Synchronization on small-world networks

Reconstruction of phase dynamics from macroscopic observations based on linear and nonlinear response theories

We propose a method to reconstruct the phase dynamics in rhythmical interacting systems from macroscopic responses to weak inputs by developing linear and nonlinear response theories, which predict the responses in a given system. By solving an inverse problem, the method infers an unknown system: the natural frequency distribution, the coupling function, and the time delay which is inevitable in real systems. In contrast to previous methods, our method requires neither strong invasiveness nor microscopic observations. We demonstrate that the method reconstructs two phase systems from observed responses accurately. The qualitative methodological advantages demonstrated by our quantitative numerical examinations suggest its broad applicability in various fields, including brain systems, which are often observed through macroscopic signals such as electroencephalograms and functional magnetic response imaging.

Synchronization enhancement subjected to adaptive blinking coupling

Synchronization holds a significant role, notably within chaotic systems, in various contexts where the coordinated behavior of systems plays a pivotal and indispensable role. Hence, many studies have been dedicated to investigating the underlying mechanism of synchronization of chaotic systems. Networks with time-varying coupling, particularly those with blinking coupling, have been proven essential. The reason is that such coupling schemes introduce dynamic variations that enhance adaptability and robustness, making them applicable in various real-world scenarios. This paper introduces a novel adaptive blinking coupling, wherein the coupling adapts dynamically based on the most influential variable exhibiting the most significant average disparity. To ensure an equitable selection of the most effective coupling at each time instance, the average difference of each variable is normalized to the synchronous solution's range. Due to this adaptive coupling selection, synchronization enhancement is expected to be observed. This hypothesis is assessed within networks of identical systems, encompassing Lorenz, Rössler, Chen, Hindmarsh-Rose, forced Duffing, and forced van der Pol systems. The results demonstrated a substantial improvement in synchronization when employing adaptive blinking coupling, particularly when applying the normalization process.

Topological data analysis of the synchronization of a network of Rössler chaotic electronic oscillators

Synchronization study allows a better understanding of the exchange of information among systems. In this work, we study experimental data recorded from a set of Rössler-like chaotic electronic oscillators arranged in a complex network, where the interactions between the oscillators are given in terms of a connectivity matrix, and their intensity is controlled by a global coupling parameter. We use the zero and one persistent homology groups to characterize the point clouds obtained from the signals recorded in pairs of oscillators. We show that the normalized persistent entropy (NPE) allows us to characterize the effective coupling between pairs of oscillators because it tends to increase with the coupling strength and to decrease with the distance between the oscillators. We also observed that pairs of oscillators that have similar degrees and are nearest neighbors tend to have higher NPE values than pairs with different degrees. However, large variability is found in the NPE values. Comparing the NPE behavior with that of the phase-locking value (PLV, commonly used to evaluate the synchronization of phase oscillators), we find that for large enough coupling, PLV only displays a monotonic increase, while NPE shows a richer behavior that captures variations in the behavior of the oscillators. This is due to the fact that PLV only captures coupling-induced phase changes, while NPE also captures amplitude changes. Moreover, when we consider the same network but with Kuramoto phase oscillators, we also find that NPE captures the transition to synchronization (as it increases with the coupling strength), and it also decreases with the distance between the oscillators. Therefore, we propose NPE as a data analysis technique to try to differentiate pairs of oscillators that have strong effective coupling because they are first or near neighbors, from those that have weaker coupling because they are distant neighbors.

Upregulation of a Small-World Brain Network Improves Inhibitory Control: An fNIRS Neurofeedback Training Study

The aim of this study was to investigate the inner link between the small-world brain network and inhibitory control. Functional near-infrared spectroscopy (fNIRS) was used to construct a neurofeedback (NF) training system and regulate the frontal small-world brain network. The small-world network downregulation group (DOWN, n = 17) and the small-world network upregulation group (UP, n = 17) received five days of fNIRS-NF training and performed the color-word Stroop task before and after training. The behavioral and functional brain network topology results of both groups were analyzed by a repeated-measures analysis of variance (ANOVA), which showed that the upregulation training helped to improve inhibitory control. The upregulated small-world brain network exhibits an increase in the brain network regularization, links widely dispersed brain resources, and reduces the lateralization of brain functional networks between hemispheres. This suggests an inherent correlation between small-world functional brain networks and inhibitory control; moreover, dynamic optimization under cost efficiency trade-offs provides a neural basis for inhibitory control. Inhibitory control is not a simple function of a single brain region or connectivity but rather an emergent property of a broader network.

How to grow an oscillators' network with enhanced synchronization

We study a way to set the natural frequency of a newly added oscillator in a growing network to enhance synchronization. Population growth is one of the typical features of many oscillator systems for which synchronization is required to perform their functions properly. Despite this significance, little has been known about synchronization in growing systems. We suggest effective growing schemes to enhance synchronization as the network grows under a predetermined rule. Specifically, we find that a method based on a link-wise order parameter outperforms that based on the conventional global order parameter. With simple solvable examples, we verify that the results coincide with intuitive expectations. The numerical results demonstrate that the approximate optimal values from the suggested method show a larger synchronization enhancement in comparison with other naïve strategies. The results also show that our proposed approach outperforms others over a wide range of coupling strengths.

