Models of epilepsy

Epilepsy is a chronic neurological disorder characterized by occurrence and recurrence of seizures; epilepsy is thus a seizure disorder. A seizure in turn is a “transient of signs and/or symptoms due to abnormal excessive or synchronous neuronal activity in the brain” (Fisher, et al., 2005). Epilepsy is the third most common neurological disorder affecting more than 50 million people worldwide (http://www.who.int/mediacentre/factsheets/fs999/en/index.html). The seizures of about 30 to 40 percent of epilepsy patients do not respond to drugs, a percentage that has remained relatively stable despite significant efforts to develop new antiepileptic medication over the past decade (http://www.epilepsyfoundation.org/about/factsfigures.cfm).

A recent and particularly powerful approach towards the understanding of seizure dynamics is the combination of electrophysiology with high-resolution fluorescent imaging. Seizure mechanisms are too complex to understand without incorporating such measurements and observations into computational models. Theoretical predictions are quantitative in nature and allow direct comparison with experiments as access to various variables and parameters becomes increasingly available. Computational models are the most suitable tools to tie the advances made at various levels in epilepsy. Indeed, epileptic seizure is an example of a phenomenon which we feel cannot be properly understood without the lens that computational theory provides (Mitra and Bokil, 2007).

There is a vast and valuable literature on computational modeling in epilepsy that we cannot completely cover in this article due to the limited scope of this review. Here we outline some of these modeling efforts and refer the interested readers to the recent text of Soltesz and Staley (2008). Specifically, we review examples from two main classes of epilepsy models: (1) macroscopic, and (2) detailed network models. An excellent review of network models literature is given in (Lytton, 2008).

## Contents

* 1 Macroscopic (mean field) models
* 1.1 Advantages and disadvantages
* 2 Biophysical network models
* 2.1 Homeostasis and epilepsy
* 2.2 Graphical representation models
* 2.3 Ion concentrations dynamics in epilepsy
* 3 Mean Field vs Biophysical Models
* 4 Summary
* 5 References
* 6 See also

## [edit] Macroscopic (mean field) models

The brain is a nonlinear system of extraordinary complexity. A cubic centimeter of brain is roughly composed of millions of neuronal and glial cells with billions of synaptic, molecular, and proteomic cascades. To mathematically model such a complex system with full details requires billions of state variables and parameters – an impossible task. Fortunately, we think that the dynamics of seizures fall into a much lower dimensional space as suggested by EEG, ECOG, and local field potential measurements. Motivated by this fact neuroscientists have developed models that are composed of fewer system variables and parameters. These macroscopic models are the mean field models, and are typically extensions of the pioneering Wilson-Cowan equations (Wilson and Cowan, 1972). They are composed of variables and parameters describing the local spatiotemporal mean activity of two subclasses of neurons – excitatory principal cells (PCs) and inhibitory interneurons (INs) - without referring to the biophysical details of individual cells. Mean field models are based on the evidence of redundancy at small volumes within brain tissues as neurons within these volumes respond in similar fashion to identical stimuli. The basic principle of these models is that the extracellular currents of the well-organized neurons sum when their synapses activate and they respond with action potentials, creating further synaptic activities, which reflects in the EEG signal that can be recorded at a certain distance from the population.

As mentioned above the simplest of macroscopic models are the two subpopulation models (excitatory and inhibitory) with both inhibitory and excitatory connections among them (Figure 1A1), while more complex models with more than two subpopulations are needed to animate various complex EEG patterns seen in experiments (Lopes da Silva, et al., 1974; Wendling, et al., 2000). Two types of transfer functions are used for modeling the activity of the populations (Figure 1A2). The linear pulse to wave transfer function transforms the action potential density of the presynaptic population to the average inhibition or excitation at the postsynaptic population. The nonlinear (sigmoid function) wave to pulse transfer function transforms the wave activity (average level of transmembrane potential) of a subpopulation into the average firing rate of action potentials of the same subpopulation. Each subpopulation gives rise to two differential equations (see Wendling et al., 2000 for details). Multiple coupled populations can be constructed by coupling the single population model shown in Figure 1A1 to other populations.

