Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh–Rose and FitzHugh–Nagumo neurons with two time delays

A memristor-coupled heterogenous neural network consisting of two-dimensional (2D) FitzHugh–Nagumo (FHN) and Hindmarsh–Rose (HR) neurons with two time delays is established. Taking the time delays as the control parameters, the existence of Hopf bifurcation near the stable equilibrium point in four cases is derived theoretically, and the validity of the Hopf bifurcation condition is verified by numerical analysis. The results show that the two time delays can make the stable equilibrium point unstable, thus leading to periodic oscillations induced by Hopf bifurcation. Furthermore, the time delays in FHN and HR neurons have different effects on the firing activity of neural network. Complex firing patterns, such as quiescent state, chaotic spiking, and periodic spiking can be induced by the time delay in FHN neuron, while the neural network only exhibits quiescent state and periodic spiking with the change of the time delay in HR neuron. Especially, phase synchronization between the heterogeneous neurons is explored, and the results show that the time delay in HR neurons has a greater effect on blocking the synchronization than the time delay in FHN neuron. Finally, the theoretical analysis is verified by circuit simulations.