AVERAGING FOR A FULLY-COUPLED PIECEWISE DETERMINISTIC MARKOV PROCESS IN INFINITE DIMENSIONS

Alexandre Genadot (1) , Michèle Thieullen (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this paper, we consider the generalized Hodgkin-Huxley model introduced by Austin in [1]. This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully-coupled Piecewise Deterministic Markov Process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that asymptotically this 'two-time-scales' model reduces to the so called averaged model which is still a PDMP in infinite dimensions for which we provide e ective evolution equations and jump rates.

Mots clés: piecewise deterministic Markov processes; fully-coupled systems; averaging principle; slow-fast systems; Hodgkin-Huxley model

Domaines: Probabilités [math.PR]

Alexandre Genadot, Michèle Thieullen. AVERAGING FOR A FULLY-COUPLED PIECEWISE DETERMINISTIC MARKOV PROCESS IN INFINITE DIMENSIONS. 2012. ⟨hal-00662268v1⟩