel technologies for contro lling and preventing the development of dynamic disorders such as mesial temporal lobe epilepsy (MTLE) has been one of the predominant focus areas in biomedical research . In recent years, the idea of controlling seizures by intervening dur ing a seizure susceptible st ate has been proposed. This dissertation takes inspiration from earlier studies focused on the concept of controlling chaos in dynamical systems and begins to in vestigate the feasibility of an automated seizure warning based (state dependent) stim ulation technique in an animal model of MTLE. The central goal in this research is to study the eff ects of state dependent stimulus delivery on brain dynamics and control. A systematic approach was taken to addre ss the question at hand. First, an animal model of MTLE that captures many of the ha llmarks of the human condition was utilized to test whether dynamical descriptors of the intracranial electroence phalogram behave in a similar fashion as observed in humans. Second, the ability of these dynamical

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xvi descriptors to warn of a seizure suscepti ble state was investigated. Third, a seizure warning and intervention scheme (based on state space regional coupling) was designed and implemented to study the effect of st ate dependent electrical stimulation. In this dissertation we also propose a novel application of nonlinear and statistical tools to study the phenomenon of epileptogenesis. Finally we propose novel optimization based chaos control methods in theoretical models emulating brain dynamics and discuss its applicability in controlling epilepsy. The results from dynamical analysis of i EEG suggest that th e rat model of MTLE can be used as a tool to investigate the dynamical transitions associated with disease mechanisms in human temporal lobe epileps y. The performance of the seizure warning algorithm based on spatiotemporal transitions se ems to be sufficient enough to be able to lend itself to a range of applications such as seizure monitori ng and control devices. Furthermore, such spatiotemporal transiti ons also seem to be a good marker of epileptogenesis. Finally, we show the effect of state dependent electrical stimulation on brain dynamics and seizure frequency in the animal model.

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1 CHAPTER 1 INTRODUCTION The advent of physical and mathematical theory of nonlinear deterministic dynamics (refer to Schuster [1] and Ott [2] fo r an overview) has introduced new concepts and powerful algorithms to analyze apparently irregular behavior – a mainstay of most physiological systems in general, including brai n electrical activity. The past two decades have seen increased application of time series analyses based on chaos theory and theory of nonlinear dynamics, which are among the mo st interesting and rapidly advancing research topics, to a broad spectrum of time series data with varying degree of success. Several experiments have shown that phy siological time series is driven by a deterministic dynamical system with a low dime nsional chaotic attractor, defined as the phase space point or set of points repr esenting the various possible steady-state conditions of a system, an equilibrium stat e or group of states to which a dynamical system converges from a set of initial conditions. This dynamic view has modified drastically the manner in which certain physio logical processes are viewed and described. Methods derived from the domain of chao s and nonlinear dynamics in tandem with powerful statistical and spectral techniques ha ve provided new theore tical and conceptual tools that allow us to capture, understa nd, and link the complex behaviors of simple systems together. Characterization and quantif ication of the dynamics of nonlinear time series are also important steps toward understanding the nature of seemingly random behavior and may aid in forecasting occurre nces of some specific events that follow temporal dynamical patterns in the time series.

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2 Early nonlinear dynamics research in the 1980s focused on identifying systems that display chaos, developing mathematical m odels to describe them, developing new nonlinear statistical methods fo r characterizing chaos, and id entifying the so-called “route to chaos” of chaotic systems. A dramatic sh ift in the focus of re search occurred around 1990 when scientists went beyond just characte rizing chaos and started investigating the possibility of controlling chaotic systems. The idea is to apply appropriately designed minute perturbations to an accessible system parameter that forces it to follow a desired behavior rather than the random, noise-like behavior indicative of chaos. The general concept of controlling chaos has captured the imagination of researchers from a wide variety of disciplines, including the field of epilepsy and seizure control and will be discussed in the next few sections. The apparent utility of time series analys is tools in capturing qualitative dynamical changes in complex physiological systems allo wing one to make better event predictions and the plausibility of combining it with cont rol techniques derived from the domain of chaos control motivated this dissertation. Th e central goal of this dissertation is to combine time series analysis methods for pred iction of epileptic seizures with controlled interventions in the form of brain electrical stimulation to drive the system to a regime associated with healthy brain dynamics and thereby provide evidence for using such a control scheme for controlling seizures. A novel application of time series analysis methods that have previously been applied for seizure detection an d prediction is also proposed as a tool for identifying markers for epileptogenesis, i.e., the process(es) that leads to the development of epilepsy. Fi nally we develop innovative techniques employing dynamical approaches and optimiza tion for controlling chaos in models

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3 emulating epileptic brain dynamics and discuss the application of su ch control techniques in controlling brain disorders such as epilepsy. Epilepsy as a Dynamic Brain Disorder The epilepsies are a family of neurologi cal disorders characterized by seizures which are transient, recurrent perturbations of normal brain function. As a chronic condition, epilepsy affects about 1% of the population in the Unite d States [3]. In contrast to the normal brain, the epileptic brain has a chronic tendency to generate recurrent spontaneous seizures. Epilepsy is often consider ed to be a dynamic di sease, continuously evolving and changing its character over time. According to Mackey and Glass [4], the term “dynamic disease” denotes states of the organism that are characterized by an abnormal temporal organization, i.e., states where the temporal organization of the system breaks down and is substituted by abnormal dynamics. Such abrupt state transitions, almost periodic dynamics, appear to be a characteristic feature of many physiological disorders. At its most fundamental leve l, the occurrence of an epile ptic seizure represents a qualitative change in the dynamics of neuronal firing [5]. This sudden change in qualitative dynamics, presumably in response to an intrin sic control parameter, is the hallmark of a dynamic disease. Dynamic diseases can arise from alteration in underlying physiological control mechanisms [4,6-11]. Analogous to these sudden qualitative changes are the bifurcations seen in mathematical models. The behavior of a system (epileptic brain) over time, including its transitions into and out of well-defined states such as interictal (seizure-free), icta l (seizure), and posticta l (following a seizure) states, is by definition, the dynamics of the system. Studying these dynamical state transitions is critical not only

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4 in better understanding the di sease process but also in deve loping therapies that adapt to the dynamic nature of the disease. Dynamical Nature of Temporal Lobe Epilepsy Temporal lobe epilepsy (TLE) was defined in 1985 by the International League against Epilepsy (ILAE) as a condition characterized by seizures that occur spontaneously and in a recurrent fashion, origin ating from the medial or latera l temporal lobe of the brain. The seizures associated with TLE consist of simple partial seiz ures without loss of awareness (with or without aura ) and complex partial seizures (i.e., with loss of awareness). This form of epilepsy was first recogni zed in 1881 by John Hughlings Jackson, who described “uncinate fits” seizures arising from the uncal part of temporal lobe and the “dreamy state.” Temporal lobe epilepsy howeve r cannot be simply described as a static disease characterized by stereotyped seizures which recurrently and spontaneously arise from the temporal lobe complex (i.e., tempor al lobe, amygdale, and hippocampus). One of the central features of TLE is that it is tran sient in nature, evolving throughout the lifetime of the epileptic patient culminating in medically intractable epilepsy, a stage in which the seizure frequency is no longer sensitive to changes in anticonvulsant medications. This evolving clinical picture suggests that TLE should be regarded as a dynamically changing disease and implies that therapeutic strategies need to take into account this facet of the disease. This dissertation will primarily deal with an animal mode l of this form of medically intractable epilepsy. Dynamic View of Neuronal Activity Neuronal networks can generate a variet y of activities, some of which are characterized by rhythmic or quasi-rhythmic signa l patterns. Brain recordings at all levels (single neuron, neuronal populations, etc.) repr esent complex signals that follow dynamic

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5 transitions, of which the statistical propert ies depend on both time and space [12]. These networks belong to a class of nonlinear systems with comple x dynamics, in the sense that variables of the network have both a strong nonlinear rang e and complex interactions amongst themselves. Characteristics of the dynamics depend strongly on small changes in the control parameters and/or the initial conditions, implying that they may exhibit chaotic behavior. Real neuronal networks be have like nonlinear complex sy stems in the sense that they display changes between states such as small-amplitude quasi -random fluctuations and large amplitude rhythmic oscillations. Moreover, such dynamical state transitions are observed in the epileptic brain during the transition between interictal and seizure states. One of the unique properties of the brain as a system is th at it shows a relatively high degree of plasticity. It can di splay adaptive responses that are essential to implement higher functions such as memory and learning. As a consequence, control parameters are essentially plastic in nature too, which implies that they can change over time depending on previous conditions. In spite of this plasti city, it is necessary for the system to stay within a stable working range in order for th e system to maintain a stable operating point. One can then conceptualize the most esse ntial difference between a normal and an epileptic network as a decrease in the dist ance between operating and bifurcation points [13]. The mathematical rigor of non linear systems theory is not always easy to translate into the biological domain due to incomple te knowledge of system parameters and dynamics, but nevertheless, the nonlinear a pproach provides the building blocks for describing and elucidating very complex be haviors at different levels of neural

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6 organization, (e.g., single neurons, small, medium and large neurona l ensemble activity). In summary, the family of attractors in phase space that characterizes complex nonlinear systems can prove valuable in describing a range of behaviors and associated neuronal activity. Transitions between attractors and transitions within attractors may serve as useful descriptors for analyzi ng state changes in the brain. Dynamic Aspect to Controlling Seizures The most important significance of identif ying epilepsy as a dynamic disorder is that it raises the possibility of devising seizure control stra tegies based on modulation of the underlying dynamics. As mentioned earlier, the difference between a normal and an epileptic brain can be formulated as a decr ease in the distance between normal operating and bifurcation points. Followi ng this line of thought we ma y consider the manifestation of epilepsy as a change in intr insic control parameters that promote a shift of the neuronal network in the direction of the bifurcation point. The precise nature of the control strategies that are to be employed will depend on the type of bifurcation that has occurred and the nature of abnormal dynamics th at arose from such a bifurcation. This dynamic nature of the disease calls fo r therapeutic approaches that can adapt to these apparently abrupt state transitions. The simplest therapeutic strategy is to apply external forcing to manipulate the altered intrinsic control parameters to a range associated with healthy dynamics. Even dynamical systems that exhibit exceedingly complex dynamical behavior such as chaos can in principle be controlled by the application of carefully timed perturbations. Another consequence of treating epilepsy as a disorder with complex dynamics that can exhi bit chaotic behavior is the application of signal processing tools to follow spatiotempor al markers to forecast these apparently abrupt transitions/bifurcations . Such tools can serve as a guide towards the design of

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7 effective treatment strategies aimed at m odulating spatiotemporal dynamics related to abnormal activity and diverting it to that asso ciated with normal operating regimes. This approach will be one of the focal points of this dissertation and shall be discussed in further detail in the secti ons and chapters to follow. Chaos Theory and Epilepsy The mathematical concept of chaos has been used to explain a variety of complex natural phenomena. Chaos is said to have been discovered in 1963 by the scientist Edward Lorenz [14], who was working on a model for long-term weather forecasting. The discovery piqued interest across the scientific research community and paved the way for extensive research in this area. Chaos has pr ovided an alternative interpretation of erratic behaviors of certain systems th at up until then had been consid ered to be random in nature. Chaotic systems, among other characteristics, can produce what appears to be random output. In dynamical systems theo ry, chaos means irregular fluctuations in a deterministic system; i.e., the system behaves irregularly be cause of its own internal structure, and not because of external forces. Chaos is also defi ned as an unpredictable behavior arising in a deterministic system because of extreme sens itivity to initial conditions. Chaos arises in a dynamical system if two arbitrarily close starti ng points diverge exponentially, so that their future behavior is eventually un predictable. Dynamical systems are deterministic if there is a unique consequence to every state and stochastic or random if there is more than one outcome chosen from some probability distri bution. A deterministic dynamical system is perfectly predictable given co mplete knowledge of the initia l condition, and is in practice always predictable in the short term. Long-term unpredictability, on the other hand, is due to the property of dependence on initial c onditions. No matter how precisely the initial conditions in these systems are measured, th e prediction of its subsequent motion goes

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8 radically wrong after a period of time. Yet anothe r definition for chaos is a trajectory that is exponentially unstable and neither periodic nor as ymptotically periodic; i.e., it oscillates irregularly without settling down. Chaos is just one type of behavior of nonlinear systems, where a nonlinear system is defined as a system whose time evolution equations are nonlinear; i.e., the dynamical variables of the system are of nonlinear form. The field of study is more properly called n onlinear dynamics, which is the study of the dynamical behavior (behavior in time) of a nonlinear system. There is an increasing body of evidence that a number of key conceptual features of nonlinear dynamical systems have particular relevance in improving understanding of spatiotemporal dynamics of epilepsy and the ep ileptogenic process. Nonlinear systems are characterized by a rich variety of dynamics incl uding bifurcations that indicate abrupt state transitions or intermittent behavior. Epile ptiform activity such as burst suppression patterns, spikes and interictal to ictal (seizure) states are all indicative of a nonlinear phenomenon. Another important property of ch aotic systems is self-organization – the property by which such systems evolve toward ordered temporal and spatial patterns [15]. The transition from chaotic to ordered behavior, or the reverse, can occur as an abrupt phase transition as a result of minute changes in the control paramete rs. Such transitions have been demonstrated in electrical activity recorded from the epile ptogenic brain [16]. Scope of Research in Epilepsy Mesial temporal lobe epilepsy (MTLE) with hippocampal sclerosis is one the most prevalent forms of epilepsy and is frequently associated with pharmacoresistance [17] The phenomenon is characterized by recurrent complex partial seizures and by extensive neurodegeneration and gliosis, notably in th e CA1 and CA3 areas of the hippocampus and in the hilus of the dentate gyrus [18,19]. In patients treated w ith modern antiepileptic drugs

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9 (AEDs), about 50% continue to experien ce seizures. Phamacoresistance to AEDs essentially defines intractable epilepsy, and has remained a major problem despite the introduction of a new generati on of AEDs. Current therapeu tic interventions for the treatment of chronic conditions such as MTLE do not take into account the variability in state of the epileptic brain. The diagnosis and treatment of epilepsy are further complicated by the fact that seizures occur spontaneously and unpredictably due to the chaotic nature of the disorder. In March 2000, leading scientists, health care providers, and leaders of voluntary health organizations came together to discuss what it would take to find a cure for epilepsy. This landmark White-House initiated conferen ce, Curing Epilepsy: Focus on the Future, was sponsored by the National Institute of Ne urological Disorders a nd Stroke (NINDS) in collaboration with the American Epilepsy Society (AES), Citi zens United for Research in Epilepsy, Epilepsy Foundation and the National Association of Epilepsy Centers. The conference encouraged scien tists nationwide to take a new look at methods for understanding and treating seizures, leading to the proposal of specific benchmarks for future research in epilepsy. Key benchmarks included i. Understanding how epilepsy develops ii. Finding ways to prevent seizures from developing in at-risk individuals iii. Finding better ways to stop seizures without side effects in those who have epilepsy This research project is part of a multidisci plinary team effort whose ultimate goal is the development of a seizure control device for the treatment of medically intractable epilepsy. The experimental scheme illustrated in Figure 1.1 shows the various experimental steps (1-5) involved in the global project, towards an important goal, i.e., an implantable

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10 seizure control device. We will now describe th e scope of this dissertation in context of a subset of the above mentioned benchmarks for epilepsy research. Figure 1-1. Development of a state-dependent seizure co ntrol device. Steps 1 to 5 indicate the various experimental leve ls involved in the research project Benchmark 1: Validation of Animal Models In the search for new treatments for epileps y, scientists usually must first conduct research on animal models that emulate the ch aracteristics of the disorder in humans. The several different types of epilepsies in huma ns mean that several different models are needed to study mechanisms and treatment optio ns for each individual class of the disease. One of the benchmarks of the conference was to determine how closely an animal model for each particular type of epilepsy resemble s the same epilepsy in humans and use the appropriate models to discover and test new ki nds of treatment. The better an animal model resembles the type of human epilepsy that it atte mpts to replicate, the more useful it would be in investigations aimed at unders tanding and curing the form of epilepsy.

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11 Several different animal models of MTLE have been developed in the recent past. The post status rat model of limbic epilepsy, originally described by Lothman et al. [20], and described in detail in following chapters, has been widely accepted as a good model of MTLE, exhibiting many of the important etiol ogical, pathological and behavioral features of MTLE in humans. In addition to these fe atures we propose to investigate the dynamical characteristics of this model to see if ch aracteristic linear and nonlinear properties associated with human TLE can also be found in this mo del. The rationale behind embarking upon this investigation is that such a validation will justify the use of this model as a biological test paradigm for preclini cal seizure forecasting and control techniques based on dynamical systems theory. Benchmark 2: Finding Ma rkers of Epileptogenesis The term epileptogenesis refers to the tran sformation of the brain to a long-lasting state in which recurrent, spontaneous seizures occur [21]. It may involve a focal area of the brain (partial epilepsy) or the entire brain simultaneously (generalized epilepsy). Epileptogenesis must be distinguished from seizure expression , which is concerned with processes that trigger and gene rate seizures, because seizur es can arise in non-epileptic brain exposed to acute insults. Another spec ific benchmark for future directions in epilepsy research is to identify markers of epileptogenesis and with the goal of using these markers in animal models to test therapy aimed at preventing the development of epilepsy. Markers of epileptogenesis represen t key targets for building an understanding of mechanisms underlying the phenomenon, anatomic vulnerability, prognosis, and response to therapy. The key qu estions to be answered when looking for markers of epileptogenesis are (i) where in the brain to sear ch for markers, (ii) what kinds of changes to look for, and (iii) when to look during a lengthy at-risk interv al [22]. We herein

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12 propose a novel application of time series anal ysis techniques for the very same purpose. The temporal dynamical changes combined with spatial dynamics of different specific brain regions offer a unique way of following the development of epilepsy in a rodent model of epilepsy. Hence, these spatiotemporal dynami cal patterns are proposed as markers of epileptogenesis Benchmark 3: Improve and Validate Seizure Forecasting Techniques Epileptic seizure prediction has steadily evolved from its conception in the 1970s, to proof-of-principle experiment s in the late 1980s and 1990s, to its current place as an area of vigorous, clinical a nd laboratory investigation. A specific benchmark for future epilepsy research was to improve and valid ate present seizure prediction methods and eventually couple it with trea tment strategies that interrupt the process before seizures begin. Studies in seizure prediction vary wide ly in their theoretica l approaches to the problem, the amount of data analyzed and valid ation of results. Some relative weaknesses in this literature are the lack of extensiv e testing on baseline data free from seizures, the lack of technically rigorous validation and quantificati on of algorithm performance in many studies, and the lack of methods for f iltering out the most useful components from multichannel and multi-feature data for forecas ting. For example, some studies, such as those by Petrosian et al., re port seizure prediction afte r analyzing one channel of electrical activity from an in tracranial depth electrode in one patient [23-25]. In these studies, using univariate techni ques no analysis of baseline data far removed from the seizure was undertaken. A potenti al pitfall of conc lusions based upon such limited data is that quantitative changes identified prior to seizure onset may not be specific to the preseizure period, but may occu r at other times as well, unr elated to epileptic events.

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13 Validation of prediction algorithms on long, cont inuous sets of clinical data, representing all states of awareness, is an important pa rt of more recent seizure prediction studies. The successful “proof-of-principle” demonstr ation of a preseizure period, validated by multiple research groups and techniques, ha s moved research in this area on to more in-depth studies exploring the temporal and spatial characteristics of the preseizure period, as well as its underlying mechanisms. Recent collaborative discussions between groups working in this area have identif ied several important related areas of investigation. First, in order to be able to assess and compare seizure prediction methods, acceptable performance metrics for these type s of studies must be agreed upon. In a recent published work on seizure prediction [ 26], investigators in the field cautioned against overstating performance results of se izure prediction algorith ms, and advised that rigorous reporting of sensitivity and specificity, among other performance criteria, are necessary to validate the util ity of different experimental approaches. Second, the need for analyzing long epochs of data free from seizures was emphasized. Without this work, eventually leading to prospect ive, online trials of prediction algorithms, the specificity of seizure precursors to the preictal period will always remain suspect. The Iasemidis and Sackellares group were the first to apply nonlinear dynamical techniques, particularly methods based upon the maximum short term Lyapunov exponent ( STLmax), for predicting seizures beginning in the late 1980s. This group has demonstrated evidence of seizure precursors in a variety of data se ts, ranging from one to multiple channels and epochs spanning minu tes to hours [27-31]. In their research, seizure onset prediction was reported from one to sixty minutes pr ior to electrographic seizure onset in some data sets. These seizure prediction studies have involved the

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14 analysis of retrospective el ectroencephalogram (EEG) da tasets acquired from human subjects undergoing intracrania l epilepsy surgery assessments. Limitations inherent to using heterogeneous data sets have prompted our group to further investigate a similar seizure prediction technique by using an an imal model that mimics human MTLE. The rationale for investigating dynamics in an animal model stems fr om our interest in ameliorating as many confounding factors as possi ble such as epilepsy type and severity, pathology, anticonvulsants, electrode placement, and overall testing conditions. We postulate that the rodent model will offe r the potential to address many of these confounds by providing the testing conditi ons upon which to conduct our controlled experiments [32] and test our hypothesis. Furthermore, prolonged datasets containing long seizure free intervals woul d overcome another one of th e concerns with regards to specificity of the measure in detecting pres eizure periods. The signi ficance of identifying and quantifying such nonlinear dynamics in this model is that it raises the possibility of developing a method to control seizures that are based on manipulations of the underlying dynamics. Benchmark 4: Design Ne w Treatment Strategies A significant portion (20-30%) of the popula tion affected with epilepsy goes on to develop chronic or medically intractable epilepsy, which is a condition in which seizures persist despite accurate diagnosis and treatment with conventional anti-epileptic drugs (AEDs). The lack of efficacy of systemically administered pharmacological agents in a large group of patients with epilepsy, together with the relatively high incidence of serious idiosyncratic or intolerable dose-dependent adverse effects including exacerbation of seizures, underscores the need to devel op novel therapies or delivery modalities. The medical, psychosocial, and economic benefits that will be derived from achieving these

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15 objectives are self-evident to patients, their car egivers, and health care providers. Surgical resection of the epileptogenic focus is currently an alternative, but in some cases this is not possible because of limitations in identifying a seizure focus. For these people, identifying new treatment modalities is even more importa nt. In this regard, animal models of pharmacoresistant epilepsy are a useful tool to study the mechanisms of chronicity and develop alternate therapie s or control mechanisms. Much of the present day therapeutic te chniques employed in the treatment of epilepsy involve ad hoc delivery of either AEDs or electrical stimulation to specific brain targets. On-demand and tempora lly selective delivery of therapy may be a safer and more efficacious method for the treatment of epilepsy. One of the central investigative topics in this dissertation involves testing the hypothesis that external timed interventions during a seizure susceptible state are able to direct the dynamical state of epileptic brain to a range associated with the healthy or normal regime. Fu rthermore, seizures have been said to be mechanisms that revert the brai n from an ordered state to a mo re chaotic state. Intuitively we can see why the maintenance of chaos in the brain is extr emely desirable in a complex system such as the brain. This dissertation also attempts to shed light on this problem by considering a simple model that emulates preseizure and seizure brain dynamics to investigate control methodologies based on optim ization that can conceivably be adapted to controlling seizures. Contributions of this Dissertation This dissertation describes work done as part of a multidisciplinary research effort towards the goal of developing improved seiz ure prediction and seizure control methods for the treatment of medically intractable epil epsy. To unlock some of the questions that plague this form of epilepsy [33], one must identify valid m odels of the disease. If the

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16 assumption about epilepsy being a dynamical dis ease is valid, then it is very important to choose appropriate animal models of human epilepsy that exhibit similar dynamics. To this end, concepts and time series analysis techniques employed in the analysis of brain electrical activity in human MTLE are adapted to an animal model of the same disorder. The goal of this endeavor was to determine whether dynamical properties as reflected in the brain electrical activity of this model were comparable to those seen in patients with intractable MTLE. Validati on of the post status animal model of MTLE against dynamical descriptors used in the analysis of human epilepsy is one of the first contributions of this dissertation. We make a case for using the model in investigating seizure prediction and control strategies based on underlying dynamics of brain electrical activity that can potentially be of use in the management of human epilepsy. A detailed description pertaining to this work can be found in Nair et al., [34,35]. The elucidation of mechanisms of epileptogenesis and pharmacoresistance, including the identification of surrogate ma rkers that could provi de faster and more predictive screening methods for patients at ri sk for developing epilepsy is one of the critical goals in epilepsy research. Emphasis has also been placed on the need to use animal models for developing markers th at can be linked to electrophysiological abnormalities in the epileptogenic brain and id entifying at-risk subjects. We describe a novel application of nonlin ear time series analysis methods and statistical techniques for following the development of epileptogenesis in an animal model of MTLE. Specifically, this dissertation outlines methods to identif y spatiotemporal transitions in the brain electrical activity during the latent period of the animal model and use them as markers of epileptogenesis. Identifying the key alterations, starting at the time of brain insult and

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17 occurring during subsequent late nt period needs to be accomplished. This is a particularly important goal because it is probable that e ffective intervention will be directed against these processes to prevent epileptogenesis afte r different forms of brain injury and that strategies directed against such proce sses may not involve st andard therapeutic approaches. This work will be pres ented in Nair et al., at the 1st North American Regional Epilepsy Congress [36]. Animal models are an excellent choice fo r testing and validating seizure prediction methods since they are relatively homogene ous in terms of etiology, pathology, epilepsy type and severity, compared to human epilepsy. Furthermore, it is possible to increase the test sample size as well as duration of recorded datasets, thereby helping to gauge a fair estimate of the performance of the prediction algorithm. Evidence for preictal transitions has been reported in experimenter induced se izures using stochastic methods [37]. Their study used a stochastic model to model tran sitions between interi ctal and drug-induced (3-mercaptopropionic acid) seizures in a ge neralized animal model. In the present dissertation, we have used a spontaneously seizing model and techniques previously employed in the analysis of human epilepsy to show seizure predictability. The algorithm was tailored to the dynamical traits of this mode l. To our knowledge this is the first report in which seizure predictability is demonstr ated in an animal model of spontaneous epilepsy. An intensive evaluation of the perf ormances of seizure prediction algorithm testing on continuous in tracranial EEG (iEEG) recordings fr om a group of 5 subjects with spontaneous seizures is reporte d in this dissertation. For the individual subject, we used the first half of seizures to train the pa rameter settings, which is evaluated by ROC (Receiver Operating Characteristic) curve anal ysis. With the best parameter setting, the

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18 algorithm was applied to all cases . The results suggest ed that this anim al model could be used as a test bed for conducting seizure wa rning based intervention experiments for seizure control. The preceding work is reflected in Nair et al. , [35] and Shiau et al., [38] There is both clinical and experimental ev idence that both direct (e.g. thalamic or hippocampal stimulation) and indirect el ectrical stimulation (e.g. vagus nerve stimulation) of the brain can have anticonvuls ant or proconvulsant effects (see Theodore and Fisher [39], for a summary). That conti nuous trains of electrica l stimulation, without knowledge of the state and sans feedback from the brain could have clinical utility is indeed encouraging. This dissertation desc ribes the design of an automated seizure warning based electrical brain stimulation pa radigm. A detailed description of both the hardware and software of the system is given. The effect of stimulation on iEEG dynamics is investigated to understand the dependence of outcomes on dynamical state of the brain. Based on the results we propose a new state dependent therapeutic scheme based on multidimensional descriptors for cont rolling brain dynamics. Specifically, the new proposed scheme takes into account bot h the spatial dynami cal coupling between brain regions and also nonlinea r cross correlations between specific brain areas. We also report the effects of a state dependent seizur e control scheme on the seizure frequency in the model. The preliminary reports suggest a strong anticonvulsant effect of such a therapeutic technique. Further details pertaining to this work can be found in Nair et al., [40], and Shiau et al., [38]. Recent investigations in human epilepsy ha ve suggested that effective modulation of brain dynamics needs new control tech niques that rely on robust prediction and adaptive optimization methods [41-43]. This dissertation will also concentrate on the

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19 theoretical problem of controllability of a sy stem property that has shown to reflect state changes in neural systems, namely the Ly apunov exponent. We first propose a model of interconnected nonlinear oscillators and dem onstrate similarities in system dynamics between the epileptic brain and the model. We then use this model to show controllability of chaos using a combination of concepts derived from dynamical and optimization theory. Specifically, we describe a ne w method based on constrained optimization techniques for controlling sp atiotemporal chaos in a coupled map lattice system that emulates brain dynamics. Adaptation of such c ontrol techniques to an epileptic brain is also described. This work has been reflected in tw o peer reviewed publ ications, Nair et al., [44], and Pardalos et al., [45] Organization of Chapters This dissertation, which employs innovative application of con cepts derived from theory of nonlinear dynamics, time series analysis, optimization and animal electrophysiology to the problem of controlling epilepsy, is di vided into seven chapters. The organization of the succeeding chapters of this dissertation is as follows. Chapter 2 gives a brief overview of current animal mode ls of TLE and a detailed description of the rat model of MTLE that has been used in this research, including model creation, experimental setup and electrophysiological char acteristics. Basic concepts of time series analysis techniques commonly used in EEG/iEEG are reviewed in Chapter 3. Additionally, background and liter ature pertaining to dynamical approaches, chaos theory and nonlinear dynamics including calculation of several dynamical descriptors are explained. The dynamical characteristics of iEEG in an animal model of MTLE are also described in detail. Chapter 4 focuses on the problem of seizure predictability in the animal model. In this chapter we give an outline of an automated seizure warning

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20 algorithm and discuss its perfor mance on long term continuous iEEG data recorded from the animal model. In Chapter 5, we introduce the problem of contro lling spatiotemporal chaos and its relevance in the field of epileps y, seizure control in particular. We also describe several chaos control technique s and finally propose a novel method of controlling spatiotemporal chaos in a theo retical model emulating EEG/iEEG dynamics using constrained optimiza tion techniques. Chapter 6 reviews the background and significance of electrical stimulation as an a lternative treatment for medically intractable epilepsy. A state dependent electrical stim ulation paradigm is then proposed and described in detail. Finally, the effects of state dependent stimulation on iEEG dynamics and seizure frequency are presented and di scussed. Concluding remarks with future directions for research ar e discussed in Chapter 7.

