Bifurcations and synchrony in a ring of delayed Wilson–Cowan oscillators

Abstract

Ring structures are crucial in network neuroscience, enabling the integration of neural information through closed loop circuits within feedback systems. Here, we use numerical bifurcation analysis to explore time delay effects on a ring of delay-coupled Wilson–Cowan masses. Investigating a low-dimensional ‘self-coupled’ version of the aforementioned system, we uncover the bifurcation structure of the synchronization manifold, and unveil a diverse array of dynamic synchronization patterns that emerge as a consequence of Hopf branch crossings and subsequent higher-order bifurcations. Analysis of the full system reveals transverse instabilities in the synchronized state for large regions of parameter space, with the ring network architecture promoting various dynamics depending on the balance between coupling strength and delay time. Under weak coupling, emergent oscillations are generally synchronous or anti-phase synchronous, with transitions between them triggered by a torus bifurcation of a periodic orbit. Regions of synchronous and anti-phase synchronous solutions are delineated by weakly chaotic borders due to the breakdown of the torus. As coupling strength increases, the bifurcation diagram displays more overlapped branching structure, resulting in increasingly complicated, multistable dynamics.