Models for a variety of physical and biological systems (such as animal gaits and the beating pattern of the leech heart)

have the form of coupled networks of systems of differential equations. One important feature of such networks is that they support solutions in which some components vary synchronously and some vary with well defined time lags (spatio-temporal symmetries).

In this lecture we build on the example of animal gaits and discuss the mathematics of spatio-temporal symmetries and how these solutions arise in coupled cell networks. We describe a global theorem that gives necessary and sufficient conditions for a given network to support solutions with given spatio-temporal symmetries.

We end with a discussion of the beating of the leech heart that when interpreted naively seems to contradict the theory --- but in fact leads to interesting mathematical questions and to notions of local symmetries and approximately symmetric periodic solutions.

This research is joint with Luciano Buono, Marcus Pivato, and Ian Stewart.