The dynamics of a forced coupled network of active elements

Abstract

This paper presents the derivation and analysis of mathematical models motivated by the experimental induction of contour phosphenes in the retina. First, a spatially discrete chain of periodically forced coupled oscillators is considered via reduction to a chain of scalar phase equations. Each isolated oscillator locks in a 1:2 manner with the forcing so that there is intrinsic bistability, with activity peaking on either the odd or even cycles of the forcing. If half the chain is started on the odd cycle and half on the even cycle (“split state”), then with sufficiently strong coupling, a wave can be produced that can travel in either direction due to symmetry. Numerical and analytic methods are employed to determine the size of coupling necessary for the split state solution to destabilize such that waves appear. Taking a continuum limit, we reduce the chain to a partial differential equation. We use a Melnikov function to compute, to leading order, the speed of the traveling wave solution to the partial differential equation as a function of the form of coupling and the forcing parameters and compare our result to the numerically computed discrete and continuum wave speeds.