Critique of the Fox-Lu model for Hodgkin-Huxley fluctuations in neuron ion channels

TL;DR

This paper critically examines the Fox-Lu model for Hodgkin-Huxley ion channel fluctuations, highlighting its limitations and providing an explicit solution for the S matrix in sodium channels, improving simulation accuracy.

Contribution

The paper offers a detailed critique of the Fox-Lu model and presents the first explicit closed-form solution for the S matrix in sodium channels, addressing a key computational challenge.

Findings

The Fox-Lu model has notable shortcomings in simulating ion channel fluctuations.

An explicit closed-form S matrix for sodium channels is derived.

The Cholesky decomposition provides a practical numerical method for the S matrix.

Abstract

Using a well known result that every FP equation has an antecedent Langevin equation LE Fox and Lu proposed such a description for ion channels in 1994. Their contraction followed the works of van Kampen and of T. Kurtz. The contraction produces a diffusion term with a state dependent diffusion matrix, D, that arises from the coupling matrix, S, in the LE. This S connected the noise terms to the channel subunit variables in the LE. Fox and Lu and many others later on observed that SS = D. Since D was determined by the contraction of the MC equations into the FP equation, this left the problem of determining the square root matrix, S, for every time step of the simulation. Since this is time consuming, Fox and Lu introduced simplified models not requiring the square root of a matrix. Subsequently, numerous studies were published that showed the several shortcomings of these simplified models. In 2011, Goldwyn et al. [6] rediscovered the overlooked original matrix dependent approach in the Fox-Lu 1994 paper. They showed that it produced results in very good agreement with the MC results. In 1991, Fox and Keizer [7] wrote a paper on an unrelated topic that utilized the work of van Kampen and of Kurtz. In that work the connection between D and S is SST = D. ST is the adjoint (transpose) of S. D remains a positive definite symmetric matrix but S need not be. Fox has reproduced the 2012 results of Orio and Soudry for potassium channels and has also found in closed form the solution for the more complicated sodium channels. The square root problem generally must be done numerically, but the Cholesky is always doable in closed form. Thereby the S matrix for sodium is given explicitly for the first time in this paper