Figure 1:

Figure 2: Basic structure of the mean field neuronal models. (A1) Two subpopulation model: PCs receive inhibitory inputs from INs and excitatory input from other PCs in the same subpopulation, while INs receive only excitatory input from PCs. Each subpopulation is characterized by two transfer functions - a linear pulse to wave and nonlinear wave to pulse (A2). The differential change in the MTLE hippocampus is taken into account by considering two interneuronal subpopulations – one projecting slow inhibition to the dendritic part of PCs and another projecting to the somatic part of PCs (B). Various behaviors exhibited by the model are summarized in (C1), where the slow dendritic inhibition is taken along the horizontal axis and fast somatic inhibition along the vertical axis. A representative trajectory from the model is shown in (C2, bottom panel) and the two inhibitory inputs in the (top panel). The model compares well with the experimental data (C3). Modified from Wendling, et al., (2002), Copyright © 2002 Federation of European Neuroscience Societies. Reproduced with permission of Blackwell Publishing Ltd.

Wendling and colleagues used the model described above to investigate mesial temporal lobe epilepsy (MTLE) (2000, 2002, 2005). The model exhibited various experimentally observed EEG patterns by varying the ratio of excitatory to inhibitory inputs. Specifically, the transition from preictal to ictal state occurred when the ratio of excitation and inhibition was increased above a certain threshold, thus supporting the hypothesis of increased relative excitation for seizure generation.

Models with more than two subpopulations were employed to explore various complex EEG patterns. For example, Wendling et al. (2002) added a third subpopulation to their model following the suggestion that various types of inhibitory projections to PCs could be impaired differentially in MTL epileptic hippocampus (Figure 1B). One IN subpopulation (dendritic projecting INs) projects to PC dendrites while the second INs subpopulation (basket cells) projects to PCs soma. The model indeed shows that transition from normal to fast ictal activity similar to experiments occurs when the two inhibition types are reduced differentially. For moderately elevated excitation the network switches from normal background activity (region b-1 in Figure 1C1) to rhythmic spikes (region b-2 in Figure 1C1) when slow dendritic inhibition is reduced while keeping the fast somatic inhibition fixed (transitions represented by black arrows in Figure 1C1). Low-voltage rapid discharges are observed when slow inhibition is reduced further (region b-3 in Figure 1C1). Finally, high amplitude paroxysmal activity is observed when slow dendritic inhibition is slightly increased and the fast somatic inhibition is reduced (region b-3 in Figure 1C1). Representative trajectories from regions (b1-b4) are shown in Figure 1C2 (bottom panel). Results from the model compares well with clinical EEG data (Figure 1C3), where the evolution of seizures through a sequence of dynamical stages is a very consistent feature of both scalp and intracranial EEG (Schiff, et al., 2005).

A similar approach was used by Suffczynski, et al. (2004) to develop a model for absence seizures that built on the model from Lopes da Silva et al. (1974) and considered four subpopulations: (1) cortical PCs, (2) cortical INs, (3) thalamocortical relay cells (TC), and (4) thalamic reticular nucleus cells (RE). It is worth mentioning that most other models in this section are cortical models whereas the model in Lopes da Silva (1974) included the thalamic component. The model includes AMPA excitatory synapses, fast GABAA, and slow GABAB inhibitory synapses. For a given set of parameters, the model shows bistability with random external input as the bifurcation parameter. Transitions between normal and seizure states occurs due to switching of the model in these two states caused by the variations in the external input. They analyzed the statistical properties of the mean duration of ictal and interictal epochs and looked at the role of various factors in changing the duration of these epochs. The duration of these epochs was most sensitive to changes in GABAA in cortex and the slope of the sigmoid transfer function for cortical INs. The model successfully demonstrated a control strategy to terminate seizures through both negative and positive current pulses. The phase space analysis in this study is an excellent example of the ease with which macroscopic models can be analyzed numerically. An excellent review of these population models is given in Lopes Da Silva, et al. (2003).

Recently, Kramer, et al. (2006) used the mean field approach to develop control strategies for human cortical electrical activity. In this study the authors modeled the cortex as a system of fourteen differential equations – one each for mean excitatory and inhibitory membrane potentials of cortical populations, and twelve equations mimicking the dynamics of excitatory and inhibitory synaptic and external inputs (from other layers). For certain parameter sets the average membrane potential of the excitatory cortical neurons bifurcates between steady state (normal activity) and limit cycle (seizure-like activity) as a function of the excitation in the system. The authors explored three control strategies to eliminate the limit cycle in the average membrane potential of the excitatory cortical neurons: (1) linear controller, (2) differential controller, and (3) filter controller. All three controllers were successful in terminating the oscillations in the model; however, the filter controller was the most successful strategy since other controllers pushed the model to a depolarized state while eliminating the limit cycle.