Synchronization in the network-frustrated coupled oscillator with attractive-repulsive frequencies

We investigate the synchronization behavior of a generalized useful mode of the emergent collective behavior in sets of interacting dynamic elements. The network-frustrated Kuramoto model with interaction-repulsion frequency characteristics is presented, and its structural features are crucial to capture the correct physical behavior, such as describing steady states and phase transitions. Quantifying the effect of small-world phenomena on the global synchronization of the given network, the impact of the random phase-shift and their mutual behavior shows particular challenges. In this paper, we derive the phase-locked states and identify the significant synchronization transition points analytically with exact boundary conditions for the correlated and uncorrelated degree-frequency distributions and their full stability analysis. We find that a supercritical to subcritical bifurcation transition occurs depending on the synchronic transition points, characterized by the power scale of the network for the correlated degree frequency and the largest eigenvalue of the network in the uncorrelated case. Furthermore, our frustrated degree-frequency distribution brings us to the classical Kuramoto model with all-to-all coupling, with β=1/2 for the correlated case and λ_{N}=1 for the uncorrelated distribution. In addition, the interplay between the network topology and the frustration forms a powerful alliance, where they control the synchronization ability of the generalized model without affecting its stability.

A new model for freedom of movement using connectomic analysis

The problem of whether we can execute free acts or not is central in philosophical thought, and it has been studied by numerous scholars throughout the centuries. Recently, neurosciences have entered this topic contributing new data and insights into the neuroanatomical basis of cognitive processes. With the advent of connectomics, a more refined landscape of brain connectivity can be analysed at an unprecedented level of detail. Here, we identify the connectivity network involved in the movement process from a connectomics point of view, from its motivation through its execution until the sense of agency develops. We constructed a "volitional network" using data derived from the Brainnetome Atlas database considering areas involved in volitional processes as known in the literature. We divided this process into eight processes and used Graph Theory to measure several structural properties of the network. Our results show that the volitional network is small-world and that it contains four communities. Nodes of the right hemisphere are contained in three of these communities whereas nodes of the left hemisphere only in two. Centrality measures indicate the nucleus accumbens is one of the most connected nodes in the network. Extensive connectivity is observed in all processes except in Decision (to move) and modulation of Agency, which might correlate with a mismatch mechanism for perception of Agency.

Epistemic Communities under Active Inference

The spread of ideas is a fundamental concern of today’s news ecology. Understanding the dynamics of the spread of information and its co-option by interested parties is of critical importance. Research on this topic has shown that individuals tend to cluster in echo-chambers and are driven by confirmation bias. In this paper, we leverage the active inference framework to provide an in silico model of confirmation bias and its effect on echo-chamber formation. We build a model based on active inference, where agents tend to sample information in order to justify their own view of reality, which eventually leads to them to have a high degree of certainty about their own beliefs. We show that, once agents have reached a certain level of certainty about their beliefs, it becomes very difficult to get them to change their views. This system of self-confirming beliefs is upheld and reinforced by the evolving relationship between an agent’s beliefs and observations, which over time will continue to provide evidence for their ingrained ideas about the world. The epistemic communities that are consolidated by these shared beliefs, in turn, tend to produce perceptions of reality that reinforce those shared beliefs. We provide an active inference account of this community formation mechanism. We postulate that agents are driven by the epistemic value that they obtain from sampling or observing the behaviours of other agents. Inspired by digital social networks like Twitter, we build a generative model in which agents generate observable social claims or posts (e.g., ‘tweets’) while reading the socially observable claims of other agents that lend support to one of two mutually exclusive abstract topics. Agents can choose which other agent they pay attention to at each timestep, and crucially who they attend to and what they choose to read influences their beliefs about the world. Agents also assess their local network’s perspective, influencing which kinds of posts they expect to see other agents making. The model was built and simulated using the freely available Python package pymdp. The proposed active inference model can reproduce the formation of echo-chambers over social networks, and gives us insight into the cognitive processes that lead to this phenomenon.

Effects of topological characteristics on rhythmic states of the D-dimensional Kuramoto model in complex networks

Synchronization is a ubiquitous phenomenon in engineering and natural ecosystems. While the dynamics of synchronization modeled by the Kuramoto model are commonly studied in two dimensions and the state of dynamic units is characterized by a scalar angle variable, we studied the Kuramoto model generalized to D dimensions in the framework of a complex network and utilized the local synchronous order parameter between the agent and its neighbors as the controllable variable to adjust the coupling strength. Here, we reported that average connectivity of networks affects the time-dependent, rhythmic, cyclic state. Importantly, we found that the level of heterogeneity of networks governs the rhythmic state in the transition process. The analytical treatment for observed scenarios in a D-dimensional Kuramoto model at D=3 was provided. These results offered a platform for a better understanding of time-dependent swarming and flocking dynamics in nature.

