Mathematical Models

Abstract

The goal of this chapter is to present certain mathematical models for circadian pacemakers, while keeping the mathematical formalism to a minimum. By necessity, our treatment must be nonrigorous, and the mathematically sophisticated reader is referred to the appropriate literature (e.g., Pavlidis, 1973a). Since most biologists are familiar with the basics of population ecology, we shall use an ecosystem as a paradigm for various concepts of oscillator dynamics. Our first order of business is to clarify the term model, since it has been used by different people to mean different things. Basically, a model is a hypothesis about how a physical system works. In general, it must have the following two essential properties: (1) it must summarize the available experimental evidence so that the description of the physical system through the model is more concise than the description through a table of experimental results; and (2) it must predict the behavior of the system under new circumstances. However, it need not say anything about the “deep” structure of the system. Strictly speaking, this structure can never be known. In many cases, it is customary to assume that it coincides with that of a very successful model, but this assumption is not really justified. For example, the acceptance of the heliocentric over the geocentric model in physics is not due to any “real truth” but to the fact that the former gives a more compact description of the planetary system and has far more successful predictions than the latter. It is still theoretically possible that one day somebody will come up with a superior geocentric model. This limitation of models has a certain implication for their value in the search for the circadian pacemaker.