An integro-differential equation for dynamical systems with diffusion-mediated coupling

Abstract

Many systems of biological interest can be modeled as pointlike oscillators whose coupling is mediated by the diffusion of some substance. This coupling occurs because the dynamics of each oscillator is influenced by the local concentration of a substance which diffuses through the spatial medium. The diffusion equation, on its hand, has a source term which depends on the oscillator dynamics. We derive a mathematical model for such a system and obtain an integro-differential equation. Its solution can be obtained by an approximation scheme for which the unperturbed solution is used to obtain a first-order solution to the coupled oscillators and so on. We present numerical results for the special case of a one-dimensional bounded domain in which the oscillators are randomly placed. Our results show the influence of the coupling parameters on some aspects of the dynamics of the coupled oscillators, like phase and frequency synchronization.