Codimension-two bursting analysis in the delayed neural system with external stimulations

Abstract

In the neural system, action potentials play a crucial role in many mechanisms of information communication. The quiescent state, spiking and bursting activities are important biological behaviors with the different neurocomputational properties. In this paper, based on the bifurcation mechanisms involved in the generation of action potentials, an interesting mathematical study of bursting behavior is obtained. The transition between the bursting and quiescence state is investigated,which shows that the time delay must be large enough for bursting behavior to occur in a delayed system. Two types of the codimension-two bifurcation, i.e., Bogdanov–Takens (BT) bifurcation and saddle-node homoclinic (SNH) bifurcation are investigated also. The bifurcation curves of the parameters and the phase portraits for the different regions are shown. The local existence of the homoclinic curve is achieved by using the center manifold reduction and normal form method. For occurrence of a periodic stimulation in the neighborhood of the SNH bifurcation, the system can switch over from an equilibrium state to an oscillatory state either through saddle-node on an invariant circle bifurcation (called circle bifurcation for simplicity) or saddle-node (SN) bifurcation, and back from the oscillatory state to the equilibrium state through the circle or homoclinic bifurcation. Complex bursting phenomena are displayed for the different values of delay couplings and stimulation intensities. Some types of bursting behaviors, such as Circle/Circle (Type II or parabolic bursting), Circle/Homoclinic, SN/Circle (triangular bursting), SN/Homoclinic (Type I or square-wave bursting), and Fold/Hopf bursting are obtained in the firing area. The results show that the different burstings are related to the delay coupling and external inputs.