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

Marius E. Yamakou, Tat Dat Tran, Hoang Duc Luu, and Jürgen Jost

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Submission date: 29. Jun. 2018 (revised version: February 2019)
Pages: 17
published in: Journal of mathematical biology, 79 (2019) 2, p. 509-532 
DOI number (of the published article): 10.1007/s00285-019-01366-z
Bibtex
with the following different title: The stochastic Fitzhugh-Nagumo neuron model in the excitable regime embeds a leaky integrate-and-fire model
MSC-Numbers: 60GXX, 92Bxx
Keywords and phrases: FitzHugh-Nagumo model, excitable regime, leaky integrate-and-fire model, random attractor, stationary distribution
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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.