Asymptotic analysis and analytical solutions of a model of cardiac excitation

Biktashev, V.N., Suckley, R., Elkin, Y.E. and Simitev, R.D. (2008) Asymptotic analysis and analytical solutions of a model of cardiac excitation. Bulletin of Mathematical Biology, 70(2), pp. 517-554. (doi:10.1007/s11538-007-9267-0)

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We describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317–352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an “Archetypal Model” with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable “caricature” of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the “caricature” right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Simitev, Dr Radostin
Authors: Biktashev, V.N., Suckley, R., Elkin, Y.E., and Simitev, R.D.
Subjects:Q Science > QA Mathematics
Q Science > QH Natural history > QH301 Biology
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Bulletin of Mathematical Biology
ISSN (Online):1522-9602
Published Online:03 December 2007

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