Advances in Applied Probability
- Adv. in Appl. Probab.
- Volume 44, Number 3 (2012), 749-773.
Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions
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.
Adv. in Appl. Probab., Volume 44, Number 3 (2012), 749-773.
First available in Project Euclid: 6 September 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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