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

TL;DR

This paper investigates how ion channel noise affects neural excitability in a stochastic Morris-Lecar model, revealing limitations of traditional Kramer's rate theory and proposing an alternative approach for analyzing spontaneous activity.

Contribution

It demonstrates the breakdown of the fast-slow separation assumption in stochastic excitable systems and introduces a new method based on maximum likelihood trajectories.

Findings

Kramer's rate theory often fails in realistic stochastic neuron models.

The assumption of a constant recovery variable during excitation is invalid in most cases.

Maximum likelihood trajectories provide a viable framework for exit time analysis.

Abstract

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