# Dynamic Modeling and Statistical Analysis

## 4.1) Information Flow in Intracranial EEG Recordings of Epilepsy Patients

Matthäus Staniek (1,2), Anton Chernihovskyi (1,2), Hannes Osterhage (1,2), Christian E. Elger (1), Klaus Lehnertz (1,2,3)

(1) Department of Epileptology, University of Bonn; (2) Helmholtz-Institute for Radiation and Nuclear Physics, University of Bonn; (3) Interdisciplinary Center for Complex Systems, University of Bonn

Mechanisms leading to the occurrence of epileptic seizures are still poorly understood. We analyzed EEG recordings using an information theoretic approach combined with concepts from symbolic dynamics in order to investigate intra- and interhemispheric interactions as well as processes of seizure generation. We here studied continuous multi-day multi-channel EEG recordings of up to now 15 patients suffering from unilateral medically intractable mesial temporal lobe epilepsy. All patients underwent invasive presurgical diagnostics and are postoperatively seizure free. EEG signals were recorded from bilateral intrahippocampal depth electrodes, each equipped with 10 contacts and implanted stereotactically along the longitudinal axis of the hippocampal formation. The transfer entropy, which quantifies the directionality of the flow of information was calculated for all channel combinations using a moving window technique. During seizure activity, we observed the mesial temporal structures in one hemisphere to be more active and driving homologous structures in the opposite hemisphere. These active structures, however, not necessarily coincided with the epileptic focus and were frequently observed in the contralateral hemisphere. During the interictal state a comparable active-passive relationship could be observed again indicating more active contralateral structures. Moreover, in many patients interactions between brain structures allowed to identify long-lasting preictal states. Measuring directionality in the human epileptic brain may provide relevant information about the location of the epileptic focus. In addition, our findings indicate the importance of brain outside of the ictal onset zone but within the epileptic network in seizure generation.

## 4.2) Influence of Network Topology on Global Synchronization in a Network of Model Neurons

Alexander Rothkegel (1,2), Marie-Therese Horstmann (1,2,3), Klaus Lehnertz (1,2,3)

(1) Department of Epileptology; (2) Helmholtz Institute for Radiation and Nuclear Physics; (3) Interdisciplinary Center for Complex Systems

We study the transition to global synchronization in a network of integrate-and-fire-neurons, which are coupled diffusively over a given network topology, as a model for seizure generation. The network dynamics is driven stochastically by a probability for spontaneous firing for each neuron. We derive criteria for the coupling strength, for which synchronization emerges in the mean network activity. We propose a necessary condition for global synchronization by requiring that each firing neuron gives rise to enough activity for at least one other neuron to fire. For low leakage currents, low refractory times, and sufficiently connected network topologies (i. e., regular networks) this condition determines a threshold for the coupling strength. At this threshold the peak heights of the mean network activity grow abruptly. We investigate the influence of different network topologies (in particular lattices, small-world and scale-free networks) on the transition to global synchronization and discuss how the coupling strength relates to general network characteristics such as mean path length or clustering coefficient.

## 4.3) Statistical Methods for Developing and Evaluating Preictal Classifiers

David E. Snyder (1), Kent Leyde (1), Gregory Dunn (1), Andrew Gardner (2)

(1) NeuroVista Corporation; (2) BioQuantix Corporation

Introduction: Evaluation of seizure prediction algorithms present many challenges. Statistical methods must prevent in-sample testing, guard against accidental correlations, and provide flexibility with respect to the timing of prediction events. To enable the development of increasingly higher performance algorithms, the methods must also provide for direct statistical comparison of alternate implementations. Methods: We propose a block-wise statistical model capable of capturing continuous as well as intermittent prediction events. Prediction alerts may be of any duration, unrestricted by the temporal block size. The test statistic is population-based and allows hypothesis testing without bias toward patients with high seizure count. Algorithm candidates may be compared either against a chance predictor, or directly against each other in a paired test using parametric or non-parametric methods. Evaluation is conducted within a k-fold cross-validation framework (k seizures) to prevent in-sample testing. Results: Utility of the model is demonstrated by comparison of algorithm candidates based on well-known EEG feature calculations. The evaluation database is comprised of continuous subdural EEG recordings of 76 EMU patients encompassing 7452 hours of data, with 301 primary seizures (clusters excluded). Algorithm superiority compared to chance prediction is demonstrated (p < 0.0001), and the distribution of alert durations shown to be approximately log-normal with median of 156 minutes. Conclusions: A block-wise temporal model allows direct statistical comparison of alternative seizure prediction algorithms, without making assumptions regarding prediction alert duration. Such a model may facilitate the ongoing development of improved algorithms.

