Bifurcation analysis for Hindmarsh-Rose neuronal model with time-delayed feedback control and application to chaos control

This paper is concerned with bifurcations and chaos control of the Hindmarsh-Rose (HR) neuronal model with the time-delayed feedback control. By stability and bifurcation analysis, we find that the excitable neuron can emit spikes via the subcritical Hopf bifurcation, and exhibits periodic or chaotic spiking/bursting behaviors with the increase of external current. For the purpose of control of chaos, we adopt the time-delayed feedback control, and convert chaos control to the Hopf bifurcation of the delayed feedback system. Then the analytical conditions under which the Hopf bifurcation occurs are given with an explicit formula. Based on this, we show the Hopf bifurcation curves in the two-parameter plane. Finally, some numerical simulations are carried out to support the theoretical results. It is shown that by appropriate choice of feedback gain and time delay, the chaotic orbit can be controlled to be stable. The adopted method in this paper is general and can be applied to other neuronal models. It may help us better understand the bifurcation mechanisms of neural behaviors.