Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

Kuramoto, Yoshio; Nishikawa, Ikuko

doi:10.1007/bf01009349

Cited by 371 publications

(234 citation statements)

References 16 publications

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“…Locking as well as the phase shift between fast and slow units are a much more general phenomenon and occur in many other circumstances (Yamaguchi and Shimizu 1984;Kuramoto and Nishikawa 1987). But while most other models of collective oscillations are based on abstract differential equations, our approach is based on a specific model of neuronal signal transmission.…”

Section: Discussionmentioning

confidence: 99%

A biologically motivated and analytically soluble model of collective oscillations in the cortex

1993

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Almtraet. A model of an associative network of spikingneurons with stationary states, globally locked oscillations, and weakly locked oscillatory states is presented and analyzed. The network is close to biology in the following sense. First, the neurons spike and our model includes an absolute refractory period after each spike. Second, we consider a distribution of axonal delay times. Finally, we describe synaptic signal transmission by excitatory and inhibitory potentials (EPSP and IPSP) with a realistic shape, that is, through a response kernel. During retrieval of a pattern, all active neurons exhibit periodic spike bursts which may or may not be synchronized ('locked') into a coherent oscillation. We derive an analytical condition of locking and calculate the period of collective activity during oscillatory retrieval. In a stationary retrieval state, the overlap assumes a constant value proportional to the mean firing rate of the neurons. It is argued that in a biological network an intermediate scenario of "weak locking' is most likely.

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Section: Discussionmentioning

confidence: 99%

A biologically motivated and analytically soluble model of collective oscillations in the cortex

1993

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“…[5] If g(ω) has compact support, then for sufficiently large K there can be a transition to a completely synchronized state with R ≈ 1. [12,13] Since the incoherent state and the partially synchronized state emerge when the initial value problem is solved numerically, each must be stable in the appropriate range of K at least in an operational sense.…”

Section: Introductionmentioning

confidence: 99%

Amplitude expansions for instabilities in populations of globally-coupled oscillators

1994

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We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable travelling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and Okuda and Kuramoto predicted stable travelling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable travelling waves results from a failure to include all unstable modes.

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“…the motions on the frequencies where the phases are locked. The proposed measure is similar to the "order parameter" which was used for the description of measure of collective synchronizations in ensembles of phase oscillators [Kuramoto & Nishikava, 1987;], but it is applicable for more complex regular and chaotic oscillators. Phase coherence was used as a chaotic synchronization measure in the work [Mormann et al, 2000] where authors used instantaneous phase approach.…”

Section: Synchronization Of Chaos In the Chain Of Oscillatorsmentioning

confidence: 99%

Developing Chaos on Base of Traveling Waves in a Chain of Coupled Oscillators With Period-Doubling: Synchronization and Hierarchy of Multistability Formation

Shabunin

Astakhov

Анищенко

2002

Int. J. Bifurcation Chaos

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The work is devoted to the analysis of dynamics of traveling waves in a chain of self-oscillators with period-doubling route to chaos. As a model we use a ring of Chua's circuits symmetrically coupled via a resistor. We consider how complicated are temporal regimes with parameters changing influences on spatial structures in the chain. We demonstrate that spatial periodicity exists until transition to chaos through period-doubling and tori birth bifurcations of regular regimes. Temporal quasi-periodicity does not induce spatial quasi-periodicity in the ring. After transition to chaos exact spatial periodicity is changed by the spatial periodicity in the average. The periodic spatial structures in the chain are connected with synchronization of oscillations. For quantity researching of the synchronization we propose a measure of chaotic synchronization based on the coherence function and investigate the dependence of the level of synchronization on the strength of coupling and on the chaos developing in the system. We demonstrate that the spatial periodic structure is completely destroyed as a consequence of loss of coherence of oscillations on base frequencies.

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Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

Cited by 371 publications

References 16 publications

A biologically motivated and analytically soluble model of collective oscillations in the cortex

A biologically motivated and analytically soluble model of collective oscillations in the cortex

Amplitude expansions for instabilities in populations of globally-coupled oscillators

Developing Chaos on Base of Traveling Waves in a Chain of Coupled Oscillators With Period-Doubling: Synchronization and Hierarchy of Multistability Formation

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