Winfree Meandering in a 2-Dimensional 2-Variable Excitable Medium

Abstract

Abstract Two-dimensional excitable media, for example the excitable version of the Belousov-Zhabotin-sky reaction, are capable of forming spiral-shaped self-sustaining rotating wave patterns (rotors). In order to explain Winfree's experimental observation of an irregular "meandering" of the rotor's core region, we present a numerical simulation of a continuous, two-variable excitable medium in two space dimensions. Two phenomena occur: 1) an irregular motion of the rotor's core; 2) a non-stationary peak inside the core region. Thus, "meandering" is obtained, together with a new phenomenon, the "peak". A sufficient condition for both phenomena is that the underlying (local) system be stiff, that is, admit two time scales for its approximate description. In this case, (1) the "reaching distance" of diffusion is small as compared with the core's radius, and (2) rotation symmetry of the core implies a gradation in local frequencies (increase toward the center), supposing radial decoupling. We propose that both constraints act together to induce a spontaneous breakdown of rotation symmetry under an increase of stiffness. Immediately (or soon) thereafter, the core is no longer synchronized by the wave circling around it; instead, excitation from the surrounding wave penetrates the core along a new path from time to time, causing a non-repetitive short-cut.