Bursting Dynamics of the 3-D Hindmarsh-Rose Neuron Model Under Periodic Excitation

In this paper, we study the bursting dynamics and mechanisms of a class of 3-D Hindmarsh-Rose neuron models with single and dual external periodic excitation terms. The aim of this study is to further enrich the compound bursting dynamics and provide guidance for the diagnosis and treatment of neurological diseases.

Initially, the stability of equilibrium and bifurcation of the system are analyzed. When two external periodic excitations exist in the system, by using the De Moivre equation to transform it into a new form of a single slow variable, and the classical fast-slow analysis method is used to study the bursting oscillations. Numerical simulations are employed to obtain the bifurcation structures of the system under single and dual excitations.

Two typical bursting oscillation modes: fold/supHopf-homoclinic bursting and fold/supHopf-fold bursting are observed under one external excitation. And the transformation of discharge rhythm has been revealed by bifurcation portrait. With two external excitations, multiple types compound bursting patterns have been revealed, such as “fold/homoclinic cycle/fold/suphopf-homoclinic/fold/homoclinic cycle” bursting. Additionally, we observed that variations in the excitation frequency can influence the delay behavior of subHopf bifurcations.

Compared with the simple bursting modes of single periodic excitation, the dual one has more abundant compound bursting mechanism. The firing mechanism of neurons studied in this paper will be helpful for the regulation of neurons.