Gap junctions and epileptic seizures--two sides of the same coin?

Multi-scale modelling of the epileptic brain: advantages of computational therapy exploration

Objective: Epilepsy is a complex disease spanning across multiple scales, from ion channels in neurons to neuronal circuits across the entire brain. Over the past decades, computational models have been used to describe the pathophysiological activity of the epileptic brain from different aspects. Traditionally, each computational model can aid in optimizing therapeutic interventions, therefore, providing a particular view to design strategies for treating epilepsy. As a result, most studies are concerned with generating specific models of the epileptic brain that can help us understand the certain machinery of the pathological state. Those specific models vary in complexity and biological accuracy, with system-level models often lacking biological details.Approach: Here, we review various types of computational model of epilepsy and discuss their potential for different therapeutic approaches and scenarios, including drug discovery, surgical strategies, brain stimulation, and seizure prediction. We propose that we need to consider an integrated approach with a unified modelling framework across multiple scales to understand the epileptic brain. Our proposal is based on the recent increase in computational power, which has opened up the possibility of unifying those specific epileptic models into simulations with an unprecedented level of detail.Main results: A multi-scale epilepsy model can bridge the gap between biologically detailed models, used to address molecular and cellular questions, and brain-wide models based on abstract models which can account for complex neurological and behavioural observations.Significance: With these efforts, we move toward the next generation of epileptic brain models capable of connecting cellular features, such as ion channel properties, with standard clinical measures such as seizure severity.

Interictal EEG source connectivity to localize the epileptogenic zone in patients with drug-resistant epilepsy: A machine learning approach

OBJECTIVE

To deconstruct the epileptogenic networks of patients with drug-resistant epilepsy (DRE) using source functional connectivity (FC) analysis; unveil the FC biomarkers of the epileptogenic zone (EZ); and develop machine learning (ML) models to estimate the EZ using brief interictal electroencephalography (EEG) data.

METHODS

We analyzed scalp EEG from 50 patients with DRE who had surgery. We reconstructed the activity (electrical source imaging [ESI]) of virtual sensors (VSs) across the whole cortex and computed FC separately for epileptiform and non-epileptiform EEG epochs (with or without spikes). In patients with good outcome (Engel 1a), four cortical regions were defined: EZ (resection) and three non-epileptogenic zones (NEZs) in the same and opposite hemispheres. Region-specific FC features in six frequency bands and three spatial ranges (long, short, inner) were compared between regions (Wilcoxon sign-rank). We developed ML classifiers to identify the VSs in the EZ using VS-specific FC features. Cross-validation was performed using good outcome data. Performance was compared with poor outcomes and interictal spike localization.

RESULTS

FC differed between EZ and NEZs (p < .05) during non-epileptiform and epileptiform epochs, showing higher FC in the EZ than its homotopic contralateral NEZ. During epileptiform epochs, the NEZ in the epileptogenic hemisphere showed higher FC than its contralateral NEZ. In good outcome patients, the ML classifiers reached 75% accuracy to the resection (91% sensitivity; 74% specificity; distance from EZ: 38 mm) using epileptiform epochs (gamma and beta frequency bands) and 62% accuracy using broadband non-epileptiform epochs, both outperforming spike localization (accuracy = 47%; p < .05; distance from EZ: 57 mm). Lower performance was seen in poor outcomes.

SIGNIFICANCE

We present an FC approach to extract EZ biomarkers from brief EEG data. Increased FC in various frequencies characterized the EZ during epileptiform and non-epileptiform epochs. FC-based ML models identified the resection better in good than poor outcome patients, demonstrating their potential for presurgical use in pediatric DRE.

Weak coupling of neurons enables very high-frequency and ultra-fast oscillations through the interplay of synchronized phase shifts

Recently, in the past decade, high-frequency oscillations (HFOs), very high-frequency oscillations (VHFOs), and ultra-fast oscillations (UFOs) were reported in epileptic patients with drug-resistant epilepsy. However, to this day, the physiological origin of these events has yet to be understood. Our study establishes a mathematical framework based on bifurcation theory for investigating the occurrence of VHFOs and UFOs in depth EEG signals of patients with focal epilepsy, focusing on the potential role of reduced connection strength between neurons in an epileptic focus. We demonstrate that synchronization of a weakly coupled network can generate very and ultra high-frequency signals detectable by nearby microelectrodes. In particular, we show that a bistability region enables the persistence of phase-shift synchronized clusters of neurons. This phenomenon is observed for different hippocampal neuron models, including Morris-Lecar, Destexhe-Paré, and an interneuron model. The mechanism seems to be robust for small coupling, and it also persists with random noise affecting the external current. Our findings suggest that weakened neuronal connections could contribute to the production of oscillations with frequencies above 1000 Hz, which could advance our understanding of epilepsy pathology and potentially improve treatment strategies. However, further exploration of various coupling types and complex network models is needed.

Antiepileptogenic and neuroprotective effect of mefloquine after experimental status epilepticus

Acquired temporal lobe epilepsy (TLE) characterized by spontaneous recurrent seizures (SRS) and hippocampal inhibitory neuron dysfunction is often refractory to current therapies. Gap junctional or electrical coupling between inhibitory neurons has been proposed to facilitate network synchrony and intercellular molecular exchange suggesting a role in both seizures and neurodegeneration. While gap junction blockers can limit acute seizures, whether blocking neuronal gap junctions can modify development of chronic epilepsy has not been examined. This study examined whether mefloquine, a selective blocker of Connexin 36 gap junctions which are well characterized in inhibitory neurons, can limit epileptogenesis and related cellular and behavioral pathology in a model of acquired TLE. A single, systemic dose of mefloquine administered early after pilocarpine-induced status epilepticus (SE) in rat reduced both development of SRS and behavioral co-morbidities. Immunostaining for interneuron subtypes identified that mefloquine treatment likely reduced delayed inhibitory neuronal loss after SE. Uniquely, parvalbumin expressing neurons in the hippocampal dentate gyrus appeared relatively resistant to early cell loss after SE. Functionally, whole cell patch clamp recordings revealed that mefloquine treatment preserved inhibitory synaptic drive to projection neurons one week and one month after SE. These results demonstrate that mefloquine, a drug already approved for malaria prophylaxis, is potentially antiepileptogenic and can protect against progressive interneuron loss and behavioral co-morbidities of epilepsy.

