Complex Bifurcations and Chaos in Simple Theoretical Models of Cardiac Oscillations a
Complex Bifurcations and Chaos in Simple Theoretical Models of Cardiac Oscillations a
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心脏振荡简单理论模型中的复杂分岔和混沌
DOI：
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发表时间：
1990
影响因子：
5.2
通讯作者：
W. Zeng
中科院分区：
文献类型：
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作者：
L. Glass;W. Zeng
人类心脏是一个复杂的器官，无论是在解剖学上还是在电学上。因此，高度过于简化的数学模型可能与完整的人类心脏中发生的事情甚至与减少的实验准备中发生的事情完全相关，这似乎令人惊讶。然而，几个小组的工作清楚地表明，简化的数学模型可能会显示出与完整心脏的实验和临床数据的定性对应关系。尽管这些对应关系的原因尚未完全阐明，但研究简单的心脏振荡模型的理论特性仍然令人感兴趣。这些模型用于固定想法，并阐明生成各种复杂动态行为所需的基本特征。当然，要了解实验和临床情况下复杂动力学的详细机制，需要分析更现实的理论模型。非线性动力学领域是数学的一个分支，它为心脏电生理学引入了全新的术语。这些术语使非数学家很难理解该领域，并且还可能导致混乱，因为某些术语（例如“混沌”）被不同的工作者使用了不同的含义。在下文中，我们将尽量减少技术术语，以便非数学家也能在不了解所有细节的情况下理解该方法和结果。然而，那些数学能力很弱或根本不存在的人可能希望只阅读本节和下一节，然后跳到结论，其中讨论了这项工作对心脏电生理学的影响。 Glass 和 Mackey 以非数学家可以理解的方式对此处使用的方法进行了更悠闲的讨论。有必要使用以下术语：稳定极限环振荡、锁相、分叉和混沌。以下是这些术语的含义。稳定的极限环振荡是微分方程（包含导数的方程，如 McAllister-Noble-Tsien 和类似的心脏活动模型 l~ , ’ -~ ）的周期解，在接近周期的初始条件下，当时间接近无穷大时，它会吸引人。例如，心脏起搏器振荡通常是
The human heart is a complex organ, both anatomically and electrically. Therefore, it may seem surprising that highly oversimplified mathematical models may be a t all relevant to what happens in the intact human heart, or even in reduced experimental preparations. Yet, work from several groups makes it clear that simplified mathematical models may show qualitative correspondences with experimental and clinical data in the intact heart. Although the reasons for these correspondences have not yet been elucidated completely, it nevertheless remains of interest to examine the theoretical properties of simple models of cardiac oscillations. Such models serve to fix ideas, and to clarify basic features needed to generate various sorts of complex dynamic behavior. Of course, an understanding of detailed mechanisms underlying complex dynamics in experimental and clinical situations requires an analysis of more realistic theoretical models. The field of nonlinear dynamics, a branch of mathematics, introduces a whole new jargon into cardiac electrophysiology. This jargon makes it difficult for nonmathematicians to follow the field, and also can lead to confusion, since some terms, for example, “chaos,” have been used with different meanings by different workers. In the following we will try to keep technical terms to a minimum so that nonmathematicians might be able to understand the approach and results without understanding all the details. Those whose mathematics is very weak or nonexistent, however, may wish to read only this and the following section and then skip to the Conclusions where the implications of this work for cardiac electrophysiology are discussed. A much more leisurely discussion of the approach used here in a style accessible to nonmathematicians is in Glass and Mackey.‘ It is necessary to use the following terms: stable limit-cycle oscillation, phase locking, bifurcation, and chaos. Here is the meaning of these terms. A stable limit-cycle oscillation is a periodic solution of a differential equation (an equation containing derivatives, as in the McAllister-Noble-Tsien and similar mode l~ , ’ -~ of cardiac activity) that is attracting as time approaches infinity for initial conditions close to the cycle. For example, cardiac pacemaker oscillations are usually