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

invariant manifold reduction;nearly linear systems;gap junctional coupling;delay differential equations;Applied Mathematics

Kobelevskiy, Ilya

University of Waterloo

关键词: invariant manifold reduction;    nearly linear systems;    gap junctional coupling;    delay differential equations;    Applied Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/3905/1/uw-thesis.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

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, γ , and the coupling time delay, τ , cause on the dynamics of the system. For small γ we use the phase model reduction technique to analyze the system behavior. We determine the stable states of the system with respect to γ and τ using the appropriate phase models, and we estimate the regions of validity of the phase models in the γ , τ plane using both analytical and numerical analysis.Next we examine asymptotic of the arbitrary conductance-based neuronal model for γ → +∞ and γ → −∞. 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 γ > 0 shows that with appropriate choice of γ 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 γ < 0 suggests the method of estimation of the boundary between ;;weak” and ;;strong” coupling.