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

Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh–Nagumo model Nils Berglund?† and Damien Landon?† Abstract We study the stochastic FitzHugh–Nagumo equations, modelling the dynamics of neuronal action potentials, in parameter regimes characterised by mixed-mode oscilla- tions. The interspike time interval is related to the random number of small-amplitude oscillations separating consecutive spikes. We prove that this number has an asymp- totically 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. Date. May 6, 2011. Revised version, April 5, 2012. Mathematical Subject Classification. 60H10, 34C26 (primary) 60J20, 92C20 (secondary) Keywords and phrases. FitzHugh–Nagumo equations, interspike interval distribution, mixed-mode oscillation, singular perturbation, fast–slow system, dynamic bifurcation, ca- nard, substochastic Markov chain, principal eigenvalue, quasi-stationary distribution. 1 Introduction Deterministic conduction-based models for action-potential generation in neuron axons have been much studied for over half a century. In particular, the four-dimensional Hodgkin–Huxley equations [HH52] have been extremely successful in reproducing the ob- served behaviour.

• large variety


• consecutive spikes


• system per- forms


• been studied


• fitzhugh–nagumo equations


• dimensional fitzhugh–nagumo


• equations can display


• noise


• system