Stability and bifurcation analysis in the coupled HR neurons with delayed synaptic connection

Abstract

This paper presents stability and bifurcations of two synaptically coupled identical Hindmarsh–Rose neurons with one time delay. The parameters we choose contribute to the single neuron exhibiting excitable behavior with a unique stable equilibrium. In the absence of time delay, we find coupling-induced oscillations, i.e., the excitable neurons can fire with different types of regular or irregular periodic spiking/bursting behaviors due to the synaptic coupling. With the help of stability and bifurcation theory, the asymptotic stability of equilibrium, fold and Hopf bifurcation are studied from the corresponding characteristic equation. In case of time delay, a detailed Hopf bifurcation analysis is given. And an explicit formula about the coupling strength and time delay for the occurrence of Hopf bifurcation is derived, based on which a series of periodic orbits generate when time delay passes through the critical value. Finally, numerical simulations are carried out for supporting our theoretical results; meanwhile, branches of Hopf bifurcation curves are plotted in the two-parameter plane.