The emergence of the deterministic Hodgkin-huxley equations as a limit from the underlying stochastic ion-channel mechanism

Tim D. Austin

Research output: Contribution to journalArticle

Abstract

In this paper we consider the classical differential equations of Hodgkin and Huxley and a natural refinement of them to include a layer of stochastic behavior, modeled by a large number of finite-state-space Markov processes coupled to a simple modification of the original Hodgkin-Huxley PDE. We first prove existence, uniqueness and some regularity for the stochastic process, and then show that in a suitable limit as the number of stochastic components of the stochastic model increases and their individual contributions decrease, the process that they determine converges to the trajectory predicted by the deterministic PDE, uniformly up to finite time horizons in probability. In a sense, this verifies the consistency of the deterministic and stochastic processes.

Original languageEnglish (US)
Pages (from-to)1279-1325
Number of pages47
JournalAnnals of Applied Probability
Volume18
Issue number4
DOIs
StatePublished - Aug 2008

Fingerprint

Ion Channels
Stochastic Processes
Markov Process
Stochastic Model
Horizon
State Space
Existence and Uniqueness
Refinement
Regularity
Trajectory
Differential equation
Verify
Converge
Decrease
Stochastic processes
State space
Markov process
Uniqueness
Time horizon
Differential equations

Keywords

  • Action potential
  • Convergence of Markov processes
  • Hodgkin-Huxley equations
  • Nonlinear parabolic PDE
  • Stochastic Hodgkin-Huxley equations

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Statistics and Probability

Cite this

The emergence of the deterministic Hodgkin-huxley equations as a limit from the underlying stochastic ion-channel mechanism. / Austin, Tim D.

In: Annals of Applied Probability, Vol. 18, No. 4, 08.2008, p. 1279-1325.

Research output: Contribution to journalArticle

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