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21 CHAPTER 2 A MODEL OF MESIAL TEMPORAL LOBE EPILEPSY Role of Animal Models in Epilepsy Research The use of animal models ha s a critical role for all m odern biomedical research. Obvious ethical constraints exist in research involving human subjects, particularly those associated with invasive techniques often required to pursue clinical investigative questions. In such cases control data can be difficult to obtain and controlling clinical control variables can also potentially be a problem. Statistical validation in research often requires large sample sizes and consequently the cost of carrying out human research can turn out to be prohibitive. Additionally, animal models offer the advantages of simplicity, reproducibility, controllability and a good unde rstanding of the anatomy and physiology. Animal models will hence most likely be a part of biomedical research for the foreseeable future. Animal models of seizures and epilepsy have a proven histor y of advancing our understanding of the physiologi cal correlates and behavioral changes associated with human epilepsy. Since th e early half of the 20th century, animal m odels have been instrumental in the discovery and screening of antiepileptic treatment strategies, particularly antiepileptic drugs (AEDs) [46]. Numerous models for epilepsy exist and all of them possess certain valid sc ientific rationale to answer specific questions. Models of epilepsy can be broadly classified into acute (provoked or reactive) and chronic . The former class includes models in which seizures are experimenter induced by some external forcing (electrical or chemical) in nave, healthy (non-epileptic) animals, while

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22 the latter means models that have been made epileptic by electrical or chemical means or by genetic alteration exhibiting spontaneous seizures. Chronic models of epilepsy can therefore be further divided into models of acquired (symptomatic) epilepsy and models of genetic (idiopathic) epilepsy. For a review of animal models of epilepsy and epileptic seizures, refer to [33,47-49]. Since it is highly likely that no one model will prove to be useful for all of the epilepsies , it is appropriate to start w ith those models that closely parallel the human condition at hand. What is a Good Animal Model? The validity of animal models of seizur es and epilepsy as models for specific human syndromes has been critiqued in the past for several reasons [49]. The need for more models and criteria for validation of models is discussed in this section. As a follow-up to the White House sponsored conf erence: Curing Epilepsy, held in March 2000, another workshop, the NINDS sponsored conference: Models for Epilepsy and Epileptogenesis, was held in March 2001 and its specific objectives were to stimulate development of new pharmacologic and non-pharmacologic therapies for control of epileptogenesis and pharmacoresistant ep ilepsy [32]. One of the conference recommendations was that models of chronic epilepsy that exhibit spontaneous seizures may be best suited for therapy discovery. Th e need for development and dissemination of technology for long term seizure monitoring and therapy administration was also emphasized. Experimental animal models, in order to be considered as go od models of human epilepsy, should exhibit seizure phenotypes di splayed by the human epileptic condition they attempt to model. The following selection criteria for a suitable animal model were outlined by Sarkisian [49]:

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23 i. The animal model should exhibit similar electrophysiological co rrelates/patterns observed in the human condition ii. The etiologies should be similar iii. If the human condition is characterized by a unique age of onset, the proposed animal model should be scaled to reflect a similar age in humans iv. The animal model should display similar pathological changes, for e.g. focal lesions or cortical dysplasia if such pa thologies are present in the human condition v. The response to standard AEDs should be similar vi. The type of seizures should be simila r in clinical phenomenology to seizures occurring in humans. Another follow-up conference to the workshops listed above: Models II – Identification and Characterization of Animal Models of Human Therapy Resistance and Epileptogenesis, was held with the goal of reviewing existing models of epilepsy and seizures and recommending the models most a ppropriate for therapy identification. At the end of the conference, the post status epileptic us (SE: defined in later sections) model of limbic epilepsy was recommended as th e best suited model for studying human pharmacoresistance and developi ng novel therapies. The reco mmendations were made on the basis of parallels that these models ha ve to human pharmacoresistant epilepsy [50]. This proved to be one of the influencing factor s in our choice of this model of epilepsy in our studies. Models of inju ry-induced epilepsy based on electrically and chemically induced SE were considered to have the greatest parallels with a common form of human drug-resistant epilepsy, the MTLE syndrome.

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24 Since we are interested in the dynamics of human epilepsy as it pertains to seizure generation as well as intervention, we are natu rally interested in finding how closely any given animal model of epilepsy mimics the dynamical properties that have been observed in human epilepsy. The better an animal model resembles the type of human epilepsy that it attempts to replicate, the more useful it would be in investigations aimed at understanding the form of ep ilepsy and developing diagnostic and therapeutic tools. Models of Temporal Lobe Epilepsy Several models that mimic complex partia l seizures observed in patients with TLE have been developed and well-characterized. In all such models, typi cally all phases of the epileptic process seen in symptomatic human TLE (initial insult epileptogenesis manifestation of spontaneous seizures) can be observed. Several pat hological findings in such animal models resemble appearance of neuronal structural damage in human pharmacoresistant TLE. These models can eith er be chemical-inducti ve or electricalinductive in nature. The popularity of such models of TLE is a ttributable to the satisfaction of a majority of the crit eria listed in the previous section. The Kindling Phenomenon Kindling, a progressive increase in electr ographic and behavioral seizure activity resulting from repetitive focal application of initially sub-convul sive short electrical stimulation to limbic structures in the brai n such as amygdale or hippocampus is widely used as a model of MTLE. Such focal stimul ation finally results in intense partial and generalized convulsive seiz ures whose clinical phenom enology and pharmacology are very similar to the human condition [47,51] . Goddard first reported on the phenomenon of kindling in the late si xties [52,53]. In further wo rk Goddard and coworkers demonstrated that kindling could also be i nduced chemically. While examining electrical

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25 stimulation of the amygdaloid complex, Godda rd recognized that the brain was changing in response to a constant stimulus, and he pr oposed that this form of plasticity could provide a useful neural model of epilepsy. The kindling phenomenon is a manifestation of the proposed theo ry that “epilepsy induces epilepsy” [33] i.e., through a pos itive feedback mechanism. Kindling is considered a network phenomenon owing to information from several sources that indicate that changes in neuronal circuits, rather than just changes at the site of stimulation, are important. The brain networks in this case can be considered to become permanently hyperexcitable resulting in repe ated partial seizures [54-56]. The initial stimulus often elicits paroxysmal activity also referred to as “afterdischarges” (AD) in the brain electrical activity. Subs equent stimulation induces progressive development of seizure activity as classified by Racine who described the kindling progression in detail, delineating the progression into 5 distinct behavioral stages from motor arrest accompanied by facial automatisms (Stage 1) to fully kindled seizures accompanied by forelimb clonus and hindlimb tonus identified by rearing and bipedal instability (Stage 5) [57]. The increased severity in behavioral manifestations al so corresponds to an increase in electrographic seizure durat ion. Kindling also leads to mo ssy fiber sprouting in the dentate gyrus of the hippocampus [58] and cell death in hilus, CA3 and CA1 pyramidal neurons [59]. Self Sustaining Status Epilepticus The World Health Organization (WHO) defines status epilepticus as “a condition characterized by an epileptic seizure that is sufficiently prolonged or repeated at sufficiently brief intervals so as to produce an unvarying and enduring epileptic condition”. Traditionally, SE is defined as 30 minutes of co ntinuous seizure activity or a

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26 series of seizures without return to full consciousness between the seizures. Many believe that a shorter period of seiz ure activity causes neuronal in jury and that seizure selftermination is unlikely after 5 minutes; some suggest times as brief as 4-5 minutes to define SE. The fundamental pathophysiology of SE involves a failure of mechanisms that normally abort an isolated seizure event. Th is failure can arise from either abnormally persistent, excessive excitation or ineffective recruitment of inhibition, or a combination of both. The relative contribution of these f actors is poorly unders tood. The temporal and spatial determinants of SE are also relativel y unknown; experimental studies suggest that there is induction of reverberating seizur e activity between, for example, hippocampal and parahippocampal structures and that seizures progress through a sequence of a distinct electrogra phic changes [60,61]. Among chronic animal models of epile psy are models in which recurrent spontaneous seizures develop af ter a period of self-sustained status epilepticus (SSSE), elicited by continuous electrical stimulation of limbic struct ures such as the hippocampus (via stimulation of the perforant pathway, the angular bundle or CA3 regions of the ventral hippocampus), and the lateral or baso lateral nucleus of the amygdale [62-64] or prolonged activity of acute convulsant induced seizures. Such models are characterized by neuropathological changes reminiscent of mesiotemporal sclerosis seen in many patients with TLE and the development of recu rrent spontaneous seizures post-SE [64]. The seizures observed in such models are either partial (“limbic”) or generalized (“motor”) and are accompanied by paroxysmal discharges in the brain electrical activity. Models of acquired epilepsy in which epil epsy develops after induction of SSSE are

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27 widely used to study the pathophysiology of TL E and in the search for alternate therapy for medically intractable epilepsy. The Continuous Hippocampal Stim ulation Model of Human MTLE The continuous hippocampal stimulation (C HS) rat model of limbic epilepsy is a spontaneous seizure model that is created by inducing prolonged seizures (SE) through direct electrical stimulation of the hippocampus. After a period of se veral weeks to a month of recovery, the animals begin to have spontane ous seizures that last for the rest of their lives [65,66]. The CHS model is considered a good model of TLE since it is characterized by spontaneous limbic seizure activity, which is the clinical hallmark of MTLE. Furthermore, in this model, both the seizure locus and th e pathological changes in the limbic system of the animals reflect what is found in the human pharmacoresistant MTLE. It has also the important features of spontan eity, chronicity, and temporal distribution of seizures associated with human TLE [67-69]. Spontaneous seizures may persist for many months, and are typically bilaterally synchronous in the hippocampus. Mechanisms The mechanisms by means of which spontan eous seizures occu r in this animal model are not yet clear. However, it has been shown that CHS causes a diminution of Gamma-Aminobutyric Acid-A (GABA-A) mediated inhibition in CA1 areas, immediately after the CHS and as well as at l ong-term, and also in the dentate gyrus [70]. On the basis of a series of experiments us ing different agonists and antagonists of both GABA-A and B receptors, Lothman [20] sugg ested that GABAergic mechanisms can influence seizure initiation and ar e critical in the termination of seizures. In a series of experiments aimed at clarifying the pathophys iology of this chronic model of TLE, a number of relevant changes observed at leas t 1 month after the SE were described: 1)

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28 Monosynaptic excitatory postsynaptic potentia ls (EPSPs) evoked in CA1 pyramidal cells, but not in the granule cell of the dentate gyrus in post-CHS tissue were always longer than those in control tissue [71]. 2) A de ficiency in GABA-A rece ptor inhibition is a major factor in this model [70]. Also in post-CHS tissue, the late inhibitory postsynaptic potentials (IPSPs), mediated by GABA-B recepto rs, were diminished [70]. In addition, the duration of the GABA-A receptor medi ated IPSPs declined in the post-CHS hippocampus when compared to controls. It is interesting to note th at as in kindling, a reduction of pre-synaptic GABA-B receptors wa s reported to occur in amygdale [72] and CA1 kindling [73]. 3) Both CA1 pyramidal cells and dentate gyrus granule cells from post-CHS tissue showed hyper-responsivene ss to electrical stimulation of the corresponding monosynaptic inputs relative to control tissue and was more robust in CA1 [74]. The cellular and molecular processes that take place after the status epilepticus and evolve during the latent pe riod, leading to the pathophysiological ch anges of MTLE and ultimately to the occurrence of spontan eous seizures, are as yet unclear. Model Preparation Experiments were performed on two month old adult male Harlan Sprague Dawley rats (n=5) weighing 210-265 g. Protocols and procedures were approved by the University of Florida Institutional Animal Care and Use Committee. Rats were anesthetized with xylaz ine (10 mg/kg, SQ) and isoflurane (1-3%) in oxygen and placed in a stereotaxic Kopf apparatus with a nose mask to continue inhalation anesthesia. Heat was supplied by a heating pad to maintain the body temperature within normal limits. Electrode Implantation Surgery . The top of the head was shaved and chemically sterilized. The skull was exposed by a mid-sa gittal incision that be gan between the eyes and extended caudally to the le vel of the ears to expose br egma and lambda. A peroxide

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29 wash was applied to remove excess soft tissu e from the skull. The approximate electrode placement locations are shown in Figure 2-2. Four 0.8 mm stainless steel screws (small parts) were placed in the skull to anchor the acrylic headset. Two were located 2 mm rostral to bregma and 2 mm laterally to either side of the midline. One was 3 mm caudal to bregma and 2 mm lateral to the midline. One of these served as a screw ground electrode. The last, which served as a sc rew reference electrode , was located 2 mm caudal to lambda and 2 mm to th e right of midline. Holes were drilled to permit insertion of 2 stainless steel bipolar tw ist electrodes (1 mm tip separation) into the left and right ventral hippocampii for electrical stimulati on and recording (AP: -5.3; left and right lateral: +/-4.9mm; vertical: -5 mm to dura) and 2 stainl ess steel monopolar recording (AP: 3.2 mm, lateral: 1mm left, vertical: -2.5 mm; AP: 1 mm, lateral: 3 mm right, vertical: -2.5 mm) electrodes in the bilateral frontal cortical hemis pheres. Electrode pins were collected into a plastic strip connector (Figure 2-1) and the entire headset was glued into place using cranioplast cement (Plastics One, Inc.). Rats were allowed to recover for a week after surgery before further procedures were performed. The typical location of the bi polar twist electrode placemen t, used to electrically stimulate the rodent brain is shown in Figure 2-3. This is sh own in a sagittal view (part A) and transverse view (part B) of the inju red excised brain of an epileptic animal. The elliptical regions marked in the images out line the location of th e electrode. The dark pixels surrounding the location of the stimulating electrod e denote the injury induced during SE. These images were adapted from published results of studies by our group aimed at studying the evolut ion of these anim als into epilepsy. (NIH-NSF grant no. 1R01EB004752-01)

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30 Figure 2-1. Animal headstage Figure 2-2. Approximate relative locations of electrodes on the animal’s skull

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31 Figure 2-3. Structural damage due to el ectrode placement and induction of SE via hippocampal stimulation Induction of SE (Hippocampal Stimulation) . The rats were electrically stimulated to induce seizures 1 week after su rgery. The stimulation target was chosen by the technician based on beha vioral response to stimulati on and intracranial EEG (iEEG) AD patterns. Hereafter the hemisphere of th e hippocampus that was stimulated will be referred to as the lesioned side and the cont ralateral hemisphere as the non-lesioned side.

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32 During stimulation and iEEG acquisition, rats were housed in specially made chambers [75]. Baseline iEEG data was collected us ing the 2 frontal monopolar electrodes and 1 branch of each hippocampal, bipolar electr ode. After recording ba seline data, the cable was changed so that an electr ical stimulus could be administ ered to one hippocampal site through a pair of bipolar twist electrodes. Th e stimulus consisted of a series of 10 sec trains (spaced 2 seconds apart) of 1 millisecond, biphasic square pulses at 50 Hz, at an intensity of 300-400 mA, for 50-70 minutes [76]. During the stimulus, a normal response was to display “wet dog shakes” and increased exploratory activity. After approximately 20 minutes, convulsive seizures (up to 1 min duration) were us ually observed about every 10 minutes. At the end of the stimulus period, the iEEG recording was observed for evidence of slow waves in al l recorded channels. If this was not the case, the stimulus was re-applied for 10 minute intervals on an other 1-3 occasions un til continual slow waves appeared after the stimulus was te rminated. Unresponsiveness towards such a stimulation protocol was observe d very rarely (less than 10% of the rats). The lack of response was attributed to inaccurate placemen t of the stimulating electrode. Only rats that were responsive to stimulation and went on to develop spontaneous seizures were included in the study. With successful seizure i nduction, the iEEG continue d to demonstrate activity below 5 Hz for 12-24 hours and intermittent a nd spontaneous electrographic seizures (30 seconds 1 minute in duration) for 2-4 hour s following an electrical stimulation session. Rats were observed for 12-24 hours after stim ulation for seizure activity, and food and water intake was monitored closely. Once their behavior stabilized, they were returned to

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33 their home room for six weeks while spontaneous seizures developed. See Figure 2-4 for model development timeline. Electrophysiological Data Acquisition . Each animal was connected through a 12channel commutator and shielded cable to th e recording system, wh ich consists of an analog amplifier (Grass Telefactor-Model 18E), a 12 bit A/D converter (National Instruments, Inc), and recording software (HARMONIE 5.2, Stellate Inc. Montreal), which was synchronized to a video unit fo r time-locked monito ring of behavioral changes. A detailed description of the reco rding setup used in this study can be found elsewhere [75]. Each channel was sampled at a uniform rate of 200 Hz and filtered using analog high and low pass filters at cutoff fr equencies of 0.1 Hz and 70 Hz, respectively. The recording system used a referential m ontage and was set to a continuous mode so that prolonged data sets containing ictal as we ll as interictal data could be collected for analysis. The saved EEG and video data was th en transferred to a 1.4 TB RAID server for future off-line review and analysis. EEG da ta set pre-processing included removal of baseline wander using a Butterworth filter. Figure 2-4. Experimental timeline

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34 Electroencephalogram a nd Seizure Description Intracranial EEG is a time-varying signal that records the summ ation of thousands of neuronal potentials around th e contact area of the elect rode. These intracranial electrodes as used in the cu rrent study, sample from much smaller volumes, are not subject to distortions introduced by the su bject’s skull and scal p and are relatively impervious to artifacts produced by eye moveme nts. These factors co ntribute to the high signal-to-noise ratio characteristic of these recordings. Figure 2-5 illustrates the patterns corresponding to interi ctal (background activity), ictal (s eizure) and pos tictal (following the seizure) periods. When a seizure starts, ne urons start firing in a synchronized fashion manifesting as abnormal rhythmic discharg es, as seen in Figur e 2-5 and Figure 2-6. Electrographic seizures in limbic epileptic rats are usually characterized by the paroxysmal onset of high frequency (greater than 5 Hz) increased amplitude discharges that show an evolutionary pa ttern of a gradual slowing of the discharge frequency and subsequent post-ictal suppre ssion. In some instances th e seizure begins with high amplitude spikes or polyspikes, followed by a brief period of electrographic suppression characterized by spike and wave theta activity and a postictal spike wave low amplitude delta. The evolutionary pa ttern and post-ictal suppres sion are key elements in determining that an event was a seizure, as artifact (especially head scratching) can have a similar appearance but lacks all of these ch aracteristics. The criteria for identifying an electrographic seizure is as follows: 1) The occurrence of repetitive spikes or spike-andwave discharges recurring at frequencies >1 Hz, or continuous polyspiking; 2) spike amplitude greater than backgr ound activity; 3) duration of continuous seizure activity greater than 10 sec. The seizures in this m odel have shown to have three classic onset patterns: 1) Non-hippocampal (amy gdale or piriform cortex ons et), 2) Hippocampal, or 3)

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35 Diffuse (apparent simultaneous onset at both hippocampal and non-hippocampal sites [76]. Figure 2-5. Sample epochs of iEEG recorded during interictal, ictal and postictal states from four different brai n sites of an MTLE model Figure 2-6. Three minutes of iEEG (demons trated by 6 sequential 30-second segments) data recorded from the left hippocampus, showing a sample seizure from an epileptic animal Figure 2-5 shows an iEEG epoch reco rded from one of the hippocampus, containing a seizure. It can be seen that the signal becomes more organized (less randomlike) as it grows in magnitude and frequency. In the iEEG, the onset of the seizure is at second 72 and is accompanied by a clear spike a nd wave discharge. About 12 seconds into the seizure, high amplitude rhythmic activity starts and this is correlated with an increase in spectral power at higher fre quencies up to 25 Hz. The postictal state is

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36 characterized by spike and slow wave activity. Th is frequency evolution is very similar to patterns seen in intracranial recordings from patients with tonic-clonic seizures. Seizure “onset” as used in this study wa s defined electrographically as the first sustained change in the iEEG clearly different from the background activity. Seizure “offset” was defined as the time at which th e rhythmic activity dies out and postictal spike and wave discharges appear. The onsets of each seizure were cl assified into two categories, ( A ) Bilateral onset with pred ominance in one side and ( B ) Bilateral. A classification scheme developed by Racine [57] for kindled seizures was used to further categorize the seizures into different grades (1-5). For clarity, the following definitions are given: Ictal period : the time from the start of the seizure to the end, covering its development and spread Preictal period : the time preceding the ictal period Postictal period : the period immediately following the ictal period Peri-ictal period : the time around the ictal period Interictal period : a period at least 5 hours away from the seizure event, used interchangeably with background or baseline Electrographic seizure onset : first sustained change in the iEEG clearly different from the background activity. Electrographic seizure offset : the time at which the r hythmic activity dies out and postictal spike and wave discharges appear. Racine Classification : o Grade 0 – Afterdischarges without motor manifestations

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37 o Grade 1 – Facial twitching (Facial clonus) o Grade 2 – Head nodding accompanied by simultaneous mastication o Grade 3 – Forelimb clonic convulsion o Grade 4 – Rearing on hindlegs and bilateral forelimb clonus o Grade 5 – Full grown generalized clonic convulsion and falling down

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38 CHAPTER 3 TIME SERIES ANALYSIS AND FEATURE EXTRACTION A time series can be defined as a record of the values of an y fluctuating quantity measured at different instances of time, us ing a consistent activity and technique of measurement. For example, one may record da ily temperatures, voltage s or currents in an electrical circuit, monthly price index of a commodity, or an electroencephalograph (EEG) that measures electrical activity (vo ltage difference) between two sites in the brain. The first three records are examples of discrete sampling whereas the fourth is an example of continuous sampling. There are two main goa ls of time series analysis: i. Drawing inferences from such series and identifying the nature of the phenomenon represented by the sequence of observations/measurements. ii. Predicting future values of the time series variable (forecasting). In both cases, the pattern of observed time series data is required so we can interpret and integrate it with other data for gaining a comprehensive understanding of the system generating the data. Regardless of the depth of our understanding and the validity of our interpretation (theory) of the phenomenon, we can extrapolate the identified pattern to predict future events with varying degrees of accuracy. One of the most common wa ys of obtaining informati on about neurophysiological systems is to study the features of the signal( s) recorded from them by using time series analysis techniques. If one is only interested in the features of a single signal, univariate analysis can perfectly carry out this task by itself. But an increasing number of experiments are being carried out in wh ich several neurophysio logical signals are

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39 simultaneously recorded (for e.g. recording electrical activity from multiple cortical regions in the brain), and the assessment of the interdepende nce between signa ls can give new insights into the functioning of the syst ems that produce them. Therefore, univariate analysis alone cannot accomplish such a task, and it is necessary to make use of multivariate analysis. Multivariate time series analysis is widely used in the domain of neurophysiology with the aim of studying th e relationship between simultaneously recorded data sources. Recently, advances in information theory, nonlinear dynamical systems theory and statistics have allowed th e study of various types of synchronization from different time series. This chapter reviews the fundamental c oncepts and well-known methods in time series analysis of electrophysiological signals , EEG and iEEG in pa rticular, for feature extraction and forecasting. The organization of th e succeeding sections of this chapter is as follows. In the first four se ctions we give some concepts of basic time series analysis methods including time domain analysis, frequency domain analysis and nonlinear dynamical analysis. The remaining sections deal with the main techniques that have been used in this research in the analysis of iEEG recorded fr om the animal model of MTLE. Time Domain Analysis Cross-Correlation This is one of the oldest and most clas sical measures of interdependence between two time series. A physical process X can be described in the time domain by the value of some quantity x ( t ) as a function of time. The cross-co rrelation function of two functions x ( t ) and y ( t ) is defined as a function of a time lag : dt t y t x Cxy) ( ) ( ) ( . (3.1)

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40 The cross-correlation of a function with itself is called its autocorrelation . In applications one usually deals with a fi nite amount of measurements offered by experimentalists. Assume that x0,, xN-1 and y0,, yN-1 are two simultaneously measured stationary time series, which have zero mean and unit variance. The estimate of crosscorrelation is then defined as a function of the time lag = ( N 1 ),, 0 ,, N 1 : 0 ), ( 0 , 1 ) ( 1 yx N i i i xyC y x N C (3.2) The cross-correlation is normalized to th e range from minus one (complete antisynchronization) to one (complet e synchronization). The value of that maximizes this function is usually taken as an estimation of the delay between the signals, under the implicit assumption that they are linearly relate d. The value of cross-correlation near zero indicates linear independence of systems. The estimator for the cross-correlation could give non-zero values for two completely line arly independent systems. That is why a significance threshold for the estimated cros s-correlation should be taken into account (e.g., the Bartlett estimator [77,78]). Nevert heless, the cross-correl ation is one of the simplest and mostly used measures of s ynchronization between two systems, although it is not sensitive to nonlinear dependencies. Autoregressive Moving Average Model ARMA or auto regressive moving aver age models are a method of modeling univariate time series data. The dependence or regression on the variable 's past values is used to make predictions of the variable's future values. A model that is derived from autocorrelated data is referred to as an autore gression. It can also be used as a tool for predicting future values in the series. Given a univariate time series of data Xt, the ARMA

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41 model can be written as a combination of two parts, an autoregressive (AR) part (3.3) and a moving average part. p i t i t i tX C X1 , (3.3) where 1,, p are the parameters of the model of order p, C is a constant and t is the error term. The ARMA mode l can be written as : p i q i i t i i t i t tX X11 , (3.4) where 1,, p are the parameters of the moving average (MA)model. Autoregressive Integrated Moving Average Models Many methods have been used to analyze and forecast nonstationary time series data. One of the well-known methods, proposed by Box and Jenkins (1976) [79], is called the autoregressive integrated moving average (A RIMA) process. It is an extension to the stationary autoregressive moving aver age (ARMA) process. ARIMA deals with nonstationary time series by using the dth difference to make the nonstationary time series a stationary ARMA process. The ARIMA method assumes that a series can be reduced to a stationary time series by differencing or de-trending. ARIMA is limited by the requirements of stationarity of the time series and normality and independence of the residuals. In other words, the statistical charac teristics of a stationary time series remain constant through time, and residuals, which are the errors betw een the observed time series and the model generated by the AR IMA method, are assumed to be caused by noise.