Regulating absence seizures by tri-phase delay stimulation applied to globus pallidus internal

In this paper, a reduced globus pallidus internal (GPI)-corticothalamic (GCT) model is developed, and a tri-phase delay stimulation (TPDS) with sequentially applying three pulses on the GPI representing the inputs from the striatal D ₁ neurons, subthalamic nucleus (STN), and globus pallidus external (GPE), respectively, is proposed. The GPI is evidenced to control absence seizures characterized by 2 Hz-4 Hz spike and wave discharge (SWD). Hence, based on the basal ganglia-thalamocortical (BGCT) model, we firstly explore the triple effects of D _l-GPI, GPE-GPI, and STN-GPI pathways on seizure patterns. Then, using the GCT model, we apply the TPDS on the GPI to potentially investigate the alternative and improved approach if these pathways to the GPI are blocked. The results show that the striatum D ₁, GPE, and STN can indeed jointly and significantly affect seizure patterns. In particular, the TPDS can effectively reproduce the seizure pattern if the D ₁-GPI, GPE-GPI, and STN-GPI pathways are cut off. In addition, the seizure abatement can be obtained by well tuning the TPDS stimulation parameters. This implies that the TPDS can play the surrogate role similar to the modulation of basal ganglia, which hopefully can be helpful for the development of the brain-computer interface in the clinical application of epilepsy.

Mean-Field Models for EEG/MEG: From Oscillations to Waves

Neural mass models have been used since the 1970s to model the coarse-grained activity of large populations of neurons. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature, and cannot hope to recreate some of the rich repertoire of responses seen in real neuronal tissue. Here we consider a simple spiking neuron network model that has recently been shown to admit an exact mean-field description for both synaptic and gap-junction interactions. The mean-field model takes a similar form to a standard neural mass model, with an additional dynamical equation to describe the evolution of within-population synchrony. As well as reviewing the origins of this next generation mass model we discuss its extension to describe an idealised spatially extended planar cortex. To emphasise the usefulness of this model for EEG/MEG modelling we show how it can be used to uncover the role of local gap-junction coupling in shaping large scale synaptic waves.

Modeling Neurotransmission: Computational Tools to Investigate Neurological Disorders

The investigation of synaptic functions remains one of the most fascinating challenges in the field of neuroscience and a large number of experimental methods have been tuned to dissect the mechanisms taking part in the neurotransmission process. Furthermore, the understanding of the insights of neurological disorders originating from alterations in neurotransmission often requires the development of (i) animal models of pathologies, (ii) invasive tools and (iii) targeted pharmacological approaches. In the last decades, additional tools to explore neurological diseases have been provided to the scientific community. A wide range of computational models in fact have been developed to explore the alterations of the mechanisms involved in neurotransmission following the emergence of neurological pathologies. Here, we review some of the advancements in the development of computational methods employed to investigate neuronal circuits with a particular focus on the application to the most diffuse neurological disorders.

Insights on the role of external globus pallidus in controlling absence seizures

Absence epilepsy, characterized by transient loss of awareness and bilaterally synchronous 2-4 Hz spike and wave discharges (SWDs) on electroencephalography (EEG) during absence seizures, is generally believed to arise from abnormal interactions between the cerebral cortex (Ctx) and thalamus. Recent animal electrophysiological studies suggested that changing the neural activation level of the external globus pallidus (GPe) neurons can remarkably modify firing rates of the thalamic reticular nucleus (TRN) neurons through the GABAergic GPe-TRN pathway. However, the existing experimental evidence does not provide a clear answer as to whether the GPe-TRN pathway contributes to regulating absence seizures. Here, using a biophysically based mean-field model of the GPe-corticothalamic (GCT) network, we found that both directly decreasing the strength of the GPe-TRN pathway and inactivating GPe neurons can effectively suppress absence seizures. Also, the pallido-cortical pathway and the recurrent connection of GPe neurons facilitate the regulation of absence seizures through the GPe-TRN pathway. Specifically, in the controllable situation, enhancing the coupling strength of either of the two pathways can successfully terminate absence seizures. Moreover, the competition between the GPe-TRN and pallido-cortical pathways may lead to the GPe bidirectionally controlling absence seizures, and this bidirectional control manner can be significantly modulated by the Ctx-TRN pathway. Importantly, when the strength of the Ctx-TRN pathway is relatively strong, the bidirectional control of absence seizures by changing GPe neural activities can be observed at both weak and strong strengths of the pallido-cortical pathway.These findings suggest that the GPe-TRN pathway may have crucial functional roles in regulating absence seizures, which may provide a testable hypothesis for further experimental studies and new perspectives on the treatment of absence epilepsy.

Dynamical mesoscale model of absence seizures in genetic models

A mesoscale network model is proposed for the development of spike and wave discharges (SWDs) in the cortico-thalamo-cortical (C-T-C) circuit. It is based on experimental findings in two genetic models of childhood absence epilepsy–rats of WAG/Rij and GAERS strains. The model is organized hierarchically into two levels (brain structures and individual neurons) and composed of compartments for representation of somatosensory cortex, reticular and ventroposteriomedial thalamic nuclei. The cortex and the two thalamic compartments contain excitatory and inhibitory connections between four populations of neurons. Two connected subnetworks both including relevant parts of a C-T-C network responsible for SWD generation are modelled: a smaller subnetwork for the focal area in which the SWD generation can take place, and a larger subnetwork for surrounding areas which can be only passively involved into SWDs, but which is mostly responsible for normal brain activity. This assumption allows modeling of both normal and SWD activity as a dynamical system (no noise is necessary), providing reproducibility of results and allowing future analysis by means of theory of dynamical system theories. The model is able to reproduce most time-frequency changes in EEG activity accompanying the transition from normal to epileptiform activity and back. Three different mechanisms of SWD initiation reported previously in experimental studies were successfully reproduced in the model. The model incorporates also a separate mechanism for the maintenance of SWDs based on coupling analysis from experimental data. Finally, the model reproduces the possibility to stop ongoing SWDs with high frequency electrical stimulation, as described in the literature.

Gap junctions set the speed and nucleation rate of stage I retinal waves

Spontaneous waves in the developing retina are essential in the formation of the retinotopic mapping in the visual system. From experiments in rabbits, it is known that the earliest type of retinal waves (stage I) is nucleated spontaneously, propagates at a speed of 451±91 μm/sec and relies on gap junction coupling between ganglion cells. Because gap junctions (electrical synapses) have short integration times, it has been argued that they cannot set the low speed of stage I retinal waves. Here, we present a theoretical study of a two-dimensional neural network of the ganglion cell layer with gap junction coupling and intrinsic noise. We demonstrate that this model can explain observed nucleation rates as well as the comparatively slow propagation speed of the waves. From the interaction between two coupled neurons, we estimate the wave speed in the model network. Furthermore, using simulations of small networks of neurons (N≤260), we estimate the nucleation rate in the form of an Arrhenius escape rate. These results allow for informed simulations of a realistically sized network, yielding values of the gap junction coupling and the intrinsic noise level that are in a physiologically plausible range.

