Dynamics of Hindmarsh–Rose diffusive system

In this paper, we discuss complex dynamic behaviors of Hindmarsh–Rose diffusive system (HRDS). Using traveling wave transformation, we firstly transform HRDS to slow-fast system with two small parameters. Applying geometric singular perturbation theory, we prove the existence of canard explosion phenomenon and relaxation oscillation in the slow-fast system. Thus, we get the existence of periodic traveling wave of HRDS. Furthermore, base on above theoretical result, we explain various firing activities of neurons.