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A simple model for excitation – contraction coupling in the heart. (English) Zbl 0925.92052
Cardiac arrhythmia, which are a major cause of heart disease by cardiac fibrillation, are far from being understood and thus controlled. Since the electrical activity of the heart triggers its mechanical activity, mechanisms based on a disorganization of the electrical excitation waves are mainly under investigation. In this context, spiral waves or scroll rings take a special importance since their possible instability or their mutual interactions can lead to a spatio-temporal disorganization of the usual regular electrical pattern. However, the robustness of these mechanisms for the generation of spatio-temporal chaos is far from being convincing and it is possible that other mechanisms could induce this chaos much more easily. One of these could be the excitation-contraction coupling which inevitably takes place during heart contraction. The main problem here is to develop a model, sufficiently complete to incorporate the main mechanisms involved in the excitation-contraction coupling, but sufficiently simple to be tractable from a numerical and mathematical point of view. This is not an easy task, since mechanisms involved in the excitation-contraction coupling are numerous, and it is not possible to consider all of them in detail. Our approach will be more macroscopic and we will consider the minimal model coupling a simple reaction-diffusion system triggering action potentials to a strain-stress equation for the cardiac tissue via the intermediate of the calcium ion. In order to test the model, we will consider simple configurations for the propagation of an action potential on a one-dimensional cardiac muscular fibre, either in isotonic contraction where fibre extremities are free to move or in isometric contraction where fibre extremities are kept fixed during the same process. In a first part we describe the elements of the excitation-contraction coupling considered here and then expose the simplified model that, we expect, keeps the important features of the coupling. Finally, we study the solutions of the model corresponding to homogeneous excitation of the tissue and to travelling action potentials on one-dimensional muscular fibres either in the isotonic or isometric conditions. We discuss the influence of the mechanoelectric feedback on the action potential profiles observed in both cases.

##### MSC:
 92C30 Physiology (general) 92C50 Medical applications (general)
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##### References:
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