Chapter 1

Introduction—Oscillators and Synchronization

Oscillation is among the simplest of dynamic behaviors to describe mathematically and has thus been conveniently used in modeling a wide variety of physical phenomena ranging from mechanical vibration to quantum mechanical behavior and even neurological systems. Certainly not the least of these is the area of electronic circuits. Many years ago, van der Pol created his classical model of an oscillator including the nonlinear saturation effects that determine the amplitude of the steady-state oscillation [9]. Soon afterward, Adler provided a simple theory of what is now known as injection locking, and coupled oscillators became a valuable design resource for the electronics engineer and the antenna designer [10]. Moreover, circuit theorists were able to apply these principles to long chains and closed rings of coupled oscillators to model biological behaviors such as intestinal and colorectal myoelectrical activity in humans [11, 12].

1.1   Early Work in Mathematical Biology and Electronic Circuits

Biologists, in trying to understand how neurons coordinate the movements of animals, have defined what is known as a “central pattern generator,” or “CPG” for short. A CPG in this context is a group of neurons that produce rhythmic or periodic signals without sensory input. Biologists have found that CPGs are conveniently modeled mathematically if treated as a set of oscillators that are coupled to each other, most often using nearest-neighbor ...

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