## 4.4) A Computational Model of Epilepsy and Response to Electrical Stimulation

Robert D. Vincent (1), Aaron Courville (2), Massimo Avoli (1), Joelle Pineau (1)

(1) McGill University; (2) Université de Montréal

We aim to develop robust control algorithms for implantable electrical stimulation devices. For this purpose we require a computational model that mimics key characteristics of the biological models we wish to use for further study. We present a simple model of a neural network that exhibits epileptiform activity. This network consists of an array of leaky integrate and fire neurons connected in a "small world" topology. We include a noise current tuned to achieve a small random spiking rate. We add an explicit representation of postictal depression effects, as well as a simple model of a stimulating electrode and a recording sensor. We demonstrate that this network responds to simulated electrical stimulation in a manner consistent with the behavior seen in biological models. We plan to use this model as a minimal-cost method for evaluation of electrical stimulation methods for epileptic seizure suppression.

## 4.5) A Computational Model of Glia-Mediated Seizure Induction

Cristina Savin, Jochen Triesch, Michael Meyer-Hermann

Frankfurt Institute for Advanced Studies

There is an increasing amount of evidence supporting a causal relation between chronic inflammation and seizures. Several proinflammatory cytokines have been studied in the context of seizure susceptibility and neuronal damage, including tumor necrosis factor alpha. It is believed that, in certain conditions, TNF-alpha can increase neural excitability and facilitate infection-related seizures [1]. The effect is potentially related to a recent discovery that links TNF-alpha to homeostatic synaptic plasticity. Specifically, acute application or long-term glial production (during chronic activity blockade) of TNF-alpha increases AMPA receptor surface expression in hippocampal neurons [2], [3]. The regulation resembles synaptic scaling, a homeostatic mechanisms which globally adjusts synaptic strengths to maintain a certain synaptic drive to neurons. It is thought to ensure the stability of the cortex throughout development and during learning. Despite having a generally beneficial role, homeostatic mechanisms were proposed in models as a cause of neural hyperexcitability and epileptogenesis [4]. We have developed a computational model of glia-mediated synaptic scaling, in a network of spiking neurons interacting with the glial tissue. It is the first model to consider the mechanisms underlying synaptic scaling and the spatial effects that can arise from the diffusion of neuromodulators. Our model reproduces experimental findings linking chronic overexpression, systemic infection, or lesions to hyperexcitability and network bursts and is consistent with the idea that chronic inflammation can increase seizure predisposition. [1] A. Vezzani and T. Granata. Brain inflammation in epilepsy: experimental and clinical evidence. Epilepsia, 46(11):1724–1743, 2005. [2] E. Beattie, D. Stellwagen, W. Morishita, J. Bresnahan, B. Ha, M. Von Zastrow, M. Beattie, and R. Malenka. Control of synaptic strength by glial TNF-alpha. Science, 295:2282–2285, 2002. [3] D. Stellwagen and R. Malenka. Synaptic scaling mediated by glial TNF-alpha. Nature, 440:1054–1059, 2006. [4] A. Houweling, M. Bazhenov, I. Timofeev, M. Steriade, and T. Sejnowski. Homeostatic synaptic plasticity can explain post-traumatic epileptogenesis in chronically isolated neocortex. Cerebral Cortex, 15:834–845, 2005.

## 4.6) Randomized EEG Analysis

Craig Savage

Dept. of Electrical Engineering, University of Melbourne and National Information Communication Technology of Australia (NICTA)

The electroencephalogram (EEG) is a useful tool in the diagnosis of many disorders, including epilepsy. Typically, EEG signals are taken from multiple locations around the head, and collected for many hours. The amount of data produced by an EEG can be sizable; in the Freiburg data set, the EEG signal from a single patient amounted to approximately eleven billion samples. One practical problem to such a large dimensional signal is to extract the relevant information. Many potential models for brain dynamics have been presented, and none fully reconstruct brain dynamics. However, recent developments in random projections, underpinned by the Johnson-Lindenstrauss Lemma [1, 2], indicate that a generically chosen projection maintains the total energy (i.e. Euclidean norm) of a generic signal. This is an amazing result, as the corresponding projections are easy to compute, are not tied to any model of the brain, and significantly reduce the dimensionality of data. This work focuses on demonstrating various properties of random projection in preserving the total energy within the EEG signal with a sparse method as proposed in [3]. Specifically, we form a random projection matrix where each of the values in a projection matrix are independent, identically distributed binary random variables. In theory, the total energy of a randomly projected signal will be preserved to within a multiplicative factor of (1±ε), independent of the original dimension. Preliminary results indicate that random projections do maintain the total energy in portions of the EEG signal, even when different numbers of channels are used. Within this work, I demonstrate approximate energy conservation between different patients and different time horizons. Furthermore, I note some potential future applications for such random projections. [1] Johnson, William and Lindenstrauss, Joram. “Extensions of Lipschitz Mappings into a Hilbert Space.” In Proceedings of the Conference in Modern Analysis and Probability. New Haven, CT. 1982. pp 189-206. [2] Dasgupta, Sanjoy and Gupts, Anupam. “An Elementary Proof of the Johnson-Lindenstrauss Lemma.” ICSI Technical Report TR-99-006. March, 1999. [3] Achliotas, Dimitris. “Database-friendly random projections: Johnson-Lindenstrauss with binary coins.” Journal of Computer and System Sciences. Vol. 66. pp. 671-687. 2003.

