A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials
A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials
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
action potential, matched asymptotic expansions, electrolyte, hodgkin–huxley model, poisson–nernst–planck equations
201-224
Richardson, G.
3fd8e08f-e615-42bb-a1ff-3346c5847b91
September 2009
Richardson, G.
3fd8e08f-e615-42bb-a1ff-3346c5847b91
Richardson, G.
(2009)
A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials.
Mathematical Medicine and Biology, 26 (3), .
(doi:10.1093/imammb/dqn027).
(PMID:19273492)
Abstract
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
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Mathematical_Medicine_and_Biology_2009_Richardson.pdf
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Published date: September 2009
Keywords:
action potential, matched asymptotic expansions, electrolyte, hodgkin–huxley model, poisson–nernst–planck equations
Organisations:
Applied Mathematics
Identifiers
Local EPrints ID: 69572
URI: http://eprints.soton.ac.uk/id/eprint/69572
ISSN: 1477-8599
PURE UUID: 72577bc4-7d9a-43ed-bd97-6d97b5b165d5
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Date deposited: 19 Nov 2009
Last modified: 14 Mar 2024 02:54
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