Contrariety and inhibition enhance synchronization in a small-world network of phase oscillators

We numerically study Kuramoto model synchronization consisting of the two groups of conformist-contrarian and excitatory-inhibitory phase oscillators with equal intrinsic frequency. We consider random and small-world (SW) topologies for the connectivity network of the oscillators. In random networks, regardless of the contrarian to conformist connection strength ratio, we found a crossover from the π-state to the blurred π-state and then a continuous transition to the incoherent state by increasing the fraction of contrarians. However, for the excitatory-inhibitory model in a random network, we found that for all the values of the fraction of inhibitors, the two groups remain in phase and the transition point of fully synchronized to an incoherent state reduced by strengthening the ratio of inhibitory to excitatory links. In the SW networks we found that the order parameters for both models do not show monotonic behavior in terms of the fraction of contrarians and inhibitors. Up to the optimal fraction of contrarians and inhibitors, the synchronization rises by introducing the number of contrarians and inhibitors and then falls. We discuss that the nonmonotonic behavior in synchronization is due to the weakening of the defects already formed in the pure conformist and excitatory agent model in SW networks. We found that in SW networks, the optimal fraction of contrarians and inhibitors remain unchanged for the rewiring probabilities up to ∼0.15, above which synchronization falls monotonically, like the random network. We also showed that in the conformist-contrarian model, the optimal fraction of contrarians is independent of the strength of contrarian links. However, in the excitatory-inhibitory model, the optimal fraction of inhibitors is approximately proportional to the inverse of inhibition strength.

Synchronization in multiplex models of neuron-glial systems: Small-world topology and inhibitory coupling

In this work, we investigate the impact of mixed coupling on synchronization in a multiplex oscillatory network. The network mimics the neural-glial systems by incorporating interacting slow ("glial") and fast ("neural") oscillatory layers. Connections between the "glial" elements form a regular periodic structure, in which each element is connected to the eight other neighbor elements, whereas connections among "neural" elements are represented by Watts-Strogatz networks (from regular and small-world to random Erdös-Rényi graph) with a matching mean node degree. We find that the random rewiring toward small-world topology readily yields the dynamics close to that exhibited for a completely random graph, in particular, leading to coarse-graining of dynamics, suppressing multi-stability of synchronization regimes, and the onset of Kuramoto-type synchrony in both layers. The duration of transient dynamics in the system measured by relaxation times is minimized with the increase of random connections in the neural layer, remaining substantial only close to synchronization-desynchronization transitions. "Inhibitory" interactions in the "neural" subnetwork layer undermine synchronization; however, the strong coupling with the "glial" layer overcomes this effect.

Time-delayed Kuramoto model in the Watts-Strogatz small-world networks

We study the synchronization of small-world networks of identical coupled phase oscillators through the Kuramoto interaction and uniform time delay. For a given intrinsic frequency and coupling constant, we observe synchronization enhancement in a range of time delays and discontinuous transition from the partially synchronized state with defect patterns to a glassy phase, characterized by a distribution of randomly frozen phase-locked oscillators. By further increasing the time delay, this phase undergoes a discontinuous transition to another partially synchronized state. We found the bimodal frequency distributions and hysteresis loops as indicators of the discontinuous nature of these transitions. Moreover, we found the existence of Chimera states at the onset of transitions.

Modulation of dynamics in a pre-existing hippocampal network by neural stem cells on a microelectrode array

Objective.Neural stem cells (NSCs) are continuously produced throughout life in the hippocampus, which is a vital structure for learning and memory. NSCs in the brain incorporate into the functional hippocampal circuits and contribute to processing information. However, little is known about the mechanisms of NSCs' activity in a pre-existing neuronal network. Here, we investigate the role of NSCs in the neuronal activity of a pre-existing hippocampalin vitronetwork grown on microelectrode arrays.Approach.We assessed the change in internal dynamics of the network by additional NSCs based on spontaneous activity. We also evaluated the networks' ability to discriminate between different input patterns by measuring evoked activity in response to external inputs.Main results.Analysis of spontaneous activity revealed that additional NSCs prolonged network bursts with longer intervals, generated a lower number of initiating patterns, and decreased synchronization among neurons. Moreover, the network with NSCs showed higher synchronicity in close connections among neurons responding to external inputs and a larger difference in spike counts and cross-correlations during evoked response between two different inputs. Taken together, our results suggested that NSCs alter the internal dynamics of the pre-existing hippocampal network and produce more specific responses to external inputs, thus enhancing the ability of the network to differentiate two different inputs.Significance.We demonstrated that NSCs improve the ability to distinguish external inputs by modulating the internal dynamics of a pre-existing network in a hippocampal culture. Our results provide novel insights into the relationship between NSCs and learning and memory.

Critical exponents in coupled phase-oscillator models on small-world networks

A coupled phase-oscillator model consists of phase oscillators, each of which has the natural frequency obeying a probability distribution and couples with other oscillators through a given periodic coupling function. This type of model is widely studied since it describes the synchronization transition, which emerges between the nonsynchronized state and partially synchronized states. The synchronization transition is characterized by several critical exponents, and we focus on the critical exponent defined by coupling strength dependence of the order parameter for revealing universality classes. In a typical interaction represented by the perfect graph, an infinite number of universality classes is yielded by dependency on the natural frequency distribution and the coupling function. Since the synchronization transition is also observed in a model on a small-world network, whose number of links is proportional to the number of oscillators, a natural question is whether the infinite number of universality classes remains in small-world networks irrespective of the order of links. Our numerical results suggest that the number of universality classes is reduced to one and the critical exponent is shared in the considered models having coupling functions up to second harmonics with unimodal and symmetric natural frequency distributions.

