System-Size Effects on the Collective Dynamics of Cell Populations with Global Coupling

Abstract

Phase-transition-like behavior is found to occur in globally coupled systems consisting of finite numbers of elements, and a theoretical explanation of this behavior is given. The system studied is a population of globally pulse-coupled integrate-and-fire cells subject to a small additive noise. As the population size is changed, the system exhibits phase-transitionlike behavior, that is, there exists a well-defined critical system size above which the system remains in a monostable state with high-frequency activity and below which a new phase characterized by the alternation of high-and low-frequency types of activity appears. The mean field motion obeys a stochastic process with a state-dependent noise, and the above described phenomenon can be interpreted as a noise-induced transition characteristic of such processes. The coexistence of high-and low-frequency activity observed in finite size systems was reported by Cohen, Soen and Braun [Physica A 249 (1998), 600] in experiments on cultivated heart cells. The present report gives the first qualitative interpretation of their experimental results. §1. IntroductionCollective dynamics of coupled dynamical elements represents a central issue of nonlinear dynamics, and has served as a subject of extensive study over the last few decades. The relevant areas to which their study has been applied include chemical reactions, 2) societies of living organisms, 3)-5) lasers, 6), 7) semiconductors, 8), 9) neural networks 10)-12) and cardiac systems. 3), 13), 14) In most theoretical studies, however, the system size is assumed infinite. While this idealization could be valid for such systems like spatially extended chemical reactions, there seem to be important practical cases in which the finiteness of the system size should be taken into account explicitly. Here we cite two existing theoretical studies on the collective dynamics of populations in which the finiteness of the system size plays a crucial role. Firstly, Daido 15) investigated the collective behavior of an inhomogeneous system of oscillators, focusing on the statistics of fluctuations close to the onset of collective motion. Secondly, Pikovsky et al. 16) studied coupled bistable elements and demonstrated numerically the existence of phase-transition-like behavior existing only for finite systems. Apart from these theoretical studies, an interesting experiment was reported recently by Cohen et al. 1) They cultivated heart cells in various population sizes and found that these cells exhibit system-size-dependent behavior. Some more details of their reports are the following. Heart cells extracted from the ventricles of neonatal rats were cultivated, and time series of their spontaneous spiking activity were recorded. Initially, these heart cells were prevented from interacting by a chemical treatment, but after some time, they spontaneously assembled to form sub-