心脏电兴奋模型中传导速度恢复的渐近性
Asymptotics of conduction velocity restitution in models of electrical excitation in the heart
摘要：
我们延长了非吉洪诺夫渐近嵌入，提出了较早的，用于计算心脏兴奋性的离子模型中的传导速度恢复曲线。传导速度恢复是可兴奋介质中最简单的非平凡空间扩展问题，在心脏组织的情况下，它是预测心律失常和颤动的重要工具。理想化的传导速度恢复曲线需要求解具有周期性边界条件的非线性特征值问题，在心脏情况下，该问题非常僵硬，并且需要使用渐近方法。我们比较了四个例子中的恢复曲线的渐近性，两个通用的可兴奋介质模型，和两个离子心脏模型。通用模型包括经典的FitzHugh-Nagumo模型及其Barkley的变体。他们被视为标准的奇异摄动技术。离子模型包括Noble（1962）模型和Beeler和Schmidter（1977）模型的简化“漫画”，其导致已知渐近结果不适用的非吉洪诺夫问题。漫画贵族模型被认为是特别小心，以证明相应的边值问题的适定性。开发的方法用于计算传导速度恢复，然后施加到Beeler-Schmidter模型。我们讨论了新的数学功能出现在心脏离子模型和可能的应用程序开发的方法。
英文摘要：
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 nontrivial 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 nonlinear 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 the Noble (1962) model and the Beeler and Reuter (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.
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