Critical and Ictal Phases in Simulated EEG Signals on a Small-World Network

Healthy brain function is marked by neuronal network dynamics at or near the critical phase, which separates regimes of instability and stasis. A failure to remain at this critical point can lead to neurological disorders such as epilepsy, which is associated with pathological synchronization of neuronal oscillations. Using full Hodgkin-Huxley (HH) simulations on a Small-World Network, we are able to generate synthetic electroencephalogram (EEG) signals with intervals corresponding to seizure (ictal) or non-seizure (interictal) states that can occur based on the hyperexcitability of the artificial neurons and the strength and topology of the synaptic connections between them. These interictal simulations can be further classified into scale-free critical phases and disjoint subcritical exponential phases. By changing the HH parameters, we can model seizures due to a variety of causes, including traumatic brain injury (TBI), congenital channelopathies, and idiopathic etiologies, as well as the effects of anticonvulsant drugs. The results of this work may be used to help identify parameters from actual patient EEG or electrocorticographic (ECoG) data associated with ictogenesis, as well as generating simulated data for training machine-learning seizure prediction algorithms.

Synchronization within synchronization: transients and intermittency in ecological networks

Transients are fundamental to ecological systems with significant implications to management, conservation and biological control. We uncover a type of transient synchronization behavior in spatial ecological networks whose local dynamics are of the chaotic, predator–prey type. In the parameter regime where there is phase synchronization among all the patches, complete synchronization (i.e. synchronization in both phase and amplitude) can arise in certain pairs of patches as determined by the network symmetry—henceforth the phenomenon of  ‘synchronization within synchronization.’ Distinct patterns of complete synchronization coexist but, due to intrinsic instability or noise, each pattern is a transient and there is random, intermittent switching among the patterns in the course of time evolution. The probability distribution of the transient time is found to follow an algebraic scaling law with a divergent average transient lifetime. Based on symmetry considerations, we develop a stability analysis to understand these phenomena. The general principle of symmetry can also be exploited to explain previously discovered, counterintuitive synchronization behaviors in ecological networks.

Synchronous patterns and intermittency in a network induced by the rewiring of connections and coupling

The connection architecture plays an important role in the synchronization of networks, where the presence of local and nonlocal connection structures are found in many systems, such as the neural ones. Here, we consider a network composed of chaotic bursting oscillators coupled through a Watts-Strogatz-small-world topology. The influence of coupling strength and rewiring of connections is studied when the network topology is varied from regular to small-world to random. In this scenario, we show two distinct nonstationary transitions to phase synchronization: one induced by the increase in coupling strength and another resulting from the change from local connections to nonlocal ones. Besides this, there are regions in the parameter space where the network depicts a coexistence of different bursting frequencies where nonstationary zig-zag fronts are observed. Regarding the analyses, we consider two distinct methodological approaches: one based on the phase association to the bursting activity where the Kuramoto order parameter is used and another based on recurrence quantification analysis where just a time series of the network mean field is required.

Random temporal connections promote network synchronization

We report a phenomenon of collective dynamics on discrete-time complex networks: a random temporal interaction matrix even of zero or/and small average is able to significantly enhance synchronization with probability one. According to current knowledge, there is no verifiably sufficient criterion for the phenomenon. We use the standard method of synchronization analytics and the theory of stochastic processes to establish a criterion, by which we rigorously and accurately depict how synchronization occurring with probability one is affected by the statistical characteristics of the random temporal connections such as the strength and topology of the connections as well as their probability distributions. We also illustrate the enhancement phenomenon using physical and biological complex dynamical networks.

Privacy-Preserving Consensus over a Distributed Network against Eavesdropping Attacks

Motivated by the increasing risk of data leaks in distributed networks, we consider the privacy-preserving problem in a consensus network in the presence of an eavesdropper who is able to intercept the data transmitted on the network. First, we introduce a consensus protocol with privacy-preserving function, and analyze its convergence and its privacy-preserving effect. Second, we propose a criterion to measure the degree of network privacy leaks in the existence of the eavesdropper. Particularly, we consider the networks with ring topology and small-world topology, where we find a suboptimal eavesdropping strategy that maximizes the probability of privacy leaks. Finally, we verify all the derived results by numerical examples.

Abnormal synchronization of functional and structural networks in schizophrenia

Synchronization is believed to play an important role in information processing of the brain. Mounting evidence supports the hypothesis that schizophrenia is related to impaired neural synchrony. However, most previous studies characterize brain synchronization from the perspective of temporal coordination of distributed neural activity, rather than network properties. Our aim was to investigate the network synchronization alterations in schizophrenia using publically available data. Resting-state functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI) were performed in 96 schizophrenia patients and 120 healthy controls. The whole-brain functional and structural networks were constructed and analyzed using graph theoretical approaches. Inter-group differences in network synchronization were investigated. Both the binary and weighted functional networks of schizophrenia patients exhibited decreased synchronizability (increased eigenratio) than those of healthy controls. With respect to the structural binary networks, schizophrenia patients showed a trend towards excessive synchronizability (decreased eigenratio). In addition, the excessive synchronizability of the structural binary networks was associated with more severe negative symptoms in schizophrenia patients. Our findings provide novel biological evidence that schizophrenia involves a disruption of neural synchrony from the perspective of network properties.

