Spase Petkoski

Stationary and traveling wave states of the Kuramoto model with an arbitrary distribution of frequencies and coupling strengths.

We consider the Kuramoto model of an ensemble of interacting oscillators allowing for an arbitrary distribution of frequencies and coupling strengths. We define a family of traveling wave states as stationary in a rotating frame, and derive general equations for their parameters. We suggest empirical stability conditions which, for the case of incoherence, become exact. In addition to making new theoretical predictions, we show that many earlier results follow naturally from our general framework. The results are applicable in scientific contexts ranging from physics to biology.

Alterations in the coupling functions between cortical and cardio-respiratory oscillations due to anaesthesia with propofol and sevoflurane.

The precise mechanisms underlying general anaesthesia pose important and still open questions. To address them, we have studied anaesthesia induced by the widely used (intravenous) propofol and (inhalational) sevoflurane anaesthetics, computing cross-frequency coupling functions between neuronal, cardiac and respiratory oscillations in order to determine their mutual interactions. The phase domain coupling function reveals the form of the function defining the mechanism of an interaction, as well as its coupling strength. Using a method based on dynamical Bayesian inference, we have thus identified and analysed the coupling functions for six relationships. By quantitative assessment of the forms and strengths of the couplings, we have revealed how these relationships are altered by anaesthesia, also showing that some of them are differently affected by propofol and sevoflurane. These findings, together with the novel coupling function analysis, offer a new direction in the assessment of general anaesthesia and neurophysiological interactions, in general.

Mean-field and mean-ensemble frequencies of a system of coupled oscillators

We investigate interacting phase oscillators whose mean field is at a different frequency from the mean or mode of their natural frequencies. The associated asymmetries lead to a macroscopic traveling wave. We show that the mean-ensemble frequency of such systems differs from their entrainment frequency. In some scenarios these frequencies take values that, counterintuitively, lie beyond the limits of the natural frequencies. The results indicate that a clear distinction should be drawn between the two variables describing the macroscopic dynamics of cooperative systems. This has important implications for real systems where a nontrivial distribution of parameters is common.

Heterogeneity of time delays determines synchronization of coupled oscillators.

Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.

Coupled Nonautonomous Oscillators

First, we introduce nonautonomous oscillator—a self-sustained oscillator subject to external perturbation and then expand our formalism to two and many coupled oscillators . Then, we elaborate the Kuramoto model of ensembles of coupled oscillators and generalise it for time-varying couplings. Using the recently introduced Ott-Antonsen ansatz we show that such ensembles of oscillators can be solved analytically. This opens up a whole new area where one can model a virtual physiological human by networks of networks of nonautonomous oscillators. We then briefly discuss current methods to treat the coupled nonautonomous oscillators in an inverse problem and argue that they are usually considered as stochastic processes rather than deterministic. We now point to novel methods suitable for reconstructing nonautonomous dynamics and the recently expanded Bayesian method in particular. We illustrate our new results by presenting data from a real living system by studying time-dependent coupling functions between the cardiac and respiratory rhythms and their change with age. We show that the well known reduction of the variability of cardiac instantaneous frequency is mainly on account of reduced influence of the respiration to the heart and moreover the reduced variability of this influence. In other words, we have shown that the cardiac function becomes more autonomous with age, pointing out that nonautonomicity and the ability to maintain stability far from thermodynamic equilibrium are essential for life.

