Persistent spatial patterns for semi-discrete models of excitable media

Excitable media have the property that in the spatially homogeneous configuration there is a globally asymptotically stable equilibrium, and no sustained non-decaying oscillations are possible. If diffusion effects are present, however, experiments seem to indicate the possibility of persistent oscillations. In more than one space dimension this may occur, apparently, even if the initial disturbance is restricted to a compact subset of an open domain in the medium. In this paper we discuss a mathematical model of this phenomenon. The model consists of a doubly infinite coupled system of ordinary differential equations, and therefore is intended to represent a spatially discrete network of interconnected “cells”.

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