#### [edit] Advantages and disadvantages

The mean field models have certain advantages over the more detailed models described in the following section. These models are suitable for looking into transitions from interictal to ictal states, and for exploring EEG analysis from epilepsy patients as the macroelectrodes used for EEG recordings represent the average local field potential arising from neuronal populations similar to the lumped models. Surgical treatments fail to stop seizures from occurring in a certain fraction of patients. Mean field models might be used for more precise identification of brain structures that belong to the seizure-triggering zone. Very often, epileptic activity spreads over quite extended regions and involves several structures (cortical and sub-cortical). Since the mean field models remain relatively simple, they represent the best alternative to physiologically describe epileptic processes occurring in 'large-scale’ systems. Further, these models are easier to analyze numerically because relatively few variables and parameters are involved. The main disadvantage of these models is that they fail to suggest molecular and cellular mechanisms of epileptogenesis and thus are unable to model therapeutics targeting molecular pathways responsible for seizures.

## [edit] Biophysical network models

Detailed biophysical network models of epilepsy are used to investigate the role of biophysical and molecular properties of single neuron and neuronal networks in causing seizure-like patterns of activity. For example, how can the shift in synaptic weights or change in conductance of certain ionic currents cause seizure-like brain storms? Firing patterns of individual neurons are mostly controlled by various ion channel conductances, synaptic inputs, and the nearby microenvironment of neurons. Thus these models look into the molecular bases of epilepsy and can suggest therapeutics based on their predictions. Despite the limitations caused by the constraints of computational power and the uncertainty of many of the details of neuronal connections and their biophysical properties, these models are one of the most useful areas of computational epilepsy. The general framework for these models is to reproduce the experimentally observed properties of neuronal networks and then to investigate the effect of various factors on the behavior of the networks. Such detailed neuronal models provide access to factors that are usually inaccessible through experimental means. These models span a range of levels starting from a single synapse to networks composed of millions of neurons (http://bluebrain.epfl.ch/).

The small-scale network models are useful for making predictions that could be generalized to larger networks. Skinner, et al. (2005) for example presented a two-cell network model and examined the behavior of this network for certain sets of parameters. They then used the same set of parameters in the larger network of several tens of neurons and observed that the large network followed the same pattern of activity as seen in the smaller network. They extracted various parameters, such as synaptic and input currents relevant for the epileptic behavior under consideration.

Figure 3:

Figure 4: Impact of variations in excitatory and inhibitory synaptic strengths (in arbitrary units, AU) on the calculated extracellular activity. The domains of activity patterns in the excitation-inhibition parameter space are shown in the left panel of (A), and representative simulated activity patterns in the right panel of (A). The arrow indicates the path that was followed to generate the seizure onset activity. Abbreviations: D-desynchronized activity, IB-irregular bursting, O-oscillatory activity, RB-regular burst-firing. A reduction of excitatory coupling strength generates oscillatory network activity that can be viewed in the compound “EEG” signal (B-E). In both (B) and (C), all excitatory connections were reduced in strength. In trace (B) the network contained bursting neurons; in (C) this cell type was absent. In (D) the strength for the synaptic connections between the excitatory populations was selectively reduced; in trace (E) only the strength of the coupling between the excitatory and inhibitory populations was reduced. Comparing traces (D) and (E) with (B), one can see that both connections are critically associated with the network bursts and oscillations observed in trace (B). In both cases, a selective reduction of part of the excitatory system abolishes the oscillatory activity; although, on weakening the excitatory to excitatory connections, some oscillatory activity is retained (D). From Van Drongelen, et al., 2005 with permission (Copyright © 2005 IEEE).