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42 We say that {yt} is an ARIMA process of order p, d, q (i.e., yt ~ ARIMA(p,d,q)), if the dth difference of yt is a stationary, invertib le ARMA process of order p and q. The model can be written as: t t dz B y B B ) ( ) 1 )( ( , (3.5) where B is the backward shift operator, { zt} is white noise, and ) ( and ) ( are polynomials of degree p and q respectively, with all root s of the polynomial equations 0 ) ( z and 0 ) ( z outside the unit circle. Autoregressive—Autoregressive Moving Average Models The autoregressive—autoregressive moving average (ARARMA) model for nonstationary time series was proposed by Parz en (1982) [80]. The model consists of a nonstationary AR part followed by a statio nary ARMA model. The Box and Jenkins ARIMA approach is a special case of the ARARMA process in which the transformation is constrained to be pure differencing operato rs. The model can be described as follows: r t r t t ty y y y ... ~ 1 1, (3.6) p j q k k t k j t jz y10) ~ ~ ( , (3.7) where zt is white noise. An ARARMA model is considered better for step forecasting [79]. Frequency Domain Analysis The Fourier Transform Time signals can be represented in many different ways depending on the interest in visualizing certain given characteristics. Amo ng these, the frequency representation is the most powerful and standard one. The main adva ntage over the time representation is that it

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43 allows a clear visualization of the periodic ities of the signal, in many cases helping to understand underlying physical phenomena. Th e Fourier transform used in frequency domain analysis, developed by Jean Baptis te Fourier (1768-1830), has found innumerable applications in mathematics, physics and natural sciences. Furthermore, the Fourier transform is computationally very attrac tive since it can be calculated by using an extremely efficient algorithm called th e Fast Fourier Tran sform (FFT) [81]. The Fourier transform describes a signal x ( t ) as a linear superposition of sines and cosines characterized by their frequency f . df e f X t xft i 2) ( ) (, (3.8) where dt e t x f Xft i2) ( ) (, (3.9) are complex valued coefficients that give the relative contributions of each frequency f . (3.9) represents the continuous Fourier transform of the signal x ( t ). It can be considered as an inner product of the signal x ( t ) with the complex sinus oidal parent function e-2 ft, i.e. fte t x f X2), ( ) (. (3.10) The inverse Fourier transf orm is given by (3.8). Let us now consider the signal as consisting of N discrete values, sampled at each time step . j Nx x x x n x 1 1 0,..., , ) (, (3.11) where xj is the sample taken at a time tj = t0+j . The discrete Fourier transform of this signal is defined as:

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44 1 0 / 2) ( ) (N n N kn ie n x k X, (3.12) and its inverse as: 1 0 / 2) ( 1 ) (N k N kn ie k X N n x, (3.13) where, the discrete frequencies are defined as: N k fk. (3.14) Note that the discrete Fourier Transform gives N=2 independent complex coefficients, thus giving a total of N values as in the original signa l and therefore being non-redundant. Clearly the frequency resolution will be N fk1 . (3.15) The infinite basis functions used in Four ier analysis are suitable for extracting frequency information from periodic, non-tran sient signals. Fourier transform, however, cannot capture the transient features in a gi ven signal and the time–frequency information is not readily seen in the transformed Four ier coefficients. The frequency spectrum of a signal as a result of the Four ier transform is not localized in time. This implies that Fourier coefficients of a si gnal are determined by the enti re signal. Consequently, if additional data are added over time, Fourier tr ansform coefficients will change. Any local behavior of a signal cannot be easily traced from its Fourier transformation. The Fourier Transform is by far the most used quantitative method for the analysis of EEG signals. Moreover it has gained popu larity as a diagnostic tool. The Fourier transform allows the separation and study of di fferent EEG rhythms, a task difficult to perform visually when several rhythms occur simultaneously. Some of the main

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45 applications of the Fourier transform in EEG analysis include topographical mapping, evoked response studies and coherence. We present an example of time frequency analysis on iEEG data recorded in the animal model of epilepsy. To get a better understanding of the temporal resolution along with the freque ncy resolution of the iEEG signal during the peri-ictal pe riod in the animal model of MTLE, a spectrogram (Figure 3-1) method was used. The spectrogram is th e squared magnitude of the windowed shorttime Fourier transform. Figure 3-1. Power spectrum (Spectrogram) of iEEG recorded from the hippocampus of an animal model of MTLE. Vertical da shed lines represent seizure onset and offset In the time-frequency plot, the seizure onset is correlated with a sudden increase in power in the 0-10 Hz range. About 12 seconds into the seizure, hi gh amplitude rhythmic activity starts and this is correlated with an increase in spectral power at higher frequencies up to 25 Hz. This activity progre ssively slows down to about 1 Hz towards the end of the seizure. The postictal state is characterized by sp ikes and slow wave activity and this is correlated with localized power distribu tion in the 0-7 Hz range. This frequency evolution is very similar to the pattern seen in depth EEG recordings from human patients

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46 The Wavelet Transform The broad concept of wavelets was introduced by Grossman and Morlet [82] in the mid 80s. The wavelet transform is an effective tool in signal pr ocessing due to its attractive properties such as time–frequency localization (obtaining a signal at particular time and frequency, or extracting features at various locations in space at different scales) and multirate filtering (differentiating the signals havi ng various frequencies) [83-85]. Using these properties one can extract the desired features from an input signal characterized by certain local properties in time and space. Wavelet tr ansforms have been used to locally and globally match sequences, and to extract features that describe properties of the sequence in various locations and varying ti me regularities. Features are extracted from the time series based on the discrete wavelet transform, wh ile local and global similar sequences are identified based on th ese feature vectors. If x ( t ) is a square integrable function of time t , then the continuous wavelet transform can be defined as: dt a b t a t x Wb a* 1 ) (, . (3.16) where 0 , , a R b a, R is the set of real numbers, the symbol ‘*’ denotes the complex conjugation and the wavelet function is defined as: a b t a tb a 1 ) (,. (3.17) The factor a / 1is used to normalize the energy so th at it stays at the same level for different values of a and b . The wavelet function a,b( t ) becomes narrower when a is increased and displaced in time when b is varied. Therefore, a is called the scaling

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47 parameter which captures the local frequency content and b is called the translation parameter which localizes the wa velet basis function at time t = b and its neighborhood. The wavelet transform is particularly eff ective for representing various aspects of signals such as trends, discontinuities, and repeated patterns where other signal processing approaches fail to capture informati on or are not as effective. It is especially powerful for non-stationary signal analysis. El ectrophysiological signals are comprised of non-stationary transient events. The Fourier transform, de scribed earlier, cannot capture the transient features in a given signal and the time–frequency information is not readily seen in the transformed Fourier coefficients . In contrast, wavelet transform is a more suitable and powerful tool for analyzing transi ent signals because bot h frequency (scales) and the time information can be obtained. Coherence One method to quantitatively measure th e linear dependency between two distant brain regions as expressed by their electrical activity is the calculati on of coherence. As described in the earlier section, the Fourier tr ansform is used to render data as a function of frequency and estimate the signal’s fr equency spectrum, also called the power spectrum. The multiplication of the Fourier transform of one signal with the complex conjugate of another signal yields the cro ss power spectrum. The coherence function is obtained by the normalization of the cross power spectrum and is basically represented as magnitude-squared coherence. The coherence C at a frequency f for two signals x and y is given by: ) ( * ) ( ) ( ) (2 2f S f S f S f Cyy xx xy xy , (3.18)

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48 where Sxy( f ) is the cross power spectral density and Sxx( f ) and Syy( f ) are the corresponding auto spectral densities. In practice, averaging techniques need to be applied to improve power spectra estimation [86]. Coherence values lie within a range of 0 to 1; 0 indicating no correlation between frequency component of the two signals and 1 implying a perfect linear relationship with constant phase shif ts between the frequency components. Figure 3-2. Coherence measure at diffe rent frequencies over time between the hippocampus and frontal cortex of an animal model of MTLE. Note the increase in coherence in the 10~ 50 Hz range during the ictal period High coherence between EEG or iEEG signals recorded at diffe rent brain regions hint at an increased functi onal interplay between underlying neuronal networks. Since the phase between frequency components of two signals may change over time, coherence may also be interpreted as a measure of the stability of pha se between the same frequency components of two simultaneously recorded EEG signals. Figure 3-2 shows the variation of coherence between the hippoc ampus and the frontal cortex during the peri-ictal period. The power spectral density of each channe l was estimated using Welch’s averaged periodogram method. The time series was divi ded into non-overlappi ng sections, each of which was de-trended and windowed by 1 sec (200 data points). The coherence between two channels was then calculated for each 5 second non-overlapping window. Detailed

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49 reviews on EEG coherence analysis can be f ound in Shaw [87], Cha llis and Kitney [88], Rappelsberger[86], Leon cani and Comi [89]. Nonlinear Information Theory Based Analysis Entropy Information theoretical measures [90, 91] like Shannon and Kolmogorov entropies are widely used to analyze nonlinear systems. In particular, they are used to characterize the degree of randomness of time sequences, and to quantify the di fference between two probability distributions. The concept of entropy in the context of information theory was first introduced by Shannon [92]. Shannon entropy, HS quantifies the probability density function (PDF) of a signal and is calculated as: i i i Sp p H log, (3.19) where i varies across all amplitude values of the signal and pi is the probability that the signal takes on amplitude value ai at any time. The base of the logarithm determines the units in which information is measured. In particular, taking the base two leads to information measured in bits. In the case of measured data, the PD F is not known and should be estimated. Moreover, it is not reasonable to take into account all amplitude values ai. The easiest way to estimate the PDF in the case of experimental data is to use the histogram method where the amplitude range of the signal is linearly divided into k bins so that the ration k / N is constant ( N is the number of signal samples). The ratio k / N characterize the average filling of the histogram. To obtain normalized values the entropy HS needs to be divided by the logarithm of k .

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50 k H HS SNORMlog . (3.20) Approximate Entropy Approximate entropy (ApEn) [93,94], a st atistic for quantifying system regularity or complexity of a time series has been widely applied in medical data. It can be used to differentiate between normal and abnormal da ta in instances where moment statistics (e.g. mean and variance) approaches fail to show a significant diffe rence. Applications include heart rate analysis in the human neonate [95,96] a nd in epileptic activity in electrocardiograms [97]. Mathematically, as part of a general theoretical framework, ApEn has been shown to be the rate of a pproximating a Markov chain process [98]. Most importantly, compared with Kolmogrov-Sinai (K-S) entropy [99], ApEn is generally finite and has been shown to classify the complexity of a system by as few as a 1000 data points via theoretical analyses of both stochastic and dete rministic chaotic processes [93,100] and clinical ap plications [101,102]. The calculation of ApEn of a signal s of finite length N is performed as follows. First fix a positive integer m and a positive real number rf. Next, from the signal s the N m +1 vectors xm( i ) = { s ( i ), s ( i +1),, s ( i + m -1)} are formed. For each i , 1 i N – m + 1, the quantity Ci m( rf) is calculated as: 1 ) ( ), ( that such of number N-m r j x i x d j Cf m m m i, (3.21) where the distance d between he vectors xm( i ) and xm( j ) is defined as: ) 1 ( ) 1 ( max ) ( ), (,..., 2 , 1 k j s k i s j x i x dm k m m. (3.22) Next the quantity m( rf) is calculated as:

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51 1 1) ( log 1 1 ) (m N i f m i f mr C m N r . (3.23) Finally the ApEn is defined as: ) ( ) ( ) , , ( ApEn1 f m f m fr r N r m . (3.24) The parameter rf corresponds to an a priori fixed distance between neighboring trajectory points and frequently, rf is chosen accordi ng to the signal’s standard deviation (SD). Hence, rf can be viewed as a filtering level and the parameter m is the embedding dimension determining the dimension of the phase space. The ApEn was used by Bruhn et al. [103,104] to analyze EEG signals in patients under general anesthesia. According to SteynRoss et al [105] the approximate entropy when applied to EEG signals reflects the intra-cortical information flow in the brain. In this dissertation we have computed the ApEn to investigate differences in outcomes of seizure warning based brain stimulations. Mutual Information The statistical dependence be tween two signals is often estimated by their mutual information (MI). This entity has close ties to Shannon entropy described earlier and in contrast to the linear cross-co rrelation, it is also sensitiv e to dependencies which are not manifested in cross-correlation. MI takes on the value 0 when the two signals are strictly independent. The MI between two discrete random variables X and Y with marginal probabilities pX( x ) = prob( X = x ) and pY( y ) = prob( Y = y ) and with joint probability p ( x,y ) = prob( X = x , Y = y ) is defined as: y x Y Xy p x p y x p y x p Y X I,) ( ) ( ) , ( log ) , ( ) , (. (3.25)

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52 For continuous variables MI can be expressed as follows: YXdxdy y f x f y x f y x f Y X I ) ( ) ( ) , ( log ) , ( ) , (. (3.26) where f ( x , y ) is the joint probability density function of X and Y , f ( x ) is the probability density function of X , and f ( y ) is the probability density function of Y . The most straightforward and widespread approach for estimating MI consists in partitioning the supports of X and Y into bins of finite size, and approximating (3.26) by the finite sum y x Y Xy p x p y x p y x p Y X I Y X I, bin) ( ) ( ) , ( log ) , ( ) , ( ) , (. (3.27) An estimator of Ibin( X , Y ) is obtained by simply counting the number of points falling into the various bins. If nx ( ny) is the number of poi nts falling into the x -th bin of X ( y -th bin of Y ), and nxy is the number of points in their intersection, then we approximate pX( x ) nx/N, pY( y ) ny/ N , and p (x, y ) nxy/ N . The bin sizes used in (3.27) do not need to be the same for all bins. Kernel techniques [106] and adaptive estimator techniques [107] are much better then estimators using fixed bin sizes. Reliable estimations of the MI often require a large amount of data, a constraint that is sometimes in conflict with the requisite of stationarity in the case of experimental data. MI has been used to study the informati on transmission between different cortical areas in Alzheimer’s disease patients [108]. The measure has also been used to study inter-hemispheric synchronizati on in animal studies but have shown poor results when compared to other measures of synchroniza tion such as phase-synchronization, crosscorrelation and the coherence f unction [109]. Despite the diffi culties inherent to their calculation, MI is a useful tool for the assessment of interdependence between

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53 experimental signals. In the present diss ertation we have used this measure of interdependence to investigate regional c oupling in the brain during automated seizure warning based interventions in an animal models of MTLE [40]. Nonlinear Chaos Theory Based Analysis One of the surprising and far-reaching mathematical discoveries of the past few decades has been the solutions of determin istic nonlinear dynamical systems may be as random (from a statistical point of view) as the sequence of outcomes in a fair coin toss [110]. This behavior is termed as deterministic chaos . Chaotic systems have an apparently noisy behavior but are in f act ruled by deterministic laws. They are characterized by their sensitivity towards init ial conditions. That means, similar initial conditions give completely different outcomes after some time. Since chaotic signals look like noise and furthermore, si nce they also have a broadband frequency spectrum, linear approaches as the ones describe d in the previous sections are sometimes not suitable for their study. Random signals generated by noise are fundamentally different from random signals generated by deterministic dynamics . The differences are revealed by dynamical analysis based on phase space reconstruction. The following sections will be organized as follows. A brief explanation of common terms and concepts used in nonlinea r dynamical systems theory is first given followed by a description of several me thods developed to calculate system characteristics such as the degree of determin ism, complexity and chaoticity. Finally, we report the application of several of these m easures on iEEG recorded from an animal model of MTLE.

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54 Glossary of Terms Phase space. Phase space is the collection of po ssible states of a dynamical system. In general, the phase space is identified w ith a topological manifold. An n-dimensional phase space is spanned by a set of n-dimens ional “embedding vectors”, each one defining a point in the phase space, thus representi ng the instantaneous state of the system. Trajectory. Trajectory or orbit of the dynamical system is the path connecting points in chronological order in phase space traced out by a solution of an initial value problem. If the state variables take real values in a conti nuum, the orbit of a continuoustime system is a curve, while the orbit of a discrete-time system is a sequence of points. Attractor. An attractor is defined as a pha se space point or a set of points representing the various possible steady-state conditions of a system; an equilibrium state or a group of states to which a dynamical sy stem converges and cannot be decomposed into two or more attractors with distinct basins of attraction . This is necessary since a dynamical system may have multiple attractors, each with its own ba sin of attraction. Informally, an attractor is a region of a dynami cal system's state spa ce that the system can enter but not leave, and which contains no sm aller such region. Thus in the long term, a dissipative dynamical system ma y settle into an attractor. Fixed point. A point to which differe nt trajectories in phase space tend to at rest is called a fixed point. Limit cycle. A limit-cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals in to it either as time approaches infinity. Strange attractor. A strange attractor is defined as an attractor that shows sensitivity to initial conditions (exponentia l divergence of neighboring trajectories) and that, therefore, occurs only in the chaotic dom ain. While all chaotic attractors are strange,

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55 not all strange attractors ar e chaotic [111]. In other word s, the chaoticity condition is necessary, but not sufficient, for th e strangeness condition of attractors. Bifurcation. A bifurcation is defined as a qu alitative change in dynamics upon a small variation in the parameters of a system . A gradually variation of a parameter in the system corresponds to the gradual variation of the solutions to the problem. Bifurcation theory is a method for studying the onset of chaos and how solutions of a nonlinear problem and their stability change as the parameters of the system vary. Degrees of freedom. In the context of this dissertation, the degree of freedom refers to the dimension of the phase space. Fractal Dimension One of the first steps in characterizing the properties of a system is to estimate the dimension of the attractor. The dimension he lps to determine the position of a point on the attractor to within a cert ain degree of accuracy. In addi tion, it gives a lower bound on the number of variables necessary to model the system. The term "fractal" was first introduced by Mandelbrot in 1983. Roughly speaking, a fractal is a set of points that when looked at smaller scales, resembles the whole set. The concept of fractal dimension (FD) that refers to a non-inte ger or fractional dimension orig inates from fractal geometry. Strange attractors often have a structure that is not simple; they are often not manifolds and actually have a highly fractured character. The dimension that is most useful takes on values that are typically not integers. Thes e non-integer dimensions are called fractal dimensions. For any attractor, the dimension can be estimated by looking at the way in which the number of points within a sphere of radius r scales as the radius shrinks to zero. The geometric relevance of this observati on is that the volume occupied by a sphere of radius r in the dimension d behaves as rd.

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56 For regular attractors, irrespective to the origin of the sphere, the dimension would be the dimension of the attractor. But fo r a chaotic attractor, the dimension varies depending on the point at which the estimation is performed. If the dimension is invariant under the dynamics of the process, we will have to average the point densities of the attractor around it. For the purpose of identifyi ng the dimension in this fashion, we find the number of points y ( k ) within a sphere around some phase space location x . This is defined by: N kx k y r N r x n1) ( 1 ) , (, (3.28) where is the Heaviside function. This counts all the points on the orbit y ( k ) within a radius r from the point x and normalizes this quantity by th e total number of points N in the data. Also , we know that the point density, ( x ), on an attractor does not need to be uniform (for a strange attractor) on the figure of the attractor. Choosing the function as n ( x ; r )q -1 and defining the function C ( q ; r ) of two variables q and r by the mean of n ( x ; r )q -1 over the attractor weighted with the natural density ( x ) yield: 1 1, 1 1) ( ) ( 1 1 ) , ( ) ( ) , ( q M k M k n n q xk y n y r K M r x n x x d r q C. (3.29) The quantity C ( q , r ) is called the correlation functi on or correlation integral on the attractor. This function measures the probabilit y that two vectors on th e attractor, selected at random, lie within a distance r of each other. M and K are some large values but not infinite. This function of two variables is an invariant on the attractor, but it has become conventional to look only at the variation of this quantity when r is small. In that limit, it is assumed that

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57 qD qr r q C) 1 () , ( , (3.30) defining the generalized fractal dimension Dq when it exists. From the above equation, Dq can be estimated in the limiting case as: r q r q C Dr qlog ) 1 ( ) , ( log lim0 . (3.31) In practice, we need to compute C ( q,r ) for a range of small r over which we can argue that the function log[ C ( q , r )] is linear in log[ r ] and then select the linear-like slope over the range. Box counting dimension The box counting dimension ( D0) is estimated as the numbe r of spheres of radius r or the number of boxes required to cover all the points in the data set. We first define the number N ( r ) as a function of r as r tends to zero, then: r q r N Drlog ) 1 ( ) ( log lim0 0 . (3.32) This can also be defined as: q qD D0 0lim. (3.33) Information dimension The information dimension ( D1) is a generalization of the capacity that takes into account the relative probab ility of cubes used to cover the dataset. D1 can be defined as: q qD D1 1lim. (3.34) Correlation dimension When q takes on the value 2, the definition of the fractal dimension, Dq, assumes a simple form that lends it to reliable computation. The resulting dimension, D2, is called

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58 the correlation dimension of the attractor [112-114] and is es timated as the slope of the log-log plot given by: r r C Drlog ) , 2 ( log lim0 2 . (3.35) The correlation dimension is easy to quantify from experimental data but very hard to quantify from time series data because of the non-existence of a linear-like slope range. D2 has proved to be very useful for characterizing the brain dynamics in different sleep states. It was found that D2 decreases in deep sleep stages, thus reflecting a synchronization of the EEG [115-118]. A decrease in D2 has been related with abnormal synchronizations of the EEG, as demonstrated in epilepsy [119-123] and other pathologies such as Alzheimer’s, dementia, Parkinson, depression, etc. State Space Reconstruction—Method of Delays A well-established technique for visua lizing the dynamical behavior of a multidimensional (multivariable) system is to generate a state space portrait of the system. A state space portrait is created by tr eating each time-dependent variable of the system as a component of a vector in a mu ltidimensional space. Each vector in the state space represents an instantaneous state of the system. These time-dependent vectors are plotted sequentially in the state space to repres ent the evolution of th e state of the system over time. For many systems, this graphical ma pping creates an object confined over time to a sub-region of the phase space. The geomet rical properties of these attractors provide information about the global state of the system . The vector reconstruction is achieved as follows: ) ) 1 ( ( ),..., ( ), ( n t x t x t x X . (3.36)

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59 where is a fixed time increment and n is the embedding dimension. Every instantaneous state of the system is theref ore represented by the vector X which defines a point in the phase-space. Figure 3-3. Ictal segment of filtered iEEG recorded from the hippocampus of an animal model of MTLE(top), and the recons tructed iEEG segment in 3-D space (bottom) The EEG or iEEG, being the output of a multidimensional system, has both spatial and temporal statistical pr operties. Components of the brain (neurons) are densely interconnected and there exists an inherent re lation between iEEG recorded from one site and the activity at other sites. This makes the iEEG a multivariable time series. The state space reconstruction of the iEEG signal can be done using the method of delays described by Takens [114]. An accurate representation of the system in state space depends upon making appropriate choices of the embedding dimension D and time delay . The choice of delay for experimental data such as ours is not a straightforward problem. Past studies have made use of the autocorrelation functi on or the average mutual information of a given time series to calculate . The first zero or the time c onstant of the autocorrelation

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60 function [124] or the first minimum of the average auto mutual information function [125] have been proposed. According to Takens, the embedding dimension D should be at least equal to (2 d + 1) in order to correctly embed of an attractor in the phase space, where d represents the fractal dimension. The measure most often used to estimate d is the phase space correlation dimension. Methods for calculating the co rrelation dimension from experimental data have been described earlier, and were employed in our work to approximate d of the epileptic attractor. Lyapunov Exponents Lyapunov exponents play a fundamental role in the characterization of dynamical systems. Lyapunov exponents provide a quantita tive indication of the level of chaos of a system. They measure the average exponentia l separation of initially nearby phase space trajectories with time. Lya punov exponents allow for the gene ralization of the linear stability analysis from pertur bations of steady state soluti ons to perturbations of timedependent solutions, and also provide a mean ingful way to characterize the asymptotic behavior associated w ith nonlinear dynamics. The Lyapunov exponents of a system are a set of invariant geometric measures which describe the dynamical content of the system. In particular, they serve as a measure of how easy it is to predict the fu ture state of the system. Lyapunov exponents quantify the rate of divergen ce or convergence of two nearby initial points of a dynamical system, in a global sense. A positive Lyapunov exponent measures the average exponential divergence of two nearby traj ectories, whereas a negative Lyapunov exponent measures exponential convergence of two nearby trajec tories. A zero Lyapunov exponent indicates the temporal continuous na ture of a flow. If a discrete nonlinear system is dissipative in nature, a positive Lyapunov exponent quantifies a measure of

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61 chaos. Consequently a system with positiv e exponents has positive entropy, in that trajectories that are initially close together move apart over time. The more positive the Lyapunov exponents are, the faster they move apart. Similarly, for negative exponents, the trajectories move together in time. A system with both a positive and negative Lyapunov exponents is said to be chaotic. In other words, Lyapunov exponents quantify the amount of linear stability or instability of an attractor or an asymptotically long orbit of a dynamical system. Given two initial conditi ons for a chaotic system, a and b , which are close together, the average values obtained in successive iterations for a and b will differ by an exponentially increasing amount. In other word s, the two sets of numbers drift apart exponentially. If this is written en for n iterations, then e is the factor by which the distance between closely related points beco mes stretched or contracted in a single iteration and is the Lyapunov exponent. In other words, at each point in the sequence, the derivative of the iterated equation is evaluated. The Lyapunov exponent is the average value of the log of the derivative. If the value is negative, the iteration is stable. Note that summing the logs corresponds to multiplying the derivatives; if the product of the derivatives has magnitude less than 1, points will attract together as they go through the iteration. In a chaotic system, at leas t one Lyapunov exponent must be positive. For n -dimensional systems, there are n Lyapunov exponents in the state space of the system, but the maximum exponent is usually the most important. The maximum Lyapunov exponent is the time constant, , in the expression for the distance between two nearby orbits, e t. If is negative, then the orbits converge in time, and the dynamical system is insensitive to initial conditions. However, if is positive, then the distance

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62 between nearby orbits grows exponentially in time, and the system exhibits sensitive dependence on initial conditions. Calculation of the Lyapunov spectrum can be derived analytically where the equations of process are known [126,127]. There are several published algorithms for performing this measurement on experimental da ta. The oldest and most tested algorithm is that by Wolf [128]; then there are estimat es based on the generation of local Jacobian metrics from Eckmann et al. [ 129], and Ellner et al. [130]. Wolf et al. [128] proposed an algor ithm for calculating the largest Lyapunov exponent. First, the phase space reconstruction is made (3.36) and the nearest neighbor is searched for one of the first embedding vect ors. A restriction must be made when searching for the neighbor: it must be suffici ently separated in time in order not to compute as nearest neighbors successive vectors of the same trajectory. Without considering this correction, Lyapunov expone nts could be spurious due to temporal correlation of the neighbors. Once th e neighbor and the initial distance ( l ) is determined, the system is evolved forward some fixed time (evolution time) and the new distance ( l ) is calculated. This evolution is repeated, calculating the successive distances, until the separation is greater than a ce rtain threshold. Then a new vector (replacement vector) is searched as close as possible to the first one, having approximately the same orientation of the first neighbor. Finally, Lyapunov exponent s can be estimated using the following formula: k i i i kt l t l t t L1 0 1) 1 ( ) ( ln ) ( 1, (3.37) where k is the number of time propagation steps.

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63 The Wolf algorithm is said to be sensitive to the number of observations as well as to the degree of measurement or system noise in the observati ons. This discovery motivated a search for new algorithm designs with improve d finite-sample properties. Abarbanel et al. [131], Ellner et al. [130], Iasemidis [132 ], Iasemidis and Sackellares [133], and McCaffrey et al. [ 134] came up with improved al gorithms for calculating the Lyapunov exponents from observed data. iEEG Dynamics in a Model of MTLE Preictal and ictal transitions have been detected in the spatiotemporal characteristics of the EEG signal in human MTLE using measures derived from nonlinear dynamical systems theory [133,135,136]. These re sults have prompted us to apply similar quantitative methods to a rat model of MT LE, as this model has several important features of the human syndrome [34,35]. The st udy mainly tests the hypothesis that ictal, preictal and postictal dynamical changes, sim ilar to those observed in humans, exist in the animal model. Data Description and Selection Test data sets include twenty-eight, 2-hour data sets from four animal subjects (mean seizure duration 78 sec) are analyzed, each containing a seizure and intracranial data (iEEG) beginning 1 hour before the seizure onset and ending approximately one hour after the seizure o ffset. We have also included a randomly chosen a seizure-free 24-hour iEEG epoch including circadian variations from each rat to investigate the specificity of findings during the period before a se izure and to compare spatial and temporal dynamical values between the interictal state and the other states (ictal, preictal and postictal) . Each 24-hour epoch was furthe r subdivided into consecutive 2-hour blocks, thus giving a total of 12 blocks from each animal (total of 48 blocks from

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64 four animals). All seizures that occurred within two we eks after the st art of data acquisition in each rat were included in the st udy. All segments were deemed sufficiently artifact-free by review of an electroencephalographer. Table 3-1. Summary of iEEG data used for dynamical analysis Seizure Duration (seconds) Inter-Seizure Interval (hours) Rat ID Number of Seizures Analyzed Mean SD Range Mean SD A 5 52.8 3 20~218 71.5 97.7 B 8 96.6 20.8 2.5~99 42.2 41 C 21 80 8.6 1.5~164 16.8 40 D 8 65 11 12.5~100 43.6 34.2 Overall 42 77.1 12.03 1.5~218 33.25 48.7 Nonlinear Energy Operator—Teager Energy The use of this measure is based in part on the reported utility of signal power (signal energy) to predict seizures and detect physiological sleep states in patients with TLE [137,138]. A nonlinear energy operator using Teager’s al gorithm has been used in seizure detection [139], studyi ng seizure propagation post onset [138], and measurement of spectral content of EEG si gnals [140]. This algorithm was presented by Kaiser who was searching for a measure of energy pr oportional to both signal amplitude and frequency [136]. This algorithm is also rec ognized as “nonlinear energy” because it has also shown to be useful for determining the instantaneous frequency and amplitude envelope of AM-FM signals [141]. For the input signal x ( n ), in its discrete form, the nonlinear energy operator is represented by:

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65 ) 1 ( ) 1 ( ) ( ) (2 n x n x n x n ET. (3.38) The nonlinear energy is an instantaneous feat ure, such that it provides one value for each value of original data. The algorithm is sensitive to both amplitude and frequency changes, and is computationally efficient, and simple to calculate. After calculating the instantaneous energy at each sample in the ti me series, a running estimate of the energy sample mean was calculated every 5 second (1000 data points) nonoverlapping epoch. Figure 3-4. iEEG signal energy. (a) The Teager energy profile of a two hour iEEG segment for four electrodes. (b) Enlarged view in the vicinity of the seizure; dotted lines represent beginning a nd end of seizure. (c) Mean ET value comparison between interictal (baseline) and ictal states

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66 The energy estimates of a two hour epoch containing a grade 5 seizure, from 4 channels calculated using Teager’s algorithm are shown in Figure 3-4a. The mean value of signal energy during the seizure is signifi cantly higher when compared to the baseline energy values computed during other periods. This high level is maintained throughout the ictal period, with a pattern related to th e different periods across the seizure. Closer inspection of the energy profiles (Figure 3-4b ) revealed that this sudden increase in energy occurred first in the hippocampal elect rodes and subsequently in the frontal electrodes. This finding was c onsistent across all seizures analyzed. Figure 3-4c shows mean energy comparison between interictal (bas eline) and ictal epochs averaged over all channels. As is evident from the figure, the mean value of ET during the ictal period is significantly larger than the interictal values. Table 3-2. Summary of Teag er Energy based seizure detection results for four rats Rat ID Mean Detection Delay (seconds) A 13.2 1.3 B 23.2 5 C 21.9 3.2 D 10.2 2.7 Overall 18.9 3.4 The energy sample means were used in th e following manner to detect seizures: (i) the sample mean for a 10 mi nute window from the start of the data set was used as the baseline interictal value, and one such es timate of the baseline value was obtained for each channel of a data set, (ii) for each ch annel, the baseline sample mean was compared to the running estimate of the sample mean and (iii) a seizure was detected when the

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67 statistical test for the equa lity of the baseline and the running sample mean failed concurrently in two or more channels. The detection time is compared with the previously recorded electrographic seizure onset time to compute the detection delay (Table 3-2). Test for Nonlinearity Previous studies have shown the existe nce of significant nonlinearities in iEEG recorded from patients with focal epile psy [142,143]. Casdagli [142] reported the existence of significant nonlinearities in huma n epileptic EEG during preictal, ictal and postictal states by comparing the correlation in tegrals estimated from the original EEG signal to that of surrogate datasets in whic h (nonlinear) phase relati ons were destroyed. Other studies have reported state dependent (wakeful rest, locomotion and seizure) changes in nonlineari ties within the EEG signal [121,144]. Theile r and Rapp [145] and Palus [146] have used parallel techniques to test for nonlinearities in scalp EEG from normal human volunteers using dynamical meas ures such as entropy and dimension. Two hour epochs of iEEG, one from each of the 4 rats were analyzed in order to test for signal nonlinearities. The iEEG se gment was first divided into non-overlapping segments of 10.24 sec duration. To demonstrat e the presence of nonlinearity, the original signal was compared to surrogate datasets ge nerated from the original signal. We used the iterative amplitude adjusted Fourier tr ansform method (TISEAN software) proposed by Schreiber and Schmitz [147,148] to generate the surrogate datasets. This method iteratively corrects deviations from the ta rget spectrum and distribution set by the measured data. The desired power spectrum is achieved by taking the Fourier transform of the original signal, replacing the squared amplitudes and then computing the inverse Fourier transform, keeping the phases unchange d. A rank-ordering is then performed to

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68 achieve the target distribution. The two step s are alternatively performed in an iterative mode. A total of 10 surrogate datasets we re obtained by repeating the procedure. The iEEG signal was embedded in phase space using the method of delays, described earlier, with an embedding dimensi on = 7 and time delay =3 (equivalent to 15 ms) to get a phase space portrait of the signa l. The correlation integral was calculated using the formulae given earlier. A Theiler correction [149] of w=50 (equivalent to 250ms) was chosen to avoid autocorrelation effe cts on the computation of the correlation integral. The correlation integral profile fr om each of the 4 electrodes was generated and compared to the ones generated from the corresponding surrogate datasets. Statistical significance was defined as: / ) ) ( (log10x r C S , (3.39) where x is the mean and is the standard deviation of the logarithm of the correlation integrals of the 10 surrogate datasets, and was estimated for each 10.24 second segment A difference was considered statistically significant if S was greater than 5. Figure 3-5. Correlation integral and statistical significance of nonlinearity (a) The values of correlation integral of the recording from the stimulated left hippocampal electrode ( solid red line) and values of 10 surrogate datasets. (b) The statistical significance S of nonlinearity in the iEEG. The white horizontal line represents 5 SD. Statistica lly significant nonlinearities are present in 65% of the total segments (10.24 sec in duration) in the iEEG epoch shown. Vertical dashed lines represent the seizure onset and offset

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69 Figure 3-5a shows the correlation integral estimates calculated from the iEEG signal obtained from the hippocampus and 10 surr ogate datasets created from the original time series, plotted as a function of time. Vi sual inspection reveals that the correlation integrals for the original iEEG signal are uniformly higher than the correlation integrals estimated from any of the 10 surrogates. Fi gure 3-5b shows the statistical significance S of nonlinearity in the same iEEG epoch. 65% of the total segm ents (10.24 sec in duration) in the iEEG epoch showed statistically significant nonlinearity ( S >5). Significant nonlinearities were id entified in all iEEG segmen ts included in the study. Estimation of Short-Term Maximum Lyapunov Exponent The method we developed for estimati on of Short Term Maximum Lyapunov Exponents ( STLmax), an estimate of Lmax for nonstationary data, is explained in detail elsewhere [128,133,135]. Herein we will presen t only a short descript ion of our method. Construction of the embedding pha se space from a data segment x ( t ) of duration T is made by the method of delays described above. An attractor is chaotic if, on the average, orbits originating from similar initial conditions (nearby points in the phase spa ce) diverge exponentia lly fast (expansion process). If these orbits belong to an attractor of finite size, they w ill fold back into it as time evolves (folding process) . The result of these two processes may be a stable topologically layered attractor. When the expansion process overcomes the folding process in some eigen-directions of the attrac tor, the attractor is called chaotic. For an attractor to be chaotic, at least the maxi mum Lyapunov exponent must be positive. If the phase space is of D dimensions, we can estimate theoretically up to D Lyapunov exponents. However, as expected, only ( d +1) of them will be real.

