Self-organization in an excitable reaction-diffusion system. II. Reduction to a coupled oscillator

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Abstract
In a previous paper [T. Ohta, A. Ito, and A. Tetsuka, Phys. Rev. A 42, 3225 (1990)] we have shown that the excited domains arrayed periodically in space in a Bonhoeffer–van der Pol–type model system exhibit an in-phase oscillation when the time evolution of the inhibitor is slow compared to that of the activator. This was obtained by the linear-stability analysis for deviation of domain boundaries around the equilibrium position. In this paper, we study the postthreshold behavior. Our main concern is to derive a coupled set of equations for the phase and the amplitude of the oscillating domains. We start with an isolated oscillating domain and take into account the interaction among the domains perturbatively. It is shown that the nearest-neighbor-interaction strength changes the sign as the domain width is changed. As a result, when the number of the interacting domains is finite, the system exhibits bistability where in-phase and antiphase oscillations can coexist.