In most models of pathological conditions such as seizures, a shift from dominant (or balanced) inhibition to dominant excitation in a neuronal network is considered to be responsible for the network transition from the preictal to ictal state (Trevelyan et al. 2006). In such conditions of imbalance between inhibition and excitation, moderate perturbations can drive a neuronal network from physiological to seizure-like activity. However, recent computational studies have questioned this idea (Van Drongelen, et al. 2005; 2007). A neocortical network consisting of 656 neurons (512 PCs and 144 INs) exhibited seizure-like behavior when the synaptic excitation was decreased (Figure 2A). In order to transition to a seizure-like state the excitation to both excitatory cells and inhibitory cells needed to be reduced (Figure 2B). Reducing each one alone was insufficient to cause the network to seizure-like behavior (Figure 2D, E). One possible explanation for this behavior could be the reduced firing rate in INs in response to reduced excitation and hence reduced relative inhibition in the network. Furthermore, the burst-firing PCs were found to be critical for observing various behaviors in the network. Without these burst-firing cells the irregular and regular spiking (increased synchrony) are both lost (Figure 2C). These findings are supported by experimental observations where brain tissue makes a transition to seizures when a small amount of CNQX (AMPA receptor blocker) is added to it. Bicuculline in mouse younger than 15 days (where GABAA is excitatory) also caused seizures in the tissue (Van Drongelen, et al. 2003). They also showed that Riluzole (persistent sodium current blocker) reduced the network burst frequency (Van Drongelen, et al. 2005). The unexpected result of this study is one example emphasizing the value of computational modeling in epilepsy, as intuition in absence of modeling would have suggested that increased excitation alone would cause seizures.

Destexhe and colleagues have done extensive work to explain absence seizures by developing detailed models of thalamocortical networks. A major finding of these efforts is the role played by the interplay between thalamocortical regions in generating absence seizures. They began with a simple model of inhibitory neurons of the thalamic reticular nucleus (RE) and thalamocortical cells (TC), with GABAA and GABAB inhibition from RE and AMPA excitation from TC, to show that cooperative GABAB responses (nonlinear response of GABAB receptors to presynaptic action potentials) can explain the effect of an anti-absence epilepsy drug clonazepam (Destexhe and Sejnowski, 1995; Destexhe, et al., 1996). This study showed that clonazepam reduces GABAB-mediated IPSPs in TC cells; however, the effect of clonazepam was not on the TC cells but instead reinforced GABAB receptors in RE cells, thus causing reduced firing in RE and hence reduced GABAB input from RE to TC.

This study also showed that the suppression of GABAB could cause the network to transition from controlled spindle oscillations to low frequency (3-4Hz) hyper-synchronized oscillations. The study however suggests that the low frequency hyper-synchronized oscillations are not sufficient to explain seizures. The typical spike-and-wave patterns of absence seizures require an interplay between both thalamus and cortex. Although seizures could be generated intracortically, the thalamus seems to be necessary for absence seizures. Spike-and-wave seizures disappear following the inactivation of the thalamus. Blocking GABAB also leads to suppression of spike-and-wave seizures.

They expanded their thalamus model to a thalamocortical network model to investigate the role played by these two regions in absence seizures. The model included RE and TC cells from thalamus, and PCs and INs from cortex (Destexhe, et al., 1998). The simulations show that blocking GABAA in cortex with the thalamus intact caused the spindle oscillations (~10Hz) to switch to spike-and-wave seizure patterns (~3Hz). Blocking GABAA in thalamus switched the spindle oscillations to low frequency oscillations (3-4Hz), and spike-and-wave seizures could not be observed. This model with different ratios of GABAA and GABAB also mimicked the relatively fast spike-and-wave seizures (5-10Hz) observed in rats.

The findings of the model were experimentally confirmed by Bal, et al. (2000) who showed that a strong corticothalamic feedback could force physiologically intact thalamic circuits to oscillate at 3Hz frequency. They observed strong synchronization in TC cells and an enhancement of burst discharges in RE, which is in agreement with the prediction that a strong GABAB response is responsible for the slow oscillations in intact thalamus.

In various seizure models, very fast oscillations having frequencies greater than 70Hz have been observed, immediately before spontaneous seizures both in vivo and in vitro (see for example Traub, et al., 2001 and Worrel, et al., 2004). Such oscillations occur in close proximity to the seizure onset sites, and might be a functional indicator of the location of the epileptic focus. Roger Traub and colleagues have performed detailed network modeling in conjunction with experimental studies to understand the mechanism of very fast oscillations and their transition to seizures (Traub, et al., 2001; Traub, et al., 2003; Traub, et al., 2005, see also Traub and Wong, 1982). Understanding these very fast oscillations is extremely important as this could provide clues to factors that initiate seizures and possible therapeutic targets. The extensive biophysical modeling of Traub’s group has unveiled the potential role played by gap-junctional coupling between PCs in shaping very fast oscillations. Based on their observations, they drew the hypothesis that the electrical coupling between the axons of PCs through gap junctions is necessary for very fast oscillations, although the kinetics of potassium channels and recurrent excitation may also play a role. Excellent reviews spanning years of the unique work of Traub’s group are given in (Traub’s et al., 2002; Traub, et al., 2004).

