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{{Missing information|[[deterministic model]]s on a micro scale (membrane potential dynamics) and [[stochastic|stochastic models]] of epileptic electrophysiological activity|date=January 2014}}
'''Mathematical modeling of electrophysiological activity in epilepsy''' is a computational method for simulating the key mechanism in the development of [[epileptic seizure]]s, namely the transition from normal [[electrophysiology|electrophysiological]] activity in the brain to abnormal [[neural synchronization|hypersynchronization]]. A similar type of hypersynchronization occurs in [[delta wave]]s during normal sleep. It is possible to estimate the rate of spread and migration of such regions of hypersynchronous [[neuron]]al activity in experimental and clinical settings using [[electroencephalography]] or [[electrocorticography]] for tracking electrophysiological activity of the brain.

== Modeling ==

Excessive, large-scale [[epilepsy|hypersynchronous neuronal activity]] in the brain is a hallmark of epilepsy.<ref>R. Fisher, W. van Emde Boas, W. Blume, C. Elger, P. Genton, P. Lee  and J. Engel (2005). "Epileptic seizures and epilepsy: definitions proposed by the International League Against Epilepsy (ILAE) and the International Bureau for Epilepsy (IBE)". Epilepsia 46 (4): 470–2. doi:10.1111/j.0013-9580.2005.66104.x. PMID 15816939.</ref> However, the analysis of large-scale electrophysiological activity during epileptic seizures is challenging, because simultaneous tracking of electrical activity in large numbers of neurons is technically difficult.<ref name="Epilepsy Book">{{pn|date=January 2014}}Epilepsy as a Dynamic Disease, edited by J. Milton and P. Jung, Biological and Medical Physics series Springer, Berlin, 2003.</ref> Moreover, electroencephalography (EEG), which is primarily used for monitoring electrophysiological activity of the brain during epileptic seizures, represents global (space-averaged) dynamical behavior of large neuronal populations. This global behavior involves millions of electrically connected, synchronized neuronal cells, and cannot be simply inferred from single-cell behavior.<ref name="Stefanescu et al.">[http://www.seizure-journal.com/article/S1059-1311(12)00224-5/fulltext Stefanescu RA, Shivakeshavan RG, Talathiemail SS. Computational models of epilepsy. Seizure: European Journal of Epilepsy, 2012; 21 (10):748-759.]</ref> A number of models have been developed for studying electrophysiological activity in epilepsy, including those based on [[Hodgkin–Huxley model|Hodgkin–Huxley-type equations]], describing changes in the flow of ionic currents across the membrane of a single cell or small groups of cells.<ref name="Stefanescu et al."/> The single-cell and small-group models are useful for studying ionic channels in the cell membrane, as well as other cellular, molecular and biochemical processes.<ref name="Stefanescu et al."/> However, when the single-cell approach is applied to model behavior of the entire brain, both theoretical analysis and numerical simulations become difficult due to large numbers of interacting variables. Furthermore, it is difficult to validate such models' predictions based on EEG data, which represent global (space-averaged) dynamical behavior of neuronal populations. 

To study the brain's behavior at the system level, Wilson and Cowan introduced a large-scale (coarse-grained mean field) approach, referred to as  [[Wilson–Cowan model]],<ref name="WC72">[http://www.cell.com/biophysj/retrieve/pii/S0006349572860685 H.R. Wilson and J.D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons.] Biophys. J., 12:1–24, 1972. PMID 4332108 </ref> which can be used for analyzing EEG patterns during epileptic seizures, as described below.

== Analysis ==

The determination of three concepts is fundamental to an understanding of hypersynchronization of neurophysiological activity at the global (system) level:<ref name="Shusterman-Troy">V. Shusterman and W. C. Troy. From baseline to epileptiform activity: A path to synchronized rhythmicity in large-scale neural networks. Phys Rev E Stat Nonlin Soft Matter Phys. 2008;77(6 Pt 1):061911 PMID 18643304</ref>

# The mechanism by which normal (baseline) neurophysiological activity evolves into hypersynchronization of large regions of the brain during epileptic seizures
<!--How can normal (baseline)  neurophysiological activity evolve into hypersynchronization of large regions of the brain during epileptic seizures?-->
# The key factors that govern the rate of expansion of hypersynchronized regions
<!--What are the key factors determining the rate of expansion of hypersynchronized regions?-->
# The electrophysiological activity pattern dynamics on a large-scale <!--This point seems a bit vague - might be useful to clarify what exactly makes patterns interesting or relevant to the model.-->
<!--How do large-scale patterns of electrophysiological activity in adjacent areas interact with each other?-->

