Mathematical modeling of electrophysiological activity in epilepsy is a computational method for simulating the key mechanism in the development of epileptic seizures, namely the transition from normal electrophysiological activity in the brain to abnormal hypersynchronization. A similar type of hypersynchronization occurs in delta waves during normal sleep. It is possible to estimate the rate of spread and migration of such regions of hypersynchronous neuronal activity in experimental and clinical settings using electroencephalography or electrocorticography for tracking electrophysiological activity of the brain.
Contents
Modeling
Excessive, large-scale hypersynchronous neuronal activity in the brain is a hallmark of epilepsy. 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. 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. A number of models have been developed for studying electrophysiological activity in epilepsy, including those based on Hodgkin–Huxley-type equations, describing changes in the flow of ionic currents across the membrane of a single cell or small groups of cells. 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. 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, 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:
- The mechanism by which normal (baseline) neurophysiological activity evolves into hypersynchronization of large regions of the brain during epileptic seizures
- The key factors that govern the rate of expansion of hypersynchronized regions
- The electrophysiological activity pattern dynamics on a large-scale
A canonical analysis of these issues, developed in 2008 by Shusterman and Troy using the Wilson–Cowan model, 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.
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.
A realistic state of baseline physiological activity has been defined, using the following two-component definition:
(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 rhythm, whereas slower (theta and delta rhythms are usually observed during deeper relaxation and sleep. To describe this general setting, a 3-variable
The variable v governs the recovery of excitation u;
Rate of expansion
The expansion of hypersynchronized regions exhibiting large-amplitude stable bulk oscillations occurs when the oscillations coexist with the stable rest state
where
How to evaluate the ratio
This system is derived using standard functions and parameter values
Since
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, has been estimated as
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.
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.