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Mathematical Medicine and Biology Advance Access originally published online on March 9, 2009
Mathematical Medicine and Biology 2009 26(3):201-224; doi:10.1093/imammb/dqn027
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials

G. Richardson{dagger}

School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK

{dagger} Email: G.Richardson{at}soton.ac.uk

Received on April 3, 2008. Revised on July 9, 2008. Accepted on November 25, 2008.

By application of matched asymptotic expansions, a simplified partial differential equation (PDE) model for the dynamic electrochemical processes occurring in the vicinity of a membrane, as ions selectively permeate across it, is formally derived from the Poisson–Nernst–Planck equations of electrochemistry. It is demonstrated that this simplified model reduces itself, in the limit of a long thin axon, to the cable equation used by Hodgkin and Huxley to describe the propagation of action potentials in the unmyelinated squid giant axon. The asymptotic reduction from the simplified PDE model to the cable equation leads to insights that are not otherwise apparent; these include an explanation of why the squid giant axon attains a diameter in the region of 1 mm. The simplified PDE model has more general application than the Hodgkin–Huxley cable equation and can, e.g. be used to describe action potential propagation in myelinated axons and neuronal cell bodies.

Keywords: action potential; matched asymptotic expansions; electrolyte; Hodgkin–Huxley model; Poisson–Nernst–Planck equations


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