Synchronization dependent on spatial structures of a mesoscopic whole-brain network

Complex structural connectivity of the mammalian brain is believed to underlie the versatility of neural computations. Many previous studies have investigated properties of small subsystems or coarse connectivity among large brain regions that are often binarized and lack spatial information. Yet little is known about spatial embedding of the detailed whole-brain connectivity and its functional implications. We focus on closing this gap by analyzing how spatially-constrained neural connectivity shapes synchronization of the brain dynamics based on a system of coupled phase oscillators on a mammalian whole-brain network at the mesoscopic level. This was made possible by the recent development of the Allen Mouse Brain Connectivity Atlas constructed from viral tracing experiments together with a new mapping algorithm. We investigated whether the network can be compactly represented based on the spatial dependence of the network topology. We found that the connectivity has a significant spatial dependence, with spatially close brain regions strongly connected and distal regions weakly connected, following a power law. However, there are a number of residuals above the power-law fit, indicating connections between brain regions that are stronger than predicted by the power-law relationship. By measuring the sensitivity of the network order parameter, we show how these strong connections dispersed across multiple spatial scales of the network promote rapid transitions between partial synchronization and more global synchronization as the global coupling coefficient changes. We further demonstrate the significance of the locations of the residual connections, suggesting a possible link between the network complexity and the brain’s exceptional ability to swiftly switch computational states depending on stimulus and behavioral context.

In a previous study, a data-driven large-scale model of mouse brain connectivity was constructed. This mouse brain connectivity model is estimated by a simplified model which only takes in account anatomy and distance dependence of connection strength which is best fit by a power law. The distance dependence model captures the connection strengths of the mouse whole-brain network well. But can it capture the dynamics? In this study, we show that a small number of connections which are missed by the simple spatial model lead to significant differences in dynamics. The presence of a small number of strong connections over longer distances increases sensitivity of synchronization to perturbations. Unlike the data-driven network, the network without these long-range connections, as well as the network in which these long range connections are shuffled, lose global synchronization while maintaining localized synchrony, underlining the significance of the exact topology of these distal connections in the data-driven brain network.

Controlling and enhancing synchronization through adaptive phase lags

We compare two methods for controlling synchronization in the Kuramoto model on an undirected network. The first is by driving selected oscillators at a desired frequency by coupling to an external driver, and the second is by including adaptive lags-or dynamical frustrations-within the Kuramoto interactions, with the lags evolving according to a dynamics as a function of the reference frequency with an associated time constant. Performing numerical simulations with random regular graphs, we find that above a certain connectivity driving via adaptive lags allows for stronger alignment to the external frequency at a lower value of the time constant compared to the corresponding coupling strength for the externally driven model. Numerical results are supported by equilibrium analysis based on a fixed-point ansatz for frequency synchronized clusters where we solve the spectrum of the associated Jacobian matrix. We find that at low connectivity the external driving mechanism is successful down to lower densities of controlled oscillators where the adaptive lag approach is Lyapunov unstable at all densities. As connectivity increases, however, the adaptive lag mechanism shows stability over similar ranges of density to the external driving and proves superior in terms of tighter splays of oscillators. In particular, the threshold for instability for the adaptive lag model shows robustness against variations in the associated time constant down to lower densities of controlled oscillators. A simple intuitive model emerges based on the interaction between splayed clusters close to a critical point.

Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators

Synchronization phenomena are of broad interest across disciplines and increasingly of interest in a multiplex network setting. For the multiplex network of coupled Rössler oscillators, here we show how the master stability function, a celebrated framework for analyzing synchronization on a single network, can be extended to certain classes of multiplex networks with different intralayer and interlayer coupling functions. We derive three master stability equations that determine, respectively, the necessary regions of complete synchronization, intralayer synchronization, and interlayer synchronization. We calculate these three regions explicitly for the case of a two-layer network of Rössler oscillators and show that the overlap of the regions determines the type of synchronization achieved. In particular, if the interlayer or intralayer coupling function is such that the interlayer or intralayer synchronization region is empty, complete synchronization cannot be achieved regardless of the coupling strength. Furthermore, for any network structure, the occurrence of intralayer and interlayer synchronization depends mainly on the coupling functions of nodes within a layer and across layers, respectively. Our mathematical analysis requires that the intralayer and interlayer supra-Laplacians commute. But, we show this is only a sufficient, and not necessary, condition and that the results can be applied more generally.

