INHIBITION-PRODUCED PATTERNING IN CHAINS OF COUPLED NONLINEAR OSCILLATORS

This paper describes the behavior of chains of oscillators in which there is both local coupling and coupling between points on the chain that are roughly a distance of a half-chain apart. The local coupling is designed to produce synchrony, while the long-distance coupling is an abstraction of inhibitory coupling: alone it would produce antiphase behavior between the oscillators directly coupled. The investigation is motivated by data concerning traveling waves of electrical activity in the nervous system of animals that swim in an undulatory manner and also by observations concerning the motor behavior of more general vertebrates in early development. The aim of the work is to show that this connectivity can give rise to waves with wavelength equal to the length of the chain, as observed in swimming animals, as well as the more complicated patterns seen in early development. The latter include ''S-waves,'' in which the two halves of the chain are each in synchrony but are oscillating in antiphase with one another. It is shown that there are families of stable solutions, including traveling waves with several wavelengths within the chain, and ''antiwaves'' with a leading or lagging oscillator near the center of the chain. Several qualitatively different solutions can be stable for the same parameter values. Changing the connectivity of the long-range coupling can alter the repertoire of possible behaviors.