Bifurcation analysis of a system of Morris-Lecar neurons with time delayed gap junctional coupling

We consider a system of two identical Morris-Lecar neurons coupled via electrical coupling. We focus our study on the effects that the coupling strength, gamma, and the coupling time delay, tau, cause on the dynamics of the system.;Next we examine asymptotic of the arbitrary conductance-based neuronal model for gamma → + infinity and gamma → -infinity. The theory of nearly linear systems developed in [30] is extended in the special case of matrices with non-positive eigenvalues. The asymptotic analysis for gamma > 0 shows that with appropriate choice of gamma the voltages of the neurons can be made arbitrarily close in finite time and will remain that close for all subsequent time, while the asymptotic analysis for gamma < 0 suggests the method of estimation of the boundary between "weak" and "strong" coupling.;For small gamma we use the phase model reduction technique to analyze the system behavior. We determine the stable states of the system with respect to gamma and tau using the appropriate phase models, and we estimate the regions of validity of the phase models in the gamma, tau plane using both analytical and numerical analysis.