Analysis of a Non-linear Partial Difference Equation, and Its Application to Cardiac Dynamics

A model of a strip of cardiac tissue consisting of a one-dimensional chain of cardiac units is derived in the form of a non-linear partial difference equation. Perturbation analysis is performed on this equation, and it is shown that regular perturbations are inadequate due to the appearance of secu...

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Bibliographic Details
Published in:Journal of difference equations and applications 2002-01, Vol.8 (12), p.1147-1169
Main Authors: Stubna, Michael D., Rand, Richard H., Gilmour, Robert F.
Format: Article
Language:English
Subjects:
Cardiac Dynamics
Non-linear
Partial Difference Equation
Singular Perturbation Method
Spatial Bifurcations
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Summary:A model of a strip of cardiac tissue consisting of a one-dimensional chain of cardiac units is derived in the form of a non-linear partial difference equation. Perturbation analysis is performed on this equation, and it is shown that regular perturbations are inadequate due to the appearance of secular terms. A singular perturbation procedure known as the method of multiple scales is shown to provide good agreement with numerical simulation except in the neighborhood of a singularity of the slow flow. The perturbation analysis is supplemented by a local numerical simulation near this singularity. The resulting analysis is shown to predict a "spatial bifurcation" phenomenon in which parts of the chain may be oscillating in period-2 motion while other parts may be oscillating in higher periodic motion or even chaotic motion.
ISSN:1023-6198
1563-5120
DOI:10.1080/1023619021000054006