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70 If we denote by L the estimate of the shortterm largest Lyapunov exponent STLmax, then: aN i j i j i aX t X t N L1 , , 2) 0 ( ) ( log 1 , (3.40) with ) ( ) ( ) 0 (, j i j it X t X X , (3.41) ) ( ) ( ) (,t t X t t X t Xj i j i . (3.42) where i. X ( ti) is the point of the fiducial trajectory ) (0t Xt with t = ti, TD t x t x t X ) ) 1 ( ( )... ( ) (0 0 0 , T denotes the transverse, and X ( tj) is a properly chosen vector adjacent to X ( ti) in the phase space. ii. ) ( ) ( ) 0 (, j i j it X t X X is the displacement vector at ti i.e., a perturbation of the fiducial orbit at ti, and ) ( ) ( ) (,t t X t t X t Xj i j i is the evolution of this perturbation after time t . iii. t i t ti ) 1 (0 and t j t tj ) 1 (0, where aN i , 1 and N j , 1 for j i. iv. t is the evolution time for Xi,j, i.e., the time one allows Xi,j to evolve in the phase space. If the evolution time t is given in seconds, then L is in bits/second. v. t0 is the initial time point of the fiducial trajectory and coincides with the time point of the first data in the data se gment of analysis. In the estimation of L , for a complete scan of the attractor, t0 should move within [0, t ]. vi. Na is the number of local Lmax's that will be estimated within a duration T data segment. Therefore, if ts is the sampling period of the time domain data, then ) 1 ( ) 1 ( D t N t N Ta s. The short term maximum Lyapunov exponent ( STLmax) was estimated using the method proposed by Iasemidis et al. [132], wh ich is a modification of Wolf’s algorithm [128]. We call the measure short term to distinguish it from those used to study autonomous dynamical systems studies. Modification of Wolf's algorithm is necessary to

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71 better estimate STLmax in small data segments that incl ude transients, such as interictal spikes. The modification is primarily in the searching procedure for a replacement vector at each point of a fiducial trajectory. For exam ple, in our analysis of the EEG, we found that the crucial parameter of the STLmax estimation procedure, in order to distinguish between the pre-ictal, the ictal and the post-ictal stages, was not the evolution time t nor the angular separation Vi,j between the evolved displacement vector Xi-1,j( t ) and the candidate displacement vector Xi-1,j(0) (as it was claimed in Frank et al. [150]. The crucial parameter is the adaptive estimation in time and phase space of the magnitude bounds of the candidate displacement vector to avoid catastrophic replacements. Results from simulation data of known attractors have shown the improvement in the estimates of L achieved by using the pr oposed modifications [132]. Our rules for estimation of the Lyapunov exponents can be stated as follows: 1. For L to be a reliable estimate of STLmax , the candidate vector X ( tj) should be chosen such that the previously evolved displacement vector X( i -1), j( t) is almost parallel to the candidate displacement vector Xi , j(0), that is, max j i j i j it X X V ) ( ), 0 (), 1 ( , , , (3.43) where Vmax should be small and , denotes the absolute value of the angular separation between two vectors and in the phase space. 2. For L to be a reliable estimate of STLmax, Xi , j(0) should also be small in magnitude in order to avoid computer overf low in the future evolution within very chaotic regions and to reduce the prob ability of starting up with points on separatrices [151]. This means, max j i j it X t X X ) ( ) ( ) 0 (,, (3.44) with max assuming small values. Therefore, the parameters to be sele cted for the estimation procedure of L are: i. The embedding dimension p and the time lag for the reconstruction of the phase space

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72 ii. The evolution time t (number of iterations Na) iii. The parameters for the selection of X ( tj), that is, Vi ; j and max iv. The duration of the data segment T Note that since only vector differences are involved in the estimation of L , any direct current (DC) present in the data segment of interest does not influence the value of L . In addition, only vector difference ra tios participate in the estimation of L . This means that also L is not influenced by the scaling of the data (as long as the parameters involved in the estimation procedure, i. e. max, do not assume absolute but relative values to the scale of every analyzed data segment). Both points above make sense when one recalls that L relates to the entropy rate of the data [152]. Selection of D and . From Takens' theorem, the embedding dimension D should be at least equal to (2 d +1) in order to correctly embed an attractor in the phase space. Of the many different methods used to estimate the dimension d of an object in the phase space, each has its own practical problems [112]. The measure most often used to estimate d is the phase space correlation dimensi on. Since capturing the ictal dynamics was central to the investigation, ictal segments were used to estimate the correlation dimension. Figure 3-6 shows a plot of r d r C d log ) ( log versus log[ r ] for an ictal segment where C(r) denotes the correlation integral and r represents the attractor size. The detailed calculation of the correlation in tegral is given in an earlier section. The curves correspond to different va lues of the embedding dimension ( D = 6, 7, 8 and 9) and a time delay of 15 milliseconds. The objective is to find a “middle” region in log[ r ] where the derivative (slope) is consistent. The figure illustrates the difficulties in establishing a clean, unsullied value of the correlation dimension for experimental data.

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73 Figure 3-6. The derivative of the correla tion function calculated for an ictal segment (2000 points) created from vector spaces of dimension D =6, 7, 8 and 9, with respect to ln( r ). We see a consistency in a broad range of ln( r ) with slopes that lie in a range of values from 3 to 5 Therefore, following the values used in hu man studies, we have used an embedding dimension D of 7 for the reconstruction of the phase space. This value of D may be too small for the construction of a phase space that can embed all interictal states of the brain, but it should be adequate for detection of th e transition of the brain toward the ictal stage if the epileptic attractor is active in its sp ace prior to the occurr ence of the epileptic seizure. The parameter should be as small enough to capture the sh ortest change (i.e., highest frequency component) present in the data. Also, should be large enough to generate (with the method of delays) th e maximum possible independence between the components of the vectors in the phase space. These two conditions are usually addressed by selecting as the first minimum of the mutual information between the components of the vectors in the phase space or as the firs t zero of the time domain autocorrelation

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74 function of the data [153]. Theoretically, since the time span ( D -1) of each vector in the phase space represents the duratio n of a state of the system, ( D -1) should be at most equal to the period of the maximum (or domin ant) frequency component in the data. For example, a sine wave (or a limit cycle) has = 1, then a D = 2 x 1 + 1 = 3 is needed for the embedding and ( D -1) = 2 should be equal to the period of the sine wave. Such a value of would then correspond to the Nyquist sampling of the sine wave in the time domain. In the case of the epileptic attractor, the highest frequency present is 70 Hz (the iEEG data are low-pass filtered at 70Hz), which means that if D = 3, the maximum to be selected is about 7 ms. However, sinc e the dominant frequency of the epileptic attractor (i.e., during the ictal period) in the animal iEEG was never more than 15 Hz, according to the above reasonin g, the adequate value of for the reconstruction of the phase space of the epileptic attractor is (7 1) = 67 ms, that is, should be about 11 ms (for more details see [76]). Selection of t. The evolution time t should not be too large, otherwise the folding process within the attractor adversely influences L . On the other hand, it should not be too small, in order for Xi , j( t ) to follow the direction of the maximum rate of information change. If there is a dominant frequency component f0 in the data, t is usually chosen as t = f0/2. Then, according to the previous arguments for the selection of D and , the value of t (( D -1) )/2, which for iEEG results in t 33msec. It has been shown that such a value is w ithin the range of values for t that can very well distinguish the ictal from the preictal state [154]. Selection of Vmax. We start with an initial Vmax(initial) = 0.1 rad . In the case that a replacement vector X ( tj) is not found with 0 | Vi , j| < Vmax(initial) and | Xij(0)| < 0.1 * max

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75 , we relax the bound for | Xij(0)| and repeat the proce ss with bounds up to 0.5* max . If not successful, we relax the bounds for | Vi , j| by doubling the value of Vmax and repeat the process with bounds for Vmax up to 1 rad . Values of Vmax larger than 0.8 rad never occurred in the procedure. If they do, the replacement procedure stops, a local L ( ti) is not estimated at ti and we start the whole procedure at the next point in the fiducial trajectory. Selection of max. In Wolf's algorithm, max is selected as: ) 0 ( max, , j i i j maxX . (3.45) where, j = 1,, N and i = 1,, Na. Thus, max is the global maximum distance betw een any two vectors in the phase space of a segment of data. This works fine as long as the data are stationary and relatively uniformly distributed in the phase space . With real data this is hardly the case, especially with the brain electrical activ ity which is strongly nonstationary and nonuniform [155-157]. Therefore, a modification in the searching procedure for the appropriate X ( tj) is essential. First, an adaptive estimation of max is made at each point X ( ti), and the estimated variable is ) 0 ( max, j i j max i,X , (3.46) where, j = 1,, N . By estimating max as above, we take care of the nonuniformity of the phase space max is now a spatially local quantity of the phase space at a point X ( ti) but not of the effect of existing nonstationarities in the data. We have attempted to solve the problem of nonstationarities by estimating max also as a temporally local quantity. Then, a more appropriate definition for max is: i j Xj i IDIST t t IDIST max i,j i ; ) 0 ( max,2 1, (3.47)

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76 and 1IDIST, (3.48) ) 1 (2 D IDIST, (3.49) where IDIST1 and IDIST2 are the lower and upper bounds for | ti – tj|, that is, for the temporal window of the local search. Thus, the search for i,max is always made temporally about the state X ( ti) and its changes within a period of the time span ( D -1) of a state. Selection of X ( tj). The replacement vector X ( tj) should be spatially close to X ( ti) in phase space (with respect to ma gnitude and angle deviation), as well as temporally not very close to X ( ti) to allow selecting X ( tj) from a nearby (but not the same) trajectory (otherwise, by replacing one state with one that shares common components would lead to a false underestimation of L ). The above two arguments are implemented in the following relations: rad V Vj i j i1 . 0 ) initial ( 0, , , (3.50) max i j i max ic X b, , ,) 0 ( , (3.51) ) 1 (3 D IDIST t tj i. (3.52) The parameter c starts with a value of 0.1 and in creases, with a step of 0.1, up to 0.5, in order to find a replacement vector X ( tj) satisfying (3.50) through (3.52). The parameter b must be smaller than c and is used to account for the possible noise contamination of the data, denoting the distance below which the estimation of L may be inaccurate (we have used b = 0:05 for our data [128,133]. The temporal bound IDIST2 should not be confused with the temporal bound IDIST3, since IDIST2 is used to find the

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77 appropriate i,max at each point X ( ti) (searching over a limited time interval), whereas IDIST3 is used to find the appropriate X ( tj) within a i,max distance from X ( ti) (searching over all possible times tj). Selection of T. For data obtained from a stationary state of the system, the time duration T of the analyzed segment of data may be large for the estimate of L to converge to a final value. For nonstationary data we have two competing requirements: we want T to be as small as possible to provide local dynamic information, but the algorithm also requires a minimum length of the data segment to stabilize the estimate of STLmax. Figure 3-7. The variation of STLmax with the length T of the data segment for data in the preictal and ictal state of a human epileptic seizure Figure 3-7 shows a typical plot for the change in STLmax with the size of the window for preictal and ictal segments fr om a human subject. After extensive sensitivity studies with EEG data in human subjects with epilepsy Ia semidis et al. [132,133] concluded that the critical parameter of the algorithm was IDIST2, i.e., the parameter that establishes a neighborhood in time at each point in fiduc ial trajectory for the estimation of the

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78 parameter i,max, which then establishes a spatial ne ighborhood for this point in the phase space (Figure 3-8). It is obvi ous that, with values of IDIST2 greater than 160 msec, one is not able to distinguish between the preictal and the ictal state of a seizure based on L . Figure 3-8. The variation of STLmax with the IDIST2 parameter for data in the preictal and ictal state of a human epileptic seizure STLmax values were calculated for the iEEG segments sampled from interictal, preictal, ictal and postictal peri ods. We first tested the hypothes es that there is a reduction in chaoticity during the ictal period – reduction in STLmax values. To test the hypotheses, average STLmax values (across all four electrodes) dur ing the interictal (one-hour segment at least 5 hours away from any seizure), preictal (one-hour segment immediately preceding a seizure), ictal (segment between ons et and offset of a seizure) and postictal (one-hour segment immediately following a seiz ure) periods were calculated and their mean values were compared to check for di fferences among different states. Nested Twoway ANOVA test was applied to test the significance of the “state” effect, where rats were considered as random blocks and seizures were nested within each rat. When “state”

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79 effect was detected to be significant (p < 0.05), multiple comparisons were applied to investigate the significance of the difference between each pair of states. Results from Short Term Maximum Lyapunov Exponent Analysis By dividing the recorded iEEG data at an electrode site into sequential nonoverlapping segments, each 10.24 sec in duration and estimating STLmax for each of these segments, profiles of STLmax over time are generated. A typical plot of STLmax over time obtained from an electrode site within th e hippocampus is shown in Figure 3-9. The exponent STLmax is positive during the whole period of recording, including the preictal state (prior to the onset of the tenth recorded seizure), approximatel y 2 minutes during the ictal state (during the seizure) and during the postictal state (after the seizure ends). The seizure onset corresponds to the maximum drop in the values of STLmax, thus the seizure can be detected from the lowest values of STLmax. However, ictal STLmax is still positive, implying a chaotic state even during the seizure. This is consistent with an interpretation of the seizure being a chaotic state with fewer degrees of freedom than before. Figure 3-9. STLmax profiles derived from an iEEG signal recorded from the hippocampus of an MTLE model. Seizure started and ended between the two vertical dashed lines. The estimation of the STLmax values was made by dividing the signal into non-overlapping segments of 10.24 sec each, using D = 7 and = 15 msec for the phase space reconstruction

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80 Figure 3-10. Mean STLmax value comparison between chan nels (hippocampal channel on the stimulated side and other channels) Figure 3-11. Comparison of mean values of STLmax among states (a) Comparison of mean values of STLmax averaged across all electrodes, during the interictal, preictal, ictal and postictal states . (b) Multiple comparison among states

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81 Figure 3-12. STLmax values during two epochs of alert and rest state in an animal model of MTLE A comparison of STLmax values among the different brain regions (Figure 3-10) revealed that the hippocampal side that was subjected to chronic stimulation during the induction of status epilepticus was more ordered (lower STLmax) when compared to all other recorded brain regions, including the co ntralateral side of the hippocampus. Mean STLmax values for different states in each of the four rats are shown in Figure 3-11a. It is clear that, for all four rats, STLmax values are the lowest duri ng the ictal period, and are the highest during the postictal period. Statistical tests confirmed these observations (state effects for STLmax are significant, p < 0.01). Multiple comparisons (Figure 3-11b) further suggested that (1) mean values during interictal and preictal periods are not significantly different (P > 0.05), (2) mean values during ictal period are significantly lower (p < 0.05) from the other three periods, and (3) mean values during postictal period are significantly higher (p < 0.05) from the other three periods. STLmax profiles from multiple electrode

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82 sites during rest and alert states are shown in Figure 3-12. Clearly the values estimated from the frontal cortical regions are closer to that of the hippocampal sites during the alert state when compared to the rest state. Estimation of Phase Space Average Angular Frequency This measure, first applie d to intracranial EEG recordings by Iasemidis [158], estimates the rate of change of the stabil ity of a dynamical system. Thus, it complements the Lyapunov exponent, which measures the local stability of the syst em. The difference in phase between two evolved states X( ti) and X( ti+ t ) is defined as i [158]. The average of the local phase differences i between the vectors in the state space is given by: N i iN11, (3.53) where N is the total number of pha se differences estimated from the evolution of X( ti) to X( ti+ t ) in the state space according to: ) ) ( ) ( ) ( ) ( arccos( t t X t X t t X t Xi i i i i . (3.54) Then the phase space average angular frequency ( ) defined by: t 1, (3.55) measures how fast a local state of the system changes on average (e.g. dividing by 2 , the rate of the change of the state of the system is expressed in sec-1 (Hz)). For estimating the average angular frequency ( ; rad/sec), the state space reconstruction was done in the same manner as that for estimating STLmax.

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83 Figure 3-13. Calculation of average angular frequency and Lmax. i represents the local phase difference between two evol ved states in the state space An illustration of the estimation of STLmax and is given in Figure 3-13. ) 0 (0x and ) 0 (kxrepresent the distance between neighboring vectors at time=0 and after an evolution time of k* t , respectively. A new search for a suita ble neighboring vector is started and both vectors are then evolved over a time k* t to compute the distance. This process is repeated for the estimation of the la rgest Lyapunov exponent. The procedure for calculation of is less computati onally intensive as there are no search criteria involved in the algorithm. values were calculated for the iEEG segm ents sampled from interictal, preictal, ictal and postictal periods. We tested the hypotheses that th e local state of the system changes more rapidly during th e ictal period – increase in values. To test the hypotheses, average values (across all four electrodes) during the interictal, preictal, ictal and postictal periods were calculated and their mean values were compared to check for differences among different states. Nested Two-way ANOVA test was applied to test the significance of the “state” effect, where ra ts were considered as random blocks and

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84 seizures were nested within each rat. When “s tate” effect was detected to be significant (p < 0.05), multiple comparisons were applied to investigate the significance of the difference between each pair of states. Results from Average Angu lar Frequency Analysis By dividing the recorded iEEG data at an electrode site into sequential nonoverlapping segments, each 10.24 sec in duration and estimating for each of these segments, profiles of over time are generated. A typical plot of over time obtained from four electrode sites is shown in Figure 3-14. The curves follow a general trend opposite to that seen in the case of STLmax, with a peak during th e seizure and sharp drop during the postictal stage. Th is pattern roughly corresponds to the typical observation of higher frequencies in the power spectrum of the original iEEG signal during the ictal period (Figure 3-1). The hippocampal channels have a consistently higher value of compared to the frontal cortical sites except at the seizure when the frontal values reach the same level as those from hippo campal regions, sometimes exceeding it during certain seizures. Figure 3-14. Sample average angular frequency profiles (a) Two hours from four brain areas in the MTLE model. Seizure onse t and offset are indicated by dashed vertical lines. (b) Enlarged view in the vicinity of a seizure

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85 Figure 3-15. Mean average angular fre quency value comparison between channels (between hippocampal channel on the stimulated side and other channels) Figure 3-16. Comparison of mean averag e angular frequency between states. (a) Comparison of mean values averaged across all el ectrodes, in the interictal, preictal, ictal and postictal states. (b) Multiple comparisons among states

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86 Figure 3-17. Average angular freque ncy values during two epochs of alert and rest state in an animal model of MTLE A comparison of values among the different brain regions (Figure 3-15) revealed that the hippocampal side that was subjected to chronic stimulation during the induction of status epilepticus was more ordered (higher ) when compared to all other recorded brain regions, incl uding the contralateral side of the hippocampus. Mean values for different states in each of the four rats are shown in Figure 3-16a. It is clear that, for all four rats, values are the highest during the ictal period, and are the lowest during the postictal period. Statistical tests c onfirmed these observations (state effects for are significant, p < 0.01). Multiple compar isons (Figure 3-16b) further suggested that (1) mean values during interictal and preictal periods are not significantly different (P > 0.05), (2) mean values during ictal period ar e significantly higher (p < 0.05) from the other three periods, and (3) mean values dur ing postictal period are significantly lower (p < 0.05) from the other three periods. profiles from multiple electrode sites during rest

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87 and alert states are shown in Figure 3-17. Clearly, as observed in the case STLmax profiles, the values estimated from the frontal cortical regions are closer to that of the hippocampal sites during the alert state when compared to the rest state. Discussion Nonlinearity and temporal dynamics The transition towards a spontaneous seiz ure appears to represent a temporal evolution towards a more ordered state as demonstrated by the changes in dynamic measures short-term largest Lyapunov exponent ( STLmax) and average angular frequency ( ) during the ictal period. The presence of highly significant nonlinearities in electrographic signals su pports the concept that the epileptogenic brai n in this model of epilepsy is a nonlinear system. The sudden incr ease in the correlation integrals of the original time series and the surrogates duri ng the postictal period is attributed to the increased autocorrelation in th e iEEG during this period [142]. In all four animals analyzed, a signi ficant reduction in iEEG chaoticity was observed during th e ictal period. STLmax values correspond to th e degree of order of a dynamical system. In the case of a complex stru cture such as the br ain, it may give an insight into the order or synchrony of underlyi ng neuronal circuitry. The lowest values of STLmax were observed during the ictal period. Th is reduction in values suggests that the underlying neuronal circuitry re sponsible for generation of the iEEG signal fire in a synchronized or organized fashion (typical neuronal behavior during a seizure). Conversely, higher STLmax values could correspond to a more random neuronal firing, independent of each other. In the case of average angular frequency, the preictal, ictal and postictal states correspond to medium, high and lower values of respectively. The

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88 highest values were observed during the ictal period, and higher values were observed during the preictal period than during the postictal period. Potential Use of Nonlinear Signal Energy for Seizure Detection Seizure detection refers to the identification of abrupt, short-term changes in the waveform morphology of the signal. In human epilepsy, ET characteristically increases during seizures [138]. Results indicate that ET in the animal model is consistently elevated above baseline energy values durin g seizure periods, suggesting that this measure could be used for detecting seizur es. A successful seizure detection algorithm using ET could eventually be coupled with se izure intervention closed-loop techniques such as the delivery of a singl e timed electrical stimulus. Similarities with Human MTLE The results presented in this study reve al a number of similarities in dynamical properties between rat MTLE model and human TLE. The presence of highly significant nonlinearities in the rodent model iEEG is sim ilar to findings in intracranial recordings obtained from humans [142,143]. The sudden increase in the values of correlation integral following a seizure is observed in both the MTLE model and human TLE, hence suggesting similarities in the postictal states. We have demonstrated that the dynamical properties of the preictal, ictal and interictal states in the animal model ar e distinctly different and can be defined quantitatively. In human TLE, seizures re present a dynamical state of increased spatiotemporal order during the ictal pe riod. Qualitative comparison of dynamical patterns in the rat model and human TLE indicat es that limbic epileptic seizures behave in a similar fashion independently of species type.

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89 CHAPTER 4 EPILEPTOGENESIS AND SE IZURE PREDICTABILITY Seizures are not abrupt tran sitions in and out of an abnormal state; instead, they follow a dynamical transition that evolve s over minutes to hours [41,158-162]. During this preictal dynamical transition, multiple re gions of the cerebral cortex progressively approach a similar dynamical state, also termed as dynamical convergence . Other investigations have confirmed the presence of a progressive preictal transition [41,123,163-166] (for a review , see [167]). The transition period is detectable by a spatiotemporal analys is of the dynamics of the electr oencephalogram. The dynamics of the preictal transition are hi ghly complex and it was observed that even in the same patient, the participating cortical regions an d the duration of the transition vary from seizure to seizure [41,166]. During the seizur e (ictal state), widespread cortical regions make an abrupt transition to a more ordered state. After the seizure, brain dynamics revert to a more disordered state (postictal state), where the multiple regions of the cerebral cortex that were in a similar dynamical stat e abruptly move to different states. This phenomenon has been termed as dynamical divergence (also referred to as resetting ). The epileptic brain repeats this series of state transitions intermittently, at seemingly irregular but, in fact, time-dependent intervals [29,168] . This implies that the transition into seizures is not a random process. Epileptic seizures of mesial temporal origin are preceded by a state transition detectable in the intracranial EEG [41,123,158166]. Evidence for a prei ctal transition has been derived from several lines of investig ation. Iasemidis, Sackellares and coworkers

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90 showed that the convergence of the short-term largest Lyapunov exponents derived from EEG signals recorded from critical electrode site s can be utilized to detect a preictal state [41,158-162] on the average of 70 minutes before the seizure onset [166]. Similar observations by applying the analysis of dyna mical phase (the angu lar velocity in the state space) were al so reported by the same group [160]. Using a measure of the relative complexity of the EEG signal, Lehnertz and Elger reported a reduction in signal complexity preceding epileptic seizures by approximately 11.5 minutes [123,164]. A group from France, Le van Quyen, Martinerie and colleagues, also showed evidence for a preictal transition before a seizure, by a pplying a measure of signal similarity on 20minute EEG recordings prior to seizures [165,169,170]. Litt, Esteller and colleagues reported an increase in the incidence of si gnal energy bursts that precede seizures by several hours [137]. More recentl y, Mormann, Andrzejak and coworkers showed that the period preceding a seizure can be characterized by a decrease in synchronization between different EEG recording sites, with a mean anticipation time of 86 minutes using the mean phase coherence method and 102 mi nutes with the maximum linear cross correlation method [171]. In the present disse rtation, we evaluate a seizure warning algorithm based on the dynamical descriptor STLmax against nave prediction methods in an animal model of MTLE that has spontaneous seizures [35]. Another important direction in epilepsy research is to identify appropriate markers for the process of epileptogenesis. Epileptogene sis and epilepsy are actually distinct parts of a dynamic process, which may vary significantly depending on the type of epilepsy. Some of the hypothesized contri butors to epilepsy and epilept ogenesis include changes in the expression of ion channels and receptors , synaptic rearrangements, inflammation and

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91 changes in glia. Some of these changes are quite discretely localized regionally. However, to date few if any of these changes ha ve been demonstrated to be as critical to the process of epileptogenesis or the cond ition of epilepsy. The development of appropriate markers for epileptogenesis is co mplicated by the multiple types of epilepsy and an at best rudimentary knowledge of the pr ocesses that lead to chronic epilepsy and the initiation of spontaneous seizures. Moreover, targets for the prevention of epileptogenesis could likely ch ange in the period that precedes the onset of the first seizure as the process develops, so that a ppropriate targets at one point may not be relevant at another. One of th e other key questions in current epilepsy research is to find suitable markers that are predictive of the risk of developing epilepsy. This is particularly significant because of the possibility of designi ng treatment to interrupt this progressive condition and may be useful for the timely id entification of key points in epileptogenesis that would be amenable to intervention. This chapter is organized as follows. We fi rst describe the sta tistical methods used in the spatiotemporal dynamical analysis of iEEG recorded from an animal model of MTLE. Second, results from the application of these spatiotemporal analysis methods will be reported and compared with results from studies in human TLE patients. In the third section, we investigate seizure predicta bility in the animal model and describe an automated seizure warning algorithm that r uns online and in real -time with the data acquisition software. Finally we propos e a novel method and application of spatiotemporal analys is involving dynamical descriptors such as STLmax in finding markers for epileptogenesis by analyzing iEEG recorded duri ng the latent period in the animal model.

