Dynamical analysis of a multiple time delays FitzHugh-Nagumo neuron system with chemical and electrical coupling

A multiple time delays FitzHugh-Nagumo (FHN) neuron system with chemical and electrical coupling is presented in this paper. Both the coupled FHN neuron systems without and with time delays are studied in detail. Firstly, the number, stability and position of equilibria of the FHN neuron system without time delay varying with the strength of chemical and electrical coupling are investigated. Secondly, under two different electrical coupling strengths, the different effects of the chemical coupling strength on the bifurcation of equilibria of the FHN neuron system without time delay are given. In particular, the codimension 2 bifurcation analysis of the FHN neuron system without time delay shows a rich bifurcation behavior of limit cycles. Furthermore, the local stability of trivial equilibrium point of the coupled FHN neuron system with delays is studied by characteristic equation method. The system exhibits a stability switching through Hopf bifurcation under the influence of time delays. Finally, the delay-dependent stability region is obtained on the parameter plane of time delays using Hopf bifurcation curves. The intersecting points of Hopf bifurcation curves indicate that the system has Hopf-Hopf bifurcation points, and then we give the dynamic behavior of the system near one of them. The system exhibits the coexistence of multiple periodic orbits of different frequencies. At each stage of the paper, numerical simulation results are illustrated to verify the correctness of the theoretical analysis.