Analysis Of Asymptotic Structure And Synchronization Of Morris-Lecar Model

The intrinsic characteristics of biologic systems are multivariable, nonlinear and strong coupled, so these systems can not be analyzed by linear methods and must be investigated through nonlinear dynamic theory with complex mathematical analysis.We use well-known asymptotical methods, which describe the fast and slow processes that occur at different moments in detail, so that we can gain a better understanding of the ways the biological systems work. Understanding asymptotic structure of systems is important for practical applications because its approximate procedures save computation time, reduce computation complexity and provide intuitive insight which does not exist in detailed model.In this paper, the asymptotical behavior of the Morris-Lecar model is studied based on Tikhonov's singular perturbation theory and Zeeman's analytical methods, and then the synchronization mechanism of the model is established in terms of the parameters in the equations using methods of mathematics. Firstly, a variety of oscillatory behaviors of the Morris-Lecar system which is subjected to different current stimulation are introduced, and the validities of them are verified by bifurcation analysis. Secondly, the superfast variable, fast variable and slow variable of the system are found out to simplify the model after embedding a small parameter. Finally, based on the mathematical methods, the attracting region of the reduced system is found, and the size of the region which globally attracts the flow can be calculated from the parameters in the equations, accordingly the synchronization mechanism in the symmetric case and asymmetric case can be established.Numerical simulations are presented to verify the validity of the theoretic analysis.The conclusions of this paper provide theory foundation for the research on conduction of acupuncture electrical signal.