Quantized Sampled-Data Control for Synchronization of Inertial Neural Networks With Heterogeneous Time-Varying Delays

This paper is concerned with the problem of synchronization for inertial neural networks (INNs) with heterogeneous time-varying delays (HTVDs) through quantized sampled-data control. The control scheme, which takes the communication limitations of quantization and variable sampling into account, is first employed for tackling the synchronization of INNs. A novel Lyapunov-Krasovskii functional (LKF) is constructed for synchronizing an error system. Compared with existing LKFs by the largest upper bound of all HTVDs, the proposed LKF is superior, since it can make full use of the information on the lower and upper bounds of each HTVD. Based on the LKF and a new integral inequality technique, less conservative synchronization criteria are derived. The desired quantized sampled-data controller is designed by solving a set of linear matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness and conservatism reduction of the proposed results.

Effects of brain-derived neurotrophic factor and noise on transitions in temporal lobe epilepsy in a hippocampal network

Brain-derived neurotrophic factor (BDNF) has recently been implicated in the modulation of receptor activation leading to dynamic state transitions in temporal lobe epilepsy (TLE). In addition, the crucial role of neuronal noise in these transitions has been studied in electrophysiological experiments. However, the precise role of these factors during seizure generation in TLE is not known. Building on a previously proposed model of an epileptogenic hippocampal network, we included the actions of BDNF-regulated receptors and intrinsic noise. We found that the effects of both BDNF and noise can increase the activation of N-methyl-D-aspartate receptors leading to excessive C a 2 + flux, which induces abnormal fast spiking and bursting. Our results indicate that the combined effects have a strong influence on the seizure-generating network, resulting in higher firing frequency and amplitude. As correlations between firing increase, the synchronization of the entire network increases, a marker of the ictogenic transitions from normal to seizures-like dynamics. Our work on the effects of BDNF dynamics in a noisy environment might lead to an improved model-based understanding of the pathological mechanisms in TLE.

Improved Stability and Stabilization Results for Stochastic Synchronization of Continuous-Time Semi-Markovian Jump Neural Networks With Time-Varying Delay

Continuous-time semi-Markovian jump neural networks (semi-MJNNs) are those MJNNs whose transition rates are not constant but depend on the random sojourn time. Addressing stochastic synchronization of semi-MJNNs with time-varying delay, an improved stochastic stability criterion is derived in this paper to guarantee stochastic synchronization of the response systems with the drive systems. This is achieved through constructing a semi-Markovian Lyapunov-Krasovskii functional together as well as making use of a novel integral inequality and the characteristics of cumulative distribution functions. Then, with a linearization procedure, controller synthesis is carried out for stochastic synchronization of the drive-response systems. The desired state-feedback controller gains can be determined by solving a linear matrix inequality-based optimization problem. Simulation studies are carried out to demonstrate the effectiveness and less conservatism of the presented approach.

How stimulation frequency and intensity impact on the long-lasting effects of coordinated reset stimulation

Several brain diseases are characterized by abnormally strong neuronal synchrony. Coordinated Reset (CR) stimulation was computationally designed to specifically counteract abnormal neuronal synchronization processes by desynchronization. In the presence of spike-timing-dependent plasticity (STDP) this may lead to a decrease of synaptic excitatory weights and ultimately to an anti-kindling, i.e. unlearning of abnormal synaptic connectivity and abnormal neuronal synchrony. The long-lasting desynchronizing impact of CR stimulation has been verified in pre-clinical and clinical proof of concept studies. However, as yet it is unclear how to optimally choose the CR stimulation frequency, i.e. the repetition rate at which the CR stimuli are delivered. This work presents the first computational study on the dependence of the acute and long-term outcome on the CR stimulation frequency in neuronal networks with STDP. For this purpose, CR stimulation was applied with Rapidly Varying Sequences (RVS) as well as with Slowly Varying Sequences (SVS) in a wide range of stimulation frequencies and intensities. Our findings demonstrate that acute desynchronization, achieved during stimulation, does not necessarily lead to long-term desynchronization after cessation of stimulation. By comparing the long-term effects of the two different CR protocols, the RVS CR stimulation turned out to be more robust against variations of the stimulation frequency. However, SVS CR stimulation can obtain stronger anti-kindling effects. We revealed specific parameter ranges that are favorable for long-term desynchronization. For instance, RVS CR stimulation at weak intensities and with stimulation frequencies in the range of the neuronal firing rates turned out to be effective and robust, in particular, if no closed loop adaptation of stimulation parameters is (technically) available. From a clinical standpoint, this may be relevant in the context of both invasive as well as non-invasive CR stimulation.

Cluster Synchronization for Interacting Clusters of Nonidentical Nodes via Intermittent Pinning Control

The cluster synchronization problem is investigated using intermittent pinning control for the interacting clusters of nonidentical nodes that may represent either general linear systems or nonlinear oscillators. These nodes communicate over general network topology, and the nodes from different clusters are governed by different self-dynamics. A unified convergence analysis is provided to analyze the synchronization via intermittent pinning controllers. It is observed that the nodes in different clusters synchronize to the given patterns if a directed spanning tree exists in the underlying topology of every extended cluster (which consists of the original cluster of nodes as well as their pinning node) and one algebraic condition holds. Structural conditions are then derived to guarantee such an algebraic condition. That is: 1) if the intracluster couplings are with sufficiently strong strength and the pinning controller is with sufficiently long execution time in every period, then the algebraic condition for general linear systems is warranted and 2) if every cluster is with the sufficiently strong intracluster coupling strength, then the pinning controller for nonlinear oscillators can have its execution time to be arbitrarily short. The lower bounds are explicitly derived both for these coupling strengths and the execution time of the pinning controller in every period. In addition, in regard to the above-mentioned structural conditions for nonlinear systems, an adaptive law is further introduced to adapt the intracluster coupling strength, such that the cluster synchronization for nonlinear systems is achieved.