Occasional long distance dispersal increases spatial synchrony of population cycles

Spatially separated populations of the same species often exhibit correlated fluctuations in abundance, a phenomenon known as spatial synchrony. Dispersal can generate spatial synchrony. In nature, most individuals disperse short distances with a minority dispersing long distances. The effect of occasional long distance dispersal on synchrony is untested, and theoretical predictions are contradictory. Occasional long distance dispersal might either increase both overall synchrony and the spatial scale of synchrony, or reduce them. We conducted a protist microcosm experiment to test whether occasional long distance dispersal increases or decreases overall synchrony and the spatial scale of synchrony. We assembled replicate 15-patch ring metapopulations of the protist predator Euplotes patella and its protist prey Tetrahymena pyriformis. All metapopulations experienced the same dispersal rate, but differed in dispersal distance. Some metapopulations experienced strictly short distance (nearest neighbour) dispersal, others experienced a mixture of short- and long distance dispersal. Occasional long distance dispersal increased overall spatial synchrony and the spatial scale of synchrony for both prey and predators, though the effects were not statistically significant for predators. As predicted by theory, dispersal generated spatial synchrony by entraining the phases of the predator-prey cycles in different patches, a phenomenon known as phase locking. Our results are consistent with theoretical models predicting that occasional long distance dispersal increases spatial synchrony. However, our results also illustrate that the spatial scale of synchrony need not match the spatial scale of the processes generating synchrony. Even strictly short distance dispersal maintained high spatial synchrony for many generations at spatial scales much longer than the dispersal distance, thanks to phase locking.

Fisher information and criticality in the Kuramoto model of nonidentical oscillators

We use the Fisher information to provide a lens on the transition to synchronization of the Kuramoto model of nonidentical frequencies on a variety of undirected graphs. We numerically solve the equations of motion for a N=400 complete graph and N=1000 small-world, scale-free, uniform random, and random regular graphs. For large but finite graphs of small average diameter the Fisher information F as a function of coupling shows a peak closely coinciding with the critical point as determined by Kuramoto's order parameter or synchronization measure r. However, for graphs of larger average diameter the position of the peak in F differs from the critical point determined by estimates of r. On the one hand, this is a finite-size effect even at N=1000; however, we show across a range of topologies that the Fisher information peak points to a transition for smaller graphs that indicates structural changes in the numbers of locally phase-synchronized clusters, often directly from metastable to stable frequency synchronization. Solving explicitly for a two-cluster ansatz subject to Gaussian noise shows that the Fisher infomation peaks at such a transition. We discuss the implications for Fisher information as an indicator for edge-of-chaos phenomena in finite-coupled oscillator systems.

Phase synchronization on spatially embedded duplex networks with total cost constraint

Synchronization on multiplex networks has attracted increasing attention in the past few years. We investigate collective behaviors of Kuramoto oscillators on single layer and duplex spacial networks with total cost restriction, which was introduced by Li et al. [Phys. Rev. Lett. 104, 018701 (2010)] and termed as the Li network afterwards. We first explore how the topology of the network influences synchronizability of Kuramoto oscillators on single layer Li networks and find that the closer the Li network is to a regular lattice, the more difficult for it to evolve into synchronization. Then, we investigate synchronizability of duplex Li networks and find that the existence of inter-layer interaction can greatly enhance inter-layer and global synchronizability. When the inter-layer coupling strength is larger than a certain critical value, inter-layer synchronization will always occur. Furthermore, on single layer Li networks, nodes with larger degrees reach global synchronization more easily than those with smaller degrees, while on duplex Li networks, due to inter-layer interaction, this phenomenon becomes much less obvious. The results are important for us to gain insight into collective behaviors of many real-world complex systems which inherently possess multiplex architecture.

Scale invariance in natural and artificial collective systems: a review

Self-organized collective coordinated behaviour is an impressive phenomenon, observed in a variety of natural and artificial systems, in which coherent global structures or dynamics emerge from local interactions between individual parts. If the degree of collective integration of a system does not depend on size, its level of robustness and adaptivity is typically increased and we refer to it as scale-invariant. In this review, we first identify three main types of self-organized scale-invariant systems: scale-invariant spatial structures, scale-invariant topologies and scale-invariant dynamics. We then provide examples of scale invariance from different domains in science, describe their origins and main features and discuss potential challenges and approaches for designing and engineering artificial systems with scale-invariant properties.

Numerical optimization of coordinated reset stimulation for desynchronizing neuronal network dynamics

Excessive synchronization in neural activity is a hallmark of Parkinson's disease (PD). A promising technique for treating PD is coordinated reset (CR) neuromodulation in which a neural population is desynchronized by the delivery of spatially-distributed current stimuli using multiple electrodes. In this study, we perform numerical optimization to find the energy-optimal current waveform for desynchronizing neuronal network with CR stimulation, by proposing and applying a new optimization method based on the direct search algorithm. In the proposed optimization method, the stimulating current is described as a Fourier series, and each Fourier coefficient as well as the stimulation period are directly optimized by evaluating the order parameter, which quantifies the synchrony level, from network simulation. This direct optimization scheme has an advantage that arbitrary changes in the dynamical properties of the network can be taken into account in the search process. By harnessing this advantage, we demonstrate the significant influence of externally applied oscillatory inputs and non-random network topology on the efficacy of CR modulation. Our results suggest that the effectiveness of brain stimulation for desynchronization may depend on various factors modulating the dynamics of the target network. We also discuss the possible relevance of the results to the efficacy of the stimulation in PD treatment.

