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.
DUAN LiXia1, LU QiShao2 & CHENG DaiZhan3 1 College of Science, North China University of Technology, Beijing 100144, China
It is crucially important to study different synchronous regimes in coupled neurons because different regimes may correspond to different cognitive and pathological states. In this paper, phase synchronization and its transitions are discussed by means of theoretical and numerical analyses. In two coupled modified Morris-Lecar neurons with a gap junction, we show that the occurrence of phase synchronization can be investigated from the dynamics of phase equation, and the analytical synchronization condition is derived. By defining the phase of spike and burst, the transitions from burst synchronization to spike synchronization and then toward nearly complete synchronization can be identified by bifurcation diagrams, the mean frequency difference and time series of neurons. The simulation results suggest that the synchronization of bursting activity is a multi-time-scale phenomenon and the phase synchronization deduced by the phase equation is actually spike synchronization.
