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September 2012 Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions
Alexandre Genadot, Michèle Thieullen
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Adv. in Appl. Probab. 44(3): 749-773 (September 2012). DOI: 10.1239/aap/1346955263

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.

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Alexandre Genadot. Michèle Thieullen. "Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions." Adv. in Appl. Probab. 44 (3) 749 - 773, September 2012. https://doi.org/10.1239/aap/1346955263

Information

Published: September 2012
First available in Project Euclid: 6 September 2012

zbMATH: 1276.60023
MathSciNet: MR3024608
Digital Object Identifier: 10.1239/aap/1346955263

Subjects:
Primary: 35K57 , 60B12 , 60J75
Secondary: 92C20 , 92C45

Keywords: averaging principle , fully coupled system , Hodgkin-Huxley model , Markov chain , neuron model , piecewise-deterministic Markov process , reaction diffusion equation , slow-fast system

Rights: Copyright © 2012 Applied Probability Trust

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Vol.44 • No. 3 • September 2012
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