Oscillation and pattern formation in a system of self-regulating cells

In this paper, a model of a system of N greater than or equal to 2 asymmetrically and nondiffusibly coupled self-regulating cells is proposed, intended to formally represent well-known basic features of biological systems organization. The model is discussed in terms of the activation/desensitization dynamics in Dictyostelium. It is shown that the cells in the ensemble are capable of stable oscillations provided the intercellular coupling is strong enough, otherwise their stationary values remain asymptotically stable. The emergence of oscillations in the one-dimensional, ring-shaped ensemble of size N greater than or equal to 2 is due to a bifurcation of Naimark-Sacker type which is proven to exist under rather mild conditions, It is shown that a supercritical as well as a subcritical bifurcation branch exist at different parameter values. Moreover, due to the dihedral symmetry of the system, the parity of N is shown to determine the multiplicity of the critical pair of eigenvalues crossing the unit-circle. The model proposed does not represent a reaction-diffusion system, it is not excitable but exhibits pattern formation under conditions under which a stable limit cycle exists. Copyright (C) 1998 Elsevier Science B.V.