Mechanism of dynamic phase transition and synchronous stability in Hindmarsh–Rose neuronal network

Abstract

The neuronal dynamics plays an important role in understanding the neurological phenomena. We study the mechanism of the dynamic phase transition and its Lyapunov stability of a single Hindmarsh–Rose (HR) neuronal model. We propose an index [Formula: see text] to express the dynamical phase of the HR neurons. When [Formula: see text] the neuron is in the pure resting state, and when [Formula: see text] the neuron closes to the pure spiking phase, while when [Formula: see text] the neuron runs in the bursting phase. Based on this method, we investigate numerically the phase diagram of the HR neuronal model in the parameter space. We find that two mechanisms governed the HR neuronal dynamic phase transition, the phase transition and crossover transition in the different regions of the parameter space. Moreover, we analyze the equilibrium point stability of the HR neuronal model based on the Lyapunov stability method. We study the synchronous stability of the HR neuronal network based on the master stability function method and give the phase diagrams of the maximum Lyapunov exponents in the parameter space of networks. The regions of the synchronous stabilities in the parameter space depend on the couplings of the HR neurons of the membrane potential and the flux of the fast ion channel between the HR neurons. These results help to understand the HR neuronal dynamics and the synchronous stability of the HR neuronal networks.