Patterns of Oscillation in Coupled Cell Systems

Coupled oscillators or coupled cell systems are used as models in a variety of physical and biological contexts. Each of these models includes assumptions about the internal dynamics of a cell (a pendulum or a neuron or a single laser) and assumptions about how the cells are coupled to each other. In a primitive sense, coupled cell systems are just moderate sized systems of ODE; for example, an eight-cell system with four-dimensional internal dynamics (such as a Hodgkin-Huxley system) yields a 32-dimensional system of ODE. In a more sophisticated sense, coupled cell systems have additional structure; we want to be able to compare the dynamics in different cells (are they synchronous, or a half-period out of phase, or do they have a more complicated phase relation?). In this paper we explore the extra structure that is associated with a coupled cell system. We argue that those permutations of the cells that are assumed to be symmetries of the cell system consititute a modelling assumption—one that in large measure dictates the kinds of equilibria and time periodic solutions that are expected in such models. We survey certain general results in the context of specific models, including locomotor central pattern generators for quadruped motion and coupled pendula. These results lead to a model for multirhythms. Coupled cell dynamics are a worthwhile subject of study and we begin here to discuss some of the fascinating features of this area.