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Asymptotic Expansions of I-V Relations via a Poisson–Nernst–Planck System
Abaid, Nicole Teresa Eisenberg, Robert S. Liu, Weishi
Abaid, Nicole Teresa
Eisenberg, Robert S.
Liu, Weishi
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Abstract
We investigate higher order matched asymptotic expansions of a steady-state Poisson–Nernst–Planck (PNP) system with particular attention to the I-V relations of ion channels. Assuming that the Debye length is small relative to the diameter of the narrow channel, the PNP system can be viewed as a singularly perturbed system. Special structures of the zeroth order inner and outer systems make it possible to provide an explicit derivation of higher order terms in the asymptotic expansions. For the case of zero permanent charge, our results concerning the I-V relation for two oppositely charged ion species are (i) the first order correction to the zeroth order linear I-V relation is generally quadratic in V; (ii) when the electro-neutrality condition is enforced at both ends of the channel, there is NO first order correction, but the second order correction is cubic in V. Furthermore (Theoremsigmo), up to the second order, the cubic I-V relation has (except for a very degenerate case) three distinct real roots that correspond to the bistable structure in the FitzHugh–Nagumo simplification of the Hodgkin–Huxley model.
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This is the published version, also available here: http://dx.doi.org/10.1137/070691322.
Date
2008-01-01
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Society for Industrial and Applied Mathematics
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Keywords
singular perturbation, matched asymptotic expansion, I-V relations
Citation
Abaid, Nicole., Eisenberg, Robert S., Liu, Weishi. "Asymptotic expansions of I-V relations via a Poisson-Nernst-Planck system." SIAM J. Appl. Dyn. Syst., 7(4), 1507–1526. (20 pages). http://dx.doi.org/10.1137/070691322.