Large Populations of Coupled Chemical Oscillators
01 Apr 1980-Siam Journal on Applied Mathematics (Society for Industrial and Applied Mathematics)-Vol. 38, Iss: 2, pp 305-316
TL;DR: In this paper, the authors study the mechanisms that underlie synchronization processes in large systems of coupled chemical oscillators and derive a nonlinear integro-differential equation that describes how the distribution of the oscillators' phases evolves in time.
Abstract: We study the mechanisms that underlie synchronization processes in large systems of coupled chemical oscillators. By synchronization, we mean the evolution from an initial state where the phases of the oscillators are distributed randomly, to a final state where all the oscillators are in phase. In the continuum limit, where there are many oscillators per unit volume, we derive a nonlinear integro-differential equation that describes how the distribution of the oscillators’ phases evolves in time. In general, this problem is very formidable. But we discover some important special cases for which there are exact solutions describing synchronization processes.
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TL;DR: In this paper, a review of the Kuramoto model of coupled phase oscillators is presented, with a rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years.
Abstract: Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.
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Abstract: We present a theoretical model which is used to explain the intersegmental coordination of the neural networks responsible for generating locomotion in the isolated spinal cord of lamprey.
536 citations
TL;DR: In this paper, a chain of weakly coupled oscillators with a linear gradient in natural frequencies is shown to exhibit frequency plateaus, or sequences of oscillators having the same frequency, with a jump in frequency from one plateau to another.
Abstract: A chain of $n + 1$ weakly coupled oscillators with a linear gradient in natural frequencies is shown to exhibit “frequency plateaus,” or sequences of oscillators having the same frequency, with a jump in frequency from one plateau to another. We first show that the equations for the coupled oscillators admit an invariant $(n + 1)$-torus on which the equations have a special form, one in which an n-dimensional subsystem is approximately invariant. We then show that when the linear gradient becomes too steep to allow phaselocking, there emerges a large-scale invariant circle in this n-dimensional system which corresponds to the existence of a pair of plateaus, and whose homotopy class within the n-torus corresponds to the position of the frequency jump. Also discussed are the effects of anisotropic and nonuniform coupling.
435 citations
TL;DR: The generalized CPG model is shown to be versatile enough that it can also generate various n-legged gaits and spinal undulatory motions, as in the swimming motions of a fish.
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180 citations
TL;DR: In this paper, the authors studied phase-locking in a network of coupled nonlinear oscillators with local interactions and random intrinsic frequencies and derived an exact expression for the probability of phase locking in a linear chain of such oscillators.
178 citations
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TL;DR: Nonlinear phenomena arising from the interaction of two oscillating systems of chemical reactions are studied experimentally, and mathematical modelling of the above phenomena failed, probably due to insufficient knowledge of a kinetic model.
105 citations
TL;DR: In this article, the interaction between a pair of coupled chemical oscillators was analyzed using singular perturbation techniques, and an equation that governs the time evolution of the phase shift was derived, which is a measure of how much the oscillators are out of phase.
Abstract: We analyze the interaction between a pair of coupled chemical oscillators. Using singular perturbation techniques, we derive an equation that governs the time evolution of the phase shift, which is a measure of how much the oscillators are out of phase. This result is the key to understanding experimental observations on coupled reactor systems. In particular, our model accounts for synchronization, and its bifurcation into rhythm splitting.
104 citations
11 citations