Hopf Bifurcations in Directed Acyclic Networks of Linearly Coupled Hindmarsh–Rose Systems

Abstract : This paper addresses the existence of Hopf bifurcations in a directed acyclic network of neurons, each of them being modeled by a Hindmarsh–Rose (HR) neuronal model. The bifurcation parameter is the small parameter corresponding to the ratio of time scales between the fast and the slow dynamics. We first prove that, under certain hypotheses, the single uncoupled neuron can undergo a Hopf bifurcation. Hopf bifurcation occurrences in a directed acyclic network of HR neurons are then discussed. Numerical simulations are carried out to observe these bifurcations and to illustrate the theoretical results.