form . In Ref. The firing rate function, which is generally accepted to have a sharply increasing sigmoidal shape, is approximated by , where H denotes the Heaviside function; is a short-time stimulus. This system has been successfully used in a wide variety of neuroscience research studies. 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. Rate of expansion The expansion of hypersynchronized regions exhibiting large-amplitude stable bulk oscillations occurs when the oscillations coexist with the stable rest state . To understand the mechanism responsible for the expansion, it is necessary to linearize the system around when is held fixed. The linearized system exhibits subthreshold decaying oscillations whose frequency increases as increases. At a critical value where the oscillation frequency is high enough, bistability occurs in the system: a stable, spatially independent, periodic solution (bulk oscillation) and a stable rest state coexist over a continuous range of parameters. When where bulk oscillations occur, "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." Therefore, when , the rate of expansion of the region is estimated by where 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., is c=22.4 mm/s. How to evaluate the ratio To determine values for it is necessary to analyze the underlying bulk oscillation which satisfies the spatially independent system This system is derived using standard functions and parameter