[3][4][5][6] 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.
The Wilson–Cowan model considers a homogeneous population of interconnected neurons of excitatory and inhibitory subtypes.
the distribution of the number of afferent synapses, a variant of function
, obtaining in this way the product at right side of the first equation of the model (with the assumption of uncorrelated terms).
where bar terms are the time coarse-grained versions of original ones.
The determination of three concepts is fundamental to an understanding of hypersynchronization of neurophysiological activity at the global (system) level:[7] A canonical analysis of these issues, developed in 2008 by Shusterman and Troy using the Wilson–Cowan model,[7] 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.
[8][9] 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.
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.
[10] Under appropriate initial conditions,[7] the excitatory component, u, dominates over the inhibitory component, I, and the three-variable system reduces to the two-variable Pinto-Ermentrout type model[11]
The firing rate function, which is generally accepted to have a sharply increasing sigmoidal shape, is approximated by
system has been successfully used in a wide variety of neuroscience research studies.
[11][12][13][14][15] 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.
The linearized system exhibits subthreshold decaying oscillations whose frequency increases as
system: a stable, spatially independent, periodic solution (bulk oscillation) and a stable rest state coexist over a continuous range of parameters.
where bulk oscillations occur,[7] "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
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."
it is necessary to analyze the underlying bulk oscillation which satisfies the spatially independent system
This system is derived using standard functions and parameter values
, Shusterman and Troy analyzed the bulk oscillations and found
(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.
[18] 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.
[18] 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,[8] has been estimated as[7]
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.
[8] 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.