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The Morris-Lecar (ML) neuronal model with current-feedback control is considered as a typical fast-slow dynamical system to study the combined influences of the reversal potential VCa of Ca2+ and the feedback current I on the generation and transition of different bursting oscillations. Two-parameter bifurcation analysis of the fast subsystem is performed in the parameter (I, VCa)-plane at first. Three important codimension-2 bifurcation points and some codimension-1 bifurcation curves are obtained which enable one to determine the parameter regions for different types of bursting. Next, we further divide the control parameter (V0, VCa)-plane into five different bursting regions, namely, the “fold/fold” bursting region R1, the “fold/Hopf” bursting region R2, the “fold/homoclinic” bursting region R3, the “subHopf/homoclinic” bursting region R4 and the “subHopf/subHopf” bursting region R5, as well as a silence region R6. Codimension-1 and -2 bifurcations are responsible for explanation of transition mechanisms between different types of bursting. The results are instructive for further understanding the dynamical behavior and mechanisms of complex firing activities and information processing in biological nervous systems.
The Morris-Lecar (ML) neuronal model with current-feedback control is considered as a typical fast-slow dynamical system to study the combined influences of the reversal potential VCa of Ca2 + and the feedback current I on the generation and transition of different bursting oscillations Two-parameter bifurcation analysis of the fast subsystem is performed in the parameter (I, VCa) -plane at first. Three important codimension-2 bifurcation points and some codimension-1 bifurcation curves are obtained which enable one to determine the parameter regions for different types of bursting. Next, we further divide the control parameter (V0, VCa) -plane into five different bursting regions, namely, the “fold / fold” bursting region R1, the “fold / Hopf” bursting region R2, the “fold / homoclinic” bursting region R3, the “subHopf / homoclinic” bursting region R4 and the “subHopf / subHopf” bursting region R5, as well as a silence region R6. Codimension- 1 and -2 bifurcations are responsible for exp lanation of transition mechanisms between different types of bursting. The results are instructive for further understanding the dynamical behavior and mechanisms of complex firing activities and information processing in biological nervous systems.