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We consider a stochastic version of an excitable system based on theMorris-Lecar model of a neuron, in which the noise originates from stochasticSodium and Potassium ion channels opening and closing. One can analyze neuralexcitability in the deterministic model by using a separation of time scalesinvolving a fast voltage variable and a slow recovery variable, whichrepresents the fraction of open Potassium channels. In the stochastic setting,spontaneous excitation is initiated by ion channel noise. If the recoveryvariable is constant during initiation, the spontaneous activity rate can becalculated using Kramer's rate theory. The validity of this assumption in thestochastic model is examined using a systematic perturbation analysis. We findthat in most physically relevant cases, this assumption breaks down, requiringan alternative to Kramers theory for excitable systems with one deterministicfixed point. We also show that an exit time problem can be formulated in anexcitable system by considering maximum likelihood trajectories of thestochastic process.

Breakdown of fast-slow analysis in an excitable system with channel noise