In this paper, a general approach for studying rings of coupled biological oscillators is presented. This approach, which is group-theoretic in nature, is based on the finding that symmetric ring networks of coupled non-linear oscillators possess generic patterns of phaselocked oscillations. The associated analysis is independent of the mathematical details of the oscillators' intrinsic dynamics and the nature of the coupling between them. The present approach thus provides a framework for distinguishing universal dynamic behaviour from that which depends upon further structure. In this study, the typical oscillation patterns for the general case of a symmetric ring of n coupled non-linear oscillators and the specific cases of three-and five-membered rings are considered. Transitions between different patterns of activity are modelled as symmetry-breaking bifurcations. The effects of one-way coupling in a ring network and the differences between discrete and continuous systems are discussed. The theoretical predictions for symmetric ring networks are compared with physiological observations and numerical simulations. This comparison is limited to two examples: neuronal networks and mammalian intestinal activity. The implications of the present approach for the development of physiologically meaningful oscillator models are discussed.
1 Introduction
Rhythmic oscillations abound in physiological systems and processes. They are involved, for example, in sleeping, locomotion, ventilation, circulation, mastication and digestion (Winfree 1980;. Currently, there is considerable interest in understanding how these oscillations are generated and controlled. Many investigators have proposed that the associated neural and/or muscular mechanisms can be modelled as systems of coupled oscillators (e.g. Cohen et al. 1988).