Asymptotics of conduction velocity restitution in models of electrical excitation in the heart

Simitev, R.D. and Biktashev, V.N. (2011) Asymptotics of conduction velocity restitution in models of electrical excitation in the heart. Bulletin of Mathematical Biology, 73(1), pp. 72-115. (doi:10.1007/s11538-010-9523-6)

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Abstract

We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh–Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified “caricature” of Noble (J. Physiol. Lond. 160:317–352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177–210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler–Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method.

Item Type:Articles
Additional Information:The original publication is available at www.springerlink.com
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Simitev, Dr Radostin
Authors: Simitev, R.D., and Biktashev, V.N.
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
Journal Abbr.:Bull. Math. Biol.
Publisher:Springer
ISSN:0092-8240
ISSN (Online):1522-9602
Published Online:01 March 2010
First Published:First published in Bulletin of Mathematical Biology
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher.

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