Self-organization in an excitable reaction-diffusion system: Synchronization of oscillatory domains in one dimension

We study the self-organizing behavior exhibited in a Bonhoffer–van der Pol–type equation in the following form for the activator u(x,t) and the inhibitor v(x,t): ετ${\mathrm{\partial }}_{\mathit{t}}$u=${\mathrm{\epsilon }}^2$ ${\mathrm{\partial }}_{\mathit{x}}^2$u+f(u)-v and ${\mathrm{\partial }}_{\mathit{t}}$v=${\mathrm{\partial }}_{\mathit{x}}^2$v+u-γv, where ε, τ, and γ are positive constants and f(u) has a cubiclike nonlinearity. The above set of equations admits a spatially periodic solution where the excited and the rest domains are arrayed alternatively. For small values of τ, these domains are found to undergo sustained oscillation. We employ a singular-perturbation method in the limit ε≪1 to derive the equation of motion for the domain boundaries (interfaces). Two interfaces separated from each other can interact through the diffusion current of v. The bifurcation from the nonoscillatory to the oscillatory state is investigated based on this interface equation of motion. We show that, when the nonoscillatory state loses its stability, an in-phase motion of domains emerges first in a wide range of the parameters.