The Study Of Dynamic Transitions And Time-delay Effects In Slow-fast Hindmarsh-rose System With Method Of Geometric Singular Perturbation

The Hindmarsh-Rose model of biological nervous system is one of the most representative slow-fast system. In one hand, due to the interaction between fast variables and slow variables, the HR system would produce kinds of complex dynamics(electrical firing patterns). In the other hand, time-delays are inevitable in Hindmarsh-Rose neural systems because of finite propagation speed of signals and finite processing time in synapses.The mechanism of typical dynamics is studied on the basis of the geometric singular perturbation theory in this paper. The vital parameters influence on many typical dynamics and the structure of the slow manifold are first investigated and state the mechanism of all kinds of dynamical transitions while these parameters vary. Then, it is found that time-delay would alter the structure of the slow manifold and furtherly transform the dynamical behaviors of system by using the method of stability switch and the analysis of bifurcation. Based on numerical simulation, it is described that time-delay leads to kinds of dynamics and dynamical transitions in the paper. The research of this paper will help to further understand the complex firing patterns in biological nervous system. Moreover, it will enrich the content of nonlinear science.