Critical phenomena in globally coupled excitable elements

Abstract

Critical phenomena in globally coupled excitable elements are studied by focusing on a saddlenode bifurcation at the collective level. Critical exponents that characterize divergent fluctuations of interspike intervals near the bifurcation are calculated theoretically. The calculated values appear to be in good agreement with those determined by numerical experiments. The relevance of our results to jamming transitions is also mentioned.PACS numbers: 82.40. Bj, 05.10.Gg, 89.75.Da,The understanding of cooperative phenomena in nonequilibrium systems is one of the most important problems in physics. In contrast to those in equilibrium systems, their nature essentially depends on the dynamical properties of the systems. This makes it difficult to perform a systematic study of the problem. Thus, it is necessary to investigate the typical cooperative phenomena in nonequilibrium systems.Recently, critical behaviors have been observed experimentally [1,2,3,4,5,6,7] and numerically [8,9,10,11,12,13,14,15] in typical examples of coupled excitable elements such as neural networks and cardiac tissues. In general, such critical behaviors are classified into several groups on the basis of the exponents of divergences. The classification enables the realization of a universality class for critical phenomena in coupled excitable elements. However, the broad distribution of the phenomena makes it difficult to elucidate the mechanism of the criticality. When we consider the role of the mean-field Ising model in theories of equilibrium statistical mechanics, it becomes apparent that it is necessary to develop an elegant method for theoretical analysis of a minimal model describing critical phenomena in coupled excitable elements.Thus, with this aim in mind, we analyze a previously proposed simple model for globally coupled excitable elements in this Letter [16]. In particular, we focus on a divergent behavior with respect to parameter change around a saddle-node bifurcation because the excitability of this model is related to the bifurcation. It should be noted that such transition properties have been studied for different excitable systems [2,8,10,11,14]. The main achievement of this Letter is a theoretical derivation of the critical exponents that characterize the singular behavior near a saddle-node bifurcation.Model: The excitable nature of a system is characterized by the existence of spikes in a time series. Mathematically speaking, spikes are described by trajectories near a homoclinic orbit in a differential equation. As a simple example, let us consider an ordinary differential equation ∂ t φ = ω−h sin φ for a phase variable φ ∈ [0, 2π], in which there exists the homoclinic orbit when ω = h. Then, when h is slightly larger than ω, a small perturbation for the fixed point φ * = sin −1 (ω/h) yields one spike. On the other hand, when h is slightly less than ω, the system shows an array of spikes with a long interspike interval. The qualitative change in the trajectories is an example of saddle-node bifurcation. By using thi...

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