94 Christiaan Huygens to Wallis Paris, [30 June]/10 July 1673

Huygens, Christiaan; Beeley, Philip; Scriba, Christoph J.

doi:10.1093/oseo/instance.00098501

Cited by 15 publications

(15 citation statements)

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“…The study of synchronization of coupled nonlinear oscillators has a history that spans several centuries, starting with Huygens' observation of synchronizing pendulum clocks [3,12]. There has been a great body of work throughout this history studying such synchronization phenomena; for reviews see [21,25,30].…”

Section: History Of Kuramoto Modelmentioning

confidence: 99%

Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model

Bronski

DeVille

Park

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We present a detailed analysis of the stability of synchronized solutions to the Kuramoto system of oscillators. We derive an analytical expression counting the dimension of the unstable manifold associated to a given stationary solution. From this we are able to derive a number of consequences, including: analytic expressions for the first and last frequency vectors to synchronize, upper and lower bounds on the probability that a randomly chosen frequency vector will synchronize, and very sharp results on the large N limit of this model. One of the surprises in this calculation is that for frequencies that are Gaussian distributed the correct scaling for full synchrony is not the one commonly studied in the literature-rather, there is a logarithmic correction to the scaling which is related to the extremal value statistics of the random frequency vector.

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Section: History Of Kuramoto Modelmentioning

confidence: 99%

Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model

Bronski

DeVille

Park

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…More than three centuries ago, Huygens first observed in his experiments on pendulum clocks that coupled nonlinear oscillators tend to synchronize [26,5]. Since then, the study of synchronized nonlinear oscillators has had a long and storied history, and along the way has inspired many ideas in modern dynamical systems theory.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory

Deville

Peskin

Spencer

2010

Math. Model. Nat. Phenom.

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Abstract. We analyze a stochastic neuronal network model which corresponds to an all-to-all network of discretized integrate-and-fire neurons where the synapses are failure-prone. This network exhibits different phases of behavior corresponding to synchrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transition as a function of the connection strength -as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.

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“…The study of synchronization of coupled nonlinear oscillators has a history that spans several centuries, starting with Huygens' observation of synchronizing pendulum clocks [1]. There has been a great body of work throughout this history studying such synchronization phenomena; for reviews see [2][3][4].…”

Section: Overviewmentioning

confidence: 99%

Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

Deville

Peskin

2011

Bull Math Biol

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We describe and analyze a model for a stochastic pulse-coupled neuronal network with many sources of randomness: random external input, potential synaptic failure, and random connectivity topologies. We show that different classes of network topologies give rise to qualitatively different types of synchrony: uniform (Erdős-Rényi) and "small-world" networks give rise to synchronization phenomena similar to that in "all-to-all" networks (in which there is a sharp onset of synchrony as coupling is increased); in contrast, in "scale-free" networks the dependence of synchrony on coupling strength is smoother. Moreover, we show that in the uniform and small-world cases, the fine details of the network are not important in determining the synchronization properties; this depends only on the mean connectivity. In marked contrast, for scale-free networks, the dynamics are significantly affected by the fine details of the network; in particular, they are significantly affected by the local neighborhoods of the "hubs" in the network.

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94 Christiaan Huygens to Wallis Paris, [30 June]/10 July 1673

Cited by 15 publications

References 0 publications

Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model

Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model

Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory

Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

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