Dynamic Analysis and Synchronization with Unidirectional Coupling Involving Energy of a HR Neuron Under Electric Field and Memristive Autapse

Abstract

Reliable neuron models play an important role in identifying the electrical activities, global bifurcation patterns, and dynamic mechanisms of neurons in complex electromagnetic environments. Considering the memristive autapse involving magnetic coupling has voltage-controlled, nonlinear, and memory, a 5-D HR neuron model containing magnetic field and electric field variables is established. Detailedly, the existence and stability conditions of the equilibrium point are determined by theoretical analysis, and the complex time-varying stability, saddle-node bifurcation, and Hopf bifurcation behaviors of the model are verified by numerical calculation. Interestingly, the system has a bistable structure consisting of quiescent state and period-1 and period-2 bursting modes near the subcritical Hopf bifurcation. It is noteworthy that the memristive autapse has a complex regulation mechanism for the bistable region so that three kinds of bistable coexisting structures and counterintuitive dynamic phenomena can be induced by appropriately adjusting the memristive autapse. Accordingly, the mechanism of positive feedback memristive autapse decreases its firing frequency, while negative feedback memristive autapse promotes its excitability was revealed by the fast-slow dynamic analysis. Extensive numerical results display that the system generally possesses period-adding bifurcation modes and comb-shaped chaotic structures. Furthermore, it is found that the firing modes and multistability regions of the system can be accurately predicted by analyzing the global dynamic behaviors of Hamilton energy. Importantly, it is verified that the unidirectional coupling controller involving energy is far more efficient and consumes less energy than electrical synaptic coupling in achieving complete synchronization with mismatched parameters.