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

Nils Berglund (1) , Damien Landon (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

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.

Mots clés: Dynamic bifurcation; Fast-slow system; Singular perturbation; Principal eigenvalue; Canard; Substochastic Markov chain; FitzHugh-Nagumo equations; Interspike interval distribution; Mixed-mode oscillation; Quasi-stationary distribution

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Nils Berglund, Damien Landon. Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model. Nonlinearity, 2012, 25 (8), pp.2303-2335. ⟨10.1088/0951-7715/25/8/2303⟩. ⟨hal-00591089v4⟩