Modulation of Connexin-36 Gap Junction Channels by Intracellular pH and Magnesium Ions

Connexin-36 (Cx36) protein forms gap junction (GJ) channels in pancreatic beta cells and is also the main Cx isoform forming electrical synapses in the adult mammalian brain. Cx36 GJs can be regulated by intracellular pH (pHi) and cytosolic magnesium ion concentration ([Mg2+]i), which can vary significantly under various physiological and pathological conditions. However, the combined effect and relationship of these two factors over Cx36-dependent coupling have not been previously studied in detail. Our experimental results in HeLa cells expressing Cx36 show that changes in both pHi and [Mg2+]i affect junctional conductance (gj) in an interdependent manner; in other words, intracellular acidification cause increase or decay in gj depending on whether [Mg2+]i is high or low, respectively, and intracellular alkalization cause reduction in gj independently of [Mg2+]i. Our experimental and modelling data support the hypothesis that Cx36 GJ channels contain two separate gating mechanisms, and both are differentially sensitive to changes in pHi and [Mg2+]i. Using recombinant Cx36 we found that two glutamate residues in the N-terminus could be partly responsible for the observed interrelated effect of pHi and [Mg2+]i. Mutation of glutamate at position 8 attenuated the stimulatory effect of intracellular acidification at high [Mg2+]i, while mutation at position 12 and double mutation at both positions reversed stimulatory effect to inhibition. Moreover, Cx36*E8Q lost the initial increase of gj at low [Mg2+]i and double mutation lost the sensitivity to high [Mg2+]i. These results suggest that E8 and E12 are involved in regulation of Cx36 GJ channels by Mg2+ and H+ ions.

On Group Synchronization for Interacting Clusters of Heterogeneous Systems

This paper investigates group synchronization for multiple interacting clusters of nonidentical systems that are linearly or nonlinearly coupled. By observing the structure of the coupling topology, a Lyapunov function-based approach is proposed to deal with the case of linear systems which are linearly coupled in the framework of directed topology. Such an analysis is then further extended to tackle the case of nonlinear systems in a similar framework. Moreover, the case of nonlinear systems which are nonlinearly coupled is also addressed, however, in the framework of undirected coupling topology. For all these cases, a consistent conclusion is made that group synchronization can be achieved if the coupling topology for each cluster satisfies certain connectivity condition and further, the intra-cluster coupling strengths are sufficiently strong. Both the lower bound for the intra-cluster coupling strength as well as the convergence rate are explicitly specified.

Electrical Activity in a Time-Delay Four-Variable Neuron Model under Electromagnetic Induction

To investigate the effect of electromagnetic induction on the electrical activity of neuron, the variable for magnetic flow is used to improve Hindmarsh–Rose neuron model. Simultaneously, due to the existence of time-delay when signals are propagated between neurons or even in one neuron, it is important to study the role of time-delay in regulating the electrical activity of the neuron. For this end, a four-variable neuron model is proposed to investigate the effects of electromagnetic induction and time-delay. Simulation results suggest that the proposed neuron model can show multiple modes of electrical activity, which is dependent on the time-delay and external forcing current. It means that suitable discharge mode can be obtained by selecting the time-delay or external forcing current, which could be helpful for further investigation of electromagnetic radiation on biological neuronal system.

Controlling mechanism of absence seizures by deep brain stimulus applied on subthalamic nucleus

Based on a classical model of the basal ganglia thalamocortical network, in this paper, we employed a type of the deep brain stimulus voltage on the subthalamic nucleus to study the control mechanism of absence epilepsy seizures. We found that the seizure can be well controlled by turning the period and the duration of current stimulation into suitable ranges. It is the very interesting bidirectional periodic adjustment phenomenon. These parameters are easily regulated in clinical practice, therefore, the results obtained in this paper may further help us to understand the treatment mechanism of the epilepsy seizure.

Gap Junctions Link Regular-Spiking and Fast-Spiking Interneurons in Layer 5 Somatosensory Cortex

Gap junctions form electrical synapses that modulate neuronal activity by synchronizing action potential (AP) firing of cortical interneurons (INs). Gap junctions are thought to form predominantly within cortical INs of the same functional class and are therefore considered to act within discrete neuronal populations. Here, we challenge that view and show that the probability of electrical coupling is the same within and between regular-spiking (RS) and fast-spiking (FS) cortical INs in 16–21 days old mice. Firing properties of these two populations were distinct from other INs types including neurogliaform and low-threshold spiking (LTS) cells. We also demonstrate that pre-junctional APs can depolarize post-junctional neurons and increase the probability of firing. Our findings of frequent gap junction coupling between functionally distinct IN subtypes suggest that cortical IN networks are much more extensive and heterogeneous than previously thought. This may have implications on mechanisms ranging from cognitive functions to modulation of pathological states in epilepsy and other neurological disorders.

Prediction of the Seizure Suppression Effect by Electrical Stimulation via a Computational Modeling Approach

In this paper, we identified factors that can affect seizure suppression via electrical stimulation by an integrative study based on experimental and computational approach. Preferentially, we analyzed the characteristics of seizure-like events (SLEs) using our previous in vitro experimental data. The results were analyzed in two groups classified according to the size of the effective region, in which the SLE was able to be completely suppressed by local stimulation. However, no significant differences were found between these two groups in terms of signal features or propagation characteristics (i.e., propagation delays, frequency spectrum, and phase synchrony). Thus, we further investigated important factors using a computational model that was capable of evaluating specific influences on effective region size. In the proposed model, signal transmission between neurons was based on two different mechanisms: synaptic transmission and the electrical field effect. We were able to induce SLEs having similar characteristics with differentially weighted adjustments for the two transmission methods in various noise environments. Although the SLEs had similar characteristics, their suppression effects differed. First of all, the suppression effect occurred only locally where directly received the stimulation effect in the high noise environment, but it occurred in the entire network in the low noise environment. Interestingly, in the same noise environment, the suppression effect was different depending on SLE propagation mechanism; only a local suppression effect was observed when the influence of the electrical field transmission was very weak, whereas a global effect was observed with a stronger electrical field effect. These results indicate that neuronal activities synchronized by a strong electrical field effect respond more sensitively to partial changes in the entire network. In addition, the proposed model was able to predict that stimulation of a seizure focus region is more effective for suppression. In conclusion, we confirmed the possibility of a computational model as a simulation tool to analyze the efficacy of deep brain stimulation (DBS) and investigated the key factors that determine the size of an effective region in seizure suppression via electrical stimulation.

Pinning Impulsive Synchronization of Reaction-Diffusion Neural Networks With Time-Varying Delays

This paper investigates the exponential synchronization of reaction-diffusion neural networks with time-varying delays subject to Dirichlet boundary conditions. A novel type of pinning impulsive controllers is proposed to synchronize the reaction-diffusion neural networks with time-varying delays. By applying the Lyapunov functional method, sufficient verifiable conditions are constructed for the exponential synchronization of delayed reaction-diffusion neural networks with large and small delay sizes. It is shown that synchronization can be realized by pinning impulsive control of a small portion of neurons of the network; the technique used in this paper is also applicable to reaction-diffusion networks with Neumann boundary conditions. Numerical examples are presented to demonstrate the effectiveness of the theoretical results.

