Stability and Bifurcation Analyses of the FitzHugh–Rinzel Model with Time Delay in a Random Network

Abstract

Due to the finite speed of signal transmission, time delay is a common phenomenon in neuronal systems. The spatiotemporal dynamics of the FitzHugh–Rinzel model with time delay and diffusion in a random network are investigated in this paper. The conditions for Turing instability and Hopf bifurcation are obtained by linear stability analysis. It is found that the stability of the system changes with the time delay. Then the critical time delay for the state transition of the system is derived. Moreover, it is shown that Turing pattern is related to the network diffusion and connection probability. The increase of the diffusion coefficient will change the spatiotemporal pattern of the system. In addition, the system will achieve firing synchronization as the connection probability increases. Finally, numerical simulation verifies the theoretical results.