Abstract:

There are a growing number of problems in biological physics involving the coupling between a piecewise deterministic dynamical system and a continuous time Markov process, which is more appropriate to be modeled as a stochastic hybrid system than a diffusion process. Specifically, we investigate the spontaneous action potential induced by channel noise in stochastic hybrid Morris-Lecar system without a saddle state both theoretically and numerically in the case of weak noise. The initiation phase of an action potential can be regarded as an event of noise induced escape, for which the optimal paths and then the quasi-potential are computed via an auxiliary Hamiltonian system. Due to the absence of the saddle point, the ghost separatrix is chosen as threshold for studying the transition events from the resting state. Through evaluating the quasi-potential on the threshold, we have found an obvious minimum that acts similarly as a saddle point. Prehistory probability distribution has been performed by improved Monte Carlo simulation, which confirmed the theoretical results for not only the initial phase but also the excitable phase. In addition, the contour line of quasi-potential as another choice of threshold selected by previous researchers has been introduced and their advantages and disadvantages are compared. Finally, the impacts on patterns of optimal paths and quasi-potential about various combinations of Na$^{+}$ and K$^{+}$ channel noise are studied thoroughly. The results shows that it is the fluctuation of K$^{+}$ channel that plays the dominant role in the process of spontaneous excitability and there exists an optimal ratio for the two channel noises which minimizes the fluctuation strength.

Key words: Morris-Lecar system, spontaneous action potential, large deviation theory, the optimal path, quasi-potential