Stochastic FitzHugh-Nagumo neuron model in excitable regime embeds a leaky integrate-and-fire model

TL;DR

This paper rigorously demonstrates how a stochastic FitzHugh-Nagumo neuron model in the excitable regime can be mathematically approximated by a leaky integrate-and-fire model, capturing its interspike interval statistics.

Contribution

It provides a complete mathematical construction linking the stochastic FHN model to a LIF model, including proofs of global attractors and firing time estimates.

Findings

Existence of a global random attractor for the stochastic FHN system.

Derivation of a radial equation representing a LIF model.

Confirmation of previous predictions relating FHN and LIF models.

Abstract

In this paper, we provide a complete mathematical construction for a stochastic leaky-integrate-and-fire model (LIF) mimicking the interspike interval (ISI) statistics of a stochastic FitzHugh-Nagumo neuron model (FHN) in the excitable regime, where the unique fixed point is stable. Under specific types of noises, we prove that there exists a global random attractor for the stochastic FHN system. The linearization method is then applied to estimate the firing time and to derive the associated radial equation representing a LIF equation. This result confirms the previous prediction in [Ditlevsen, S. and Greenwood, P. (2013). The Morris-Lecar neuron model embeds a leaky integrate-and-fire model. Journal of Mathematical Biology, 67(2):239-259] for the Morris-Lecar neuron model in the bistability regime consisting of a stable fixed point and a stable limit cycle.