Nonsmooth Bifurcations of Mean Field Systems of Two-Dimensional Integrate and Fire Neurons

Abstract

Mean field systems have recently been derived that adequately predict the behaviors of large networks of coupled integrate-and-fire neurons [W. Nicola and S.A. Campbell, J. Comput. Neurosci., 35 (2013), pp. 87-108]. The mean field system for a network of neurons with spike frequency adaptation is typically a pair of differential equations for the mean adaptation and synaptic gating variable of the network. These differential equations are nonsmooth, and, in particular, are piecewise smooth continuous (PWSC). Here, we analyze the smooth and nonsmooth bifurcation structure of these equations and show that the system is organized around a pair of co-dimension-two bifurcations that involve, respectively, the collision between a Hopf equilibrium point and a switching manifold, and a saddle-node equilibrium point and a switching manifold. These two co-dimension-two bifurcations can coalesce into a co-dimension-three nonsmooth bifurcation. As the mean field system we study is a nongeneric piecewise smooth continuous system, we discuss possible regularizations of this system and how the bifurcations which occur are related to nonsmooth bifurcations displayed by generic PWSC systems.