Most recently, experimental evidence from Trevelyan (2008; unpublished data, 2009) suggests that high frequency inhibitory activity may contribute to such high frequency field oscillations. Such activity may be the cellular signature of an inhibitory ‘veto’, whose breakdown may enable seizures to propagate from their focus of inception (Trevelyan et al. 2006, 2007).

Computational modeling tied to experiments will work together in the coming years to sort out the role and origins of high frequency oscillations in seizures.

#### [edit] Homeostasis and epilepsy

Homeostatic scaling is a synaptic plasticity mechanism that scales up or down the excitation in a single neuron, or entire network, to tune its excitability to certain target values (see Marder and Prinz, 2002; Turrigiano, 2008 for review). For example, if the frequency of a network drops below a certain target frequency the network raises its excitability to achieve this target frequency. Homeostasis thus maintains a balanced excitatory-inhibitory environment in the brain. However, there is theoretical evidence that homeostatic plasticity could be a possible mechanism for post-traumatic epileptogenesis. Within few hours of head injury many patients with deep intracranial wounds display clinical seizures. Houweling, et al. (2005) investigated the role played by homeostatic plasticity in developing post-traumatic epileptogenesis. They constructed a network model that mimics traumatic-induced deafferentiation by reducing the excitatory inputs. Homeostatic plasticity was taken into account by making the synaptic strengths variable as a function of average firing rate of the network. The deafferentiated network has reduced excitability and thus reduced firing rate in PCs. They set the target frequency of PCs at 5Hz, and allowed homeostasis to act through an increase in excitability and simultaneously a 50% decrease in inhibition. They found that increased homeostasis could cause epileptic type bursts in PCs. The burst rate, spikes per burst, and velocity of bursts increases with increasing homeostasis.

Following the strategy in (Houweling, et al., 2005), Flavio, et al. (2008) explored in detail the contribution of deafferentiation level and various other factors in developing post-traumatic epilepsy. The authors constructed a cortical network model to show that homeostatic plasticity in networks with deafferentation beyond a certain degree could cause pathological periodic discharges with similar patterns as seen in pathological conditions. As above, deafferentation after trauma was modeled as a reduction in the excitatory inputs in the network. Homeostatic plasticity, while trying to maintain the target frequency, by enhancing the excitatory inputs and reducing inhibition could cause periodic discharges in the network. The study found a negligible role of inhibitory down regulation in generating these discharges – they were mostly regulated by excitation. To get to the target frequency, the intact cells increased their frequency beyond the post-deafferented state, while the deafferented cells increased their firing rate in order to recover the mean target firing rate. They showed that the transition to pathological conditions due to homeostatic plasticity is a nonlinear phenomenon.

#### [edit] Graphical representation models

The functional models described above consider the synaptic interactions among various neuronal types to be the primary determining factor of the functional state of neuronal networks. Seizure-like behavior, for example, is typically modeled as the transition from dominant inhibition to dominant excitation. These kinds of models ignore the role played by the topology of the network. Graphical representation models take into consideration the topological details of the network along with functional details. The main focus of graphical representation models in epilepsy has been the effect of changes in network topology on the stability of networks.

In the surgically removed hippocampus from patients with temporal lobe epilepsy (TLE), major changes in the anatomy of dentate gyrus were observed (Babb, et al., 1991). These changes include cell death and formation of new synaptic connections as axons sprout. To investigate the role of anatomical changes in causing spontaneous seizures, Lytton and coworkers developed a network model of the dentate gyrus (Lytton, et al., 1998). The model consisted of granule cells (the principal excitatory cells located in the granule layer), inhibitory aspiny interneurons, and excitatory mossy cells (that reside largely within the hilus). They further supposed that the cells that die mostly include mossy cells and aspiny interneurons. The axons of granule cells sprout (mossy fiber sprouting1) into areas denervated by the cell loss and mostly form synapses to other granule cells. This study found that mossy fibers sprouting along with disinhibition is necessary to produce repeated firing in the granule cells characteristic of seizures.