Autaptic regulation of electrical activities in neuron under electromagnetic induction

Realistic neurons may hold complex anatomical structure, for example, autapse connection to some internuncial neurons, which this specific synapse can connect to its body via a close loop. Continuous exchanges of charged ions across the membrane can induce complex distribution fluctuation of intracellular and extracellular charged ions of cell, and a time-varying electromagnetic field is set to modulate the membrane potential of neuron. In this paper, an autapse-modulated neuron model is presented and the effect of electromagnetic induction is considered by using magnetic flux. Bifurcation analysis and sampled time series for membrane potentials are calculated to investigate the mode transition in electrical activities and the biological function of autapse connection is discussed. Furthermore, the Gaussian white noise and electromagnetic radiation are considered on the improved neuron model, it is found appropriate setting and selection for feedback gain and time delay in autapse can suppress the bursting in neuronal behaviors. It indicates the formation of autapse can enhance the self-adaption of neuron so that appropriate response to external forcing can be selected, this biological function is helpful for encoding and signal propagation of neurons. It can be useful for investigation about collective behaviors in neuronal networks exposed to electromagnetic radiation.

Stimulus-induced Epileptic Spike-Wave Discharges in Thalamocortical Model with Disinhibition

Epileptic absence seizure characterized by the typical 2–4 Hz spike-wave discharges (SWD) are known to arise due to the physiologically abnormal interactions within the thalamocortical network. By introducing a second inhibitory neuronal population in the cortical system, here we propose a modified thalamocortical field model to mathematically describe the occurrences and transitions of SWD under the mutual functions between cortex and thalamus, as well as the disinhibitory modulations of SWD mediated by the two different inhibitory interneuronal populations. We first show that stimulation can induce the recurrent seizures of SWD in the modified model. Also, we demonstrate the existence of various types of firing states including the SWD. Moreover, we can identify the bistable parametric regions where the SWD can be both induced and terminated by stimulation perturbations applied in the background resting state. Interestingly, in the absence of stimulation disinhibitory functions between the two different interneuronal populations can also both initiate and abate the SWD, which suggests that the mechanism of disinhibition is comparable to the effect of stimulation in initiating and terminating the epileptic SWD. Hopefully, the obtained results can provide theoretical evidences in exploring dynamical mechanism of epileptic seizures.

Dynamic transition of neuronal firing induced by abnormal astrocytic glutamate oscillation

The gliotransmitter glutamate released from astrocytes can modulate neuronal firing by activating neuronal N-methyl-D-aspartic acid (NMDA) receptors. This enables astrocytic glutamate(AG) to be involved in neuronal physiological and pathological functions. Based on empirical results and classical neuron-glial "tripartite synapse" model, we propose a practical model to describe extracellular AG oscillation, in which the fluctuation of AG depends on the threshold of calcium concentration, and the effect of AG degradation is considered as well. We predict the seizure-like discharges under the dysfunction of AG degradation duration. Consistent with our prediction, the suppression of AG uptake by astrocytic transporters, which operates by modulating the AG degradation process, can account for the emergence of epilepsy.

Computational modeling of epileptiform activities in medial temporal lobe epilepsy combined with in vitro experiments

In this paper, we propose a comprehensive computational model that is able to reproduce three epileptiform activities. The model targets a hippocampal formation that is known to be an important lesion in medial temporal lobe epilepsy. It consists of four sub-networks consisting of excitatory and inhibitory neurons and well-known signal pathways, with consideration of propagation delay. The three epileptiform activities involve fast and slow interictal discharge and ictal discharge, and those activities can be induced in vitro by application of 4-Aminopyridine in entorhinal cortex combined hippocampal slices. We model the three epileptiform activities upon previously reported biological mechanisms and verify the simulation results by comparing them with in vitro experimental data obtained using a microelectrode array. We use the results of Granger causality analysis of recorded data to set input gains of signal pathways in the model, so that the compatibility between the computational and experimental models can be improved. The proposed model can be expanded to evaluate the suppression effect of epileptiform activities due to new treatment methods.

Anti-kindling Induced by Two-Stage Coordinated Reset Stimulation with Weak Onset Intensity

Abnormal neuronal synchrony plays an important role in a number of brain diseases. To specifically counteract abnormal neuronal synchrony by desynchronization, Coordinated Reset (CR) stimulation, a spatiotemporally patterned stimulation technique, was designed with computational means. In neuronal networks with spike timing–dependent plasticity CR stimulation causes a decrease of synaptic weights and finally anti-kindling, i.e., unlearning of abnormally strong synaptic connectivity and abnormal neuronal synchrony. Long-lasting desynchronizing aftereffects of CR stimulation have been verified in pre-clinical and clinical proof of concept studies. In general, for different neuromodulation approaches, both invasive and non-invasive, it is desirable to enable effective stimulation at reduced stimulation intensities, thereby avoiding side effects. For the first time, we here present a two-stage CR stimulation protocol, where two qualitatively different types of CR stimulation are delivered one after another, and the first stage comes at a particularly weak stimulation intensity. Numerical simulations show that a two-stage CR stimulation can induce the same degree of anti-kindling as a single-stage CR stimulation with intermediate stimulation intensity. This stimulation approach might be clinically beneficial in patients suffering from brain diseases characterized by abnormal neuronal synchrony where a first treatment stage should be performed at particularly weak stimulation intensities in order to avoid side effects. This might, e.g., be relevant in the context of acoustic CR stimulation in tinnitus patients with hyperacusis or in the case of electrical deep brain CR stimulation with sub-optimally positioned leads or side effects caused by stimulation of the target itself. We discuss how to apply our method in first in man and proof of concept studies.

Reverberation of excitation in neuronal networks interconnected through voltage-gated gap junction channels

Voltage-dependent gap junction gating contributes to reverberation in neuronal circuits.

