Oscillator-Generated Wave Phenomena and Central Pattern Generators

In Chapter 11 we saw how diffusion, chemotaxis and convection mechanisms could generate spatial patterns; in Volume II we discuss mechanisms of biological pattern formation extensively. In Chapter 13, and Chapter 1 and Chapter 13, Volume II we show how diffusion effects, for example, can also generate travelling waves, which have been used to model the spread of pest outbreaks, travelling waves of chemical concentration, colonization of space by a population, spatial spread of epidemics and so on. The existence of such travelling waves is usually a consequence of the coupling of various effects such as diffusion or chemotaxis or convection. There are, however, other wave phenomena of a quite different kind, called kinematic waves, which exhibit wavelike spatial patterns, which depend on the coupling of biological oscillators whose properties relating to phase or period vary spatially. The two phenomena described in this chapter are striking, and the models we discuss are based on the experiments or biological phenomena which so dramatically exhibit them. The first involves the Belousov-Zhabotinskii reaction and the second, which is specifically associated with the swimming of, for example, lamprey and dogfish, illustrates the very important concept of a central pattern generator. The results we derive here apply to spatially distributed oscillators in general.