CORRESPONDENCE BETWEEN DISCRETE AND CONTINUOUS MODELS OF EXCITABLE MEDIA - TRIGGER WAVES

We present a theoretical framework for relating continuous partial dif ferential equation (PDE) models of excitable media to discrete cellula r automata (CA) models on a randomized lattice. These relations establ ish a quantitative link between the CA model and the specific physical system under study. We derive expressions for the CA model's plane wa ve speed, critical curvature, and effective diffusion constant in term s of the model's internal parameters (the interaction radius, excitati on threshold, and time step). We then equate these expressions to the corresponding quantities obtained from solution of the PDEs (for a fix ed excitability). This yields a set of coupled equations with a unique solution for the required CA parameter values. Here we restrict our a nalysis to ''trigger'' wave solutions obtained in the limiting case of a two-dimensional excitable medium with no recovery processes. We tes ted the correspondence between our CA model and two PDE models (the Fi tzHugh-Nagumo medium and a medium with a ''sawtooth'' nonlinear reacti on source) and found good agreement with the numerical solutions of th e PDEs. Our results suggest that the behavior of trigger waves is actu ally controlled by a small number of parameters.