Volume 12 Supplement 1

Twentieth Annual Computational Neuroscience Meeting: CNS*2011

Open Access

Extension of the Kuramoto model to encompass time variability in neuronal synchronization and brain dynamics

BMC Neuroscience201112(Suppl 1):P313

DOI: 10.1186/1471-2202-12-S1-P313

Published: 18 July 2011

The Kuramoto model (KM) is extended to incorporate at a basic level one of the most fundamental properties of living systems – their inherent time-variability. In building the model, we encompass earlier generalizations of the KM that included time-varying parameters in a purely physical way [1, 2] together with a model introduced to describe changes in neuronal synchronization during anæsthesia [3], as one of the many experimentally confirmed phenomena [4, 5] which this model should address. We thus allow for the time-variabilities of both the oscillator natural frequencies and of the inter-oscillator couplings. The latter can be considered as describing in an intuitive way the non-autonomous character of the individual oscillators, each of which is subject to the influence of its neighbors. The couplings have been found to provide a convenient basis for modeling the depth of anæsthesia [3].

Non-autonomous natural frequencies in an ensemble of oscillators, on the other hand, have already been investigated and interpreted as attributable to external forcing [6]. Our numerical simulations have confirmed some interesting, and, at first sight counter-intuitive, dynamics of the model for this case, and have also revealed certain limitations of this approach. Hence, we further examine the other aspects of the frequencies’ time-variability. In addition, we apply the Sakaguchi extension (see [3] and the references therein) of the original KM and investigate its influence on the system’s synchronization. Furthermore, we propose the use of a bounded distribution for the natural frequencies of the oscillators. A truncated Lorentzian distribution appears to be a good choice in that it allows the Kuramoto transition to be solved analytically: the resultant expression for the mean field amplitude matches perfectly the results obtained numerically.

The work to be presented helps to describe time-varying neural synchronization as an inherent phenomenon of brain dynamics. It accounts for the experimental results reported earlier [4] and it extends and complements a previous attempt [3] at explanation.

Authors’ Affiliations

Department of Physics, Lancaster University


  1. Rougemont J, Felix N: Collective synchronization in populations of globally coupled phase oscillators with drifting frequencies. Phys. Rev. E. 2006, 73: 011104-10.1103/PhysRevE.73.011104.View ArticleGoogle Scholar
  2. Taylor D, Ott E, Restrepo JG: Spontaneous synchronization of coupled oscillator systems with frequency adaptation. Phys. Rev. E. 2010, 81: 046214-10.1103/PhysRevE.81.046214.View ArticleGoogle Scholar
  3. Sheeba JH, Stefanovska A, McClintock PVE: Neuronal synchrony during anesthesia: A thalamocortical model. Biophys. J. 95: 2722-2727. 10.1529/biophysj.108.134635.
  4. Musizza B, Stefanovska A, McClintock PVE, Palus M, Petrovcic J, Ribaric S, Bajrovic FF: Interactions between cardiac, respiratory and EEG-δ oscillations in rats during anæsthesia. J. Physiol. 2007, 580: 315326-10.1113/jphysiol.2006.126748. Bahraminasab A, Ghasemi F, Stefanovska A, McClintock PVE, Friedrich R: Physics of brain dynamics: Fokker–Planck analysis reveals changes in EEG δ–θ interactions in anæsthesia, New Journal of Physics 2009, 11: 103051View ArticleGoogle Scholar
  5. Rudrauf D, et al.: Frequency flows and the time-frequency dynamics of multivariate phase synchronization in brain signals,. Neuroimage. 2006, 31: 209-227. 10.1016/j.neuroimage.2005.11.021.View ArticlePubMedGoogle Scholar
  6. Choi MY, Kim YW, Hong DC: Periodic synchronization in a driven system of coupled oscillators. Phys. Rev. E. 1994, 49: 3825-3832. 10.1103/PhysRevE.49.3825.View ArticleGoogle Scholar


© Petkoski and Stefanovska; licensee BioMed Central Ltd. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.