Nonlocal coupling among oscillators mediated by a slowly diffusing substance

A general theory is presented for the coupling among nonlinear oscillators mediated by a diffusing chemical substance. We extend a model originally developed by Kuramoto, who supposed that the diffusion characteristic time is much shorter than the oscillator main period, such that diffusion occurs very fast. We eliminate this constraint and consider diffusion to have an arbitrary characteristic time, by solving exactly the diffusion equation using suitable Green functions. We present results in one, two and three dimension, with and without boundary conditions.