Bifurcations in the Wilson-Cowan Equations with Nonsmooth Firing Rate

The Wilson-Cowan (WC) equations represent a common means of studying the dynamics between excitatory and inhibitory populations of neurons. A number of recent experimental papers suggest that the cerebral cortex lies in the so-called inhibitory stabilized network (ISN) state, which means that there is an equilibrium state that intersects the middle branch of the excitatory nullcline [H. Ozeki et al., Neuron, 62 (2009), pp. 578-592]. A commonly used simplification replaces the smooth firing rate function with the Heaviside step function. In this paper, we explore the consequences of the nonsmooth approximation on the dynamics for WC systems that have such a middle-branch equilibrium in their smooth analogue. In the nonsmooth system, the inhibitory stabilized state corresponds to a pseudo focus (or focal crossing) occurring at the intersection of two switching manifolds-the subject of current work in nonsmooth systems theory. To study the dynamics we introduce and use techniques from Filippov systems and differential inclusions. We show nonsmooth equivalents of the Hopf, saddle-node on an invariant circle, and homoclinic bifurcations. We then compare these solutions with trajectories of both the smooth and piecewise linear systems, which show similar qualitative but differing quantitative bifurcating behavior. In addition, we present several nonstandard bifurcations, such as linear sliding modes, tangencies, and grazing solutions, that arise as a result of the discontinuities in the system.