We combined Hodgkin–Huxley equations and gating models of gap junction (GJ) channels to simulate the spread of excitation in two-dimensional networks composed of neurons interconnected by voltage-gated GJs. Each GJ channel contains two fast and slow gates, each exhibiting current–voltage (I-V) rectification and gating properties that depend on transjunctional voltage (V_j). The data obtained show how junctional conductance (g_j), which is necessary for synchronization of the neuronal network, depends on its size and the intrinsic firing rate of neurons. A phase shift between action potentials (APs) of neighboring neurons creates bipolar, short-lasting V_j spikes of approximately ±100 mV that induce V_j gating, leading to a small decay of g_j, which can accumulate into larger decays during bursting activity of neurons. We show that I-V rectification of GJs in local regions of the two-dimensional network of neurons can lead to unidirectional AP transfer and consequently to reverberation of excitation. This reverberation can be initiated by a single electrical pulse and terminated by a low-amplitude pulse applied in a specific window of reverberation cycle. Thus, the model accounts for the influence of dynamically modulatable electrical synapses in shaping the function of a neuronal network and the formation of reverberation, which, as proposed earlier, may be important for the development of short-term memory and its consolidation into long-term memory.

Pinning Synchronization of Directed Networks With Switching Topologies: A Multiple Lyapunov Functions Approach

This paper studies the global pinning synchronization problem for a class of complex networks with switching directed topologies. The common assumption in the existing related literature that each possible network topology contains a directed spanning tree is removed in this paper. Using tools from M -matrix theory and stability analysis of the switched nonlinear systems, a new kind of network topology-dependent multiple Lyapunov functions is proposed for analyzing the synchronization behavior of the whole network. It is theoretically shown that the global pinning synchronization in switched complex networks can be ensured if some nodes are appropriately pinned and the coupling is carefully selected. Interesting issues of how many and which nodes should be pinned for possibly realizing global synchronization are further addressed. Finally, some numerical simulations on coupled neural networks are provided to verify the theoretical results.

Critical Roles of the Direct GABAergic Pallido-cortical Pathway in Controlling Absence Seizures

The basal ganglia (BG), serving as an intermediate bridge between the cerebral cortex and thalamus, are believed to play crucial roles in controlling absence seizure activities generated by the pathological corticothalamic system. Inspired by recent experiments, here we systematically investigate the contribution of a novel identified GABAergic pallido-cortical pathway, projecting from the globus pallidus externa (GPe) in the BG to the cerebral cortex, to the control of absence seizures. By computational modelling, we find that both increasing the activation of GPe neurons and enhancing the coupling strength of the inhibitory pallido-cortical pathway can suppress the bilaterally synchronous 2-4 Hz spike and wave discharges (SWDs) during absence seizures. Appropriate tuning of several GPe-related pathways may also trigger the SWD suppression, through modulating the activation level of GPe neurons. Furthermore, we show that the previously discovered bidirectional control of absence seizures due to the competition between other two BG output pathways also exists in our established model. Importantly, such bidirectional control is shaped by the coupling strength of this direct GABAergic pallido-cortical pathway. Our work suggests that the novel identified pallido-cortical pathway has a functional role in controlling absence seizures and the presented results might provide testable hypotheses for future experimental studies.

Gap Junctions as Common Cause of High-Frequency Oscillations and Epileptic Seizures in a Computational Cascade of Neuronal Mass and Compartmental Modeling

High frequency oscillations (HFO) appear to be a promising marker for delineating the seizure onset zone (SOZ) in patients with localization related epilepsy. It remains, however, a purely observational phenomenon and no common mechanism has been proposed to relate HFOs and seizure generation. In this work we show that a cascade of two computational models, one on detailed compartmental scale and a second one on neural mass scale can explain both the autonomous generation of HFOs and the presence of epileptic seizures as emergent properties. To this end we introduce axonal–axonal gap junctions on a microscopic level and explore their impact on the higher level neural mass model (NMM). We show that the addition of gap junctions can generate HFOs and simultaneously shift the operational point of the NMM from a steady state network into bistable behavior that can autonomously generate epileptic seizures. The epileptic properties of the system, or the probability to generate epileptic type of activity, increases gradually with the increase of the density of axonal–axonal gap junctions. We further demonstrate that ad hoc HFO detectors used in previous studies are applicable to our simulated data.

Control of absence seizures induced by the pathways connected to SRN in corticothalamic system

The cerebral cortex, thalamus and basal ganglia together form an important network in the brain, which is closely related to several nerve diseases, such as parkinson disease, epilepsy seizure and so on. Absence seizure can be characterized by 2-4 Hz oscillatory activity, and it can be induced by abnormal interactions between the cerebral cortex and thalamus. Many experimental results have also shown that basal ganglia are a key neural structure, which closely links the corticothalamic system in the brain. Presently, we use a corticothalamic-basal ganglia model to study which pathways in corticothalamic system can induce absence seizures and how these oscillatory activities can be controlled by projections from the substantia nigra pars reticulata (SNr) to the thalamic reticular nucleus (TRN) or the specific relay nuclei (SRN) of the thalamus. By tuning the projection strength of the pathway "Excitatory pyramidal cortex-SRN", "SRN-Excitatory pyramidal cortex" and "SRN-TRN" respectively, different firing states including absence seizures can appear. This indicates that absence seizures can be induced by tuning the connection strength of the considered pathway. In addition, typical absence epilepsy seizure state "spike-and-slow wave discharges" can be controlled by adjusting the activation level of the SNr as the pathways SNr-SRN and SNr-TRN open independently or together. Our results emphasize the importance of basal ganglia in controlling absence seizures in the corticothalamic system, and can provide a potential idea for the clinical treatment.

Carbenoxolone pretreatment and treatment of posttraumatic epilepsy

Gap junction blocking agents can inhibit spontaneous discharge frequency in cells. We established a rat model of posttraumatic epilepsy induced using ferric ions. Rats were intraperitoneally injected with carbenoxolone, 20 mg/kg, prior to and 30 minutes after model establishment, once a day for 14 consecutive days. Immunohistochemistry showed glial cell proliferation around a cortical focus and significantly increased connexin expression in posttraumatic epilepsy. However, carbenoxolone pretreatment or treatment significantly reduced connexin expression in the cortex, inhibited glial fibrillary acidic protein expression and ameliorated seizure degree in rats. These findings indicate that large amounts of glial cell proliferation and abnormal gap junction generation play a role in posttraumatic epilepsy, and that carbenoxolone may prevent and treat this disease.

