The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability

Abstract

We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter ε uncouples the system at ε = 0. Using a normal form for N = 2 identical systems undergoing Hopf bifurcation, we explore the dynamical properties. Matching the normal form coefficients to a coupled Wilson-Cowan oscillator network gives an understanding of different types of behaviour that arise in a model of perceptual bistability. Notably, we find bistability between in-phase and anti-phase solutions that demonstrates the feasibility for synchronisation to act as the mechanism by which periodic inputs can be segregated (rather than via strong inhibitory coupling, as in existing models). Using numerical continuation we confirm our theoretical analysis for small coupling strength and explore the bifurcation diagrams for large coupling strength, where the normal form approximation breaks down.