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From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators
 Phys. D
, 2000
"... The Kuramoto model describes a large population of coupled limitcycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, w ..."
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Cited by 164 (4 self)
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The Kuramoto model describes a large population of coupled limitcycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory, bifurcation theory, and plasma physics. © 2000 Elsevier Science B.V. All rights reserved.
Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous States, and Locking
 NEURAL COMPUTATION
, 2000
"... An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrateandfire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analy ..."
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Cited by 139 (24 self)
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An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrateandfire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived. The response of a population of noisy integrateandfire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized `locked' solutions is analyzed.
Draguhn A. Neuronal oscillations in cortical networks
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Cited by 136 (1 self)
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The following resources related to this article are available online at
Resonance and the Perception of Musical Meter
 CONNECTION SCIENCE
, 1994
"... Many connectionist approaches to musical expectancy and music composition let the question of "What next?" overshadow the equally important question of "When next?". One cannot escape the latter question, one of temporal structure, when considering the perception of musical meter ..."
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Cited by 115 (5 self)
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Many connectionist approaches to musical expectancy and music composition let the question of "What next?" overshadow the equally important question of "When next?". One cannot escape the latter question, one of temporal structure, when considering the perception of musical meter. We view the perception of metrical structure as a dynamic process where the temporal organization of external musical events synchronizes, or entrains, a listener's internal processing mechanisms. This article introduces a novel connectionist unit, based upon a mathematical model of entrainment, capable of phase and frequencylocking to periodic components of incoming rhythmic patterns. Networks of these units can selforganize temporally structured responses to rhythmic patterns. The resulting network behavior embodies the perception of metrical structure. The article concludes with a discussion of the implications of our approach for theories of metrical structure and musical expectancy.
The calculi of emergence: Computation, dynamics, and induction
 Physica D
, 1994
"... Defining structure and detecting the emergence of complexity in nature are inherently subjective, though essential, scientific activities. Despite the difficulties, these problems can be analyzed in terms of how modelbuilding observers infer from measurements the computational capabilities embedded ..."
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Cited by 89 (14 self)
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Defining structure and detecting the emergence of complexity in nature are inherently subjective, though essential, scientific activities. Despite the difficulties, these problems can be analyzed in terms of how modelbuilding observers infer from measurements the computational capabilities embedded in nonlinear processes. An observer’s notion of what is ordered, what is random, and what is complex in its environment depends directly on its computational resources: the amount of raw measurement data, of memory, and of time available for estimation and inference. The discovery of structure in an environment depends more critically and subtlely, though, on how those resources are organized. The descriptive power of the observer’s chosen (or implicit) computational model class, for example, can be an overwhelming determinant in finding regularity in data. This paper presents an overview of an inductive framework — hierarchicalmachine reconstruction — in which the emergence of complexity is associated with the innovation of new computational model classes. Complexity metrics for detecting structure and quantifying emergence, along with an analysis of the constraints on the dynamics of innovation, are outlined. Illustrative examples are drawn from the onset of unpredictability in nonlinear systems, finitary nondeterministic processes, and
On partial contraction analysis for coupled nonlinear oscillators
 technical Report, Nonlinear Systems Laboratory, MIT
, 2003
"... We describe a simple but general method to analyze networks of coupled identical nonlinear oscillators, and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized)results on synchroni ..."
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Cited by 73 (37 self)
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We describe a simple but general method to analyze networks of coupled identical nonlinear oscillators, and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized)results on synchronization, antisynchronization and oscillatordeath. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positivedefinite diffusion coupling, it can be shown that synchronization always occur globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in “flocks ” of oscillators or dynamic elements.
On the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators
, 2005
"... We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for alltoall networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using ..."
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Cited by 72 (9 self)
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We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for alltoall networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.
Synchronization Induced by Temporal Delays in PulseCoupled Oscillators
, 1995
"... We derive return maps and phase diagrams to identify mechanisms of synchronization of pulsecoupled oscillators and emphasize the importance of temporal delays and inhibitory coupling. Optimum synchronization between two oscillators is obtained for inhibitory coupling, where the presence of delays g ..."
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Cited by 57 (0 self)
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We derive return maps and phase diagrams to identify mechanisms of synchronization of pulsecoupled oscillators and emphasize the importance of temporal delays and inhibitory coupling. Optimum synchronization between two oscillators is obtained for inhibitory coupling, where the presence of delays gives rise to stable inphase synchronization, while oscillators with excitatory coupling only synchronize with a phase lag. In large ensembles of globally coupled oscillators the delayed interaction leads to new collective phenomena like synchronization in multistable clusters of common phases for inhibitory coupling, while for excitatory coupling a mechanism of emerging and decaying synchronized clusters prevails. Printed in Physical Review Letters, 74 (9), 15701573 (1995). PACS numbers: 05.45.+b,87.10.+e Typeset using REVT E X Synchronization of coupled oscillators is a widespread phenomenon occuring in physics [1], chemistry [2] and biology [3]. Theoretical efforts towards a mathemati...
Behavioral coordination, structural congruence and entrainment in a simulation of acoustically coupled agents
 Adaptive Behavior
, 2000
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