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

Nils Berglund and Damien Landon
Nonlinearity 25:2303-2335 (2012)

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.

Mathematical Subject Classification: 60H10, 34C26 (primary) 60J20, 92C20

Keywords and phrases: FitzHugh-Nagumo equations, interspike interval distribution, mixed-mode oscillation, singular perturbation, fast-slow system, dynamic bifurcation, canard, substochastic Markov chain, principal eigenvalue, quasi-stationary distribution.

 

Journal Homepage Published article: 10.1088/0951-7715/25/8/2303 MR2946187 Zbl1248.60059

PDF file (848 Kb) hal-00591089 arXiv/1105.1278