### Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions

#### Abstract

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). 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-scale' model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

#### Article information

Source
Adv. in Appl. Probab., Volume 44, Number 3 (2012), 749-773.

Dates
First available in Project Euclid: 6 September 2012

https://projecteuclid.org/euclid.aap/1346955263

Digital Object Identifier
doi:10.1239/aap/1346955263

Mathematical Reviews number (MathSciNet)
MR3024608

Zentralblatt MATH identifier
1276.60023

#### Citation

Genadot, Alexandre; Thieullen, Michèle. Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions. Adv. in Appl. Probab. 44 (2012), no. 3, 749--773. doi:10.1239/aap/1346955263. https://projecteuclid.org/euclid.aap/1346955263

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