Social Synchrony on Complex Networks

Social synchrony (SS) is an emergent phenomenon in human society. People often mimic others which, over time, can result in large groups behaving similarly. Drawing from prior empirical studies of SS in online communities, here we propose a discrete network model of SS based on four attributes: 1) depth of action; 2) breadth of impact, i.e., a large number of actions are performed with a large group of people involved; 3) heterogeneity of role, i.e., people of higher degree play more important roles; and 4) lastly, emergence of phenomenon, i.e., it is far from random. We analyze our model both analytically and with simulations, and find good agreement between the two. We find this model can well explain the four characters of SS, and thus hope it can help researchers better understand human collective behavior.

Effective Subnetwork Topology for Synchronizing Interconnected Networks of Coupled Phase Oscillators

A system consisting of interconnected networks, or a network of networks (NoN), appears diversely in many real-world systems, including the brain. In this study, we consider NoNs consisting of heterogeneous phase oscillators and investigate how the topology of subnetworks affects the global synchrony of the network. The degree of synchrony and the effect of subnetwork topology are evaluated based on the Kuramoto order parameter and the minimum coupling strength necessary for the order parameter to exceed a threshold value, respectively. In contrast to an isolated network in which random connectivity is favorable for achieving synchrony, NoNs synchronize with weaker interconnections when the degree distribution of subnetworks is heterogeneous, suggesting the major role of the high-degree nodes. We also investigate a case in which subnetworks with different average natural frequencies are coupled to show that direct coupling of subnetworks with the largest variation is effective for synchronizing the whole system. In real-world NoNs like the brain, the balance of synchrony and asynchrony is critical for its function at various spatial resolutions. Our work provides novel insights into the topological basis of coordinated dynamics in such networks.

Small-world networks exhibit pronounced intermittent synchronization

We report the phenomenon of temporally intermittently synchronized and desynchronized dynamics in Watts-Strogatz networks of chaotic Rössler oscillators. We consider topologies for which the master stability function (MSF) predicts stable synchronized behaviour, as the rewiring probability (p) is tuned from 0 to 1. MSF essentially utilizes the largest non-zero Lyapunov exponent transversal to the synchronization manifold in making stability considerations, thereby ignoring the other Lyapunov exponents. However, for an N-node networked dynamical system, we observe that the difference in its Lyapunov spectra (corresponding to the N - 1 directions transversal to the synchronization manifold) is crucial and serves as an indicator of the presence of intermittently synchronized behaviour. In addition to the linear stability-based (MSF) analysis, we further provide global stability estimate in terms of the fraction of state-space volume shared by the intermittently synchronized state, as p is varied from 0 to 1. This fraction becomes appreciably large in the small-world regime, which is surprising, since this limit has been otherwise considered optimal for synchronized dynamics. Finally, we characterize the nature of the observed intermittency and its dominance in state-space as network rewiring probability (p) is varied.

Multiplexing topologies and time scales: The gains and losses of synchrony

Inspired by the recent interest in collective dynamics of biological neural networks immersed in the glial cell medium, we investigate the frequency and phase order, i.e., Kuramoto type of synchronization in a multiplex two-layer network of phase oscillators of different time scales and topologies. One of them has a long-range connectivity, exemplified by the Erdős-Rényi random network, and supports both kinds of synchrony. The other is a locally coupled two-dimensional lattice that can reach frequency synchronization but lacks phase order. Drastically different layer frequencies disentangle intra- and interlayer synchronization. We find that an indirect but sufficiently strong coupling through the regular layer can induce both phase order in the originally nonsynchronized random layer and global order, even when an isolated regular layer does not manifest it in principle. At the same time, the route to global synchronization is complex: an initial onset of (partial) synchrony in the regular layer, when its intra- and interlayer coupling is increased, provokes the loss of synchrony even in the originally synchronized random layer. Ultimately, a developed asynchronous dynamics in both layers is abruptly taken over by the global synchrony of both kinds.

Self-organisation of small-world networks by adaptive rewiring in response to graph diffusion

Complex networks emerging in natural and human-made systems tend to assume small-world structure. Is there a common mechanism underlying their self-organisation? Our computational simulations show that network diffusion (traffic flow or information transfer) steers network evolution towards emergence of complex network structures. The emergence is effectuated through adaptive rewiring: progressive adaptation of structure to use, creating short-cuts where network diffusion is intensive while annihilating underused connections. With adaptive rewiring as the engine of universal small-worldness, overall diffusion rate tunes the systems' adaptation, biasing local or global connectivity patterns. Whereas the former leads to modularity, the latter provides a preferential attachment regime. As the latter sets in, the resulting small-world structures undergo a critical shift from modular (decentralised) to centralised ones. At the transition point, network structure is hierarchical, balancing modularity and centrality - a characteristic feature found in, for instance, the human brain.

