Stability and Hopf Bifurcation Analysis in Hindmarsh–Rose Neuron Model with Multiple Time Delays

Abstract

In this paper, the dynamical behaviors of a single Hindmarsh–Rose neuron model with multiple time delays are investigated. By linearizing the system at equilibria and analyzing the associated characteristic equation, the conditions for local stability and the existence of local Hopf bifurcation are obtained. To discuss the properties of Hopf bifurcation, we derive explicit formulas to determine the direction of Hopf bifurcation and the stability of bifurcated periodic solutions occurring through Hopf bifurcation. The qualitative analyses have demonstrated that the values of multiple time delays can affect the stability of equilibrium and play an important role in determining the properties of Hopf bifurcation. Some numerical simulations are given for confirming the qualitative results. Numerical simulations on the effect of delays show that the delays have different scales when the two delay values are not equal. The physiological basis is most likely that Hindmarsh–Rose neuron model has two different time scales. Finally, the bifurcation diagrams of inter-spike intervals of the single Hindmarsh–Rose neuron model are presented. These bifurcation diagrams show the existence of complex bifurcation structures and further indicate that the multiple time delays are very important parameters in determining the dynamical behaviors of the single neuron. Therefore, these results in this paper could be helpful for further understanding the role of multiple time delays in the information transmission and processing of a single neuron.