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92 Statistical Test for Spatiote mporal Dynamical Analysis Although a great deal is now known about low dimensional chaos, the erratic behavior of dynamical systems described by a few variables is not we ll understood about systems where the number of degrees of freed om becomes very large. Typically such systems show disorder in both space and tim e and are said to exhibit spatiotemporal chaos. Spatiotemporal chaos occurs when th e system of coupled dynamical systems gives rise to dynamical behavior that exhibits both sp atial disorder (as in rapid decay of spatial correlations) and temporal diso rder (as in nonzero Lyapunov e xponents). One of the traits of the system under considerati on (brain) is that it has a sp atial extent and, as such, information about the transition of the syst em towards the ictal state should also be included in the interactions of its spatial co mponents. The preictal transition, progressive convergence of STLmax profiles, is another evidence of spatiotemporal chaos in the brain (shown in Figure 4-1). Having estimated the temporal profiles of dynamical descriptors such as STLmax or at individual cortical sites, and as the brain proceeds towards the ictal state, the temporal evolution of the stab ility of each cortical site is quantified. The spatial dynamics of this transition are captured by considering relationship of the dynamical descriptor between different cortical sites. For example, if a similar transition occurs at different cortical sites, the descri ptors of the involved sites are expected to converge to similar values prior to the tran sition. Such participa ting sites have been termed as “critical sites” and such a convergen ce as “dynamical entrainment”. More specifically, in order for this convergence to ha ve a statistical meaning, we have allowed a period over which the mean of the differences of the descriptor values at two sites is estimated. We have used 60 descriptor valu es (i.e. moving windows of approximately 10

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93 minutes at each electrode site) to test th e dynamical convergence at the 0.01 statistical significance level. We employ the T-index as a measure of dynamical convergence of descriptor profiles over time. We will now consider the example with the dynamical descriptor being the short term largest Ly apunov exponent. The calcu lation of a pair-T statistic is described as follows: For channels i and j let their descriptor values in a window Wt of 60 points be: } ,..., , {59 max 1 max max t i t i t i t iSTL STL STL L, (4.1) } ,..., , {59 max 1 max max t j t j t j t jSTL STL STL L, (4.2) and } ,..., , { } ,..., , {59 max 59 max 1 max 1 max max max 59 1 t j t i t j t i t j t i t ij t ij t ij t j t i t ijSTL STL STL STL STL STL d d d L L D (4.3) then, the pair-T statistic at time window Wt between channels i and j is calculated by: 60 d t ij t ijD T , (4.4) where t ijD and d are the average value and the sample standard deviation of } ,..., , {59 1 t ij t ij t ij t ijd d d D. The thus defined T-index fo llows a t-distribution with N-1 degrees of freedom. For the estimation of the Ti,j(t) indices in our data we used N = 60 (i.e., average of 60 differences of STLmax exponents between sites i and j per moving window of approximately 10 minute duration).with Therefor e, a two-sided t-test with N -1(= 59) degrees of freedom, at a statistical significance level should be used to test the null hypothesis, Ho: “brain sites i and j acquire identical STLmax values at time t”. In this experiment, we set = 0:01, the

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94 probability of a type I error, or better, the probability of falsely rejecting Ho if Ho is true, is 1%. For the T-index to pass this test, the Ti,j(t) value should be within the interval [0,2.662]. Figure 4-1. T-inde x calculated from STLmax values of groups of electrodes (indicated in figure legend) showing convergence/di vergence before/after a seizure The T-index profiles for all possible co mbinations (groups) of electrodes (N=11) were visually analyzed to check for dynamical transitions associated with the peri-ictal period. If any group of elect rodes exhibited a convergence in dynamical values before a seizure, the seizure was considered to be accompanied by a preictal convergence. If any group of electrodes exhibited a divergence in dy namical values after a seizure, the seizure was considered to be accompanied by a postic tal divergence. An imaginary seizure point was fixed at the middle of each interictal epoch and a simila r approach was employed to check for transitions before and af ter the imaginary seizure point. To test the differences in spatiotemporal interactions between brain areas, among different states (the same four states described earlier in ch apter 3), we compared mean Tindex values (for both STLmax and ) from a group of three electrode sites including the

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95 stimulated hippocampal site and the two frontal electrode sites. This selection was based on our overall working hypotheses postulated in earlier studies [16] that (1) “dynamical characteristics of electrical signals generated by the epileptogenic hippocampus differ from those generated by similar but healthy region s” and (2) “the epileptogenic hippocampus initiates or participates in a seizure if a c ondition of spatiotemporal dynamical entrainment of a critical mass of interconnected regions of temporal and frontal cortex are met.” The same statistical analysis was applie d here as described in chapter 3. Figure 4-2. T-index calculated from aver age angular frequency values of groups of electrodes (indicated in figure lege nd) showing convergence/divergence before/after a seizure Quantitative estimation of co nvergence and divergence of dynamical values among different electrode sites revealed that ther e was a significant preictal convergence and postictal divergence in dynamical values betw een the frontal and temporal brain regions. Figure 4-1 and 4-2 show examples of the T-index profiles calculated from STLmax and values respectively, of a pair of channels including one hippocampal and one frontal site, over a period of 2 hours, from each of the 4 rats (Refer to Chapter 3 for data description

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96 and summary). This pattern shows the convergence of STLmax and values among the specified electrode sites (indicated in figure) before a seizure and th e divergence after the seizure. From a dynamical perspective, this suggests an increase in dynamical interactions between the brain sites during the preictal period and a postictal dynamical dissociation (reduced interaction) of these brain sites following a seizure. A sudden significant divergence during the postictal stage indicates the “resetting” feature of the seizure (the seizure restores the pre-seizure convergence to a more normal state) [161]. The summary of test results for preictal convergence and po stictal divergence is given in table 4-1. Table 4-1. Summary of test results obtained from spatiote mporal analysis in four animals Seizures with preictal convergence (T-index) Seizures with postictal divergence (T-index) Rat ID Number of Seizures STLmax STLmax A 5 5 5 5 5 B 8 7 7 7 8 C 7 7 5 6 7 D 8 7 6 8 8 Overall 28 26(92.8%) 23(82.1%) 26(92.8%) 28(100%) To test the above observations, we compar ed mean T-index values obtained from interictal, preictal, ictal and po stictal periods (please refer data description in Chapter 3 for sampling times). Figures 4-3a and 4-4a illustrated the mean T-index values for STLmax and , respectively. T-index values were calculated from a triad of electrodes consisting of the hippocampal electrode and a pair of frontal el ectrodes. Consistent pattern observed across all four rats in T-index of STLmax (Figure 4-3a) was that the me an T-index values decreased from interictal to preictal and to ictal peri od, and reached the high est values during the postictal period. This pattern can also be observed in T-index values of from three rats,

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97 however, with less significant differences (F igure 4-4a). Statistical tests of these observations revealed that there exists a significant state effect on T-index (of both STLmax and ). Figure 4-3. Comparison of mean T-index values calculated from STLmax between states. (a) Comparison among mean T-index values calculated from the stimulated hippocampal and a pair of frontal electro des, during the interictal, preictal, ictal and postictal states. (b) Multiple comparisons among mean values of Tindex for STLmax during interictal, preictal, ictal and postictal states

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98 Figure 4-4. Comparison of mean T-index values calcu lated from average angular frequency between states. (a) Comparison among mean T-index values calculated from the stimulated hippocampal and a pair of frontal electrodes, during the interictal, preictal, ictal and postictal states. (b) Multiple comparisons among mean values of T-index for during interictal, preictal, ictal and postictal states Multiple comparisons in m ean T-index values of STLmax (Figure 4-3b) revealed that mean T-index value obtained from each of the four periods wa s significantly different from the other three periods (p < 0.05), with an ascendi ng order from ictal to pr eictal to interictal to postictal. Multiple comparisons among mean T-index values of (Figure 4-4b) from different periods suggested that mean Tindex value obtained from ictal period was

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99 significantly smaller than those at the other th ree periods (p < 0.05). However, differences among interictal, preictal and postictal pe riods were not significant (p > 0.05). Results from specificity analys is on 48 interictal iEEG segm ents (Table 4-2) revealed that seventeen percent (8/48) of interictal segments analyzed sh owed a preictal like convergence in STLmax values and ten percent (5/48) showed a postictal like divergence in STLmax values. However, on more careful examina tion of the interictal data we found that six out of the eight interictal segments that showed a pre-ictal like convergence of STLmax values were less than 5 hours away (mean = 2. 84 hrs) from a seizure. The postictal-like divergence in STLmax values in four interictal segmen ts reflected morphol ogical changes in the iEEG similar to what is observed after an actual seizure. The specificity results for spatial behavior of values has not been shown but w ill be discussed in the following section. Table 4-2. Summary of STLmax analysis on interictal segments from four rats Rat ID Number of Interictal Segments Segments with preictallike transition Segments with postictallike divergence A 12 1 1 B 12 2 1 C 12 3 1 D 12 2 2 Overall 48 8 (16.7%) 5 (10.4%) Seizure Predictability in an Animal Model of MTLE Methods based on quantitativ e dynamical descriptors such as short term Lyapunov exponent STLmax have been used to forewarn of epileptic seizures in human TLE [160,166,172]. The present stud y investigates the applicat ion of such a method to a controlled animal model to determine if the spontaneous seizures that characterize this model may be predicted in advance of the actual onset. The seizure warning algorithm

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100 used in this study is based on the conve rgence of the dynamical descriptor (STLmax) values among critical electrode sites as the brain approaches a seizure susceptible state and a divergence in the values immediately following a seizure. We have used this approach since our earlier inve stigations using this model re vealed similar preictal and postictal transitions as described above (refe r to earlier sections of the chapter). We evaluated the MTLE model to determine its suitability for th e investigation of closed-loop intervention schemes that would apply a therapeutic in tervention based on dynamical and physiological state reflected in the iEEG. The study employs a seizure warning algorithm based on the preictal transi tions mentioned earlier, to investigate the predictability of seizures in the MTLE model. The predictability power is defined as the difference of warning performance between the employed seizure warning algorithm and null warning schemes (periodic and random). After estimating the sensitivity and false warning rate (FWR), two performance measur es were utilized: (1) the area above the ROC curve (AAC) and (2) the fr action of time under false warn ings (FTF) with at least 80% sensitivity required for each rat. Since the performance depends on the pre-set seizure warning horizon (SWH), predictability results from a wide range of SWH from 1 to 6 hours are presented. Description of Data Long term continuous iEEG recordings from 5 animals with a total of 48 spontaneous seizures were included in this st udy. We have selected these rats based on duration of recordings (at leas t two weeks), and number of seiz ures (at least 5 seizures). The mean total duration of recordings was approximately 19 days and the mean seizure interval was a little over 2 days (~ 49.7 hours). A summary of the test dataset is given in Table 4-3. Seizures were identified by review of technician logs, visual scanning of the

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101 recordings, and automated seiz ure detection algorithms. The seizures were confirmed and classified by a board-certified electroence phalographer who also made an independent determination of the time and anatomical location of electrographic seizure onsets. Table 4-3. Summary of iEEG data obtained from five animals used for seizure predictability study Rat Duration of iEEG recordings (days) Number of seizures Range of Interseizure interval (hours) Mean Interseizure interval (hours) R-1 14.3 7 3.54 ~ 115.6 52.3 R-2 31.3 8 20.56 ~ 217.7 98.46 R-3 15.7 10 2.82 ~ 74.3 30.23 R-4 14.6 8 12.44 ~ 76.25 43.53 R-5 19.1 15 2.98 ~ 186.37 23.77 Total 95.0 48 2.82 ~ 217.70 49.7 Automated Seizure Warning Algorithm The short-term maximum Lyapunov exponent was utilized to extract the nonlinear dynamical characteristics of th e iEEG signals over time for each recording channel. The STLmax convergence is quantified by the average of pair-T statistics (T-index) over all pairs among the group of channels. Based on the STLmax/T-index profiles over time, we evaluate an automated seizure warning algor ithm that involves the following steps (flow chart is demonstrated in Figure 4-5): 1. Selection of anticipated critical groups of channels. Based on the STLmax profiles from all recording channels before and after the first available seizure in the recording per rat, ASWA selects the most critical groups of iEEG channels for prospective monitoring. Theref ore, the first seizure in the record must be manually indicated to initiate the algorithm (Step 202 in Figure 4-5). On ce the algorithm is initiated, the ASWA automatically selects the iEEG channels to be employed for warning of the subsequent seizures (Step 203). The channel selection is performed automatically,

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102 based on the T-index values. The critical groups of electrode channels are defined as the groups of channels which maximize the value Tpostictal Tpreictal, where Tpostictal is the average T-index in the 10 minute window follo wing the offset of the first seizure and Tpreictal is from the 10 minute window preceding the first onset. The channels thus selected are named critical electrode sites. In this application, three groups of critical electrode channels are selected for use in predicting subsequent seizures. This parameter (number of groups) was selected based on a wa rning sensitivity analysis from a set of seizures in the first three rats (f irst four seizures from each rat). 2. Monitoring T-index profiles and detecting convergence transitions. The average T-index values of these groups ar e monitored forward in time (moving window of 10.24 seconds at a ti me) (Step 204). A convergence (entrainment transition) is detected when the average T-index value for any of the three groups falls below a dynamically adapted critical threshold (Ste ps 205 and 206). The adaptive threshold includes a “deadzone” with an upper threshold UT and a lower threshold LT (Step 205): The upper threshold UT is determined as follows: if the current T-index value is greater than max20 (that is the maximum T-index value in the past 20 mi nute interval), UT is set equal to max20, otherwise, the algorithm continues se arching sequentially in time to identify UT. Once UT is identified, the lower threshold LT is set equal to UT -D, where D is a user controlled variable in T-index units. After determining UT and LT, a convergence (entrainment transition) is detected if an average T-index curv e is initially above UT and then gradually (at least 30 minutes of traveling time) drops below LT (step 206). Once a convergence (entrainment transition) is detect ed, the algorithm returns to step 205 to search for a new UT to be used for detection of the next transition. We thereafter use the

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103 notation ij TU and ij TL to denote the ith group of channels whose average T-index satisfies the necessary conditions for detection of the jth transition. It has to be mentioned that the parameters of 20 minutes for determining UT and 30 minutes for traveling time are purely empirical. Figure 4-5. Automated seizure warning algorithm

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104 3. Warn of an impending seizure. After a convergence (entrainment transition) is detected, the algorithm will determine whethe r a seizure warning should be issued (step 207). In this algorithm, if a transition is de tected within the warning horizon from the previous warning, the transition is considered as part of a cluster of transitions (due to a possible cluster of impending seizures) and a new warning is not issued. Thus, a seizure warning defines mathematically the beginni ng of a new dynamical iEEG state called the preictal transition. Statistical Evaluation of Seizure Predictability Estimating seizure warning ROC curves and performance index. To evaluate the performance of a warning scheme, a se izure warning horizon (SWH) needs to be specified because it is practic ally impossible to predict the exact time when an event will occur. The SWH applied here is similar to the “seizure occurren ce period” defined by Winterhalder et al. [173-175], which actually means a short “intervention preparation period”. After the issue of a warning, a warning is considered as correct if the event occurs within the preset SWH. If no event occurs within the window of the SWH, the warning is classified as a fa lse warning. Since there is no t yet a method to determine the optimal SWH for a given warning algorithm, in this study, we have investigated the performance with a wide range of SWH from 1 ~ 6 hours. Sensitivity is defined as the probability of an event being correctly warned and the FWR as the number of false warning per unit time. A common practice in comparing diagnostic methods is to let the sensitivity and the specificity vary together and use their relation, called the receiver operating charac teristic (ROC) curve, to evaluate their performance. Based on the estimations of sensitivity and FWR, we generated seizure warning ROC (SWROC) curve from each test rat.

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105 Based on the SWROC curve, we utilized the area above the curve (AAC) as a performance index to quantify the overall perf ormance of the seizure warning scheme. It was calculated by: 0) ( 1 dx x f AAC, (4.5) where y = f(x), with x and y being the FWR and sensitivity, respectively. Smaller AAC indicates better seizure warning performance. In the seizure warning a pplication, since it is less important to evaluate the performance when sensitivity is low, we have estimated AAC with seizure warning sensitivity above 50%. Another performance index employed is the fraction of time under false warnings (FTF) with warning sensitivity at least 80%. This index provi des a simple measure that indicates, once a detection rate is established, what is the percentage of time the subject will be under false warning. FTF provides more clinical meaning, in particular if the subject is going to be subjec t to some form of seizure abortive stimulation as many groups are anticipating. Predictability power analysis. Similar to any statistical hypothesis testing, the performance under a null seizure warning scheme has to be estimated (from the same test subjects) in order to estimate the predictability power of a test seiz ure warning algorithm. In this study, we utilized a periodic and a random warning scheme as null warning schemes. The periodic and random warning schemes are simple and intuitive. The periodic scheme gives seizure warnings with a fixed time interval . The random prediction scheme warns of an event according to an expon ential distribution with a fixed mean. It is important to mention that, for the random wa rning scheme, since it essentially is a

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106 random process, each point in ROC curve wa s estimated as the mean sensitivity and mean FWR from 100 Monte Carlo simulations. After obtaining the performance indices (PI) based on the estimated SWROC curves, we define predictability power (PP) as the difference in PI between the null warning scheme and the ATSWA, normalized by the PI of the null warning scheme. In other words, this PP estimates the improveme nt of performance provided by a warning algorithm over the null warning scheme. Naive orithm A Naive orithm API PI PI PPlg lg . (4.6) The PP will be zero when the test algorithm perf orms at chance (null) level, and will be one when the algorithm performs perfectly. Th e normalization takes care of the fact that the performance of the null predictor may vary as a function of the SWH, and therefore provides a measure independent of SWH. Results of Predictability Study The ROC curves for each animal were estimated from three test warning schemes using 6 different warning horizons. Figure 46 illustrates the SWROC curves from the first rat. Closer inspection of these curves shows a consistent superior performance of the ASWA (solid line) when compared to the two null seizure warning schemes (dashed and dotted line), with the lower FWR values for al most the entire range of sensitivities. A summary of seizure warning performance (FWR with sensitivity at least 80%) as a function of SWH is given in Table 4-4. Th is performance characterizes the overall (all rats) FWR and seizure warning time (the aver age of the period from the true warnings issued by the algorithm up to the onset of the subsequent seizures) for ASWA when a sensitivity of 80% or better was required fo r each rat. With SWH = 1 hour, an FWR of

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107 0.267 per hour (approximately 1 false predicti on per 3.8 hours) with the mean seizure warning time 30.7 minutes was observed, wh ile the corresponding FW R for periodic and for random warning scheme are 0.777 and 0.827 per hour, resp ectively. The FWR decreases when the SWH increases. With SWH = 3 hours, ASWA performed an FWR of 0.116 per hour (approximately 1 false predicti on per 8.6 hours) with the mean seizure warning time 69.5 minutes, while the corres ponding FWRs for periodic and for random warning schemes are 0.242 and 0.265 per hour, respectively. For the purpose of estimating predictabi lity power, for each animal, each SWROC curve was translated into two performance indices (AAC and FTF). The overall indices are shown in Figure 4-7. Corresponding to the above observations, the performance indices obtained from ASWA were smaller th an those from the two null seizure warning schemes in each of the 6 SWHs. The predictability powers of ASWA ove r periodic and over random warning scheme derived from AAC and FTF with diffe rent SWHs are shown in Figure 4-8. From AAC performance index, ASWA improved th e performance over the random scheme by almost 60% and over the periodic scheme by 55%, with SWH = 1 hour. From the FTF performance index, the improvement was 68% over random scheme and was 66% over periodic scheme, with the same SWH. The PP decreased when the SWH increased. This indicates that the predictability of ASWA over chance is more significant when the SWH is shorter (i.e., more accurate seizure warning). It is worth noting that the predictability pow er derived from FTF is larger than from AAC. Since FTF is estimated from the minimu m FWR when the warning sensitivity is at least 80% while AAC is estimated from all the FWRs when the sensitivity is over 50%,

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108 this suggests that ASWA provides more improvement over the null warning schemes (or chance) when only higher sensitivities are considered. Figure 4-6. Estimated SWROC curves derived from recordings in the first rat (R-1) for ASWA and two null seizure warning sc hemes: solid line = ASWA; dashed line = periodic, and dotted line = rand om. The warning horizons applied are from 1 to 6 hours. SWROC curves were smoothed by regression method Table 4-4. Evaluation of overall warning perf ormance of the three test algorithms when sensitivity is set larger than 80% per rat (Sensitivity and mean prediction time are from ASWA. Periodic and Random schemes have the similar outcomes for these two characteristics, but with much higher FWRs) SWH (hrs.) Sensitivity FWR/hr (ASWA) Mean Prediction Time (mins.) FWP/hr (Periodic) FWR/hr (Random) 1 37/43 = 86.1% 0.267/hr 30.7 ( 18.4) 0.777/hr 0.827/hr 2 37/43 = 86.1% 0.164/hr 54.9 ( 37.7) 0.344/hr 0.402/hr 3 37/43 = 86.1% 0.116/hr 69.5 ( 47.1) 0.242/hr 0.265/hr 4 37/43 = 86.1% 0.080/hr 108.5 ( 60.2) 0.171/hr 0.201/hr 5 37/43 = 86.1% 0.076/hr 133.4 ( 84.6) 0.130/hr 0.161/hr 6 37/43 = 86.1% 0.057/hr 176.9 ( 96.6) 0.107/hr 0.134/hr

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109 Figure 4-7. Overall performance indices (AAC and FTF) from ASWA and two null warning schemes with respect to different seizure warning horizons Figure 4-8. Predictability power of ASWA derived from AAC and from FTF, with respect to different seizure warning ho rizons: solid line = ASWA vs. periodic scheme, and dashed line = ASWA vs. random scheme

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110 Discussion The ability to predict an impending seizure well ahead of its clinical or electrographic onset provide s an opportunity for new di agnostic and therapeutic applications that could revolutionize the me dical and surgical management of epilepsy. Control of seizures through electrical, ma gnetic stimulation of the brain or its connections, or pulsed pharm acological intervention is a rapidly developing field. Stimulation via implanted neurostimulators in the brain is currently considered a promising new form of treatm ent (functional neurosurgery) for various brain disorders (e.g. epilepsy, Parkinson’s and other neurodegene rative diseases). A reliable prediction of seizures in epilepsy, as well as of critical tr ansitions in other neurol ogical disorders, may provide the answer to the question of when and where to stimulate in the brain to accomplish the disruption of the route toward these undesirable tran sitions (closed loop control schemes). The results of this study suggest that it may be possible to predict spontaneous seizures in the model of MTLE, using simila r techniques. The technique used in this study and results suggest that the MTLE model e xhibits similar state transitions from the interictal to the ictal state as that found in human patients with epilepsy and that it may be possible to warn of an impending seiz ure by monitoring the convergence of STLmax values among different brain regions. Further, this transition from in terictal to preictal state may be detected several minut es before the actual seizure onset. Evidence for preictal transitions has been reported in experimenter induced seizures using stochastic methods [37]. This study used a stochastic model to model transitions between interictal and drug-induced (3-mercap topropionic acid) seizures in a generalized animal rat model. In the present study, we have used a spontaneously seizing model and

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111 techniques previously employed in the anal ysis of human epilepsy to show seizure predictability. To our knowledge this is the fi rst report in which seizure predictability is demonstrated in an animal model of spontaneous epilepsy. It is very encouraging to note that seizure warning may be possible not onl y in experimental seizures, but also in spontaneously occurring epilepsy that has ma ny of the characterist ics associated with human TLE. We have demonstrated the usefulness of an automated seizure warning scheme in a model of intractable human MTLE by using a pharmacoresistant limbic epileptic rat in which seizures occur spontaneously, intermittent ly and at varied intervals. In addition, EEG data sets previously used to test the algorithm were obt ained from patients undergoing pre-surgical evalua tion and hence had short inte r-seizure intervals (mean 10 hours). Such data sets have less likelihood of having enough interi ctal intervals for reliable estimation of false warning rates. The mean inter-seizure interval in this animal model is much longer than the human cases analyzed in previous studies by our group, and therefore the evaluation of the algorithm in this model could be more meaningful with respect to clinical applications su ch as portable seizure monitoring devices. Seizures in the MTLE model appear to occur after sustained dynamical convergence in dynamical states of different cortical areas and are often followed by a sudden dynamical divergence [34]. It has been postulated that seizures may be intrinsic mechanisms that serve to reset the brain to a normal state. This theory has lead us to hypothesize that it may be possible to use external intervention schemes to achieve dynamical resetting of the epileptic brain by reversing the preictal convergence among critical brain regions [161]. A timed intervention, for e.g. an electrical stimulus or a dose

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112 of an anticonvulsant drug deliv ered during the preict al state may serve to reset the brain and consequently delay the actual seizure ev ent, if not completely avoid it. Future experiments that employ an automated seizur e warning scheme coupled with external intervention schemes (both electrical and ch emical) delivered during the preictal period to test this hypothesis will be explained in subsequent chapters. We hence propose this model as a good choice for developing seizur e prediction as well as closed-loop control schemes for use in the treatment of human refractory epilepsy. In summary, the results suggest that the dynamics governing the transition from the interictal state to the seizure state in this model are similar to those observed in human TLE. This phenomenon may be detectable usin g other measures such as the stochastic approach used in drug-induced seizures as me ntioned earlier and need to be investigated in this model. A seizure warning system utilizing such a model could be used to activate pharmacological or physiological interventions designed to prevent an impending seizure. The performance of the algorithm seems to be sufficient enough to be able to lend itself to a range of applications such as seizur e monitoring and interv ention/control devices. However, the performance needs to be tested in prospective studies using a larger cohort of animals. Spatiotemporal Transitions during Epileptogenesis The development of chronic ep ilepsy after a precipitating brain insult (injury) is often associated with a delay or latent period of weeks to several years (1). During the latent period, there is progr essive axonal and dendritic pl asticity, significant neuronal death, marked gliosis, and molecular reor ganization of receptors and ion channels. Collectively, these and other neurobiologic ch anges are thought to contribute to increased network hyperexcitability and ultimately to the expression of clinical seizures. The latent

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113 period provides a “window of opportunity” for intervention with novel therapies aimed at preventing the development of epilepsy. Such antiepileptogenic therapies may be entirely different from those that are effective in the symptomatic treatment of seizures associated with chronic epilepsy. Patients with TLE usually have had an in itial precipitating injury during early childhood. However, epilepsy does not develop in all subjects that have undergone such early traumatic insults. We have also observe d in the rodent model of MTLE, that only a subset of the animals used in the study go on to develop spontaneous seizures following the status epilepticus stage and a subsequent latent period. The development of markers that may serve to denote the risk for epilepsy is of particular importance for the early identification of at risk candi dates and possible inte rvention to prevent the manifestation of epilepsy. Identification of critical and realistic time points for intervention is essential in the prevention of epilepsy. Some of the relevant questions at hand include: When are risk-factors for the eventual development of epilepsy sufficiently clear and of high enough predictive value to warrant treatment w ith drugs that could have potential side effects? When is clinical identi fication of risk factors possible? The best markers of epileptogenesis to date have been in the post status epilepticus models, in which the early a ppearance of neuronal loss has a high predictive value for later development of epilepsy. In this dissertation, we propose a novel application of dynamical spatiotemporal transitions as dynamical markers in predicting as well as following the process of epileptogenesis in a rodent MTLE model. Progressive preictal convergence and postictal divergence of dynamical descriptors such as STLmax have been reported in human TLE as well as animal mode ls of MTLE [34]. We hypothesize that the

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114 transition from the interictal st ate (seizure resistant) to a seizure involves an interim transition to a preictal (s eizure prone state) charac terized by dynamical properties intermediate between those of the interictal and ictal states. Based on this hypothesis, we anticipated that epis odic convergence of STLmax values, such as those seen prior to seizures, would begin to occu r in the MTLE model in animal s that developed seizures after induction of status epilepticus, but not in those animals that did not develop spontaneous seizures. We also hypothesize that the frequency of these transitions increases progressively from the time of initia l insult to the subsequent manifestation of epilepsy. Description of Data Long term continuous iEEG recordings fr om 5 animals that were subjected to continuous hippocampal stimulation were includ ed in this study. The mean total duration of recordings was approximately 229.7 hours. A summary of the test dataset is given in Table 4-5. The first seizure in each animal was identified by review of technician logs, visual scanning of the recordings, and au tomated seizure detection algorithms. The seizures were confirmed and classified by an electroencephalographer who also made an independent determination of the seizure time and electrographic onset location. Table 4-5. Summary of data a nd results from epileptogenesis study Rat ID Total Duration Analyzed (hours) Number of Transitions Slope A* 232.9 31 slope=0.09, p=0.0005 B* 142.5 2 slope=0.07, p=0.0228 C 307.7 0 N.A. D 217.5 0 N.A. E 248.06 0 N.A. * Rats that developed seizures follo wing continuous hippocampal stimulation

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115 Study Methods The animals were connected to an automa ted system that monitored the T-index calculated from STLmax values estimated from multiple brain regions. Electrode groups including the stimulated hippocampus and the c ontralateral frontal cortex were selected for monitoring during the latent period. A conve rgence transition was defined as a drop in T-index from a preset upper threshold (UT=10) to a lower threshold with a minimum traveling time of 1 hour (LT=2.662). The time leading to the first seizure was divided into blocks of 24 hours and the frequency of tran sitions in each block was calculated. The null hypothesis is rejected if th e regression slope is found to be significantly positive (p<0.05). Results and Discussion Figure 4-9. Transitions in Ra t A* during the latent period showing a linearly increasing trend in the incidence of transiti ons leading up to the first seizure

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116 Figure 4-10. Transitions in Rat B* during the latent period showing an abrupt increase in the incidence of transitions leading up to the first seizure Two out of the four animals developed s pontaneous seizures after CHS. The first showed a progressive increase in frequency of spatiotemporal transitions leading up to the first seizure (recording time before se izure~232.9 hrs, slope=0.09, p=0.0005), the second animal showed an abrupt increase in incidence immediately preceding the seizure (recording time before seizure~142.5 hr s, slope=0.07, p=0.0228). No dynamical transitions were observed in the three animal s that did not develop spontaneous seizures. There is general consensus that there exists a need for identifying reliable biochemical, pharmacological or electrophysiol ogical elements as surrogate markers of epileptogenesis. Such markers may provide an indication of the likelihood of chronic seizure development following a potentially ep ileptogenic insult and al so would facilitate rapid screening of new compounds with an tiepileptic/antiepileptogenic potential. Traditionally, epileptogenesis is thought of as a cascade of dynami c biological events altering the balance between excitatory and i nhibitory mechanisms in neuronal networks.

