first equation of the model (with the assumption of uncorrelated terms). Same rationale can be done for inhibitory cells, obtaining second equation. Simplification of the model assuming time coarse graining When time coarse-grained modeling is assumed the model simplifies, being the new equations of the model:: where bar terms are the time coarse-grained versions of original ones. 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 spatially dependent extension of the classical Wilson–Cowan model can be utilized. Under appropriate initial conditions, the excitatory component, u, dominates over the inhibitory component, I, and the three-variable system reduces to the two-variable Pinto-Ermentrout type model The variable v governs the recovery of excitation u; and determine the rate of change of recovery. The connection function is positive, continuous, symmetric, and has the typical