Ring Populations

Abstract

My intent in this chapter is to direct your attention to several idealizations of rhythmic behavior in collections of many similar ring devices. It turns out that some of the peculiar limitations on the behavior of simple clocks do not apply to populations of simple clocks. Here we also encounter our first example in which a phase singularity emerges from an idealized model of the structure and mechanism of a rhythmic system. The chapter is divided into four sections:

• A. Collective rhythmicity in a population without interactions among constituent clocks. This is mainly about phase resetting by a stimulus.

• B. Collective rhythmicity in a population whose individuals are all influenced by the aggregate rhythmicity of the community. This is mainly about mutual synchronization and opposition to it.

• C. Spatially distributed simple clocks without interactions. This is mostly about patterns of phase in space.

• D. Ring devices interacting locally in space. This is mostly about waves.

If thou (dear reader) art wearied with this tiresome method of computation, have pity on me, who had to go through it seventy times at least, with an immense expenditure of time . . .

Johannes Kepler, 1609, Astronomia Nova, Chapter 16