Systems biology, complexity, and the impact on antiepileptic drug discovery

The number of available anticonvulsant drugs increased in the period spanning over more than a century, amounting to the current panoply of nearly two dozen so-called antiepileptic drugs (AEDs). However, none of them actually prevents/reduces the post-brain insult development of epilepsy in man, and in no less than a third of patients with epilepsy, the seizures are not drug-controlled. Plausibly, the enduring limitation of AEDs' efficacy derives from the insufficient understanding of epileptic pathology. This review pinpoints the unbalanced reductionism of the analytic approaches that overlook the intrinsic complexity of epilepsy and of the drug resistance in epilepsy as the core conceptual flaw hampering the discovery of truly antiepileptogenic drugs. A rising awareness of the complexity of epileptic pathology is, however, brought about by the emergence of nonreductionist systems biology (SB) that considers the networks of interactions underlying the normal organismic functions and of SB-based systems (network) pharmacology that aims to restore pathological networks. By now, the systems pharmacology approaches of AED discovery are fairly meager, but their forthcoming development is both a necessity and a realistic prospect, explored in this review.

Control effects of stimulus paradigms on characteristic firings of parkinsonism

Experimental studies have shown that neuron population located in the basal ganglia of parkinsonian primates can exhibit characteristic firings with certain firing rates differing from normal brain activities. Motivated by recent experimental findings, we investigate the effects of various stimulation paradigms on the firing rates of parkinsonism based on the proposed dynamical models. Our results show that the closed-loop deep brain stimulation is superior in ameliorating the firing behaviors of the parkinsonism, and other control strategies have similar effects according to the observation of electrophysiological experiments. In addition, in conformity to physiological experiments, we found that there exists optimal delay of input in the closed-loop GPtrain|M1 paradigm, where more normal behaviors can be obtained. More interestingly, we observed that W-shaped curves of the firing rates always appear as stimulus delay varies. We furthermore verify the robustness of the obtained results by studying three pallidal discharge rates of the parkinsonism based on the conductance-based model, as well as the integrate-and-fire-or-burst model. Finally, we show that short-term plasticity can improve the firing rates and optimize the control effects on parkinsonism. Our conclusions may give more theoretical insight into Parkinson's disease studies.

Neuronal networks and energy bursts in epilepsy

Epilepsy can be defined as the abnormal activities of neurons. The occurrence, propagation and termination of epileptic seizures rely on the networks of neuronal cells that are connected through both synaptic- and non-synaptic interactions. These complicated interactions contain the modified functions of normal neurons and glias as well as the mediation of excitatory and inhibitory mechanisms with feedback homeostasis. Numerous spread patterns are detected in disparate networks of ictal activities. The cortical-thalamic-cortical loop is present during a general spike wave seizure. The thalamic reticular nucleus (nRT) is the major inhibitory input traversing the region, and the dentate gyrus (DG) controls CA3 excitability. The imbalance between γ-aminobutyric acid (GABA)-ergic inhibition and glutamatergic excitation is the main disorder in epilepsy. Adjustable negative feedback that mediates both inhibitory and excitatory components affects neuronal networks through neurotransmission fluctuation, receptor and transmitter signaling, and through concomitant influences on ion concentrations and field effects. Within a limited dynamic range, neurons slowly adapt to input levels and have a high sensitivity to synaptic changes. The stability of the adapting network depends on the ratio of the adaptation rates of both the excitatory and inhibitory populations. Thus, therapeutic strategies with multiple effects on seizures are required for the treatment of epilepsy, and the therapeutic functions on networks are reviewed here. Based on the high-energy burst theory of epileptic activity, we propose a potential antiepileptic therapeutic strategy to transfer the high energy and extra electricity out of the foci.

Computational model of erratic arrhythmias in a cardiac cell network: the role of gap junctions

Cardiac morbidity and mortality increases with the population age. To investigate the underlying pathological mechanisms, and suggest new ways to reduce clinical risks, computational approaches complementing experimental and clinical investigations are becoming more and more important. Here we explore the possible processes leading to the occasional onset and termination of the (usually) non-fatal arrhythmias widely observed in the heart. Using a computational model of a two-dimensional network of cardiac cells, we tested the hypothesis that an ischemia alters the properties of the gap junctions inside the ischemic area. In particular, in agreement with experimental findings, we assumed that an ischemic episode can alter the gap junctions of the affected cells by reducing their average conductance. We extended these changes to include random fluctuations with time, and modifications in the gap junction rectifying conductive properties of cells along the edges of the ischemic area. The results demonstrate how these alterations can qualitatively give an account of all the main types of non-fatal arrhythmia observed experimentally, and suggest how premature beats can be eliminated in three different ways: a) with a relatively small surgical procedure, b) with a pharmacological reduction of the rectifying conductive properties of the gap-junctions, and c) by pharmacologically decreasing the gap junction conductance. In conclusion, our model strongly supports the hypothesis that non-fatal arrhythmias can develop from post-ischemic alteration of the electrical connectivity in a relatively small area of the cardiac cell network, and suggests experimentally testable predictions on their possible treatments.

Reduced chemical and electrical connections of fast-spiking interneurons in experimental cortical dysplasia

Aberrant neural connections are regarded as a principal factor contributing to epileptogenesis. This study examined chemical and electrical connections between fast-spiking (FS), parvalbumin (PV)-immunoreactive (FS-PV) interneurons and regular-spiking (RS) neurons (pyramidal neurons or spiny stellate neurons) in a rat model of prenatal irradiation-induced cortical dysplasia. Presynaptic action potentials were evoked by current injection and the elicited unitary inhibitory or excitatory postsynaptic potentials (uIPSPs or uEPSPs) were recorded in the postsynaptic cell. In dysplastic cortex, connection rates between presynaptic FS-PV interneurons and postsynaptic RS neurons and FS-PV interneurons, and uIPSP amplitudes were significantly smaller than controls, but both failure rates and coefficient of variation of uIPSP amplitudes were larger than controls. In contrast, connection rates from RS neurons to FS-PV interneurons and uEPSPs amplitude were similar in the two groups. Assessment of the paired pulse ratio showed a significant decrease in synaptic release probability at FS-PV interneuronal terminals, and the density of terminal boutons on axons of biocytin-filled FS-PV interneurons was also decreased, suggesting presynaptic dysfunction in chemical synapses formed by FS-PV interneurons. Electrical connections were observed between FS-PV interneurons, and the connection rates and coupling coefficients were smaller in dysplastic cortex than controls. In dysplastic cortex, we found a reduced synaptic efficiency for uIPSPs originating from FS-PV interneurons regardless of the type of target cell, and impaired electrical connections between FS-PV interneurons. This expands our understanding of the fundamental impairment of inhibition in this model and may have relevance for certain types of human cortical dysplasia.

