### 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 language | English (US) |
---|---|

Pages (from-to) | 1279-1325 |

Number of pages | 47 |

Journal | Annals of Applied Probability |

Volume | 18 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2008 |

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### 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.

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 18, no. 4, pp. 1279-1325. https://doi.org/10.1214/07-AAP494

}

TY - JOUR

T1 - The emergence of the deterministic Hodgkin-huxley equations as a limit from the underlying stochastic ion-channel mechanism

AU - Austin, Tim D.

PY - 2008/8

Y1 - 2008/8

N2 - 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.

AB - 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.

KW - Action potential

KW - Convergence of Markov processes

KW - Hodgkin-Huxley equations

KW - Nonlinear parabolic PDE

KW - Stochastic Hodgkin-Huxley equations

UR - http://www.scopus.com/inward/record.url?scp=52949122900&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52949122900&partnerID=8YFLogxK

U2 - 10.1214/07-AAP494

DO - 10.1214/07-AAP494

M3 - Article

VL - 18

SP - 1279

EP - 1325

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 4

ER -