Other excellent examples of graphical representation models are those studied in Santhakumar et al. (2005), and Dyhrfjeld-Johnsen et al. (2007), where the authors investigated the role of hilus cell loss2 and mossy fiber (granule cell axons) sprouting in the epileptic brain. These network models are based on the dentate gyrus model of MTLE and consists of eight cell types: granule cells (excitatory), mossy cells (excitatory), basket cells (inhibitory), hilar interneurons with axonal projections to the perforant-path termination zone, axo-axonic cells (inhibitory), hilar interneurons with axonal projections to commissural-associational pathway termination zone, interneuron selective cells, and molecular layer interneurons with axonal projections to the perforant-path termination zone.

Figure 5: Effect of mossy fiber sprouting and mossy cell loss on the activity of dentate excitatory network. (A) Spike raster plots showing the activity of granule cells from topographic-ring networks illustrate the spread of perforant-path-evoked activity with increasing mossy fiber sprouting. (B) Granule cell spike-time raster evoked by stimulation of 100 granule cells in disinhibited dentate gyrus incorporating axonal conductance delays. Networks were simulated with 0%, 50%, and 100% mossy cell loss. (C) Summary plot of the granule cell activity with varying degrees of mossy cell loss and sprouting in networks with non-spontaneously active mossy cells. Because the network activity increased with the degree of mossy fiber sprouting, the granule cell activity in each plot was normalized to the firing without mossy cell loss (average number of spikes during the 1st second after stimulation: 10% sprouting: 4.61; 15%: 13.33; 25%: 121.95; 50%: 162.81; 10% sprouting with delay. (D) Summary histogram of the effect of mossy cell loss on the number of granule cells activated by input stimulus with 10% mossy fiber sprouting. From Santhakumar, et al. (2005) with permission (Copyright © 2005 American Physiological Society).

As shown in Figure (3A) mossy fiber sprouting significantly enhanced the excitability of a network with 500+ cells. Hilus cell loss on the other hand had little effect on the network activity (average firing of granule cells and spread of network activity) with sprouting > 10% (100% sprouting corresponds to the densest number of synaptic connections between granule cells experimentally observed in epileptic dentate gyrus). However, for 10% sprouting, the reduction in hilus cells decreases both the average granule cell firing (Figure 3B, C) and number of granule cells involved in the network activity (Figure 3D). This result indicates that once the degree of mossy fiber sprouting is above a certain limit the network can sustain seizure-like activity regardless of the number of hilus cells present. Furthermore, mossy cell loss is neither necessary nor sufficient to increase excitability, since deletion of mossy cells consistently resulted in decreased excitability in model networks and impeded the propagation of network hyperexcitability. This behavior is in contrast to previous assumptions where the reduction in mossy cells was considered to cause increased excitability, since mossy cells excite basket cells that in turn inhibit granule cells.

Following Santhakumar et al. (2005), a large-scale model of over a million nodes and over a billion synaptic connections was constructed to study the structural characteristics of the epileptic dentate gyrus network and the role of these structural factors on network excitability (Dyhrfjeld-Johnsen et al., 2007). This study demonstrated that small world characteristics of the dentate graph (low path length, high clustering coefficient3) increases during sclerosis4 until levels above 90% sclerosis where the network transformed into a regular graph (high path length and high clustering coefficient). No change in the path length with only sprouting (no hilus cell loss) indicates that the hilus cell loss (mainly mossy cells) plays a major role in changing the path length of the network. On the other hand, the clustering coefficient in the case where only sprouting is considered follows a very similar course to the case where both sprouting and hilus cell loss is considered, indicating that sprouting is the main determinant of the changes in clustering coefficient. Network hyperexcitability increased by increasing sclerosis up to 80% and then started decreasing. The decrease in hyperexcitability started at the same point where the network topology converted from small world network to normal network. This biphasic behavior in the hyperexcitability cannot be explained by only sprouting as the hyperexcitability increased monotonically up to maximum level with increasing sprouting. To summarize, during sub-maximal sclerosis sprouting is the primary determinant of change in network topology. At maximal sclerosis the hilus cells loss (specifically moss cells) is the primary determinant for transforming network topology, and thus the sprouting is not sufficient to compensate for the loss of connections due to hilus cells loss. Other factors such as the number of inhibitory cells, axonal delay, and stimulus pattern did not substantially change the outcome of this study.

