CONSTRUCTION AND CUSTOMIZATION OF STABLE OSCILLATION MODELS IN BIOLOGY

Oscillations exist at all levels of biological systems and are often crucial for their proper functioning. Among the various types of oscillations, limit cycles have received particular attention for more than one hundred years. Specifically, theorems have been established that characterize whether a system might have the capability of exhibiting limit cycles. However, the practical application of these theorems is usually cumbersome and there are hardly any guidelines for devising de novo models that exhibit limit cycles of a desired form. In this paper, we propose a simple method for constructing and customizing stable limit cycles in two-dimensional systems according to desired features, including frequency, amplitude, and phase shift between system variables. The method is based on inverting a criterion proposed by Lewis for characterizing oscillations in two-dimensional S-system models. First, we execute comprehensive simulations that result in a set of over 2000 prototype limit cycles. Second, we show with examples how these prototypes can be further customized to adhere to predetermined specifications. This two-step process is fast and efficacious, especially when one considers the paucity of alternative methods. Finally, we illustrate how one may create systems with more complex dynamics by modulating the prototypes with external input signals.