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

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.Comment: 36 page

First passage time problems and related computational methods

Motivated by the interest of first passage time problems in neurobiology and in a variety of applied fields, we study the solution of first passage time equations for the Wiener and the Ornstein Uhlenbeck process both in the general case of time dependent threshold functions. Some computational results obtained by two different numerical methods are reported and briefly discussed