#### [edit] Ion concentrations dynamics in epilepsy

In physiological states assuming constant extra- and intracellular ion concentrations may be a reasonable assumption. However, in epilepsy where neurons exhibit intense firing, relying on this assumption is especially subject to question. During neuronal activity the extracellular potassium and intracellular sodium concentrations increase causing more positive transmembrane reversal potential for potassium current and less positive reversal potential for sodium current respectively. Due to the smaller extracellular space as cells swell following activity, the extracellular buildup of potassium can sufficiently enhance network excitability to cause spontaneous neuronal activity. While the experimental investigation of the role played by dynamic ion concentrations in epileptogenesis is limited, several theoretical groups have considered the effect of dynamic extra- and intracellular ion concentrations on the excitability of single neuron and neuronal networks (Bazhenov, et al., 2004; Cressman, et al., 2009; Park, et al., 2006; Somjen, et al., 2008, Ullah, et al., 2009). Single neurons have been shown to exhibit spontaneous “seizures” due to dynamic potassium and sodium concentrations when glial potassium buffering is impaired causing an extracellular potassium buildup (Cressman, et al., 2009; see also Somjen, et al., 2008; Kager, et al., 2000; Kager, et al., 2007). A similar dynamics is observed in network models where a given network makes a transition between physiological and seizure-like states depending on the glial buffering strength (Ullah, et al., 2009). Thus the balance between excitatory and inhibitory synaptic inputs is not the only reason that a normal network can switch to seizure activity - the ionic microenvironment plays a significant role.

Figure 6: Bifurcation diagram of single cell potassium dynamics (A) and the observed seizure-like discharges during the limit cycle of the bifurcation diagram (B, top panel). Bottom panel in (B) shows oscillations in potassium (solid line) and sodium (dotted line) concentrations. From Cressman, et al., (2009) with permission (Copyright © 2009 Springer).

Recently, Cressman, et al. (2009) performed a detailed bifurcation analysis of potassium dynamics in a single cell model. They extended the Hodgkin-Huxley equations by adding variable potassium and sodium concentrations (and hence variable reversal potentials for potassium and sodium currents). In the model sodium membrane currents and ATP-dependent pumps working against chemical gradients control sodium concentration inside the cell. Similarly, potassium membrane currents cause a buildup of potassium concentration in the extracellular volume, which is either buffered by the glial network surrounding the cell, diffuses away into the extracellular microenvironment, or is pumped back into the cell by the ATP-dependent pump. For a given set of parameters the extracellular potassium and intracellular sodium concentrations switch from steady state to an oscillatory mode following a change in potassium concentration in the bath solution (Figure 4A). During this limit cycle, the neuronal membrane potential exhibits seizure-like discharges where each seizure lasts for several seconds. Similar limit cycles are observed when the glial strength of potassium buffering is considered as a bifurcation parameter. The findings of Cressman, et al. (2009) were extended to the neuronal network of the hippocampus where glial buffering impairment proved crucial for the network’s transition from normal memory retaining states to seizure-like behaviors (Ullah, et al., 2009). This and other studies (cited above) show that dynamic ion concentrations, specifically potassium, could be a crucial ingredient in the dynamics of seizures. We have recently shown through detailed mathematical modeling that taking into account the dynamics of extra - and intracellular ion concentrations is necessary for reproducing the firing interplay between INs and PCs during seizure-like events observed by Ziburkus, et al. (2006, Ullah, et al., unpublished data).

It has recently been shown that such potassium dynamics could be placed within a control engineering approach (Ullah and Schiff, 2009). The use of ensemble Kalman filters (Schiff and Sauer, 2008) for such biological nonlinear dynamics offer powerful model-based approach to assimilate data from neuronal recordings. This gives us the ability to track and control phenomena such as seizures. Such control methods are only as good as the ability of the computational models of epilepsy are to replicate seizure dynamics. As this review should make clear, the increasing fidelity of such computational models are offering us substantially improved frameworks for future experimental and clinical model-based observation and control of seizures (Ullah and Schiff, 2009; Sauer and Schiff, 2009).

