Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model

TL;DR

This paper analyzes the stochastic FitzHugh-Nagumo model, revealing that the interspike intervals follow an asymptotically geometric distribution, with results applicable across various noise levels and providing insights into neuronal firing patterns.

Contribution

It establishes a rigorous connection between mixed-mode oscillations and interspike interval statistics by proving geometric distribution of small oscillations and deriving eigenvalue bounds.

Findings

Interspike intervals follow an asymptotically geometric distribution.

Eigenvalue bounds are provided in the small-noise regime.

The model's parameters influence the distribution of mixed-mode patterns.

Abstract

We study the stochastic FitzHugh-Nagumo equations, modelling the dynamics of neuronal action potentials, in parameter regimes characterised by mixed-mode oscillations. The interspike time interval is related to the random number of small-amplitude oscillations separating consecutive spikes. We prove that this number has an asymptotically geometric distribution, whose parameter is related to the principal eigenvalue of a substochastic Markov chain. We provide rigorous bounds on this eigenvalue in the small-noise regime, and derive an approximation of its dependence on the system's parameters for a large range of noise intensities. This yields a precise description of the probability distribution of observed mixed-mode patterns and interspike intervals.