Kesmia M, Boughaba S, Jacquir S
Département de Mathématiques, Université de Constantine I, Constantine, Algeria.
Le2i, FRE CNRS 2005, Arts et Métiers, Université Bourgogne Franche-Comté, Dijon, France.
J Math Biol. 2019 Apr;78(5):1529-1552. doi: 10.1007/s00285-018-1318-7. Epub 2019 Jan 1.
The aim of this work is the analysis of the nonlinear dynamics of two-dimensional mapping model of cardiac action potential duration (2D-map APD) with memory derived from one dimensional map (1D-map). Action potential duration (APD) restitution, which relates APD to the preceding diastolic interval (DI), is a useful tool for predicting cardiac arrhythmias. For a constant rate of stimulation the short action potential during alternans is followed by a longer DI and inversely. It has been suggested that these differences in DI are responsible for the occurrence and maintenance of APD alternans. We focus our attention on the observed bifurcations produced by a change in the stimulation period and a fixed value of a particular parameter in the model. This parameter provides new information about the dynamics of the APD with memory, such as the occurrence of bistabilities not previously described in the literature, as well as the fact that synchronization rhythms occur in different ways and in a new fashion as the stimulation frequency increases. Moreover, we show that this model is flexible enough as to accurately reflect the chaotic dynamics properties of the APD: we have highlighted the fractal structure of the strange attractor of the 2D-map APD, and we have characterized chaos by tools such as the calculation of the Lyapunov exponents, the fractal dimension and the Kolmogorov entropy, with the next objective of refining the study of the nonlinear dynamics of the duration of the action potential and to apply methods of controlling chaos.
这项工作的目的是分析从一维映射（1D映射）导出的具有记忆的心脏动作电位持续时间二维映射模型（2D映射APD）的非线性动力学。动作电位持续时间（APD）恢复，即将APD与前一个舒张间期（DI）相关联，是预测心律失常的有用工具。对于恒定的刺激速率，交替期间的短动作电位之后是较长的DI，反之亦然。有人提出，DI的这些差异是APD交替发生和维持的原因。我们将注意力集中在由刺激周期的变化和模型中特定参数的固定值产生的观察到的分岔上。这个参数提供了关于具有记忆的APD动力学的新信息，例如文献中先前未描述的双稳态的出现，以及随着刺激频率增加同步节律以不同方式和新的方式出现的事实。此外，我们表明该模型足够灵活，能够准确反映APD的混沌动力学特性：我们突出了2D映射APD奇异吸引子的分形结构，并且我们通过诸如计算李雅普诺夫指数、分形维数和柯尔莫哥洛夫熵等工具来表征混沌，其下一个目标是完善对动作电位持续时间的非线性动力学的研究并应用混沌控制方法。