## [edit] Mean Field vs Biophysical Models

The detailed network models are best suited for understanding the molecular and cellular bases of epilepsy and thus are well positioned to suggest therapeutics that could target molecular pathways. Macroscopic models, on the other hand, are more appropriate for describing epileptic processes occurring on large-scale (see section 1.1). Due to the substantial complexity of neuronal structures, relatively few variables and parameters can be accessed at any time experimentally. Although biophysically explicit modeling is the primary technique to look into the role played by experimentally inaccessible variables in epilepsy, the usefulness of detailed biophysical models is limited by constraints in computational power, uncertainties in detailed knowledge of neuronal systems, and the required simplification for the numerical analysis. Unlike the lumped models, detailed network models are much more difficult to analyze numerically for a range of parameters as their dynamics take place in many dimensional state space. An intermediate 'across-scale' approach, establishing relationships between sub-cellular/cellular variables of detailed models and "aggregated" parameters governing macroscopic models, would be a very useful strategy to cover the gaps between these two modeling approaches.

## [edit] Summary

Here we have reviewed some of the modeling efforts that have been made in recent years to unveil the mystery of epilepsy. Due to several experimentally and clinically observed epilepsies, the modeling work spans a wide range from the single synapse to networks of millions of neurons, and from lumped two variable models to detailed biophysical models involving tens of thousands of variables and parameters. We began with the relatively simple mean field models and showed how these models could explain various EEG signals recorded during seizures and interictal to ictal transitions. We then discussed more detailed network models with examples involving various levels of complexity and biophysical detail. The role of homeostatic plasticity and topological factors in epileptogenesis was discussed. Finally, we outlined the latest developments concerning the involvement of dynamic ion concentrations in seizures, and some recent work placing such findings in a control engineering framework. Stochastic models are another important category of computer models in epilepsy that we omitted in this review. These probabilistic models are directed towards the prediction of seizure onset and are well reviewed in Lehnertz, et al. (2007).

Despite an extraordinary amount of interest in understanding the dynamics of seizures, we still lack a unifying dynamical definition of what a seizure is (Soltesz and Staley, 2008). The extraordinary variety of experimental preparations and human epilepsies makes the quest for unifying principles especially difficult. Epilepsy is a good example of a dynamical disease where theory and computation must work hand-in-hand with experiment to bring us a deeper understanding and more rational therapeutics.

Footnotes

1Mossy fibers sprouting is modeled by increasing the number of synaptic connections between granule cells and their synaptic targets.

2Hilus cell loss contains both excitatory and inhibitory cell loss.

3Path length is defined as the number of steps required connecting any given two nodes in a network, while clustering coefficient is a property of a node in a network. Roughly speaking it tells how well connected the neighborhood of the node is. If the neighborhood is fully connected, the clustering coefficient is 1 and a value close to 0 means that there are hardly any connections in the neighborhood.

4Sclerosis is defined as the combined effect of sprouting and hilus cell loss. For example, 50% sclerosis is equal to 50% sprouting plus 50% hilus cell loss.

## [edit] References

Babb, T., Kupfer, W., Pretorius, J., Crandall, P., Levesque, M. (1991) Synaptic reorganization by mossy fibers in human epileptic fascia dentate. Neurosci. 42:341-363.

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Internal references

* John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.

* Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.

* Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.

* Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.

* Paul L. Nunez and Ramesh Srinivasan (2007) Electroencephalogram. Scholarpedia, 2(2):1348.

* Roger D. Traub (2006) Fast oscillations. Scholarpedia, 1(12):1764.

* Eugene Roberts (2007) Gamma-aminobutyric acid. Scholarpedia, 2(10):3356.

* Tamas Freund and Szabolcs Kali (2008) Interneurons. Scholarpedia, 3(9):4720.

* Bertil Hille (2008) Ion channels. Scholarpedia, 3(10):6051.

* Howard Eichenbaum (2008) Memory. Scholarpedia, 3(3):1747.

* Richard H. Granger and Robert A. Hearn (2007) Models of thalamocortical system. Scholarpedia, 2(11):1796.

* Peter Jonas and Gyorgy Buzsaki (2007) Neural inhibition. Scholarpedia, 2(9):3286.

* Rodolfo Llinas (2008) Neuron. Scholarpedia, 3(8):1490.

* Kendall E. Atkinson (2007) Numerical analysis. Scholarpedia, 2(8):3163.

* Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

* Alain Destexhe (2007) Spike-and-wave oscillations. Scholarpedia, 2(2):1402.

* Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

* David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.

* Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.

* S. Murray Sherman (2006) Thalamus. Scholarpedia, 1(9):1583.

## [edit] See also

Models of thalamocortical system, Spike-and-wave oscillations