## The Annals of Applied Probability

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

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

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 21 July 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1216677123

**Digital Object Identifier**

doi:10.1214/07-AAP494

**Mathematical Reviews number (MathSciNet)**

MR2434172

**Zentralblatt MATH identifier**

1157.60066

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60K99: None of the above, but in this section 92C20: Neural biology

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aoap/1216677123