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Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions

Part of: Parabolic equations and systems Markov processes Physiological, cellular and medical topics Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  04 January 2016

Alexandre Genadot  and
Michèle Thieullen
Alexandre Genadot*
Affiliation:
Université Pierre et Marie Curie
Michèle Thieullen*
Affiliation:
Université Pierre et Marie Curie
*
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris 6, UMR 7599, Case courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France.
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris 6, UMR 7599, Case courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France.
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Abstract

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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.

Keywords

Piecewise-deterministic Markov processfully coupled systemaveraging principlereaction diffusion equationslow-fast systemMarkov chainneuron modelHodgkin-Huxley model

MSC classification

Primary: 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case) 60J75: Jump processes 35K57: Reaction-diffusion equations
Secondary: 92C20: Neural biology 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 749 - 773
Copyright
© Applied Probability Trust 

Footnotes

This work was supported by the Agence Nationale de la Recherche through the project MANDy, Mathematical Analysis of Neuronal Dynamics, ANR-09-BLAN-0008-01.

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