Abstract
Biological systems offer many interesting examples of oscillations, chaos, and bifurcations. Oscillations in biology arise because most cellular processes contain feedbacks that are appropriate for generating rhythms. These rhythms are essential for regulating cellular function. In this tutorial review, we treat two interesting nonlinear dynamic processes in biology that give rise to bursting, spiking, chaos, and fractals: endogenous electrical activity of excitable cells and Ca2+ releases from the Ca2+ stores in nonexcitable cells induced by hormones and neurotransmitters. We will first show that each of these complex processes can be described by a simple, yet elegant, mathematical model. We then show how to utilize bifurcation analyses to gain a deeper insight into the mechanisms involved in the neuronal and cellular oscillations. With the bifurcating diagrams, we explain how spiking can be transformed to bursting via a complex type of dynamic structure when the key parameter in the model varies. Understanding how this parameter would affect the bifurcation structure is important in predicting and controlling abnormal biological rhythms. Although we describe two very different dynamic processes in biological rhythms, we will show that there is universality in their bifurcation structures.
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