Advances in Applied Probability

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

Alexandre Genadot and Michèle Thieullen

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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

Permanent link to this document
https://projecteuclid.org/euclid.aap/1346955263

Digital Object Identifier
doi:10.1239/aap/1346955263

Mathematical Reviews number (MathSciNet)
MR3024608

Zentralblatt MATH identifier
1276.60023

Subjects
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.) [See also 80A30]

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

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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