A Combinatorial Problem Arising in the Study of Reaction-Diffusion Equations

We study a discrete model based on the observed behavior of excitable media. This model has the basic properties of an excitable medium, that is, a threshold phenomenon, at refractory period, and a globally stable rest point. We are mainly interested in two dimensional periodic patterns. We characterize the initial conditions which lead to such patterns, by introducing a basic invariant, the “winding number of a continuous cycle.”