Dynamics of a Reduced System Connected to the Investigation of an Infinite Network of Identical Theta Neurons

Abstract

We consider an infinite network of identical theta neurons, all-to-all coupled by instantaneous synapses. Using the Watanabe–Strogatz Ansatz, the mathematical model of this infinite network is reduced to a two-dimensional system of differential equations. We determine the number of equilibria of this reduced system with respect to two characteristic parameters. Furthermore, we discuss the stability properties of each equilibrium and the possible bifurcations that may take place. As a result, the occurrence of exotic higher codimension bifurcations involving a degenerate center is also unveiled. Numerical results are also presented to illustrate complex dynamic behaviour in the reduced system.