Effects of core-periphery structure on explosive synchronization

We investigate the explosive synchronization in networks with core-periphery structure. The effects of different patterns of core-periphery networks on explosive synchronization are studied by altering network structural parameters. With the variation of two core-periphery structural parameters, the relative connection strength between core and peripheral nodes, and the relative connection strength among peripheral nodes, we find distinct roles played by structural parameters in the path toward explosive synchronization. Our results show that the order parameter of periphery is closer to that of the core in the synchronization phase with the increment of connections between core and peripheral nodes. In addition, we find that sparser the connections among peripheral nodes are, the easier the whole dynamic network is to reach explosive synchronization. We also discover that if the number of connections between core and periphery scales vary sublinearly with the network size, there exists a novel two-jump behavior of the order parameter of the whole network. Moreover, as the level of the sublinearity increases, the order parameter starts to oscillate in a decaying manner, rather than being increasing monotonically and slowly as in the case of normal explosive synchronization when the coupling strength exceeds a critical threshold. Hence in this regime, it becomes increasingly difficult for the network to maintain stable explosive synchronization even though the underlying network topology is connected.

Permutation Entropy of State Transition Networks to Detect Synchronization

The dynamic behavior of many physical, biological, and other systems, are organized according to the synchronization of chaotic oscillators. In this paper, we have proposed a new method with low sensitivity to noise for detecting synchronization by mapping time series to complex networks, called the ordinal partition network, and calculating the permutation entropy of that structure. We show that this method can detect different kinds of synchronization such as complete synchronization, phase synchronization, and generalized synchronization. In all cases, the estimated permutation entropy decreases with increased synchronization. This method is also capable of estimating the topology of the network graph from the time series, without knowledge of the dynamical equations of individual nodes. This approach has been applied for the two identical and nonidentical coupled Rössler systems, two nonidentical coupled Lorenz systems, and a ring of coupled Lorenz96 oscillators.

Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs

The hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 836733 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law–tailed synchronization durations, with τt ≃ 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τt ≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τt ≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.

An oscillator ensemble model of sequence learning

Learning and memorizing sequences of events is an important function of the human brain and the basis for forming expectations and making predictions. Learning is facilitated by repeating a sequence several times, causing rhythmic appearance of the individual sequence elements. This observation invites to consider the resulting multitude of rhythms as a spectral ‘fingerprint’ which characterizes the respective sequence. Here we explore the implications of this perspective by developing a neurobiologically plausible computational model which captures this ‘fingerprint’ by attuning an ensemble of neural oscillators. In our model, this attuning process is based on a number of oscillatory phenomena that have been observed in electrophysiological recordings of brain activity like synchronization, phase locking and reset as well as cross-frequency coupling. We compare the learning properties of the model with behavioral results from a study in human participants and observe good agreement of the errors for different levels of complexity of the sequence to be memorized. Finally, we suggest an extension of the model for processing sequences that extend over several sensory modalities.

Phase-response analysis of synchronization for periodic flows

We apply phase-reduction analysis to examine synchronization properties of periodic fluid flows. The dynamics of unsteady flows is described in terms of the phase dynamics, reducing the high-dimensional fluid flow to its single scalar phase variable. We characterize the phase response to impulse perturbations, which can in turn quantify the influence of periodic perturbations on the unsteady flow. These insights from phase-based analysis uncover the condition for synchronization. In the present work, we study as an example the influence of periodic external forcing on an unsteady cylinder wake. The condition for synchronization is identified and agrees closely with results from direct numerical simulations. Moreover, the analysis reveals the optimal forcing direction for synchronization. Phase-response analysis holds potential to uncover lock-on characteristics for a range of periodic flows.