The Annals of Applied Probability

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

Tim D. Austin

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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.

Article information

Ann. Appl. Probab., Volume 18, Number 4 (2008), 1279-1325.

First available in Project Euclid: 21 July 2008

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60K99: None of the above, but in this section 92C20: Neural biology

Hodgkin–Huxley equations stochastic Hodgkin–Huxley equations action potential convergence of Markov processes nonlinear parabolic PDE


Austin, Tim D. The emergence of the deterministic Hodgkin–Huxley equations as a limit from the underlying stochastic ion-channel mechanism. Ann. Appl. Probab. 18 (2008), no. 4, 1279--1325. doi:10.1214/07-AAP494.

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