Resolving Molecular Contributions of Ion Channel Noise to Interspike Interval Variability through Stochastic Shielding

TL;DR

This paper develops a mathematical framework to decompose ion channel noise contributions to spike timing variability, extending stochastic shielding to more realistic conditions and providing accurate approximations for interspike interval variance.

Contribution

It introduces a method to resolve individual molecular transition contributions to spike timing variability and extends stochastic shielding to current-clamp conditions.

Findings

The IPI variance closely approximates the ISI variance within a few percent.

The proposed theory is exact in the small noise limit and accurate for physiologically relevant noise levels.

The ISI variance depends on the spike-detection voltage threshold.

Abstract

The contributions of independent noise sources to the variability of action potential timing has not previously been studied at the level of individual directed molecular transitions within a conductance-based model ion-state graph. The underlying connection provides an important example of how mathematics can be applied to study the effects of unobservable microscopic fluctuations to macroscopically observable quantities. We study a stochastic Langevin model and show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean--return-time (MRT) phase reduction developed in (Cao et al. 2020, SIAM J. Appl. Math) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike-detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt et al. 2012, Phys. Rev. Lett.) from the stationary voltage-clamp case to the current-clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. We show that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike-detection threshold, both in vitro and in silico.