  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.
