Synchronization of phase oscillators with heterogeneous coupling: A solvable case

Paissan, G. H.; Zanette, Damián H.

doi:10.1016/j.physd.2007.10.016

Cited by 26 publications

(17 citation statements)

References 14 publications

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“…In our theory, the brain uses multiple phases and metastable regimes to integrate the activity of diverse and heterogenously connected parts into a functional dynamics, or in other words, to encode and communicate information. A focus of this theory is to understand how such dynamics comes about (Ernst et al, 1998; Saraga et al, 2006; Paissan and Zanette, 2008; Jirsa and Kelso, 2000; Jirsa, 2008) and what it means at various levels from the cell (Markram et al, 1997) to the whole system (Varela et al, 2001). Brain coordination dynamics thus lies at the intersection of where neuroscience meets complexity.…”

Section: Discussionmentioning

confidence: 99%

Brain coordination dynamics: True and false faces of phase synchrony and metastability

Tognoli

Kelso

2009

Progress in Neurobiology

126

124

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Understanding the coordination of multiple parts in a complex system such as the brain is a fundamental challenge. We present a theoretical model of cortical coordination dynamics that shows how brain areas may cooperate (integration) and at the same time retain their functional specificity (segregation). This model expresses a range of desirable properties that the brain is known to exhibit, including self-organization, multi-functionality, metastability and switching. Empirically, the model motivates a thorough investigation of collective phase relationships among brain oscillations in neurophysiological data. The most serious obstacle to interpreting coupled oscillations as genuine evidence of inter-areal coordination in the brain stems from volume conduction of electrical fields. Spurious coupling due to volume conduction gives rise to zero-lag (inphase) and antiphase synchronization whose magnitude and persistence obscure the subtle expression of real synchrony. Through forward modeling and the help of a novel colorimetric method, we show how true synchronization can be deciphered from continuous EEG patterns. Developing empirical efforts along the lines of continuous EEG analysis constitutes a major response to the challenge of understanding how different brain areas work together. Key predictions of cortical coordination dynamics can now be tested thereby revealing the essential modus operandi of the intact living brain.

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Section: Discussionmentioning

confidence: 99%

Brain coordination dynamics: True and false faces of phase synchrony and metastability

Tognoli

Kelso

2009

Progress in Neurobiology

126

124

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“…Models in which the summand contains the symmetric combination λ i λ j as in (20) are well-known, see for example Daido [30] (1987) where the corresponding parameters s i are regarded as random variables. The model (20) has previously been considered also in [31], but only for positive values of the coupling constants.…”

Section: Amplitude Dependencementioning

confidence: 99%

Conformist–contrarian interactions and amplitude dependence in the Kuramoto model

Lohe¹

2014

Phys. Scr.

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Abstract. We derive exact formulas for the frequency of synchronized oscillations in Kuramoto models with conformist-contrarian interactions, and determine necessary conditions for synchronization to occur. Numerical computations show that for certain parameters repulsive nodes behave as conformists, and that in other cases attractive nodes can display frustration, being neither conformist nor contrarian. The signs of repulsive couplings can be placed equivalently outside the sum, as proposed in Phys. Rev. Lett. 106 (2011) 054102, or inside the sum as in Phys. Rev. E 85 (2012) 056210, but the two models have different characteristics for small magnitudes of the coupling constants. In the latter case we show that the distributed coupling constants can be viewed as oscillator amplitudes which are constant in time, with the property that oscillators of small amplitude couple only weakly to connected nodes. Such models provide a means of investigating the effect of amplitude variations on synchronization properties.

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“…Remarkably, the only heterogeneity of the classical Kuramoto model lies in their natural frequency, while the dispersion of frequencies competes with the attractive coupling K, in a way that a phase transition to synchronization takes place when the coupling strength is strong enough, or the frequency distribution g(ω) is sufficiently narrow. A straightforward extension of the current model is to add a new component of heterogeneity into the coupling strength, which is a natural attribute in the realistic systems [9][10][11][12][13][14]. For examples, when the phase oscillators are set on complex networks, the network properties strongly impact the route to synchronization, moreover, this structure heterogeneity is equivalent to the coupling heterogeneity in a mean-field form under suitable approximation [15,16].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of phase oscillators with generalized frequency-weighted coupling

Gao

Xiang

et al. 2016

Phys. Rev. E

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We generalize the Kuramoto model for the synchronization transition of globally coupled phase oscillators to populations by incorporating an additional heterogeneity with the coupling strength, where each oscillator pair interacts with different coupling strength weighted by a genera; function of their natural frequency. The expression for the critical coupling can be straightforwardly extended to a generalized explicit formula analytically, and s self-consistency approach is developed to predict the stationary states in the thermodynamic limit. The landau damping effect is further revealed by means of the linear stability analysis and resonance poles theory above the critical threshold which turns to be far more generic. Furthermore, the dimensionality reduction technique of the Ott-Antonsen is implemented to capture the analytical description of relaxation dynamics of the steady states valid on a globally attracting manifold. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.

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Synchronization of phase oscillators with heterogeneous coupling: A solvable case

Cited by 26 publications

References 14 publications

Brain coordination dynamics: True and false faces of phase synchrony and metastability

Brain coordination dynamics: True and false faces of phase synchrony and metastability

Conformist–contrarian interactions and amplitude dependence in the Kuramoto model

Dynamics of phase oscillators with generalized frequency-weighted coupling

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