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117 The term applies to any of the progressi ve biochemical, anatomic, and physiologic changes leading up to recurr ent seizures. Progressive changes are suggested by the existence of a so-called silent (latent) pe riod, often years in duration, between CNS infection, head trauma or febrile seizur es and the later appe arance of epilepsy. Understanding these changes is ve ry important in prev enting the onset of epilepsy. In this dissertation we study the phenomenon of epile ptogenesis as a progressive condition exhibiting characteristic dynamical transitions as reflected in the electrophysiological activity. An important goal is to understand mechanis ms of epileptogenesis that incorporate information from various different levels of organi zation that range from molecular (e.g., altered gene expression) to macrostructural (e.g., altered neuronal networks). The possibilities are so diverse that sorting out which mechanisms are causal, correlative, or consequential (detrimental or compensatory) is a primary re search direction. Markers of epileptogenesis represent key targets for building an understanding of the mechanisms of epilepsy, anatomic vulnerability, prognosis, and response to th erapy. Several techniques including imaging and gene array technique s are currently employed in the study of epileptogenesis. Magnetic resonance imagi ng (MRI), for example, is one of the techniques used to predict the developmen t of TLE [176]. Fast ripples at 250-600 Hz associated with interictal spikes curren tly provide information on localization and epileptogenicity of the tissue involved in thei r generation. We propose an alternate, novel technique based on dynamical transitions (interactions) betw een different brain regions that would complement existing technique s employed in the study of epileptogenesis

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118 In this preliminary study, spatiotemporal dynamical transitions were found only in animals that developed spontaneous seizures. In these animals, the time of onset or frequency of transitions corresponded to the occurrence of the first spontaneous seizure. These findings support the concept that these transitions represent a preictal (seizure susceptible state). We propose that from this state, the brain may move into the ictal state, or revert back to the interictal state. Furt hermore the results suggest that spatiotemporal transitions involving convergence of dynamical descriptors such as STLmax among brain regions may be a marker for epileptogenesi s. Coupling MRI techniques with/or other predictive techniques such as the one described in this dissertation to clinical outcome in early preclinical studies could represent a us eful experimental appr oach for identifying those therapies that prevent or modify the development of epilepsy. Given recent advances at the molecular and genetic level and developments in seizure prediction and brain imaging, a “cure” (or at least the prev ention) of some forms of acquired epilepsy would appear to be a reasonable expect ation for the not-so-distant future. In summary, we propose that dynamical analysis of EEG/iEEG via powerful nonlinear techniques, combined with the superi or spatial resolution of imaging, may offer powerful integrated tools for monitoring ep ileptogenesis. One challenge will be to develop preclinical model systems that can be used in early proof-ofconcept studies that ultimately support more timely and costly human studies.

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119 CHAPTER 5 CONTROLLING CHAOS: THEORY, MODE LS AND APPLICATIONS TO BRAIN DISORDERS Motivation for Controlling Spatiotemporal Chaos Controlling spatiotemporal chaos in mu ltidimensional systems with distributed parameters is a problem that has attracted increased attention over the past few decades due to its broad applications in physical, chemical and biological systems. One of the important and challenging areas for the appl ication of chaos cont rol techniques is the problem of treatment of neurol ogical disorders such as epilepsy. Works of Freeman and his collaborators [177] sugge st that from a dynamical sy stems point of view, brain functionality is linked with changes in dynamics under influe nce of external stimuli – external signals controlling the dynamics. Further, the mechanisms are such that switching between various attractors in stat e space is accomplished by suitable signals. There are clearly control mechanisms involved and such mechanisms need to be studied in detail to get a better unde rstanding of dynamical controllab ility. Chaos offers rich and versatile ways of control in cases where simple phenomena cannot. Characterization of brain activity (EEG), using measures of chaos has been very useful in providing information about the dynami cal state of the epileptic brain. Studies in humans [41,158-162] and animal models [34, 35] of epilepsy suggest that occurrence of spontaneous seizures correlates with the evol ution of the brain to a more temporally ordered state. This has been demonstrated by changes in gross system properties such as Lyapunov exponents calculated from EEG record ed from multiple brain regions. It has

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120 been postulated that such spatiotemporal transitions occur due to self organizing transitions in the epileptic brain that drives it from chaos to order. Furthermore, seizures have been considered to be i nherent resetting mechanisms of the brain to revert it back from order to the chaotic regime [16]. It is hence obvious as to why the maintenance of chaos in such biological systems is extremely desirable because of its implications from a therapeutic or control point of view. More recently the concep t of “anticontrol” schemes, where the goal is the maintenance of chaos, has been the topic of much investigation and several algorithms have been devised to realize this objective [178-180].The design and adaptation of such chaos control algorith ms to dynamical systems that exhibit exceedingly complex dynamics, such as th e brain, is not a trivial challenge. Goals of Chaos Control In general control always means influencing a system in such a way that it behaves in a desired manner. In the context of contro lling spatiotemporal systems, we need to consider some of the goals to strive for. Formation of specific spatial or spatio temporal patterns with desired or prescribed characteristics. Stabilization of desired ki nd of behavior, such as sp atiotemporal orbits and trajectories. Synchronization/Desynchronization – in ce rtain cases it may be desirable to obtain coherent operation of the whole spa tial structure or a part of the cells only (Maintenance or suppression of chaos). Efficient and fast switching between va rious attractors in the phase space. Such (intrinsic) switching has been proposed as a mainstay of dynamical systems such as the brain. Removal of certain type of be havior (e.g. spiral waves). Cluster stabilization – which calls for stabilizing a small cluster in the multidimensional space while the surroundi ng medium has to operate in a chaotic mode.

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121 Biological systems are inherently adaptive in nature and therefore require adaptive control techniques [4]. Recent investigations in human ep ilepsy have suggested that effective modulation of brain dynamics needs new control techniques that are based on robust prediction and adaptive optimization sc hemes [41-43]. In this chapter we will mainly concentrate on the theoretical problem of controllability of a system property that has shown to reflect state changes in neur al systems, namely the Lyapunov exponent. Possible real world implications of contro lling the Lyapunov spectru m include being able to control the convergence and divergence of this entity as quantif ied by the statistical measure T-index among different regions in an epileptic brai n, which has been shown to be predictive of a seizure susceptible state. We reformulate the prob lem of controllability of the Lyapunov exponent as an optimization problem in which we try to estimate the control parameters by minimizing the error f unction calculated from the global Lyapunov exponents of a system. Computational Models in EEG Analysis Need for Computational Models Stochastic behavior occurring in phenom ena formally described by deterministic equations has attracted much interest recently. As a simple case, one can consider spatially homogeneous systems with compli cated temporal dependence described by ordinary differential equations. From the study of simple systems of nonlinear differential equations and from various simple mappings , much has been understood about chaotic temporal behavior. In such systems the time dependence of the quantities describing the state of the system can be very differe nt depending on the value of some control parameters. In addition to constant, pe riodical, and quasi-p eriodical temporal

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122 dependence, random-like behavior (called determ inistic chaos) can also appear even in very simple nonlinear dynamical systems. On the other hand, when one is interested in complex dependency in space, it is necessary to consider partial differential e quations (PDEs) or various discrete models. PDEs involve state variables that are conti nuous in time and space, but because of this feature the related large scale numerical investigations are very difficult to realize. A possible alternative strategy to understanding spa tiotemporal complexity is to investigate simple chaotic maps assigned to the points of a lattice and coupled together via some rule. Discrete models make the simulations easier and at the same time some general aspects of PDEs may be retained. It is wi dely accepted that from the study of discrete systems one can obtain useful information about the nature of processes involving complex spatiotemporal behavior, includi ng fluid flows and pattern formation. Hence, as a first step, we need to identi fy a simple model that emulates system dynamics as seen in the epileptic brain to investigate control methodologies that can conceivably be adapted to a more complex system. As far as multichannel EEG is concerned, methods aimed at evaluating signal interdependencies and dynamical interactions have a particular interest. This remains a difficult task, since couplings between neural groups are nonlinear and non st ationary. In epilepsy, for instance, both their degree and direction (uni directional/bidirectional) have been shown to change with time as the seizure develops. Consequently, fo r relevant information to be extracted from EEG signals, processing methods must take these non stationary and nonlinear properties into account. Some of the problems to be addres ses are: i) statistical parameters sensitive to signal interdependencies and corresponding me thods of estimation have to be chosen,

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123 ii) candidate estimators have to be evaluated for their statistical performances and iii) provided quantities (a crosscorrelation coefficient between two signals, for example) must be related to physiological knowledge (an anatomical-functi onal coupling between the two structures that generate the signals, fo r example) in order to be interpreted [181]. Modeling may be a way to ad dress some of the above listed problems. Generally computational models potentially useful in the study of EEG dynamics can be divided into two classes: “External models” based on generic mathemati cal description of signals or signal dynamics and “Physiologically constrained models” that are aimed at representing the underlying neur ophysiological processes at th e origin of generation of these signals [181]. If relying on a neurophysio logical basis, models can allow signal analysis to go beyond the simple external description of real data and provide a framework in which observed dynamics ma y be interpreted with respect to the knowledge encoded in the models. Figure 51 shows a general classification of the computational models that have been used in dynamical analysis of EEG signals. Figure 5-1. A simple cla ssification of models used in EEG dynamical analysis Coupled Map Lattice Models Coupled map lattice (CML) systems, fi rst introduced by Kaneko [182,183] are simple and popular models for studying spatiote mporal behavior of systems and seem to

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124 be gaining popularity as tools for modeling complex phenomena in physics, engineering, biology, chemistry, social sciences, economic s, etc. The local dynamics of coupled map lattice is given by a nonlinear map and a coup ling factor. The problem of synchronization and controlling spatiotemporal chaos in such lattice systems has been investigated earlier in literature [184-189] . Such models are conceptually interesting sin ce the notion of coupling is explicitly taken into account. However some interrogations remain about the relevance of such simulation models in the fi eld of EEG and brain el ectrical activity. We have chosen a globally coupled lattice system and present numerical results with respect to the controllability of spatiotemporal chao s in this system. The skepticism was mainly based on observations that even though the simu lated signals in such coupled oscillators are nonlinear, the exhibited dynamics did not resemble those observed in EEG signals. In this dissertation we report results contrary to these observations and show similarities in both spatial and temporal dynamical profile s of simulated signals and intracranial recordings obtained from the epileptic brai n. Based on these observations we propose the coupled lattice model as a good conceptual model of EEG dynamics and a test bed for developing nonlinear methods aimed at controlling spatio temporal chaos. A coupled map lattice is a N-dimensional network of in terconnected units where each unit evolves in time through a map (or r ecurrence equation) of the discrete form: ) (1k kX F X , (5.1) where Xk denotes the field value (N -dimensional vector) at the indicated time k. In the case of a globally coupled map, with a global (mean field) coupling factor , the dynamics can be rewritten as: L j k j j k n n k nx f L x f x1 1) 1 ( , (5.2)

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125 where n and j are the labels of lattice sites (j n). The term L indicates over how many neighbors we are averaging and it is sometimes referred to as coordination number. The local N-dimensional map is assumed to be chaotic. Completely synchronous chaotic states are possible with this model when corresponding N-dimensional manifolds are attracting or stable. The criterion for stabilit y of this synchronization manifold has been derived in [186]. Further stabil ity analysis of synchronized pe riodic orbits in coupled map lattices can be found in [187]. Varying and L we can change the extent of spatial correlations, from systems with local interactio ns to systems with long-range interactions. These systems typically exhibit spatially and/ or temporally chaotic behavior, the control of which is very desirable because of its poten tial real-life applicati ons. Several strategies have been proposed to contro l the collective spat iotemporal dynamics of such systems. We will consider the example of a simple logistic map as the individual nonlinear oscillators in the CML, given by: ) 1 (1k k k kx x x . (5.3) Calculation of Lyapunov exponent When the objective is to maintain a desired level of chaoticity, a natural choice for the monitored property of the controlled sy stem is the Lyapunov exponent. The global Lyapunov exponent for a discrete one dimensional system ) (1k kx f x can be defined by: 1 0 ') ( ln 1 limN k k Nx f N. (5.4) In order to study the evolution of Ly apunov exponents of a coupled map lattice system as described by equation (5.2), we firs t introduce the Jacobian matrix as follows:

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126 k L k L k k L k k L k k L k L k k k k k k k k L k k k k k k k kx x x x x x x x x x x x x x x x x x x x x x x x1 3 1 2 1 1 1 1 2 3 1 2 2 1 2 1 1 2 1 1 3 1 1 2 1 1 1 1 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (5.5) The Lyapunov exponents of the system are cal culated from the eigenvalues of the above matrix. If the eigenvalues of the k are {1 k, 2 k, 3 k,..., L k} then the local Lyapunov exponents are given by n k = log| n k|; (n = 1,2,...,L). (5.6) Spatiotemporal dynamics in coupled map lattices Experimental studies in rodent models of epilepsy [34] have used iEEG recordings from four to six electrodes placed in frontal a nd temporal regions of the animal brain. We have therefore considered a CML model with five non-identical logi stic maps [190]. Each cell is connected to ev ery other cell in th e lattice with the same magnitude of connectivity, determined by the global coupling term . The system parameters 1 5 were chosen randomly as 3.96, 3.8, 3.75, 3.83 and 3.9. The coupling term was varied from a value of 0.10 to 0.14 to study the dynamical behavior in both the spatial and temporal regimes. Figure 5-2 shows the cha nges in spatiotemporal patterns as we increase the value of the parameter . For illustration purposes we have only shown the amplitude and Lyapunov exponent profiles of a single cell. The remaining cells exhibit a similar pattern. As we increase the value of gradually (Figure5-2d), the amplitude plot (Figure 5-2a) becomes more ordered and we can also see a drop in the average Lyapunov

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127 exponent (Figure 5-2b) calculated as a running mean from the same time series, suggesting a more ordered state. To investigate the dynamical interaction between the cells in the lattice, we computed the statistical measure called T-index, described in the previous chapter, which quantifies convergence or di vergence in the values of Lyapunov exponents estimated from multiple time series. The T-index at time t between cells i and j is defined as in (4.4). We have chosen a moving window lengt h of 100 points in our simulations. Figure 5-2c shows the average T-index profile calcul ated over all 5 cells in the CML and we see a gradual convergence among individual Ly apunov exponent profiles reflected by a decrease in the T-index value at higher values of coupling. It was also observed that increasing the value of beyond a certain value, kept the spatiotemporal amplitude pattern periodic but the Lyapunov exponent prof iles become more complex. To further describe the similarities between EEG dynamics and model dynamics we compared the changes in spatiotemporal patterns in both cases. Another example is shown in Figure 5-3 illustrating the changes in spatiotemporal patterns as we increase the value of the coupling parameter . For illustration purposes we have only shown the amplitude and Lyapunov exponent profiles of a single cell (ce ll 1). The remaining cells exhibit a similar pattern. As we increase the value of gradually as shown in Figure 5-3d, the amplitude plot, shown in Figure 5-3a becomes more or dered and we can also see a drop in the Lyapunov exponents (calculated as a runni ng mean) from the same time series, suggesting a more ordered state as illustrated in Figure 5-3b. Figure 5-3c shows the mean Lyapunov exponent profile calculated over all 5 cells in the CML. We can observe a gradual fall in the values of this global measure with increasing values of coupling.

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128 The average Lyapunov exponent calculated from a single channel of iEEG recorded in a rodent model of epilepsy is s hown in Figure 5-4b. It can be observed that the Lyapunov exponent profile shows a drop at the seizure as illustrated by high amplitude, rhythmic discharges in Figure 5-4a. Figure 5-2. Change in amplitude and spatiote mporal behavior in a five cell coupled map lattice with varying coupling. (a ) Amplitude spectrum and (b) Lyapunov exponent profile of a single cell; (c) Spatiotemporal dynamical profiles (Tindex) estimated from a five cell CML with (d) varying global coupling

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129 Figure 5-3. Change in amplitude and Lyapunov exponents in a coupled map lattice system with varying coupling. (a) Amplitude spectrum and (b) Lyapunov exponent profile of a single cell; (c) Mean Lyapunov exponent profile (L=5) estimated from a five cell CML with (d) varyi ng global coupling Figure 5-4. Change in amplitude and Lya punov exponents in the iEEG from an animal model of MTLE. (a) Amplitude spectrum; (b) Average Lyapunov exponent profile of a single channel of iEEG calculated from an animal model of epilepsy. Note the fall in Lyapunov e xponent corresponding to the seizure at the end of the epoch The similarity in dynamics of the CML a nd the iEEG segment are obvious in terms of the chaoticity prof iles as in there is a drop in va lues of the Lyapunov exponent as the amplitude spectrum in each case becomes more rhythmic and ordered. Furthermore we also observe that the average T-index al so falls corresponding to a convergence in Lyapunov exponent profiles when the osci llators become more synchronized

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130 (corresponding to regular rhythm in the simula ted signal). This is very similar to the phenomenon observed in the epileptic brain where, as described earlier, the T-index begins to fall and reaches a very low value during the seizure (rhythm ic discharges in the brain). This denotes that th e participating brain regions converge towards a similar dynamical state. In the case of the CML model, the convergence denotes that the individual oscillators approach a common dynamical state [44]. The remainder of this chapter is organi zed as follows. We start by introducing the problem of control of a dynamical system a nd the need for optimization based techniques for choosing the optimal feedb ack parameters. In the next section, we present some general feedback schemes for controlling the mean Lyapunov exponent from a system of globally coupled nonlinear maps. We also desc ribe a constrained optimization technique to solve for the optimal feedback paramete rs. We then present some numerical and experimental results and finally propose a learning scheme based on optimized feedback parameter selection to simulate dynamics of an epileptic brain using a globally coupled map lattice system. We think that insights gained from investigation into controlling spatiotemporal chaos in these models will help in formula ting similar control ideas in more complex systems such as the epileptic brain. Summary of Chaos Control Techniques During many years, control theory was pr imarily devoted to the study of linear systems. Recently a surge of interest in nonl inear controls, in part icular controls of chaotic systems, is reinven ting the discipline [190]. One of the landmark discoveries in control theory was the use of feedback. In feedback control, the difference between the desired response and the system output (i.e. the error) is brought back to the input of the

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131 controller that is designed to minimize the er ror. In this section we will review chaos controls and present our pr evious experience in the c ontrol of chaotic systems. The empirical evidence indicating the possi ble relevance of chaos to brain function was first obtained by Walter J. Freeman, th rough his work on the large-scale collective behavior of neurons in the perception of olfactory perception [ 191,192]. His findings suggested that a separate chaotic attractor is maintained for each stimulus and the act of perception consists of a transition of the sy stem from the domain of influence of one attractor to another. In a chaotic attractor, the system state may be, at any given time, infinitesimally close to any one of the infin ite periodic attractors but due to the highly unstable nature of the period ic orbits, the periodicity is never manifested over a measurable period of time. These characteri stics have attracted many researchers to the area, and many important works in the area of chaotic control were recently reviewed in [193]. The possibility that chaos can be controll ed efficiently using feedback schemes was described by Ott, Grebogi and Yorke (OGY method) in 1990 [194], i.e., converting the chaotic behavior of a system to a time-pe riodic one. A special ca se of the OGY method was introduced by Hunt [195], termed as occas ional proportional feedback which is used for stabilization of the amplitude of a lim it cycle and is based on estimating the local maxima (or minima) of the output. A modifi cation of proportional pe rturbation feedback (PPF) called stable manifold placement (SMP), which is simpler and more robust than PPF has been described in [196]. This met hod requires fewer assumptions to be made about the system parameters and has been us ed in modification of bursting behavior in hippocampal slices [197]. However, the c ontrol application is dependent upon the

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132 dynamics (since the perturbation is synchr onized with the intrinsic fly by around the unstable point). It also requires knowledge about the system, which most often is unavailable. An alternative is to modify the chaotic dynamics themselves into periodic orbits. During recent years there has been increasing interest in the method of time-delayed feedback [198] in which the control input is fe d by the difference between the current state and the delayed state. The delay time is determined as the period of the unstable periodic orbit to be stabilized. Hence, th e control input vanish es when the unstable periodic orbit is stabilized. In addition, this method require s no preliminary calculation of the unstable periodic orbit. Hence, it is si mple and convenient for controlling chaos. Applications of this method re ported include stabilization of coherent modes of lasers [199,200], control of cardiac conduction mo del [201], paced excitable oscillator described by Fitzhugh-Nagumo model widely used in physiology [202]. In addition, a robust local controller, the decen tralized delayed feedback control, has been proposed in [203] showing some advantages relative to th e conventional delayed f eedback control. As another robust controller, a simple feedback control design method was proposed in [204] by using some idea borrowed from the state observer approach. Especially, this method was demonstrated to be highly robust agains t system parametric variations in system parameters. Alternatives to anal ytical control algorithms consid ered to control chaos involve some intelligent control techniques, e.g ., neural networks [205-207], fuzzy control [208,209], etc. Specifically, neuro-genetic cont rollers for chaotic sy stems were proposed in [210] using large control adjustments and in [211] using small perturbations without a

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133 priori knowledge of the dynamics, even in the presence of significant noise. Recently, the cerebellar model articulation controller (CMAC) has been adopted in [212] for the control of unknown chaotic systems, and the weights of the CMAC were online tuned by an adaptive law based on the Lyapunov sense. Adaptive feedback control [178] is another method proposed by Ramaswamy et al. for steering a chaotic dynamical system to a target state with desired characteristics. The algorithms was designed to maintain a desired level of chaoticity and to achieve a target value of the local Lyapunov exponent or a loca l stretching rate. Th e adaptive control is effected by the additional dynamics wherein a control parameter is basically some value of stiffness times the difference between cu rrent and desired dynamics. The anticontrol technique, which is rapid, powerful and r obust, extends adaptive control methods for obtaining periodic orbits. The me thod has been applied to cont rolling neuronal spikes in in silico models [213]. In this dissertation, we w ill concentrate on this form of feedback control and describe a novel application of optimization methods in choosing the feedback control parameters. Adaptive Feedback Control The main idea behind controlling dynamical sy stems is to control apparently abrupt and intermittent transitions between dynamical modes of operation that are the mainstay of nonlinear chaotic systems. Some of the goals to be met while controlling spatiotemporal systems include formati on of specific spatiotemporal patterns, stabilization of behavior, synchronization/ desynchronization, suppression/enhancement of chaos, etc. The goal behind this adaptive f eedback strategy is to control some specific property of the system. The controllers are app lied in the feedback loops associated with

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134 every cell in the lattice structure, based on th e internal states of the system. The control input U* to the system can be defined as follows: ) * ( *k k kG U , (5.7) where G is the stiffness of control and * and are the desired and estimated values of the system property respectively. The target va lue of the system property can either be a constant or a time varying function. In the case of a multidimensional system, could be a global property of the system or so me property of individual subsystems. Additive Control For the system described by (5.1), this control strategy is implemented by the following general dynamical equations: , ) (1k k kU X F X (5.8a) ). * (k kG U (5.8b) For the logistic map as given by (5.3) the c ontrol dynamics can be written in the form: ) * ( ) 1 (1k k k kg x x x . (5.9) where g specifies the control stiffness for a single oscillator. Here the target value of the global Lyapunov exponent is a constant as opposed to a time varying function. The optimal value of the control parameter g needs to be worked out in any practical implementation of this strategy. For a coupled map lattice we use the mean global Lyapunov exponent as described by (5.10). L n k n kL1 __1 . (5.10)

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135 Figure 5-5. Probability distributions of finite step Lyapunov exponents of a logistic map with parameter = 3.9 controlled via a dditive control strategy Shown in Figure 5-5 is an implementation of simple additive control as described by (5.9) for a target * = 0.3. The probability distribution for finite time step Lyapunov exponents with different values of stiffness g is shown for = 3.9. The optimal value for the parameter g was numerically found to be 0.055. The solid line in the figure corresponds to the probability distribution of Lyapunov exponents when g = 0 or when there is no control. We observe that this type of control is very sensitive to the value of the stiffness parameter, wherein small va riations from the optimal value can be ineffective in driving the system to a desired level of chaoticity. Multiplicative Control Consider the system described by (5.1) with an additional controlled variable V. Here the control is implemented by changing th is variable using a feedback method. Let us consider a lattice with controlled maps

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136 ) , (1k k kV X F X, (5.11) where kVis a multiplicative control. For multip licative control of the lattice it is necessary to study a controllability problem . Suppose that the lattice includes only a single map with a scalar multip licative control that can be described by the following equation: ,... 3 , 2 , 1 , ) , (1 1 1 1 k X BU A U X F Xk k k k k, (5.12) where k X is the state vector, 1 kUis the scalar control, and B A, are real constant matrices of appropriate dimensions. If n=} 0 { n is the punctured n-space, then k k k k k i i kU U U U X X B U A X ] ,..., [ ); , (10 1 0. (5.13) Definition 5.1. A lattice is said to be controllable on n if for anynX X ,2 1, there exists a positive integer s and finite control sequence sUsuch that) , (1 2sU X X , where is a mapping factor. The main result is stated in the following theorem. Theorem 5.1. The lattice system given by (5.12) is controllable on n if there exist positive integers P, and Q such that for allnX we have: a) X X AP, b) rank HQ(X) = n, where HQ(X) = [BAQ-1X, ABAQ-2X,..., AQ-1BX]. The feedback control algorithm can be described by the following equations: ), , (1k k kU X F X (5.14)

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137 ). * (1 R U Uk k (5.15) where R specifies the control stiffness in th is case. This scheme, as with the previous one, is adaptive in nature in that the parameters that determine the nature of dynamics adapt themselves to yield the desired dynamics. This type of feedback has also been termed as “dynamic feedback control” in literature [178]. We demonstrate the implementation of this control strategy in bot h single and coupled map logistic maps, the monitored property being the mean Ly apunov exponent in the latter case. Figure 5-6. Multiplicative Contro l: Variation of the parameter as a function of iteration step for * = 0.30, and stiffness a) g = 0.001, and b) g = 0.02. The different curves correspond to different initial . Probability distributions of finite step Lyapunov exponents for for 0 = 4.0 and stiffness (c) g = 0.001, and d) g = 0.01 Figure 5-6 illustrates the feedback control strategy, for a target * = 0.3. Since there can be several values of the controlled parameter (corresponding to several different attractors) which gives the desired value of the Lyapunov exponent, the actual value of

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138 the controlled parameter takes depends on the stiffness of control, and initial conditions. The fluctuations in the controlled parameter are proportional to the value of the stiffness, converging to a single value for small sti ffness while exhibiting large variations for higher values of stiffness as seen in Figures 5-6a and 5-6b. The proba bility distribution of finite step Lyapunov exponents for both low and high values of the s tiffness parameter is shown in Figures 5-6c and 5-6d. As can be seen, even though the adaptive control works differently for small and large stiffness, the desired value of is maintained. Figure 5-7. Probability dist ributions of finite step m ean Lyapunov exponents of a CML model with five non-identical logist ic maps controlled via multiplicative control strategy, *[1,,5] = 0.3 Figure 5-7 illustrates the probability dist ribution of the mean Lyapunov exponent of a five cell CML model with five non-id entical logistic maps for a target *n = 0.3 for all n using a multiplicative control strategy. Additive with Multiplicative Control The third and final control strategy described in this chapter is a combination of both additive and multiplicative control. The implementation of control in this case

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139 follows naturally from the above two schemes and can be described by the following dynamics: ), * ( ) , (1k k k kG V X F X (5.16) ). * (1k k kR V V (5.17) Figure 5-8. Probability dist ributions of finite step m ean Lyapunov exponents of a CML model with five non-identical logistic maps contro lled via a combination of multiplicative and additive control strategy, *[1,,5] = 0.3 Figures 5-8 illustrates the probability di stribution of the mean Lyapunov exponent of a five cell CML model with five nonidentical logistic maps for a target *n = 0.3 for all n using a combination of additive and multiplicative control strategies. Optimization of Feedback Parameters Next we introduce a performance function th at can also be termed as an error function given below (5.18) that calculates the error between the target value of the system property and the computed value at ea ch time step. The goal of optimization is to minimize this error by choosing the most optimal feedback parameter for the system. If

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140 we choose the mean Lyapunov exponent as the monitored system property, then the function to be minimized is as follows: k *, (5.18) where * is the target value of the Lyapunov e xponent and for a coupled map lattice we can define the mean Lyapunov exponent as given by (5.10) The application of optimization techni ques in control of dynamical systems involves minimizing the error function de scribed by (5.18). The objective of the optimization technique in our examples is to find the optimal stiffness parameter that gives a constrained minimum of the error function. Consider the multiplicative control strategy as described in the earlier sect ion. We define the linear inequality RminRRmax and proceed to find iteratively, the value of R that gives the minimum of (5.18).The problem is formulated as follows: ) ( minRR subject to max minR R R . (5.19) We have used the Matlab optimization toolbox to solve this optimization problem. A sequential quadratic programming (SQP) method is used to solve this minimization problem. In this method, the f unction solves a quadrat ic programming (QP) sub-problem at each iteration. An estimate of the Hessian of the Lagrangian is updated at each iteration. A line search is performed usi ng a merit function similar to that proposed by [214,215]. A more detailed explanation of the optimization function can be found in the Matlab optimization toolbox documentation.

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141 Figure 5-9. Error calculat ed as a difference between mean global Lyapunov exponents and target Lyapunov exponents of a two cell CML with varying stiffness of control.R11 and R22 refer to control stiffness corresponding to cell 1 and cell 2 respectively We present the results from an iterative constrained optimization technique (Figure 5-9) to select the optimal parameters for a two dimensional lattice. The figure illustrates the variation of error with different co mbinations of the stiffness parameter R which in this case is a two dimensional diagonal ma trix. The choice of using optimization techniques to select the optim al stiffness is obvious since in the example shown, the error tends to decrease with increasing stiffness but is not quite entirely proportional to the stiffness values. In this particular example we were able to find two local minima for the error function. Application to Brain Disorders We have described different control tech niques for controlling spatiotemporal chaos in coupled map lattice systems. The natu re of control techniques is such that they are self-adjusting in nature and adapt themselves to direct the system into a desired chaotic regime. This method can be applied to systems with hidden variables and hence

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142 may be plausible in control design for a highl y complex system such as the brain where not all variables are known. We have also demonstrated a constrained optimization technique using minimization of an error func tion to select optimal control parameters. Such an optimization could be very useful in designing tighter and more controlled experiments. Figure 5-10. Schematic diagram for seizure c ontrol: control parameters of the stimulator are determined by a control algorithm utilizing the brain state (dynamical descriptor) Our goal is to seek a fully automated, effective and safe close loop control methodology that will be able to increase th e time between seizures , or in the best scenario, eliminate them, without interfering with the cognitive stat e of the subject. To this end, dynamical descri ptors of brain state (e.g. STLmax) need be estimated in real time from several brain sites and used as the observable output for the closed loop control scheme (Figure 5-10). In this work we c onsider a macroscopic modeling of the brain dynamics for the following reasons: (1) Our pr evious work shows that global dynamical descriptors of brain activity can be used to pr edict temporal lobe seizures, (2) This level of analysis simplifies the modeling. Indeed, we reason that if the epileptic brain is not always seizing, it is becau se it stabilizes in a tempor ary “healthy” operating point.

