Received on May 19, 1994; revised on June 13, 1994

ABSTRACT Biological systems provide many interesting examples of oscillations, chaos, and forks. In biology, oscillations occur because most cellular processes contain feedback that is appropriate for rhythm. These rhythms regulate cell function to. In this review of the tutorial, we will discuss two interesting nonlinear dynamic processes in biology that cause cell rupture, spikes, chaos, and fractals: endogenous electrical activity of excitable cells and by hormones.And the release of Ca in non-excitable cells induced by neurotransmitters. First, we will show that each of these complex processes can be described by a simple and elegant mathematical model. Then, we will show how to use bifurcation analysis to gain insight into the mechanisms involved in neuronal and cellular oscillations. Through the bifurcation diagram, we explain how to convert a peak into a blast through a complex dynamic structure when the key parameters in the model change. Understanding the effect of this parameter on the bifurcation structure is important for predicting and controlling biorhythm abnormalities. Although we describe two very different dynamic processes in the biological rhythm, we will prove that their bifurcation structure is universal.

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摘要 生物系统提供了许多振荡、混沌和分叉的有趣例子。在生物学中，振荡的产生是因为大多数细胞过程包含适合产生节律的反馈。这些节律对调节细胞功能至关重要。在本教程的回顾中，我们将讨论生物学中两个有趣的非线性动态过程，它们会导致细胞破裂、尖峰、混沌和分形:可兴奋细胞的内源性电活动和由激素和神经递质诱导的不可兴奋细胞中Ca的释放。首先，我们将展示这些复杂过程中的每一个都可以用一个简单而优雅的数学模型来描述。然后，我们将展示如何利用分叉分析来深入了解神经元和细胞振荡所涉及的机制。通过分岔图，我们解释了当模型中的关键参数发生变化时，如何通过一种复杂的动态结构将峰值转化为爆破。了解这一参数对生物分叉结构的影响，对于预测和控制生物节律异常具有重要意义。虽然我们描述了生物节律中两个非常不同的动态过程，但我们将证明它们的分叉结构具有普遍性。

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