## 4.7) Emergence of Spreading Hyperexcitability in Diffusively Coupled FitzHugh-Nagumo Systems

Markus Dahlem, Felix Schneider, Gerald Hiller, Philipp Hövel, Eckehard Schöll

Institut für Theoretische Physik, Technische Universität Berlin, Germany

We investigate the emergence of excitability in diffusively coupled FitzHugh-Nagumo systems and demonstrate how to protect neurons adjacent to a hyperexcitable core against recruitment into this pathological state. Our efforts focus on spatially extended systems and on two coupled discrete neural populations. We determine the parameter regime in which transient wave forms emerge in the spatially extended system and show how control can minimize the volume of invaded tissue. In the discrete population model, we investigate effects of time-delayed feedback schemes on noise-induced cooperative dynamics of the ensemble of neural populations.

## 4.8) Ordinal EEG Analysis

Karsten Keller, Mathieu Sinn

Institute of Mathematics, University of Lübeck

Ordinal time series analysis is a new approach to the qualitative investigation of long and complex time series. The idea behind it is to consider the order relation between the values of a time series instead of the values themselves. Roughly speaking, a given time series is transformed into a series of so called ordinal patterns describing the up and down in the original series. Then the distribution of ordinal patterns obtained is the base of the analysis. Here we discuss applicability of ordinal time series analysis to the analysis of EEG data. In particular, we demonstrate how Cluster Analysis (CA) of distributions of ordinal patterns in EEG time series can be used for the classification and discrimination of basic brain states.

## 4.9) Symbolic Analysis of Multivariate Data

Anton Chernihovskyi (1,2), Matt Staniek (1,2), Lei Wang (1,2), Christian E. Elger (1), Klaus Lehnertz (1,2,3)

(1) Department of Epileptology, University of Bonn; (2) Helmholtz - Institute for Radiation and Nuclear Physics, University of Bonn; (3) Interdisciplinary Center for Complex Systems, University of Bonn

Many natural systems exhibit complex patterns in their output signals that reflect the underlying dynamics. By symbolization, raw signals can be transformed into a series of discretized symbols that – being a considerably simplified representation of the data – might still contain enough information about the underlying dynamics. By utilizing information-theoretic measures (entropy rate, mutual information, and transfer entropy) we analyze statistical correlations of spatiotemporal binary patterns derived from multivariate signals. In many cases, such an extremely simplified representation suffices to identify changes in the system dynamics. To illustrate our approach we study a network of coupled chaotic oscillators that contains two interacting clusters with a time-varying degree and direction of interaction. We show that our approach allows to qualitatively identify the induced changes in the simulated dynamics. Using our approach, we analyzed intracranial multi-channel, multi-day EEG recordings from epilepsy patients who underwent presurgical evaluation. Preliminary findings indicate that changes in the degree and direction of inter(intra)-hemispheric interactions appear to be related to ictogenesis. If findings can be validated for a larger patient group, the underlying simplicity of a binary representation of multivariate data may enable the development of a miniaturized analysis device using special-purpose digital signal processors (DSP) or analog hardware such as Cellular Neural Networks (CNN).

## 4.10) Simultaneous Analysis of Population Spikes and Single Neuron Activity In Vivo at the Onset of Seizure

Jeremiah D. Mitzelfelt (1), Wenjuan Yan (2), Jose C. Principe (2), Justin C. Sanchez (3)

(1) Department of Neuroscience; (2) Department of Electrical and Computer Engineering, University of Florida; (3) Department of Pediatrics, Division of Neurology, University of Florida

We present here an analysis of the underlying mechanisms related to interictal spikes (IS) and their contribution to the process of seizure onset. Since the role of population discharges for impending seizures is still largely unknown, we have developed a technique to simultaneously monitor and correlate the activity of single neurons and ISs using chronic microelectrode array electrophysiology. The firing rates of ISs and single neurons (interneurons and pyramidal cells) in the CA3 region of the hippocampus was compared 1.5 hr prior to seizure onset for 10 seizures in rats with stimulation induced spontaneous temporal lobe seizures. During this interval, both cell types exhibited burst-like neuromodulations with the interneurons producing the highest firing rate (mean 1.357 spikes/s), with a statically significant peak in activity 490 s before seizure onset and the pyramidal cells producing a lower firing rate (mean 0.566 spikes/s), with a peak in activity 170 s before seizure onset. Cross-correlation between neuronal firing and ISs indicated that the increases in firing were not synchronous with the population spikes possibly indicating one is driving the other. The cross-correlation between pyramidal cell and IS modulation yielded a periodic trend with first maxima at an IS lag of 600 s while interneurons produced a maxima in correlation at an IS lag of 520 s. Interestingly, no ISs occurred within 160 s of seizure onset while single neuron modulation was still elevated. The multiscale analysis presented here may provide new markers for seizure prediction.