Extension of the Kuramoto model to encompass time variability in neuronal synchronization and brain dynamics
The Kuramoto model (KM) is extended to incorporate at a basic level one of the most fundamental properties of living systemstheir 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 anaesthesia  , as one of the many experimentally confirmed phenomena [4, 5] which this model should address. We
... hus 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 anaesthesia  . Non-autonomous natural frequencies in an ensemble of oscillators, on the other hand, have already been investigated and interpreted as attributable to external forcing  . 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  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  and it extends and complements a previous attempt  at explanation. References 1. Rougemont J, Felix N: Collective synchronization in populations of globally coupled phase oscillators with drifting frequencies. Phys. Rev. E 2006, 73:011104. 2. Taylor D, Ott E, Restrepo JG: Spontaneous synchronization of coupled oscillator systems with frequency adaptation. Phys. Rev. E 2010, 81:046214. 3. Sheeba JH, Stefanovska A, McClintock PVE: Neuronal synchrony during anesthesia: A thalamocortical model. Biophys. J 95:2722-2727. 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 anaesthesia. J. Physiol 2007, 580:315326, 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 anaesthesia, New Journal of Physics 2009, 11: 103051. 5. Rudrauf D, et al: Frequency flows and the time-frequency dynamics of multivariate phase synchronization in brain signals,.