A [[canonical analysis]] of these issues, developed in 2008 by Shusterman and Troy using the Wilson–Cowan model,<ref name="Shusterman-Troy"/> predicts qualitative and quantitative features of epileptiform activity. In particular, it accurately predicts the propagation speed of epileptic seizures (which is approximately 4–7 times slower than normal brain wave activity) in a human subject with chronically implanted electroencephalographic electrodes.<ref name="Towle-Milton Book">V. L. Towle, F. Ahmad, M. Kohrman, K. Hecox, and S. Chkhenkeli, in ''Epilepsy as a Dynamic Disease'', pp. 69–81</ref><ref>J. G. Milton. Mathematical Review: From baseline to epileptiform activity: A path to synchronized rhythmicity in large-scale neural networks by V. Shusterman and W. C. Troy (Phys. Rev. E 77: 061911). Math. Rev. 2010: 92025.</ref>

===Transition into hypersynchronization===

The transition from normal state of brain activity to epileptic seizures was not formulated theoretically until 2008, when a theoretical path from a baseline state to large-scale self-sustained oscillations, which spread out uniformly from the point of stimulus, has been mapped for the first time.<ref name="Shusterman-Troy"/>

A realistic state of baseline physiological activity has been defined, using the following '''two-component definition''':<ref name="Shusterman-Troy"/>

'''(1)''' A time-independent component represented by subthreshold excitatory activity E and superthreshold inhibitory activity I.

'''(2)''' A time-varying component which may include singlepulse waves, multipulse waves, or periodic waves caused by spontaneous neuronal activity.

This baseline state represents activity of the brain in the state of relaxation, in which neurons receive some level of spontaneous, weak stimulation by small, naturally present concentrations of neurohormonal substances. In waking adults this state is commonly associated with [[Alpha wave|alpha rhythm]], whereas slower ([[Theta wave|theta]] and [[Delta wave|delta]] rhythms  are usually observed during deeper relaxation and sleep. To describe this general setting, a 3-variable <math> (u,I,v) </math> spatially dependent extension of the classical [[Wilson–Cowan model]] can be utilized.<ref name="WC-paper2">H. R. Wilson and J. D. Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13, 55,1973. PMID 4767470</ref> Under appropriate initial conditions,<ref name="Shusterman-Troy"/> the excitatory component, u, dominates over the inhibitory component, I, and the three-variable system reduces to the two-variable Pinto-Ermentrout type model<ref name="Pinto and G. B. Ermentrout-2001">D. Pinto and G. B. Ermentrout. Spatially Structured Activity in Synaptically Coupled Neuronal Networks: I. Traveling Fronts and Pulses. SIAM J. Appl. Math. 62, 206;2001.</ref>

<math>{\partial u  \over \partial t}=u-v+             \int_{R^2}\omega(x-x',y-y')f(u-\theta)\,dxdy + \zeta(x,y,t),</math>

<br />
<math>{\partial v  \over \partial t}=\epsilon (\beta u-v).</math>

The variable v governs the recovery of excitation u; <math> \epsilon>0 </math> and <math> \beta>0 </math> determine the rate of change of recovery. The connection function <math> \omega(x,y) </math> is positive, continuous, symmetric, and has the typical form <math> \omega=Ae^{-\lambda\sqrt {-(x^2+y^2)}}</math>.<ref name="Pinto and G. B. Ermentrout-2001"/> In Ref.<ref name="Shusterman-Troy"/> <math>(A,\lambda)=(2.1,1).</math> The firing rate function, which is generally accepted to have a sharply increasing sigmoidal shape, is approximated by <math> f(u-\theta)=H(u-\theta) </math>, where H denotes the Heaviside function; <math> \zeta(x,y,t) </math> is a short-time stimulus. This <math> (u,v) </math> system has been successfully used in a wide variety of neuroscience research studies.<ref name="Pinto and G. B. Ermentrout-2001"/><ref name="Folias_Bressloff_1">S. E. Folias and P. C. Bressloff, Breathing pulses in an excitatory neural network. SIAM J. Appl. Dyn. Syst. 3,: 378-407, 2004.</ref><ref name="Folias_Bressloff_2">S. E. Folias and P. C. Bressloff, Breathers in two-dimensional neural media, Phys Rev Lett. 2005 Nov 11;95(20):208107. PMID 16384107 </ref><ref name="Kilpatrick_Bressloff">Z. P. Kilpatrick and P. C. Bressloff. Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network. Physica D, 239, 547-560, 2010.</ref><ref name="Laing_Troy">C. R. Laing and W. C. Troy. PDE methods for non-local models. SIAM J Appl Dyn Syst. (Vol 2, no. 3, pp. 487-516, 2003).</ref> In particular, it predicted the existence of spiral waves, which can occur during seizures; this theoretical prediction  was subsequently confirmed experimentally using optical imaging of slices from the rat cortex.<ref name="Spiral-wave paper">X. Huang, W. C. Troy, Q. Yang, H. Ma, C. R. Laing, S. J. Schiff and J. Y. Wu. Spiral waves in disinhibited mammalian neocortex. J Neurosci. 2004 Nov 3;24(44):9897-902.</ref>