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143 Therefore, it will be easier to keep th e brain state close to this neighborhood than modeling of the brain dynamics themselves. However, the difficulties with this approach are that (1) the macroscopic variables may not be specific enough to quantify the underlying dynamics that cause epilepsy, (2) the causal relation between stimulation and changes in dynamical variables may be too w eak to be quantified (and perhaps be many to one). However we submit that the only way to answer these questions is to do the research due to the lack of mathemati cal understanding about brain function. We have however to be very cautious and realize the enormous difference in complexity and knowledge between the model sy stems being used in nonlinear controls and the distributed nature of a system such as the brain and the lack of obvious models and parameters to be controlled. For exam ple in the case of choosing the value of a correlate of R (stiffness parameter) one should have knowledge about the dynamic range of this value when it comes to application for seizure control and this in turn involves a large set of experiments for empirical de termination of parameters. Successful applications of control theory to brain dynamics must address in a comprehensive fashion at least the following items: (i) parsimonious variable selection for inputs, outputs and desired behavior, (ii) cons truction of dynamical models of the brain tissue and its interactions, (iii) appropri ate simplifying hypothesis to establish performance bounds, (iv) physiologic realism in the specifications . The feedback control scheme faces many challenges in epileptic seizure control. Ho w can the desired response, i.e. the normal brain state, be defined? Whic h is the brain output to compos e the error? Where to actuate in the brain (control input) give n that an error is available? How can we stimulate the

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144 brain tissue to take the system to the desi red operating point while being physiologically benign and transparent to normal brain function? Figure 5-11. Phase portrait of STLmax of a rodent seizure (grade 5) The epileptic brain during the normal state or interictal state seems to work at a relatively constant (Figure 5-11) short term Lyapunov exponent (STLmax) value (the control parameter) which means that a homeo static equilibrium point for the healthy brain dynamics may exist, as has been postulated by Haken am ong other researchers [216]. It is therefore plausibl e that the dynamical equation for brain dynamics can be written as a parametric function of the state x(t) and the homeostatic equilibrium d. The practical difficulty is to find d, but since this is a single constant parameter, we will conduct an efficient Fibonacci search to find an appropriate value. Tw o alternatives will be pursued to estimate d. The first alternative is to use the average STLmax during long periods as the desired response. The second alte rnative that could be pursued instead of a determined d is what has been called a “dead zone controller” where th e control loop will

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145 be open until the STLmax falls below a predetermined value (e.g. the value used for the seizure warning alarm). At that point, we use the value of the T-index as d, i.e. the system will try to stabilize the STLmax at that value. The second method has the advantage of avoiding any stimulation when the STLmax is within the normal region. Figure 5-12. Proposed learni ng algorithm for emulating e volution of Lyapunov exponent in a complex system with a coupled map lattice using feedback control and optimization. CO refers to the constrained optimization block It is well known from control theory th at the knowledge of the model makes the controller much easier to build and more accura te and robust. These properties extend to the design of controllers for nonlinear systems, taking the chaotic system to basically any point in state space, reliably and fast. Howe ver, when compared with direct control, model based control requires an extra step of system identification. Therefore, the optimality of the control actions comes at a pr ice of more complex implementation. In the specific case of seizure control, this means th at model based control requires an explicit

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146 model of the spatiotemporal brain dynamics, which is non trivial, but offers the possibility of the best performance (fewer seizures) and the most parsimonious control actions (best control efficiency) which are th e more appealing in terms of the patient’s quality of life. Therefore the a preliminary st ep would be to develop models for system identification. For dynamic modeling (i.e. system identif ication of chaotic time series), the primary method is iterative pr ediction. The results of this method have given very exciting results for real signals (laser instability and sea cl utter, for example), but they have not been applied to EEG. The high dimens ionality of the system (brain) and its nonstationary nature (i.e. time varying system parameters) are enormous difficulties. This is one of the reasons we propose here to m odel the dynamics of the EEG descriptors (STLmax is chosen as an example in the project, but others may be explored) via a learning model, which we expect to be easier due to the intrinsic simplification that occurs when we use order parameters We therefore propos e a learning algorithm in which a coupled map lattice system can be used to model the dynamical evolution of Lyapunov exponents (or any other property for that matter) in a complex system (Figure 5-12). The algorithm involves generating an error function between th e target Lyapunov exponent profile of the complex system and some nonlinear tran sformation of estimat ed lattice Lyapunov exponent values. The error is used to generate an optimized feedback input to the lattice. Such a learning algorithm can be used in de veloping realistic mode l of complex system dynamics and hence make the models more useful in the study and control of such complex systems.

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147 CHAPTER 6 A STATE DEPENDENT BRAIN STIMULATION PARADIGM FOR SEIZURE CONTROL Background and Significance Of the roughly two million people with ep ilepsy in the USA and Canada, about 30%-40% continue to have seizure despit e antiepileptic drug th erapy [217]. Although resective surgery is an option in such cases , the majority of patients with uncontrolled epilepsy will either not have access to surgic al therapy due to the marked limitation in availability of human and tec hnical resources and high techni cal complexity and cost or be poor candidates for surgical resection due to ubiquity or resulti ng debilitating effects. Consequently there is an imperative need to develop more practical and efficacious therapies and delivery modalities with fewer adverse effects and lower costs than those in existence. Brain stimulation in particular , is one form of therapy that has been proposed as an alternative to drug therapy. Electrical stimulation has the advantage of being both sufficiently reversible as well as adjustable in patients who would have otherwise been considered to be candidates for surgery. Se veral electrical stimulation protocols have been reported in both human epilepsy as well as animal models of epilepsy. Such techniques have been designed to directly mo dulate neuronal firing or to interfere with the synchronization of neuronal populations. Both subthreshold currents as well as superthreshold currents have been used to inhibit neurona l activity. There have been reports of uniform [218] as well as locali zed [219] DC electric fields attenuating the

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148 bursting in hippocampal slices, although th e former has been found to be highly orientation dependent. Others have investigated into anticonvulsant effects of low frequency periodic stimulation. Jerger and Sc hiff [220] reported a re duction in frequency of occurrence of tonic phase seizure episodes in the CA1 regions of hippocampal slices using for frequencies of 1.0 and 1.3 Hz. Schi ff et al. also employed low-frequency pulsed stimuli, whose timing was derived from a chaos control algorithm, with the aim of reducing the periodicity of high-potassium activity in the CA3 region. Their results showed that the system could be made more periodic or more chaotic by using a strategy of anti-control. However, it is not known to what extent the neuronal firing of the cells that generate the epileptic events was affected by the stimulus. Most of the stimulation protocols used in clinical studies th ough with the exception of vagus nerve stimulation (VNS) have em ployed frequencies upwards of 50 Hz. VNS uses output currents up to 3 mA, pulse width of 250~500 sec, and frequencies between 10 and 50 Hz (5 Hz for long term stimulati on). High frequency stimulation (>50 Hz) on the other hand, has been used in clinical sett ings to treat the symptoms of epilepsy for the past couple of decades. Stimulation targets for epilepsy have included the cerebellum, the caudate nucleus, the hippocampus, the thalamus-including the centromedian, anterior and subthalamic nuclei-the vagus nerve and the ep ileptic focus itself. Recent animal studies have begun to shed light on the mechanism of action of high frequency stimulation. Electrical stimulation of the anterior nucleus of the th alamus has been shown to have an anticonvulsant effect on PTZ induced seizures in rats [221]. Current levels between 300 and 1000 A at 100 Hz were found to have an anticonvulsant effect while low frequency stimulation of the same target was not effective in inhibiting seizures.

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149 Stimulation of the subthalamic nucleus us ing a 5 second high fre quency (130 Hz) train has been found to interrupt ongoing absence seiz ures in animal seizure models [222]. The effect of subthalamic nucleus stimulation ha s been reported to be frequency dependent [223]. Frequencies of 130 Hz increased clonic seizure threshold indicating an anticonvulsant effect while stimulation at 260 Hz did not cha nge the threshold. Stimulation at 800 Hz was found to slightly lo wer the threshold but the changes were not significant. Trigeminal nerve stimulation [224] at frequenc ies greater than 50 Hz has been found to reduce PTZ induced seizure activ ity although it can be quite a challenge to extend this to the human clinical cases as the nerve is involved in transmitting both somatosensory and pain information from the head. The caudate nucleus is another structure that has been explored as a target for stimulation for epilepsy. The effects of stimulation of the caudate nucleus were f ound to be frequency de pendent [225]. 10-100 Hz stimulation was inhibitory while 100 Hz stimulation increased seizure frequency. Low frequency conditioning stimulation of the epileptic focus (direct stimulation) has been found to suppress kindling caused by 60 Hz stim ulation, afterdischar ge duration and also seizure intensity. Goodman and coworkers showed that preemptive delivery of low frequency stimulation decreases the inciden ce of kindled afterdis charges significantly [226]. Clinical studies in patients with epilepsy have shown the antiepileptic effect of cerebellar stimulation. Contro lled [227] and uncontrolled [228] studies have reported improvement in a subset of patients included with the former citing a positive result in a high percentage of cases. Velasco and coll eagues reported improvement in seizure frequency and EEG spiking af ter bilateral stimulation of the centromedian nucleus.

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150 Typical stimulation parameters ranged fr om 60-130 Hz, 2.5-5.0 V and 0.2-1.0 ms duration [229-232]. Bilateral ante rior thalamic stimulation in five patients with various seizure types was found to cause a significa nt reduction in seizure frequency, with a mean reduction of almost 54% [233]. The ob served benefits however did not differ between stimulation-on and stimulation-off periods, suggesting the presence of a placebo effect. In other studies bilateral high fre quency stimulation of the anterior nucleus produced no observable changes in background EEG or in the interictal spike frequency [234]. Studies by Velasco [235] and colleag ues with hippocampal stimulation have revealed the inhibitory na ture of subacute continuou s hippocampal stimulation. Continuous high frequency stimul ation (130 Hz), at low-in tensity (200-400mA) of the hippocampus produced complete blockage of c linical seizures (bot h complex partial or associated to generalized tonic-clonic) and also significant reduction in epileptiform activity at the epileptic focus. However the au thors stated that appropriate interpretation of results would require studies in extracellular and intracellular recordings in humans. Similar studies have been conducted us ing amygdalohippocampal stimulation with comparable results [236]. Current applications of d eep brain stimulation for treatment of TLE as described above, do not take into account the potential improvements in the treatment paradigm that could be achieved with a more careful design of stimulus a nd timing parameters. Contingent or closed-loop s timulation techniques may be well suited to overcome some of the limitations of current therapies. Automated seizure detection based electrical stimulation modalities are currently in place and are already being tested in clinical settings. High frequency amygdalohippocampal and anterior thalamic stimulation in

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151 patients with MTLE, triggered by a seizure detection system, has been found to have some beneficial results [237,238]. These studies were designed with the goal of aborting or reducing the intensity of seizure activity and the evaluation of trials was based only on seizure intensity. A more desirable system for management of patients susceptibl e to seizures would be fully automated and capable of not only detecting a seizure, but of predicting the occurrence of a seizure as well. Such a sy stem would provide th erapeutic electrical stimulation capable of not only aborting a seiz ure in progress, but also of increasing the time between seizures and ideally eliminating them altogether, without interfering with the cognitive state of the subject. Such a system would be a marked improvement over state-of-the-art procedures in current clinical use, which are directed only at aborting seizures, not to preventing their re-occurrence in a s ubject prone to seizures. A critical feature of any automated seiz ure prediction/prevention system is the control mechanism by which the system determines when and where in the brain to deliver an electrical stimulus. Existing seiz ure intervention systems are controlled by what can be called a “nave” control methodology, meaning that they are either limited to measuring the results of the stimulation (sei zure severity and occurrence), or, when in closed loop, are triggered by a seizure occu rrence itself. Certain methods under investigation using in vitro paradigms comprising brain s lices are more sophisticated, utilizing control concepts involving chaos theory. Unfortunately, these experimental control systems have not been very successful, possibly due to the microscopic level chosen for the control variables. From the fo regoing, it is apparent that there is a clear need for improved control systems for use in fully automated closed-loop systems for

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152 seizure prediction and intervention. In this chapter, we will describe an alternate closed loop strategy which takes into account the dynamical state of the brain and delivers therapy during a seizure susceptible state with the goal of preventing seizures from even occurring in the first place. This chapter is organized as follows: The first section gives a brief review of the postulated and established mech anisms of some popular electrical stimulation methods. The second section will deal with acute hippoc ampal stimulation studies and explain the rationale behind our choice of the hippocampus as the stimulation target. The third section will describe in deta il the design and working of an automated state dependent (Seizure warning based) hippocampal stimul ation paradigm. Results from such an automated therapeutic strategy will be discusse d in the fourth section. Section five will describe the post hoc analysis on the stimulati on data to carefully investigate and identify the dynamical effects of regi onal coupling in therapeutic effects of such brain stimulations. Finally we will discuss the insights gained from such experiments and lay out future goals for this line of research. Neuronal Activity Modulation via Electrical Stimulation Uniform DC Electric Fields Electric fields generated endogenously by the nervous system can directly modulate neuronal activity [239] and are also important from a functional perspective [239-241]. It has long been r ecognized that electrical fields can also influence the excitation threshold of neurons [242-244].Uniform fields when applied with large field electrodes placed on the cortical surface for in vivo or across the tissue for in vitro studies have shown that DC electric fields (with amplitude s similar to those generated endogenously) can produce both excitation or i nhibition of neuronal activity depending

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153 on the orientation of the applied field with respect to the dendritic tree. A cathode placed near the basal dendrites produ ces depolarization of the de ndrites and cell body, thereby generating excitation. However, an anode placed in the same location produces hyperpolarization in that regi on and can inhibit electrical activity [245].The maginitude of electric fields required to modify the activity of a neuron is much smaller than the magnitude required for inducing neuronal firing from a resting state [242]. The geometry of pyramidal cells in the neocortex and th e hippocampus favor modulation of neuronal activity via electrical fields. Gluckman and co -workers [246] demonstrated that external DC electric fields applie d to brain tissue suppressed epileptiform activity. Localized DC Electric Fields Localized fields generated by point sour ces have the advantage of not being dependent on the orientation of neurons with respect to the elect ric field lines. They hence have been proposed to be more effec tive than uniform electric fields since the second spatial derivative of the extracellular voltage near the elec trode can produce large transmembrane currents [247,248]. The effects of local DC fields (subthreshold current levels) on spontaneous epilep tic events, analyzed in in vitro hippocampal slice preparations [249] included inhibition al ong with a suppression of spontaneously occurring epileptiform activity . The mechanisms underlying this suppression was an inhibitory polarization effect caused by tran smembrane currents generated by the applied field. Low level localized DC curre nts have also been tested in in vivo models such as the kindling model of epilepsy [250] and the m echanisms are still unde r investigation. The effects of DC stimulation were found to be reversible and th at stimulation itself did not result in anatomically evident damage.

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154 There is little or no evidence to suggest that either localized or uniform field stimulation is more effective in suppressing epileptiform activity in any epilepsy model. Both methods can completely suppress spontaneous activ ity generated in the lowcalcium, penicillin, and high-potassium models. The advantage of field electrodes is that they do not need to be in direct contact with the target tissue thereby minimizing electrochemical damage. The volume of tissue affected by the electric field is limited only by the size of the stimulating electrode. Clinical implementation of this technique would require that two large elec trodes be placed on either si de of the stimulation target, for example the hippocampus. Low Frequency Stimulation Low frequency stimulation in the test fre quency range of 0.1 – 10 Hz applied to the the Cornu Ammonis region of the hippocampus was fo und to reduce the frequency of occurrence of tonic phase seizure events for a very narrow input frequency range (1.0 – 1.3 Hz) [220]. Low frequency stimulation by magnetic stimulation has also been implemented with some degree of efficacy in pa tients with medically intractable epilepsy. Long-term depression (LTD) has been suggested as a mechanism underlying the effect of low-frequency stimulation. The term stands for a synaptic plasticity phenomenon first observed in the hippocampal sl ices whereby orthodromic stim ulation at low frequency (1 Hz) generates a long-lasting decrease in syna ptic efficacy [251], and is observed only at low frequencies. High Frequency Stimulation High-frequency stimulation in the freque ncy range of 50 to 200 Hz can activate neural tissue as well as induce secondary eff ects on CNS function, such as extracellular potassium accumulation [252,253] not observed during low-frequency stimulation.

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155 Bikson et al. showed in an in vitro model that high-frequenc y stimulating inhibits neuronal firing by inducing e fflux of potassium and bloc king depolarization. They proposed that an increase in extracellular potassium could t hus mediate suppression of epilepsy by high-frequency stimulation. Sinc e an increase in extracellular potassium levels cannot be maintained indefinitely, intermittent stimulation or triggered stimulation was proposed as a more effective method. High-frequency stimula tion applied globally using scalp electrodes (electro convulsive therapy) or via implanted electrodes targeting specific CNS (deep brain stimulation) or peri pheral nervous system (PNS) structures, is used in clinical settings to treat the symp toms of epilepsy. Using these paradigms, both stimulation and post-stimulati on anticonvulsant effects have been reported. Furthermore, the antiepileptic effects of hi gh-frequency stimulation have also been characterized in several in vitro and in vivo animal epilepsy models. An Emerging View of Seizure Control Previous studies have suggested that seizures may not just be abrupt transitions into and out of an abnormal state, but may actually follow a dynamical transition that evolves over time periods ranging from minutes to hours. Seizures are manifestations of recruitment of brain sites in an abnormal hypersynchroni zation. The onset of such recruitment occurs long before a seizure an d progressively culminates into a seizure. Therefore, seizures appear to be bifurca tions of a neural network that involves a progressive coupling of the focus with the nor mal brain sites during a preictal period that may last days to tens of minutes. The pe riod following a seizure reveals a sudden divergence in these descriptors thereby s uggesting that seizures may be intrinsic mechanisms to revert the brain to a normal (interictal) state. During this postictal state, time-irreversible resetting of the preictal dynamical recruitment is typically observed via

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156 a hysteretic loop. The preictal and postictal pe riods could be mathematically defined and, therefore, detected from the electroence phalogram. This dynamical view leads to a characterization of the seizure it self as a mechanism that is intrinsic to the epileptic brain to reset the preictal convergence when a cri tical mass of sites, or a mass of specific, critical sites, is recruited. The significance of this emerging view of seizures as dynamical resetting (divergent) mechanisms is that it opens up new avenues for controlling epilepsy. Sackellares and Iasemidis [167] postulate d that seizures are epiphenomena and a successful therapy should concentrate on th e resetting of the intermittent functional recruitment of the focus with the “normal” brain sites long before the occurrence of a seizure. According to this hypothe sis, seizures do not occur as long as there is no need to reset the brain. External interventions deli vered during the preictal period could aid in disrupting the transition to a seizure thereby alleviating the need for a seizure to happen. Electromagnetic stimulation and/ or administration of anti-epil eptic drugs at the beginning of the preictal period to disrupt the obser ved entrainment of normal brain with the epileptogenic focus may result to a significant reduction of epileptic seizures. Figure 6-1 illustrates this concept of dynamical divergen ce towards a “healthy” operating region by a control input on observed entrainment (c onvergence) among brain areas (“unhealthy” regime). The development of an automated seizur e warning algorithm as described in Chapter 4 that is capable of analyzing multi-channel EEG signals on-line in real-time offers the exciting possibility of using control devices to activate a brief, well-tolerated, therapeutic intervention in order to abort/prevent an impending seizure. We postulate that

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157 Figure 6-1. A desired effect of a controller based on resetting theory. u is the control input and y* is the target range of the controlled parameter the delivery of an electrical stimulation intervention during the preictal transition will prove to be an effective method for seizure co ntrol. We have named this system a “statedependent seizure control system”. Using rodent models of MTLE, we will analyze recordings from implanted electrodes on-line and automatically de liver an electrical stimulus to the chosen target, based on the automated seizure warning algorithm. A detailed explanation of such a control scheme will be given in later sub-sections. Acute Hippocampal Stimulation The hippocampus was chosen as the stimul ation target in this study due to a number of reasons. The hippocampus has been found to play a major role in the genesis and propagation of seizure in humans. There ex ists sufficient clinical evidence to believe that temporal lobe seizures are initiated from and/or pr opagated through the hippocampus [254,255]. Structural damage, most fre quently in the hippocampus and adjacent parahippocampal structures ha s been found in the brains of epileptic patients. This pathological condition has come to be known as hippocampal or mesial temporal sclerosis. Further evidence supporting the role of the hi ppocampus in epilepsy was the finding that excision of this structure could pr event further seizures in select patients

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158 [256]. Hippocampal slices even when isolated from the rest of the brain retain their propensity for generating epileptiform discha rges. The hippocampus has been extensively studied in animal models for its role in ki ndling and kindled responses. In the rat model of limbic epilepsy, even though there was no fixed pattern of seizure onset, the hippocampal structure played an important role in either generation or the propagation of the seizures [76]. Moreover, the hippocam pus is a central part of the classic Papez circuit (that approximates the hippocampo-mammillo-c ortical circuit). Hence the hippocampus becomes a natural choice as an intervention ta rget for controlling or disrupting seizure activity. Subacute chronic high frequency electrical stimulation applied at the hippocampus and parahippocampal cortex [235,257] has been shown to have an anticonvulsant effect producing complete blockage of clinical seizures and also causes significant reduction in epileptic activity at the epil eptic focus. Similar studies have been conducted using amygdalohippocampal stimulation with comp arable results [236]. Other unblinded studies have shown beneficial but smaller effects and long-term effects of hippocampal stimulation without adverse effects [258]. Velasco and colleagues suggested that the antiepileptic effect of acute hippocampal stimulation could invol ve among others, (i) appearance of an electroposit ive DC shift and monomorphi c 1.5/delta activity; (ii) increased threshold and decreased duration of the hippocampal-induced afterdischarge; and (iii) flattening of the recovery cycles of hippocampa l responses evoked by pair-pulse amygdaloid stimulation. Hence it was proposed that AHS also produces extratemporal inhibitory effects that may regulate initiati on and propagation of temp oral lobe seizures

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159 [259]. These promising reports further motivated us to select the hippocampal structure as the site of intervention in the seizure control experiments described in this dissertation. State Dependent Brain Electrical Stimulation There is general recognition that electrical stimulation of brain structures involved in the TLE loop can affect the seizure outcome. In contrast to previous ad hoc approaches to combinations of pulse amplitudes and tim ings for brain stimulation, the paradigm described herein incorporates methods derive d from control theory for exploiting the time dimension, based on measured values of a control variable. Expe rience from control theory applied to other fields has shown that in complex systems, the control input normally has a narrow dynamic range, and its timing and strength profoundly affect the global dynamics. In complex dynamical systems, the control action must be precisely determined from the present dynamical stat e and the “error”, which is better achieved with closed loop control. The paradigm relates generally to methods of controlling delivery of electrical stimulation in a closed loop system for wa rning of a seizure susceptible state and intervention. More particularly, operation of th e system (i.e., control of the delivery of therapeutic electrical stimuli aimed at interrupting, delaying , or preventing the occurrence of an impending seizure) is dependent upon the st ate of the brain area to be treated (i.e., is “state-dependent”). This sophisticated le vel of control is achie ved by detecting abnormal seizure-related electrophysiol ogical characteristics of brai n activity during the preictal state of an epileptic seizure. Successful intervention, provided in the form of appropriate electrical stimulation, relies on accurate de tection of particular electrophysiological patterns of brain wave activity as determin ed from EEG/iEEG recording signals that exhibit identifiable changes during the pre-seiz ure (preictal) state. Because particular

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160 seizure-associated patterns of brain electrica l activity are registered in the controller system in advance of an impending seizure, th e system is capable of predicting a seizure susceptible state, and interven ing in advance of its progressi on from the preictal state to the state of seizure. An overview of an exem plary closed loop seizure control systems in accordance with this paradigm is pres ented schematically in Figure 6-2. Figure 6-2. An automated seizure warni ng based state dependent closed-loop seizure control system In the closed loop control system of the paradigm, particular seizure-associated features (dynamical descriptors) of the br ain electrical activity, as detected in electrophysiological recordings, are characte rized using automated algorithms and the processed information is used to provide i nput to a controller that interfaces with an electrical stimulator. The stimulat or is used to deliver an appropriate electrical stimulus to either the affected areas of the epileptic brain from which the abnormal patterns arise or to another strategic target th at plays an important role in the modulation of neuronal dynamical activity. The closed loop system is interfaced with multip le electrical leads (electrodes) implanted in areas of the ep ileptic brain. Based on feedback from the electrodes, information is processed by the controller and electrical stimulation is delivered in a precisely tailored fa shion to selected electrodes. Accordingly, the paradigm constitutes a closed loop neuroprosthetic system for seizure warning and therapeu tic intervention. The system includes a detection module

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161 that detects and collects elec trophysiological information detectable by sensors placed in neural structures in a subject. Also included in the neuroprosthetic system is an analysis module that evaluates the acquired electrophys iological information, and in real-time extracts features associated with a pre-seizure state in one or more monitored neural structures in the subjec t. From the nature of the extract ed features, the analysis system determines when electrical stimulus intervention is requi red. Included in the control paradigm is an electrical stimulation inte rvention system that provides electrical stimulation output signals having desired stim ulation frequency and stimulation intensity to a neural structure. The realization of the closed loop neuropros thetic system in the present dissertation is achieved by an algorithm for automate d seizure warning (ASWA). The ASWA can include algorithms for a performing variet y of functions, includi ng but not limited to programs for dynamical analysis of EEG signals, for selecti on of particular electrode groups registering a seizure-associated st ate for further monitoring and delivery of therapeutic stimulation, and for statistical pa ttern recognition. However in this study we have not used algorithms to dynamically select particular electrode s or electrode groups for stimulus delivery as the electrodes have been pre-selected based on effects of acute hippocampal stimulation studies discussed earlie r. Future experiments incorporating such a dynamical selection are planned. The system hence provides a method for preventing or delaying a seizure. System Description The basic blocks constituting the experi mental setup for an automated seizure warning based stimulation is illustrated in Figure 6-3 and consisted of two main parts: (1) Data acquisition and stimulation and (2) Data Analysis and Control. The animals are

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162 housed in specially made chambers (Figure 64) and each animal is connected to a long term acquisition system via analog amplifiers (Grass Telefactor).The amplified iEEG signals is then transferred to an acquisiti on computer running the acquisition software (Stellate Harmonie 5.2) via a 12 bit A/D converter (National Instruments Inc.). A parallel processing, custom written applic ation running with the acquisition software streams the data into a secondary unlocked fi le which is written directly into a RAID server along with original iEEG signal files and video. This secondary file is then read by the automatic seizure warning algorithm (describ ed in Chapter 4) that runs on a separate PC and generates control signa ls for activating the stimulation device. All communication between PC1, RAID server and PC2 are via Et hernet connection. The control signals are fed to a custom made switching box using a CK1601 PC parallel po rt relay board, the schematic and figure of which is given in Figu re 6-5. On receiving the control input from PC2, the hippocampal electrodes are disconnect ed from the amplifier and are connected to the stimulator, while all other channels are blanked (shunted to ground), to prevent damage to the amplifiers. The control signals are also fed to the trigger of the stimulating device (Model 2100, AM Systems Inc.) that de livers a pulse train to the animal’s brain via the stimulating electrodes. A delay is provid ed in the software to make sure that the channels have been switched to the right m ode before the stimuli is delivered. A basic flowchart of the software control is provided in Figure 6-6. The channels are switched back to their original recordi ng mode once the stimulus train is delivered. The data pins of the parallel port are used to control the switching of channels from Record to Stimulate mode and vice versa. The cont rol pins are used to control the trigger of the pulse

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163 stimulator. The procedure is repeated each time the algorithm meets the user defined conditions for intervention. Figure 6-3. Block schematic of components and design of an automated seizure warning based electrical stimulation para digm in the animal model

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164 Figure 6-4. Long -term iEEG/Video Monitoring and Stimulation Station Figure 6-5. The CK1601 PC parallel port relay board with 8 terminals

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165 Figure 6-6. Flow chart of the softwa re control for automated stimulation Stimulation Methodology and Control Experiments were performed on two month old male Harlan Sprague Dawley rats (n=5) weighing 210-265 g. The first three anim als were used to test the effect of hippocampal stimulation on dynamical converge nce among brain areas and the last two

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166 animals were used to evaluate the effects of stimulation on seizure frequency. Protocols and procedures were approved by the University of Florida Institutional Animal Care and Use Committee. The short term stimulation experiments were conducted in the Children’s Miracle Network An imal Neurophysiology Laborator y at the University of Florida and the long-te rm ASWA based experiments were conducted at the Animal Care Services facility in the McKnight Brain Institu te, University of Florida. Each animal first underwent a procedure for determ ining its afterdischarge thre shold. Biphasic square wave pulse trains were delivered us ing bipolar electrodes in th e ventral hippocampus, with the two prongs of the electrode acting as the anode and cathode. With the following stimulation parameters constant, (i) freque ncy = 125 Hz, (ii) train duration =10 seconds, and (iii) pulse width = 400 seconds (to ensure sufficient current spread), the output current intensity was increased from an initial low value in small increments (10 ~ 20 A) until ADs were observed in the simultaneously recorded iEEG. Monitored channel pairs included a hippoc ampal channel and its contralateral frontal cortical channel (i.e . left hippocampus-right front al and right hippocampus-left frontal). When the T-index calculated from either pair crossed an upper threshold (UT = 5, significance level, < 0.00001) to a lower threshold (LT = 2.662, significance level, = 0.01), the ASWA (see Figure 45 and Figure 6-8), issued a stimulating pulse train via the bipolar electrode in the hippocampal side subjected to continuous stimulation during the model creation phase (see chapter 2). The critical thresholds were chosen from tdistributions based on earlier human studies [172]. The pulse train has the characteristics: output current intensit y = 50-150 A (well below afte r-discharge threshold of each animal); frequency = 125 Hz; pulse width = 400 s and train duration = 10 s. The animals

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167 showed no perceivable change in behavior dur ing delivery of stimulus. These parameters were partly based on previously reporte d anticonvulsant effects in human studies [235,236]. AHS Effects on iEEG Dynamics Acute hippocampal stimulation exhi bited no discernable effects on STLmax values when entrainment was not observed. The deliver y of an external electrical stimulus with the parameters outlined in the previous section when the state space coupling (T-index) reached a critical low threshold, more ofte n than not a significant amount of resetting (rise in T-index) was observed (Figure 6-9). During non-automated delivery of these external interventions, it wa s observed that the time within which the stimulus was delivered after a threshold crossing seemed to be a major factor in the ability of these stimulations to be able to produce a dynami cal divergence. Longer wait periods after a threshold crossing seemed to be less e ffective in producing a dynamical divergence (resetting) of monito red brain sites. Table 6-1. Summary of stimulation results Subject NStim D ND IIB (Nsz) IIS (Nsz) 1 12 9 3 2 12 11 1 3 12 8 4 4* 8 5 3 2.7 1 (11) 7.2 1.3 (4) 5* 25 18 7 10.6 12.4 (13) 32.1 9.7 (3) **Seizure frequency compared in these animals; D-Divergent; ND-Non-divergent; IIBBaseline interseizure interval in hours; IIS-Stimulation phase interseizure interval in hours. A total of 36 stimulations (see Table 6-1) were delivered to the first set of rats (N=3) to evaluate the dynamical effects of high frequency stimulation. A reversal of dynamical convergence (rise in T-index) in the monitored electrode pair was observed 77.7% (28/36) of the time. This reversal of convergence was observed to be sustained for

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168 a significantly long period of time following the stimulation (see Figure 6-10), thereby eliminating the possibility of stimulus artif act in the iEEG being responsible for the observed phenomenon. Diffused suppression of elec trographic activity was also observed following divergent stimulations (Figure 6-7). Eight out of the 36 stimulations had very little effect on the dynamical measures (no di vergence), and no disc ernable changes in electrographic background activity followed th ese stimulations. The changes in EEG and STLmax due to a timed electrical stimulus are sh own in Figure 6-7. In some cases a failed stimulation (no divergence) was followed by a seizure; an ex ample of such an event is shown in Figure 6-11. Figure 6-7. iEEG (top), STLmax (middle) and T-index (bottom) plots before and after a stimulus. Shaded area repres ents stimulation artifact

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169 Figure 6-8. Automated seizure warning system showing a warning prior to the onset of a seizure. Black arrow indicates the beginning of the seizure Figure 6-9. Automated seizure warning system showing an example of stimulation of the hippocampus after a warning. Note the re setting (rise in T-index) after the stimulation. Shaded area represents stimulus artifact

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170 Figure 6-10. Sustained reversal of dynamical convergence (T-index) among hippocampal and frontal cortical br ain regions by acute hippocampal stimulation Figure 6-11. Automated seizure warning system showing a seizure warning and stimulation. Note that the stimulation was not able to reset the T-index and a seizure ensued shortly AHS Effects on Seizure Frequency We also evaluated the effect of AHS on seizure frequency in two animals, comparing a baseline seizure frequency befo re AHS and the seizure frequency during the stimulation phase. We found a significant redu ction in seizure fre quency (increase in

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171 interseizure interval) during the stimulation phase. The mean seiz ure interval in both animals was increased by ar ound a factor of 3 (Table 6-1) suggesting a strong anticonvulsant effect of AHS in this animal model. Stimulations in these two animals were also observed to be eith er divergent or non-divergent in nature. All stimulations that were followed by the occurrence of a seizure shortly afterwar ds were observed to be nondivergent in nature. Figure 6-12. Seizure distribution before and after a stimulus block. Note the significant reduction in inter-seizure in terval during the stimulus block, compared to the pre-stimulus and post-stimulus blocks. The difference of mean inter-seizure intervals between pre-stimulus and poststimulus blocks was not significant (p > 0.5, by Wilcox Rank-Sum test) In the animals included in the s timulation study, a convergence in STLmax values (dynamical convergence) was followed by a se izure after a mean duration of 21.6 minutes. Stimulation following a dynamical en trainment, when the T-index was below a lower threshold (LT = 2.662), was found to delay th e occurrence of a seizure by 203.7.1 minutes. In the example illustrate d by Figure 6-12, the mean inter-seizure interval was 2.7 hrs during no stimula tion and 7.2.3 hrs during state dependent stimulation.

