In computational neuroscience, the Wilson–Cowan model describes the dynamics of interactions between populations of very simple excitatory and inhibitory model neurons. It was developed by H.R. Wilson and Jack D. Cowan and extensions of the model have been widely used in modeling neuronal populations. The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal populations to stimuli. Because the model neurons are simple, only elementary limit cycle behavior, i.e. neural oscillations, and stimulus-dependent evoked responses are predicted. The key findings include the existence of multiple stable states, and hysteresis, in the population response.
Contents
Mathematical description
The Wilson–Cowan model considers a homogeneous population of interconnected neurons of excitatory and inhibitory subtypes. The fundamental quantity is the measure of the activity of an excitatory or inhibitory subtype within the population. More precisely,
Sensitive cells
Proportion of cells in refractory period (absolute refractory period
Proportion of sensitive cells (complement of refractory cells)
Excited cells
Subpopulation response function based on
Subpopulation response function based on the distribution of afferent synapses per cell (all cells have the same threshold)
Average excitation level
where
Excitatory subpopulation expression
Complete Wilson–Cowan model
Time Coarse Graining
Isocline Equation
Sigmoid Function
Application to epilepsy
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.