dorsal/arxiv
View SchemaAsymptotic analysis and analytical solutions of a model of cardiac excitation
| Authors | Vadim N. Biktashev, Rebecca Suckley, Yury E. Elkin, Radostin D. Simitev |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0607047 |
| URL | https://arxiv.org/abs/q-bio/0607047 |
Abstract
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, 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.
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"abstract": "We describe an asymptotic approach to gated ionic models of single-cell\ncardiac excitability. It has a form essentially different from the Tikhonov\nfast-slow form assumed in standard asymptotic reductions of excitable systems.\nThis is of interest since the standard approaches have been previously found\ninadequate to describe phenomena such as the dissipation of cardiac wave fronts\nand the shape of action potential at repolarization. The proposed asymptotic\ndescription overcomes these deficiencies by allowing, among other non-Tikhonov\nfeatures, that a dynamical variable may change its character from fast to slow\nwithin a single solution. The general asymptotic approach is best demonstrated\non an example which should be both simple and generic. The classical model of\nPurkinje fibers (Noble, 1962) has the simplest functional form of all cardiac\nmodels but according to the current understanding it assigns a physiologically\nincorrect role to the Na current. This leads us to suggest an ``Archetypal\nModel\u0027\u0027 with the simplicity of the Noble model but with a structure more\ntypical to contemporary cardiac models. We demonstrate that the Archetypal\nModel admits a complete asymptotic solution in quadratures. To validate our\nasymptotic approach, we proceed to consider an exactly solvable ``caricature\u0027\u0027\nof the Archetypal Model and demonstrate that the asymptotic of its exact\nsolution coincides with the solutions obtained by substituting the\n``caricature\u0027\u0027 right-hand sides into the asymptotic solution of the generic\nArchetypal Model. This is necessary, because, unlike in standard asymptotic\ndescriptions, no general results exist which can guarantee the proximity of the\nnon-Tikhonov asymptotic solutions to the solutions of the corresponding\ndetailed ionic model.",
"arxiv_id": "q-bio/0607047",
"authors": [
"Vadim N. Biktashev",
"Rebecca Suckley",
"Yury E. Elkin",
"Radostin D. Simitev"
],
"categories": [
"q-bio.TO"
],
"title": "Asymptotic analysis and analytical solutions of a model of cardiac excitation",
"url": "https://arxiv.org/abs/q-bio/0607047"
},
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