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172 Post hoc Analysis Since preliminary analyses revealed th at acute hippocampal stimulation was not always followed by a reversal of observed dynamical convergence among brain regions, we performed a retrospective investigation to look for diffe rences between divergent and non-divergent stimuli. To investigate the differences we compared multiple dynamical measures estimated from the iEEG data recorded just prior to the delivery of the stimulus. A bivariate measure of interdependence (average mutual information), also considered as a nonlinear cross correlation f unction, between electrodes in the monitored pairs was estimated. The mutual information (M I) was calculated according to (3.26) (see Chapter 3). The average MI between all mon itored pairs was calculated for a 5 minute window immediately prior to the time of s timulus delivery. We used the same window length of 2048 points as that used in the estimation of STLmax. We acknowledge accuracy limitations in the histogram method of calcu lating probability density functions from experimental data [125]. The MI was calculate d using standard functions available in the MATLAB software package. We also compared univariate measures including STLmax and Approximate Entropy (ApEn) for all brain regions t oo investigate differences betw een states before divergent and non-divergent stimulations in the MTLE model. The estimation of ApEn was done according to (3.24) with the same window length of 10.24 seconds. Both STLmax and ApEn values were embedded in a three dimensi onal space and plotted to see if they could be classified into separate clusters in stat e space. For illustration purposes we have shown the 3-D plots of both dynamical values for multiple stimulations in a single animal.

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173 Figure 6-13. Comparison of mutual info rmation values before divergent and nondivergent stimulations Figure 6-14. Comparison of STLmax values before divergent and non-divergent stimulations

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174 Figure 6-15. Comparison of approximate entropy values before divergent and nondivergent stimulations In all 5 animals included in the study, the average MI among monitored brain regions was significantly diffe rent for divergent and non-divergent stimulations. The average MI estimated from a 5 minute EEG epoch immediately before stimulations was found to be significantly higher for divergent stimulations (F igure 6-13). Comparisons of raw dynamical values from each electrode suggest ed that the dynamical state of the brain regions were significantly diffe rent before divergent and nondivergent stimulations as can be clearly seen in Figures 6-14 and 6-15. Discussion Results of the present study suggest subs tantial effects of high frequency acute hippocampal stimulation on both iEEG dynamics as reflected by the changes in values of T-index calculated from STLmax. Preliminary findings also suggest that hippocampal

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175 stimulation based on changes in EEG dynamics, seemed to also have an anticonvulsant effect as evidenced by the decrease in seiz ure frequency and longe r seizure free periods in this spontaneously seizing animal mode l of MTLE. We suggest that this observed anticonvulsant effect could involve dynamical divergence or reversal of dynamical convergence among brain regions. The difference in MI values as well as the short term maximum Lyapunov exponent values and approxi mate entropy values calculated before divergent and non-divergent stimul ations suggest that more in formation about the state of the system (brain) could ai d in the design of more e ffective control strategies. It has been suggested that therapeutic in terventions with the goal of dynamically resetting the brain early in th e preictal period could alleviat e the need for a seizure to occur. The ability of AHS to cause a modulat ory effect on EEG dynamics similar to that seen after the occurrence of a seizure and the observation of longer seizure free periods after such changes, leads us to postulate that the antic onvulsant nature of hippocampal stimulation may involve reversal of a preictal dynamical stat e. Moreover, even though all non-divergent stimulations were not followe d by a seizure, it wa s observed that all stimulations immediately followed by a seizure shortly thereafter failed to reset or reverse the dynamical convergence in the monitored elec trode sites. This further leads us to believe that reversal of dynamical convergen ce among brain areas play s a significant role in the ability of AHS to preven t the occurrence of a seizure. We speculate that the reasons for all non-divergent stimulations not being followed by a seizure could be that the dynamical convergence seen among brain region s that triggered the stimulus may have been false positives.

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176 Suppression of electrogra phic epileptiform activity in the hippocampus has been shown in both in vivo and in vitro animal st udies as well as human hippocampal slices [252,260,261]. In this study we have observe d similar suppression in electrographic activity corresponding to divergen t stimulations. It would be interesting to study if the divergence was also accompanied by phase de synchronization as has been reported with in vitro hippocampal studies [262]. Failure to cause a reversal in a subset of AHS trials raises qu estions about avenues for improvement such as more careful desi gn of stimulation parameters, choice of alternate locations, all of whic h could potentially play signifi cant roles in the outcome of such interventions. The difference in MI values before diverg ent and non-divergent values suggests that automated stimulati on based on more than one descriptor could potentially help achieve the desired state more consistently. Pr evious studies have proposed and shown the utility of seizure detection and forecasting based on multiple descriptors [263]. Whether taking into account the MI among brain areas along with other dynamical measures such as T-index of STLmax values would provide a better indicator of a seizure susceptible state needs to be inve stigated. One could also envision a control strategy that continuousl y monitors the responses to interv entions and continues to repeat the process until a desired condi tion is met or the target stat e of the brain is reached. More sophisticated control methods such as model based schemes and adaptive intervention schemes that continuously adap t control parameters, targets, according to feedback from the brain could also potentially be useful. Due to the ad hoc ways that the stimula tion has been performed so far, this is considered strong suggestive evidence that more sophisticated methods exploiting the

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177 time dimension (i.e. the timing of the stimul ation) and stimulation based on measured values of control variables will be successful . More recently studies have used continuous EEG monitoring and pattern rec ognition algorithms to identify the onset of seizures and apply hippocampal electrical stim ulation with the goal of abor ting or reducing the seizure activity. However it would be more desirable if one could prevent seizure activity from even beginning. We propose that timed acute hippocampal stimulation combined with an online state monitoring system that monitors multiple dynamical descriptors could be a useful therapeutic alternative fo r controlling seizures in TLE.

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178 CHAPTER 7 CONCLUDING REMARKS AND FUTURE DIRECTIONS Summary Brain Dynamics in the MTLE Model Gradual dynamical transitions in spatiotempor al characteristics in the iEEG, similar to those seen in human temporal lobe epile psy can be observed in an animal model of spontaneous limbic epilepsy. The transition towards a spontaneous seizure appears to represent a temporal evolution towards a mo re ordered state as demonstrated by the changes in dynamic measures shortterm largest Lyapunov exponent (STLmax) and average angular frequency ( ) during the ictal period. Furthe r, the results suggest that the signal processing measures used in this study may prove useful in identifying seizure prone states in spontaneously seizing, fr eely moving rodent models of MTLE. Such identification may be possibl e by monitoring the converg ence or divergence of the nonlinear measure, STLmax and to a lesser extent by monitoring changes in calculated from multiple brain areas. It is interesting to note that dynamical values between different brain areas notably diverge followi ng seizures. This fi nding suggests that seizures somehow “reset” brain dynamics, as manifest by significant change in spatiotemporal measures using dynamical analysis. The above observations are consistent with those previously described in human TLE [36,154-158]. Preliminary results from quant itative EEG analysis indicate that it may be possible to use the rat model of limbi c epilepsy as a tool to investigate the dynamical transitions associated with disease mechanisms in human TLE. The

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179 observations also suggest th at the MTLE model may be a good model for developing algorithms that can eventually be used in the analysis of human epile psy. Similarities in spatiotemporal dynamical properties between the MTLE model and humans, as revealed from quantitative nonlinear EEG analysis may reflect a similarity in underlying network properties and their role in the gene sis and expression of these seizures. Nonlinearity and temporal dynamics EEG (iEEG) is a complex signal whose st atistical properties depend on both time and space [13]. The characteristics of an iE EG signal such as, the existence of limit cycles, bursting behavior, amplitude dependent frequency behavior, hysteresis, frequency multiplication, are among the list of typical properties of a nonlinear system (Jansen, 1991). The presence of highly si gnificant nonlinearities in electrographic signals supports the concept that the epileptogenic brain in th is model of epilepsy is a nonlinear system. The sudden increase in the corre lation integrals of the orig inal time series and the surrogates during the postictal period is attributed to the increased autocorrelation in the iEEG during this period [138,139]. In all animals analyzed, a significant re duction in iEEG chaoticity was observed during the ictal period. STLmax values correspond to the de gree of order of a dynamical system. In the case of a complex structure such as the brain, it may give an insight into the order or synchrony of underlying neur onal circuitry. The lowest values of STLmax were observed during the ictal period. This reduction in values suggests that the underlying neuronal circuitry re sponsible for generation of the iEEG signal fire in a synchronized or organized fashion (typical neuronal behavior during a seizure). Conversely, higher STLmax values could correspond to a more random neuronal firing, independent of each other. In the case of average angular frequency, the preictal, ictal

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180 and postictal states correspond to medium, high and lower values of respectively. The highest values were observed during the ictal period, and higher values were observed during the preictal period than during the postictal period. This pattern roughly corresponds to the typical observation of highe r frequencies in the original iEEG signal during the ictal state, and lower iEEG frequencies during the postictal state. Spatiotemporal dynamics Perhaps the most exciting discovery to em erge from nonlinear quantitative analysis of the EEG time-series in human temporal lobe epilepsy is that seizures are preceded by spatiotemporal dynamical changes whic h occur several minutes before the electroencephalographic seiz ure onset. In all animals, pre-seizure periods are characterized by a gradual convergence (transi tion from a high T-inde x value to a low Tindex value) among specific electrode site s followed by a high degree of dynamical divergence (increasing T-inde x) during the postictal period. From a mathematical perspective, this observation suggests that seizures emul ate the formation of selforganizing spatiotemporal pa tterns. From a dynamical pe rspective, the low T-index values during the ictal period, reflects a stat e of increased dynamical interaction between the hippocampus and the frontal cortex du ring a seizure [16]. From a biological standpoint it may suggest that the hippocampus recruits the fr ontal cortex progressively resulting in a seizure. The convergence of STLmax values was observed to be more specific to grade 5 seizures in this model than convergence in values. The T-index profiles calculated from values exhibit rapid and frequent fluctuations corresponding to sudden morphological changes in iEEG. A retrospectiv e visual correlation of variations in

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181 values with raw iEEG revealed that this m easure was sensitive to most state changes in the iEEG including changes preceding the grade 5 seizures analyzed. Therefore, we conclude that STLmax would be a more useful parameter than in any seizure warning algorithm developed for this model. Similarities with human MTLE The results presented in this study reve al a number of similarities in dynamical properties between rat MTLE model and human temporal lobe epilepsy. The presence of highly significant nonlinearities in the rodent model iEEG is similar to findings in intracranial recordings obtai ned from humans. The sudden increase in the values of correlation integral following a seizure is observed in both the MTLE model and human TLE, hence suggesting similarities in the postictal states. The analysis of spatial and temporal dyna mical patterns of long-term intracranial EEG recordings, recorded for clinical purpose s in patients with medically intractable temporal lobe epilepsy has demonstrated preictal transitions characterized by progressive convergence of dynamical measures (STLmax) at specific anatomical areas. These dynamical changes have been attributed to spatial interactions or synchronization between the underlying nonlinear components (neurons) of the brain. We observe similar dynamical changes during the preictal, ictal and postictal states in the MTLE model. A two sample test for equality of proportions of seizures that exhibited a detectable preictal dynamical convergence of STLmax (MTLE rats=26/28; Humans=53/58) and (MTLE rats=26/28; Humans=21/24) va lues revealed no significa nt difference between humans and the MTLE rats analyzed (p-value = 0.8143 for STLmax and 0.856 for ).

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182 Seizure Predictability in the MTLE Model Studies using methods derived from non linear system analyses, applied to electroencephalogram recordings from intracr anial depth electrodes implanted in patients during evaluation for epilepsy surgery, anticipated seizures with prediction times of several minutes to hours [154162]. Application of simila r techniques to the MTLE model has revealed similar state transitions from the interictal to the ictal state as those found in human patients with epilepsy. Th e results from inves tigation into brain dynamics in this model of MTLE suggested that it may be possible to warn of an impending seizure by monitori ng the convergence of dy namical descriptors among different brain regions. Further, this transiti on from interictal to preictal state may be detected several minutes before the actual seizure onset. In the present study, we have used a s pontaneously seizing model and techniques previously employed in the analysis of human epilepsy to show seizure predictability. To our knowledge this is the first report in which seizure predictability is demonstrated in an animal model of spontaneous epilepsy. It is very encouraging to not e that seizure warning may be possible not only in ex perimental seizures [34], but also in spontaneously occurring epilepsy that has many of the char acteristics associated with human MTLE. We have demonstrated the usefulness of an automated seizure warning scheme in a model of intractable human MTLE by using a pharmacoresistant limbic epileptic rat in which seizures occur spontaneously, intermittent ly and at varied intervals. In addition, EEG data sets previously used to test the algorithm were obt ained from patients undergoing pre-surgical evalua tion and hence had short inte r-seizure intervals (mean 10 hours). Such data sets have less likelihood of having enough interi ctal intervals for reliable estimation of false warning rates. The mean inter-seizure interval in this animal

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183 model is much longer than the human cases analyzed in previous studies by our group, and therefore the evaluation of the algorithm in this model could be more meaningful with respect to clinical applications su ch as portable seizure monitoring devices. Markers of Epileptogenesis The results from the application of techniques developed for studying seizure predictability to iEEG recordings during the latent period of the MTLE model revealed that the spatiotemporal transitions involvi ng different brain regions may be used as markers for studying epileptogenesis. Furtherm ore these spatiotemporal transitions could potentially be used as predictors for the development of epilepsy. Epilepsy research during the past years has focu sed on developing markers of epileptogenesis based mainly on imaging techniques. In this dissertation we propose a nove l application of statistical and nonlinear methods in the study of the process of evolution into epilepsy [36]. Coupling imaging techniques with other predictive techniques such as the one described in this dissertation to clinical outcome in early preclinical and clinical studies could represent a useful experimental approach for identifying therapeutic ap proaches geared at preventing or modifying the development of epilepsy. Optimization Based Control Strate gies and Modeling EEG Dynamics In this thesis we have described a coupled map lattice model for emulating preseizure brain dynamics. This is accomplis hed by interconnecting similar nonlinear oscillators and varying the global coupling term appropriately to get the desired dynamics. We have described different contro l techniques for controlling spatiotemporal chaos in these coupled map lattice systems. The nature of control techniques is such that they are self-adjusting in nature and adapt them selves to direct the system into a desired chaotic regime. This method can be applied to systems with hidden variables and hence

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184 may be plausible in control design for a highl y complex system such as the brain where not all variables are known. We have also demonstrated a constrained optimization technique using minimization of an error func tion to select optimal control parameters. Such an optimization could be very useful in designing tighter and more controlled experiments. Such modeling techniques coul d be used for designing model based control techniques that will be discussed in later sections of this chapter. State Dependent Therapeutic Inte rvention for Seizure Control Preliminary results from state dependent electrical stimulation of the hippocampus revealed significant effects on EEG dynamics as reflected by changes in spatiotemporal measures of the recorded br ain electrical activity. Since an anticonvulsant effect was observed in cases where the stimulations were able to successfully cause dynamical divergence among brain areas, we postulate that reported anticonvuls ant effects of high frequency acute hippocampal stimulation may involve change in system dynamics. Currently the mechanisms by which high freque ncy stimulation are ei ther not well known or have been explained on the basis of neuronal chemical imbalances and neurophysiological inhibition and reduced firing rate. To our knowledge this is the first report on experimental data s uggesting that the anticonvulsan t nature of high frequency stimulation could involve dynamical changes in the epileptic brain. Furthermore, based on results from post hoc analysis of stimulation data, we propose an improvement to the automated seizure warning based state de pendent intervention by including multiple dynamical descriptors to improve the timing of stimulus delivery a nd thereby potentially improve the efficacy of such alternative therapeutic approaches.

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185 Future Directions The results from seizure predictability studies suggest that the dynamics governing the transition from the interictal state to the se izure state in this model are similar to those observed in human TLE. This phenomenon may be detectable using other measures such as the stochastic approach used in drug-induced seizures as mentioned earlier and need to be investigated in this model. The perfor mance of the algorithm seems to be sufficient enough to be able to lend itself to a range of applications such as seizure monitoring and intervention/control devices. Howe ver, the performance needs to be tested in prospective studies using a larger cohort of animals. Furthermore, the utility of dynamical techniques in developing markers for epileptogenesis n eeds to be further explored. Other dynamical descriptors may prove to be equally if not more useful in identifying precursors for the development of epilepsy. Further investigat ion needs to be conducted to study if combining dynamical techniques as described in this thesis with sophisticated imaging techniques would improve current dia gnosis of epilepsy and epileptogenesis. Typical dynamical studies involving human intracrani al EEG recordings have employed 28 to 32 electrode channels. Using micro-electrode arrays [264] in future experiments may thereby improve spatial samp ling of the epileptogenic process as it unfolds over time. It could be possible to sa mple from a wider range of brain areas and gain a better understanding of seizure pr ogression and subtle dynamic interactions between nearby sites. Additional investigatio ns into the underlying neurobiology in the MTLE model could help us understand the basic mechanisms responsible for the dynamical changes in the epileptic brain. Controlling spatiotemporal chaos has found applicatio ns in several physical, chemical, and biological systems. The goals of control have been outlined in [265]. This

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186 dissertation describes several ad aptive control strategies for controlling chaos in coupled lattice models that emulate epileptic brai n dynamics. We have however to be very cautious and realize the enormous differen ce of complexity and knowledge between the model systems being used in nonlinear controls and the distributed nature of a system such as the brain and the lack of obvious models and parameters to be controlled. Successful applications of control theory to brain dyn amics must address in a comprehensive fashion at least the following items: (i) parsimonious variable selection, (ii) construction of dynamical models of the brain tissue and its interactions, (iii) appropriate simplifying hypothesis to esta blish performance bounds, (iv) physiologic realism in the specifications. The Lya punov exponent may be too gross a system parameter when it comes to application in brain disorders and hence one may need to identify other dynamical descriptors or comb ination of descriptors to achieve robust control and for feasible applications in cont rolling bran disorders such as epilepsy. As part of future research investigating such optimization based control approaches, we also propose to develop realistic model of comple x system dynamics using a learning scheme as described in chapter 5 and hence make the models more useful in the study and control of such complex systems. The state dependent therapeutic interven tion paradigm as described in this dissertation offers may potential areas of improvement. The analysis system as described herein, could also analyze co llected electrophysiological info rmation following electrical stimulation intervention, to assess the short-term effects of the stimulation intervention therapy and to provide outputs to maintain or modify such stimulation intervention. The closed loop system could further include an el ectrode array suitable for implantation in or

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187 on a subject's head, configured to selectively detect el ectrophysiological information detectable by electroencephalography, and to ou tput electrical stimulation. Typically the electrode array could be configured so as to create a plurality of channels. Electrical stimulation output signals having desired stim ulation frequency and stimulation intensity can be directed to one or more of the plural ity of channels, in which it is predicted or determined that there is the onset of an epileptic state. A major area of future research in this line of work relates to investigations into its means for self controlling automated operation of the paradigm, in particular with regard to determining when and where to deliver a therapeutic electrical stimulus to prevent or delay a susceptible area of the brain from transi tioning from a preictal state to a seizure. The algorithms and incorporated methods of the paradigm could address this issue in several ways. The control of the timing, frequency and intensity of the electrical stimulation intervention output signals could be determined by a direct control method in which a control law is derived from the state of the neural structure, or alternatively it could be decided by a model that utilizes macroscopic modeling of the dynamics of spatiotemporal parameters in the brain to predict seizures. Typically the model would quantify aspects of local chaoticity associated with seizure activity that is detected in a neural structure being monitored. Various othe r control modalities need to be investigated including but not limited to hybrid continuou s-discontinuous control schemes, global nonlinear dynamic modeling, and multiple switching local linear modeling

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188 APPENDIX PROOF OF THEOREM 5.1 The set of all n x n matrices forms an n2 dimensional metrix space under the metric TY X Y X tr Y X) )( (2 . The class of orthogonal matrices I XX X n OT : ) ( is a closed subspace of ; and since n X for ) (n O X , O(n) is a compact set. Then the sequence ,...} 2 , 1 , { k APkof orthogonal matrices has a convergent subsequence } {iPkA, where we can take ki+1>ki. By the Couchy property in the space , it is easy to show thatI Ai iPk Pk i 1lim. Then for some m, X X I APm ) ( for allnX . The last L columns of ) (1X HL are obtained by multiplying each column of ) (X HL by A , which is nonsingular from 5.1(a) . The following results follow by induction: under 5.1(a) and 5.1(b), for all nX and all integers L Q, rank HL(X) = rank HP(X) = n. Our next step is to show local controllability. Let X be any initial state. We can choose P such that AP is orthogonal and the Jacobian matrix ) ( ) , (0X H U U XP U p pp has rank n. By the implicit function theorem, there exists 1>0 (depending on X) and positive such that the P-dimensional box: } 1 , ; { P i U U Bi P P ,

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189 is mapped by onto an open n -dimensional region in n containing the point APX= (X ,OP), and this region contains a sphere ) 2 , (1X X AP with center at APX and radius X12. We can find m1 = m1( 1) such that X X X APm 11( . The set of special control sequences of length 1Pmsuch that the initial sequenceP PB Uand iU = 0, 1Pm i P , has the property that ), , ( ) 2 , ( ) , (1 11U X X X A X XPm where the second inclusion follows from orthogonality and control 1PmU such that ) , (1PmU X Y . Given our present state, we can define the set of source points Sk by: } ), , ( : { i k kU U Z X Z S . Since A is a nonsingular matrix, we can choose a small enough such that if | Uk| < . Then ( A + UkB)-1 exists. Next we can consider th e system run backward in time as 1 1 1) ( k k kX B U A X , for1 1) ( , B U A Z S Zk. ) (1X B U Ak A-P is orthogonal and the Jacobian ) ( | ) / (0X A H A U Zk k k U kk , for P k , has rank n ; so our preceding argument works again and we obtain an 2 > 0 depending on X and m2 = m2( 2) such that if X z X2 there exists a control 1Pmu for which ). , (2PmU X X Let = min ( 1, 2). Then the sphere ) , (X X has the property that any point in it can be connected to any other by a control sequence whose length P ( m1+m2) depends on X . Now let us consider any two points X , Y in n. We can find an arc which is either a straight line segment or, at worst, a once-broken line segment joining X and Y and not

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190 passing through the “puncture” at the origin. Consider the set of spheres X X X), , ( where we note that depends on X but is always positive. is compact, so this open covering of has a finite subcovering of spheres with centers X1,...,X say, on ) , ( ;1 1X X X and ) , ( X Y Y can then be connected by using a control sequence with 1 2 1)) ( ) ( (i i iX m X m N steps. This completes the proof of Theorem 5.1.

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191 LIST OF REFERENCES 1. H.G. Schuster, Deterministic Chaos , Weinheim: VCH–Verlag, 1989. 2. E. Ott, Chaos in Dynamical Systems , Cambridge, UK: Cambridge University Press, 1993. 3. W.A. Hauser, L.T. Kurland, “The ep idemiology of epilepsy in Rochester, Minnesota, 1935 -1967,” Epilepsia , vol. 16, pp. 1-66, 1975 4. M.C. Mackey, L. Glass, “Oscillation and chaos in physiological control systems,” Science , vol. 197, pp. 287289, 1977. 5. J.G. Milton, D. Black, “Dynamic dis eases in psychiatry and neurology,” CHAOS , vol. 5, pp. 8-13, 1995. 6. J. Belair, U. an der Heiden, L. Gla ss, J.G. Milton, “Dynamical diseases: Identification, temporal aspects and treatment strategies for human illness,” CHAOS , vol. 5, pp. 1-7, 1995. 7. L. Glass, M.C. Mackey, “Pathological c onditions resulting from instabilities in physiological control systems,” Ann. N.Y. Acad. Sci. , vol. 316, pp. 214-235, 1979. 8. M. Mackey, U. an der Heiden, “Dyna mical diseases and bifurcations: Understanding functional disorder s in physiological systems,” Funkt. Biol. Med. , vol. 156, pp. 156, 1982. 9. M.C. Mackey, J.G. Milton, “Dynamical diseases,” Ann. N.Y. Acad. Sci. , vol. 504, pp. 16-32, 1987. 10. J.G. Milton, “Epilepsy: multistability in a dynamic disease,” in Self-organized biological dynamics and nonlinear control , J. Walleczek, Ed. New York: Cambridge University Press, pp. 374-386, 2000. 11. J.G. Milton, M.C. Mackey, “Periodic haem atological diseases: Mystical entities or dynamical disorders?,” J. Royal College of Physicians of London , vol. 23, pp. 236241, 1989. 12. M. Le Van Quyen, J. Martinerie, V. Navarro, M. Baulac, F.J. Varela, “Characterizing neurodynamic changes before seizures,” J. Clin. Neurophysiol. , vol. 18, pp. 191-208, 2001.

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212 BIOGRAPHICAL SKETCH Sandeep P. Nair was born on November 24, 1978, in Thiruvananthapuram, India. He received his B.Tech. with Distinction in electronics engineering from the Cochin University of Science and Technology, Coch in, India, in 2000. From 2000 to 2001 he worked as a Programmer Analyst in Cogni zant Technology Solutions (CTS), Chennai India. In 2001, he joined the Department of Electrical and Computer Engineering at the University of Florida and received his M.S. degree in August 2003. He then joined the J. Crayton Pruitt Family Department of Biomedical Engineering to pursue a Ph.D. in biomedical signal processing an d neural engineeri ng. Under the guidance of Dr. J. Chris Sackellares in the Brain Dyna mics Laboratory and Dr. Panos M. Pardalos in the Center for Applied Optimization, his re search is in the area of br ain dynamics and control with special emphasis on epilepsy. The goal is to develop a state depe ndent seizure control paradigm and design innovative control methodologies based on nonlinear systems theory and optimization. In 2006 he was a recipient of the Outs tanding Achievement Award instituted by the J. Crayton Pruitt Fa mily Department of Biomedical Engineering and the Alpha-One Graduate Fellowship fr om the Alpha-One Foundation. His research interests include neurophysiology, neural s timulation, dynamical control, chaos theory, and nonlinear dynamics in biomedical engineer ing applications. His studies are funded by the NIH, National Institute of Biomedical Imaging and Bioengineering, the Department of Veterans Affairs, the Alpha-One f oundation and the Children’s Miracle Network