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Hypoelliptic stochastic FitzHugh–Nagumo neuronal model: Mixing, up-crossing and estimation of the spike rate

The FitzHugh-Nagumo is a well-known neuronal model that describes the generation of spikes at the intracellular level. We study a stochastic version of the model from a probabilistic point of view. The hypoellipticity is proved, as well as the existence and uniqueness of the stationary distribution. The bi-dimensional stochastic process is $\beta$-mixing. The stationary density can be estimated with an adaptive non-parametric estimator. Then, we focus on the distribution of the length between successive spikes. Spikes are difficult to define directly from the continuous stochastic process. We study the distribution of the number of up-crossings. We link it to the stationary distribution and propose an estimator of its expectation. We finally prove mathematically that the mean length of inter-up-crossings interval in the FitzHugh-Nagumo model is equal to its up-crossings rate. We illustrate the proposed estimators on a simulation study. Different regimes are explored, with no, few or high generation of spikes.