Sparse short-distance connections enhance calcium wave propagation in a 3D model of astrocyte networks

Traditionally, astrocytes have been considered to couple via gap-junctions into a syncytium with only rudimentary spatial organization. However, this view is challenged by growing experimental evidence that astrocytes organize as a proper gap-junction mediated network with more complex region-dependent properties. On the other hand, the propagation range of intercellular calcium waves (ICW) within astrocyte populations is as well highly variable, depending on the brain region considered. This suggests that the variability of the topology of gap-junction couplings could play a role in the variability of the ICW propagation range. Since this hypothesis is very difficult to investigate with current experimental approaches, we explore it here using a biophysically realistic model of three-dimensional astrocyte networks in which we varied the topology of the astrocyte network, while keeping intracellular properties and spatial cell distribution and density constant. Computer simulations of the model suggest that changing the topology of the network is indeed sufficient to reproduce the distinct ranges of ICW propagation reported experimentally. Unexpectedly, our simulations also predict that sparse connectivity and restriction of gap-junction couplings to short distances should favor propagation while long–distance or dense connectivity should impair it. Altogether, our results provide support to recent experimental findings that point toward a significant functional role of the organization of gap-junction couplings into proper astroglial networks. Dynamic control of this topology by neurons and signaling molecules could thus constitute a new type of regulation of neuron-glia and glia-glia interactions.

Bidirectional control of absence seizures by the basal ganglia: a computational evidence

Absence epilepsy is believed to be associated with the abnormal interactions between the cerebral cortex and thalamus. Besides the direct coupling, anatomical evidence indicates that the cerebral cortex and thalamus also communicate indirectly through an important intermediate bridge–basal ganglia. It has been thus postulated that the basal ganglia might play key roles in the modulation of absence seizures, but the relevant biophysical mechanisms are still not completely established. Using a biophysically based model, we demonstrate here that the typical absence seizure activities can be controlled and modulated by the direct GABAergic projections from the substantia nigra pars reticulata (SNr) to either the thalamic reticular nucleus (TRN) or the specific relay nuclei (SRN) of thalamus, through different biophysical mechanisms. Under certain conditions, these two types of seizure control are observed to coexist in the same network. More importantly, due to the competition between the inhibitory SNr-TRN and SNr-SRN pathways, we find that both decreasing and increasing the activation of SNr neurons from the normal level may considerably suppress the generation of spike-and-slow wave discharges in the coexistence region. Overall, these results highlight the bidirectional functional roles of basal ganglia in controlling and modulating absence seizures, and might provide novel insights into the therapeutic treatments of this brain disorder.

Epilepsy is a general term for conditions with recurring seizures. Absence seizures are one of several kinds of seizures, which are characterized by typical 2–4 Hz spike-and-slow wave discharges (SWDs). There is accumulating evidence that absence seizures are due to abnormal interactions between cerebral cortex and thalamus, and the basal ganglia may take part in controlling such brain disease via the indirect basal ganglia-thalamic pathway relaying at superior colliculus. Actually, the basal ganglia not only send indirect signals to thalamus, but also communicate with several key nuclei of thalamus through multiple direct GABAergic projections. Nevertheless, whether and how these direct pathways regulate absence seizure activities are still remain unknown. By computational modelling, we predicted that two direct inhibitory basal ganglia-thalamic pathways emitting from the substantia nigra pars reticulata may also participate in the control of absence seizures. Furthermore, we showed that these two types of seizure control can coexist in the same network, and depending on the instant network state, both lowing and increasing the activation of SNr neurons may inhibit the SWDs due to the existence of competition. Our findings emphasize the bidirectional modulation effects of basal ganglia on absence seizures, and might have physiological implications on the treatment of absence epilepsy.

Bursting synchronization dynamics of pancreatic β-cells with electrical and chemical coupling

Based on bifurcation analysis, the synchronization behaviors of two identical pancreatic β-cells connected by electrical and chemical coupling are investigated, respectively. Various firing patterns are produced in coupled cells when a single cell exhibits tonic spiking or square-wave bursting individually, irrespectively of what the cells are connected by electrical or chemical coupling. On the one hand, cells can burst synchronously for both weak electrical and chemical coupling when an isolated cell exhibits tonic spiking itself. In particular, for electrically coupled cells, under the variation of the coupling strength there exist complex transition processes of synchronous firing patterns such as "fold/limit cycle" type of bursting, then anti-phase continuous spiking, followed by the "fold/torus" type of bursting, and finally in-phase tonic spiking. On the other hand, it is shown that when the individual cell exhibits square-wave bursting, suitable coupling strength can make the electrically coupled system generate "fold/Hopf" bursting via "fold/fold" hysteresis loop; whereas, the chemically coupled cells generate "fold/subHopf" bursting. Especially, chemically coupled bursters can exhibit inverse period-adding bursting sequence. Fast-slow dynamics analysis is applied to explore the generation mechanism of these bursting oscillations. The above analysis of bursting types and the transition may provide us with better insight into understanding the role of coupling in the dynamic behaviors of pancreatic β-cells.

Modern concepts of seizure modeling

Seizures are complex phenomena spanning multiple spatial and temporal scales, from ion dynamics to communication between brain regions, from milliseconds (spikes) to days (interseizure intervals). Because of the existence of such multiple scales, the experimental evaluation of the mechanisms underlying the initiation, propagation, and termination of epileptic seizures is a difficult problem. Theoretical models and numerical simulations provide new tools to investigate seizure mechanisms at multiple scales. In this chapter, we review different theoretical approaches and their contributions to our understanding of seizure mechanisms.

How to suppress undesired synchronization

Examples of synchronization can be found in a wide range of phenomena such as neurons firing, lasers cascades, chemical reactions, and opinion formation. However, in many situations the formation of a coherent state is not pleasant and should be mitigated. For example, the onset of synchronization can be the root of epileptic seizures, traffic congestion in networks, and the collapse of constructions. Here we propose the use of contrarians to suppress undesired synchronization. We perform a comparative study of different strategies, either requiring local or total knowledge, and show that the most efficient one solely requires local information. Our results also reveal that, even when the distribution of neighboring interactions is narrow, significant improvement is observed when contrarians sit at the highly connected elements. The same qualitative results are obtained for artificially generated networks and two real ones, namely, the Routers of the Internet and a neuronal network.

Computational models of neuron-astrocyte interaction in epilepsy

Astrocytes actively shape the dynamics of neurons and neuronal ensembles by affecting several aspects critical to neuronal function, such as regulating synaptic plasticity, modulating neuronal excitability, and maintaining extracellular ion balance. These pathways for astrocyte-neuron interaction can also enhance the information-processing capabilities of brains, but in other circumstances may lead the brain on the road to pathological ruin. In this article, we review the existing computational models of astrocytic involvement in epileptogenesis, focusing on their relevance to existing physiological data.