Effects of nonlocality on the dynamics of large assemblies of
oscillatory and excitable units

Yoshiki Kuramoto

Kyoto University

A certain class of biological populations could be imagined as a system composed of similar dynamical units of oscillatory/excitable nature whose activities are strongly influenced by some diffusive chemicals produced by the individual units themselves. This picture may apply, though with varying degrees, to such systems as social amoeba Dictyosterium discoideum, suspension of yeast cells under glycolytic oscillation, assemblies of circadian pacemaker cells in suprachiasmic nucleus, and some neural populations under the influence of neuromodulators. In this connection, we propose a universal class of three-component reaction-diffusion model in which the first two components, assumed nondiffusive, constitute a large assembly of noninteracting dynamical units, while the third component, assumed strongly diffusive, serves itself as a coupling agent among these units. Here effective nonlocality in coupling arises in the nondiffusive subsystem as a result of elimination of the diffusive variable. We find that by changing the extent of space- and time nonlocalities, the types of dynamical behavior exhibited are extremely rich. Specifically, we discuss rotating spirals without phase singularity, self-sustained pacemakers in a homegenous medium, and fractalized turbulent field with strong intermittency, which are something that would hardly arise in conventional reaction-diffusion systems.