Bifurcation Analysis and Synchronization Control of e-HR Neuron Model with Time Delay

Derived from the Hindmarsh-Rose (HR) neuron model, the extended Hindmarsh-Rose (e-HR) neuron model with time delay is constructed considering the effect of intracellular ion transfer and the inherent slow current delay on the change of neuron membrane potential. Firstly, the stability of the model is analyzed by using the Routh-Hurwitz criterion, and the conditions for the existence of Hopf bifurcation are derived by using the Hopf bifurcation theorem. Secondly, through numerical simulation, it is found that synchronization occurs in the limited range of delay value without control. Therefore, this paper introduces a nonlinear feedback control scheme, designed based on a Lyapunov-Krasovskii functional with a double integral term. The derived sufficient conditions for the synchronization of the master-slave neuron model with self-feedback delay are presented. Finally, the results of theoretical analysis are verified by numerical simulation and the effectiveness and viability of the controller are proved.