Key result
In mathematical models of cardiac excitability, suppressing tissue excitability below a critical limit blocks active propagation and causes front dissipation, explaining re-entrant wave termination.
In mathematical models of cardiac excitability, suppressing excitability leads to wave dissipation that does not recover, which accounts for breakups and self-termination of re-entrant waves.
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Small-parameter analysis sharpens multi-scale cardiac excitability models; extends singular perturbation theory for computational research.
We analyse small parameters in selected models of biological excitability, including Hodgkin-Huxley (Hodgkin & Huxley 1952 J. Physiol.117, 500-544) model of nerve axon, Noble (Noble 1962 J. Physiol.160, 317-352) model of heart Purkinje fibres and Courtemanche et al. (Courtemanche et al. 1998 Am. J. Physiol.275, H301-H321) model of human atrial cells. Some of the small parameters are responsible for differences in the characteristic time-scales of dynamic variables, as in the traditional singular perturbation approaches. Others appear in a way which makes the standard approaches inapplicable. We apply this analysis to study the behaviour of fronts of excitation waves in spatially extended cardiac models. Suppressing the excitability of the tissue leads to a decrease in the propagation speed, but only to a certain limit; further suppression blocks active propagation and leads to a passive diffusive spread of voltage. Such a dissipation may happen if a front propagates into a tissue recovering after a previous wave, e.g. re-entry. A dissipated front does not recover even when the excitability restores. This has no analogy in FitzHugh-Nagumo model and its variants, where fronts can stop and then start again. In two spatial dimensions, dissipation accounts for breakups and self-termination of re-entrant waves in excitable media with Courtemanche et al. kinetics.
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Biktasheva et al. (2006) studied Cardiac excitability (mathematical models). Mathematical modeling of excitability was evaluated. In mathematical models of cardiac excitability, suppressing tissue excitability below a critical limit blocks active propagation and causes front dissipation, explaining re-entrant wave termination.
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