Dynamical Analysis Of Two Kinds Of Morris-lecar Model

With the continuous development and improvement of computational neuroscience,the firing patterns and the bifurcation theory of various neuron models have been widely studied, for example, the Hodgkin-Huxley model, the Morris-Lecar model, the Hindmarsh-Rose model and so on, this lay the foundation for us to study deeply of the nervous system’s information coding and cognitive function. In this paper, we study the dynamic properties of a modified Morris-Lecar model and Morris-Lecar-like model in spinal lamina I neurons, the main conclusions as follows.For the modified Morris-Lecar model, firstly, using numerical simulation, we study of the membrane capacitance’s effect on the firing patterns, we get rich firing patterns. And then combined with the diagram of ISIs, when regulate membrane capacitance values, the firing patterns caused by the change within a certain range of the time scale factor and maximal conductance of calcium ion have a very big difference, appearing firing, bursting firing. Between these firing patterns, several typical nonlinear dynamical bifurcation phenomenon are investigated, such as period doubling bifurcation, inverse period doubling bifurcation, period adding bifurcation, the simulation results may give us some help in further understanding the relationship between signal encoding features and the combination of capacitance and the ion channels. In the end, we study the two-dimensional Morris-Lecar-like model. Using numerical simulation, we obtain the corresponding saddle-node bifurcation curve, the Hopf bifurcation curve and the homoclinic bifurcation curve near the Bogdanov-Takens points. At the same time, we consider add slow variable for having rich firing patterns, and get five bursting modes according to the number of Hopf bifurcation points in the upper equilibrium curve of the fast subsystem.