Phase-lags in large scale brain synchronization: Methodological considerations and in-silico analysis

Architecture of phase relationships among neural oscillations is central for their functional significance but has remained theoretically poorly understood. We use phenomenological model of delay-coupled oscillators with increasing degree of topological complexity to identify underlying principles by which the spatio-temporal structure of the brain governs the phase lags between oscillatory activity at distant regions. Phase relations and their regions of stability are derived and numerically confirmed for two oscillators and for networks with randomly distributed or clustered bimodal delays, as a first approximation for the brain structural connectivity. Besides in-phase, clustered delays can induce anti-phase synchronization for certain frequencies, while the sign of the lags is determined by the natural frequencies and by the inhomogeneous network interactions. For in-phase synchronization faster oscillators always phase lead, while stronger connected nodes lag behind the weaker during frequency depression, which consistently arises for in-silico results. If nodes are in anti-phase regime, then a distance π is added to the in-phase trends. The statistics of the phases is calculated from the phase locking values (PLV), as in many empirical studies, and we scrutinize the method’s impact. The choice of surrogates do not affects the mean of the observed phase lags, but higher significance levels that are generated by some surrogates, cause decreased variance and might fail to detect the generally weaker coherence of the interhemispheric links. These links are also affected by the non-stationary and intermittent synchronization, which causes multimodal phase lags that can be misleading if averaged. Taken together, the results describe quantitatively the impact of the spatio-temporal connectivity of the brain to the synchronization patterns between brain regions, and to uncover mechanisms through which the spatio-temporal structure of the brain renders phases to be distributed around 0 and π. Trial registration: South African Clinical Trials Register: http://www.sanctr.gov.za/SAClinicalbrnbspTrials/tabid/169/Default.aspx, then link to respiratory tract then link to tuberculosis, pulmonary; and TASK Applied Sciences Clinical Trials, AP-TB-201-16 (ALOPEXX): https://task.org.za/clinical-trials/.

Effects of multimodal distribution of delays in brain network dynamics

Large-scale modeling of the brain is defined by the local oscillatory dynamics that are superimposed on an architecture based on a comprehensive map of neural connections in the brain - connectome [1]. Besides coupling strengths, time-delays due to transmissions via tracts are crucial features of a connectome. They represent a proxy of the spatial structure (the tract lengths) to the temporal dynamics. Thus, the most straightforward approach to model brain dynamics in space and time is to concatenate oscillatory nodes to a connectome-based network. The analysis that we performed on the experimentally derived connectome suggests that the tract lengths - distances between different brain nodes, thus the time delays, follow a multimodal distribution. Here, we investigated the conceptual implementation of multimodal distributions of discrete time delays of network links, and its effects on the mean-field dynamics. Because of the analytical tractability, the Kuramoto oscillator describes the temporal dynamics of each node, and the links between the nodes are symmetric but heterogeneous. Hence, we analyze synchronization in populations of phase oscillators [2], which have the same distribution of natural frequencies and coupling strengths, but their structure is defined solely by their different intra- and inter-population delays. Assuming a same overall distribution of time delays, several cases are investigated: from fully random distribution, to two delays-imposed structures of subpopulations, Figure ​Figure11. Figure 1 Schematic representation of the delay-imposed structure of population of oscillators: with different inter and same intra delays in A; same inter and intra delays in B; and random distribution of the delays, in C. For all scenarios, mean-field dynamics are analytically obtained [3] and numerically confirmed. Moreover, boundaries and stabilities of different low-dimensional solutions are also investigated. These reveal a split of phase dynamics in different clusters, which can be phase shifted, or even non-stationary with different time-varying frequencies of synchronization and order parameters for the clusters. In summary, the large-scale spatial organization of the brain is integrated in a network model. Using this model, we present the effects of the multimodal distribution of time delays and the structure that they impose on the network dynamics such as synchronization. Hence, we stress the role of the spatial organization of the brain that is reflected through the different time-delays between different parts of the brain in the formation of spatiotemporal dynamics.

Introducing time-varying parameters in the Kuramoto model for brain dynamics

The mean field dynamics of the Kuramoto model (KM) with time-dependent (TD) parameters is described, and the response in the adiabatic and non-adiabatic limits is explained. The observed low-frequency, homogeneity-dependent filtering is discussed, together with the possible implications to the modelling of the brain dynamics.

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 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 anaesthesia [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 anaesthesia [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.