===Rate of expansion===
The expansion of hypersynchronized regions exhibiting large-amplitude stable bulk oscillations occurs when the oscillations coexist with the stable rest state <math>(u,v)=(0,0)</math>. To understand the mechanism responsible for the expansion, it is necessary to linearize the <math> (u, v) </math> system around <math>(0,0)</math> when <math>\epsilon>0</math> is held fixed. The linearized system exhibits subthreshold decaying oscillations whose frequency increases as <math>\beta</math> increases. At a critical value  <math>\beta^{*}</math> where the oscillation frequency is high enough, bistability occurs in the <math>(u,v)</math>  system: a stable, spatially independent, periodic solution (bulk oscillation) and a stable rest state coexist over a continuous range of parameters. When <math>\beta\ge\beta^{*}</math> where bulk oscillations occur,<ref name="Shusterman-Troy"/> "The rate of expansion of the hypersynchronization region is determined by an interplay between two key features: (i) the speed '''c''' of waves that form and propagate outward from the edge of the region, and (ii) the concave shape of the graph of the activation variable '''u''' as it rises, during each bulk oscillation cycle, from the rest state '''u'''=0 to the activation threshold. Numerical experiments show that during the rise of 
'''u''' towards threshold, as the rate of vertical increase slows down, over time interval <math>\Delta t,</math> due to the concave component, the stable solitary wave emanating from the region causes the region to expand spatially at a '''Rate''' proportional to the wave speed. From this initial observation it is natural to expect that the proportionality constant should be the fraction of the time that the solution is concave during one cycle." Therefore, when <math>\beta\ge\beta^{*}</math>, the rate of expansion of the region is estimated by<ref name="Shusterman-Troy"/>

<math> Rate =(\Delta t/T)*c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1) </math>

where <math>\Delta t </math> is the length of subthreshold time interval, '''T''' is period of the periodic solution;  '''c''' is  the speed of waves emanating from the hypersynchronization region. A realistic value of '''c''', derived by Wilson et al.,<ref name="Wilson-Nature">H. R. Wilson, R. Blake and S. H. Lee. Dynamics of travelling waves in visual perception. Nature 2001, 412, 907-910.</ref> is '''c'''=22.4&nbsp;mm/s.

'''How to evaluate the ratio <math>\Delta t/T?</math>''' To determine values for <math> \Delta t/T </math> it is necessary to analyze the underlying bulk oscillation which satisfies the spatially independent system

<math>{{du} \over {dt}}=u-v+H(u-\theta),</math>

<br />
<math>{{dv} \over {dt}}=\epsilon (\beta u-v).</math>

This system is derived using standard functions and parameter values <math> \omega=2.1e^{-\lambda\sqrt {-(x^2+y^2)}}</math>, <math> \epsilon=0.1</math> and <math> \theta=0.1 </math><ref name="Shusterman-Troy"/><ref name="Pinto and G. B. Ermentrout-2001"/><ref name="Folias_Bressloff_1"/><ref name="Folias_Bressloff_2"/> Bulk oscillations occur when <math> \beta \ge \beta^{*}=12.61</math>. When <math>12.61 \le \beta \le 17</math>, Shusterman and Troy analyzed the bulk oscillations and found <math>0.136 \le \Delta t/T \le 0.238</math>. This gives the range

<br />
<math> 3.046 mm/s \le Rate \le 5.331 mm/s~~~~~~~~~~~~(2) </math>

<br />
Since <math>0.136 \le \Delta t/T \le 0.238</math>, Eq. (1) shows that the migration '''Rate''' is a fraction of the traveling wave speed, which is consistent with experimental and clinical observations regarding the slow spread of epileptic activity.<ref name="Epilepsy Book"/> This migration mechanism also provides a plausible explanation for spread and sustenance of epileptiform activity without a driving source that, despite a number of experimental studies, has never been observed.<ref name="Epilepsy Book"/>

===Comparing theoretical and experimental migration rates===

The rate of migration of hypersynchronous activity that was experimentally recorded during seizures in a human subject, using chronically implanted subdural electrodes on the surface of the left temporal lobe,<ref name="Towle-Milton Book"/> has been estimated as<ref name="Shusterman-Troy"/>

<br />
<math>Rate \approx 4 mm/s</math>,

<br />
which is consistent with the theoretically predicted range given above in (2). The ratio '''Rate/c''' in formula (1) shows that the leading edge of the region of synchronous seizure activity migrates approximately 4–7 times more slowly than normal brain wave activity, which is in agreement with the experimental data described above.<ref name="Towle-Milton Book"/>

To summarize, mathematical modeling and theoretical analysis of large-scale electrophysiological activity provide tools for predicting the spread and migration of hypersynchronous brain activity, which can be useful for diagnostic evaluation and management of patients with epilepsy. It might be also useful for predicting migration and spread of electrical activity over large regions of the brain that occur during deep sleep ([[Delta wave]]), cognitive activity and in other functional settings.

==References==
{{reflist|30em}}

[[Category:Epilepsy]]
[[Category:Disorders causing seizures]]
[[Category:Neuroscience]]
[[Category:Neural networks]]
[[Category:Neurophysiology]]
[[Category:Electrophysiology]]
[[Category:Computational neuroscience]]