Development of structural correlations and synchronization from adaptive rewiring in networks of Kuramoto oscillators

Synchronization of non-identical oscillators coupled through complex networks is an important example of collective behavior, and it is interesting to ask how the structural organization of network interactions influences this process. Several studies have explored and uncovered optimal topologies for synchronization by making purposeful alterations to a network. On the other hand, the connectivity patterns of many natural systems are often not static, but are rather modulated over time according to their dynamics. However, this co-evolution and the extent to which the dynamics of the individual units can shape the organization of the network itself are less well understood. Here, we study initially randomly connected but locally adaptive networks of Kuramoto oscillators. In particular, the system employs a co-evolutionary rewiring strategy that depends only on the instantaneous, pairwise phase differences of neighboring oscillators, and that conserves the total number of edges, allowing the effects of local reorganization to be isolated. We find that a simple rule-which preserves connections between more out-of-phase oscillators while rewiring connections between more in-phase oscillators-can cause initially disordered networks to organize into more structured topologies that support enhanced synchronization dynamics. We examine how this process unfolds over time, finding a dependence on the intrinsic frequencies of the oscillators, the global coupling, and the network density, in terms of how the adaptive mechanism reorganizes the network and influences the dynamics. Importantly, for large enough coupling and after sufficient adaptation, the resulting networks exhibit interesting characteristics, including degree-frequency and frequency-neighbor frequency correlations. These properties have previously been associated with optimal synchronization or explosive transitions in which the networks were constructed using global information. On the contrary, by considering a time-dependent interplay between structure and dynamics, this work offers a mechanism through which emergent phenomena and organization can arise in complex systems utilizing local rules.

Optimizing interconnections to maximize the spectral radius of interdependent networks

The spectral radius (i.e., the largest eigenvalue) of the adjacency matrices of complex networks is an important quantity that governs the behavior of many dynamic processes on the networks, such as synchronization and epidemics. Studies in the literature focused on bounding this quantity. In this paper, we investigate how to maximize the spectral radius of interdependent networks by optimally linking k internetwork connections (or interconnections for short). We derive formulas for the estimation of the spectral radius of interdependent networks and employ these results to develop a suite of algorithms that are applicable to different parameter regimes. In particular, a simple algorithm is to link the k nodes with the largest k eigenvector centralities in one network to the node in the other network with a certain property related to both networks. We demonstrate the applicability of our algorithms via extensive simulations. We discuss the physical implications of the results, including how the optimal interconnections can more effectively decrease the threshold of epidemic spreading in the susceptible-infected-susceptible model and the threshold of synchronization of coupled Kuramoto oscillators.

Bifurcations and Singularities for Coupled Oscillators with Inertia and Frustration

We prove that any nonzero inertia, however small, is able to change the nature of the synchronization transition in Kuramoto-like models, either from continuous to discontinuous or from discontinuous to continuous. This result is obtained through an unstable manifold expansion in the spirit of Crawford, which features singularities in the vicinity of the bifurcation. Far from being unwanted artifacts, these singularities actually control the qualitative behavior of the system. Our numerical tests fully support this picture.

Small-World Brain Networks Revisited

It is nearly 20 years since the concept of a small-world network was first quantitatively defined, by a combination of high clustering and short path length; and about 10 years since this metric of complex network topology began to be widely applied to analysis of neuroimaging and other neuroscience data as part of the rapid growth of the new field of connectomics. Here, we review briefly the foundational concepts of graph theoretical estimation and generation of small-world networks. We take stock of some of the key developments in the field in the past decade and we consider in some detail the implications of recent studies using high-resolution tract-tracing methods to map the anatomical networks of the macaque and the mouse. In doing so, we draw attention to the important methodological distinction between topological analysis of binary or unweighted graphs, which have provided a popular but simple approach to brain network analysis in the past, and the topology of weighted graphs, which retain more biologically relevant information and are more appropriate to the increasingly sophisticated data on brain connectivity emerging from contemporary tract-tracing and other imaging studies. We conclude by highlighting some possible future trends in the further development of weighted small-worldness as part of a deeper and broader understanding of the topology and the functional value of the strong and weak links between areas of mammalian cortex.

Network-complement transitions, symmetries, and cluster synchronization

Synchronization in networks of coupled oscillators is known to be largely determined by the spectral and symmetry properties of the interaction network. Here, we leverage this relation to study a class of networks for which the threshold coupling strength for global synchronization is the lowest among all networks with the same number of nodes and links. These networks, defined as being uniform, complete, and multi-partite (UCM), appear at each of an infinite sequence of network-complement transitions in a larger class of networks characterized by having near-optimal thresholds for global synchronization. We show that the distinct symmetry structure of the UCM networks, which by design are optimized for global synchronizability, often leads to formation of clusters of synchronous oscillators, and that such states can coexist with the state of global synchronization.

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

We present an analysis of conditions under which the dynamics of a frustrated Kuramoto-or Kuramoto-Sakaguchi-model on sparse networks can be tuned to enhance synchronization. Using numerical optimization techniques, linear stability, and dimensional reduction analysis, a simple tuning scheme for setting node-specific frustration parameters as functions of native frequencies and degrees is developed. Finite-size scaling analysis reveals that even partial application of the tuning rule can significantly reduce the critical coupling for the onset of synchronization. In the second part of the paper, a codynamics is proposed, which allows a dynamic tuning of frustration parameters simultaneously with the ordinary Kuramoto dynamics. We find that such codynamics enhance synchronization when operating on slow time scales, and impede synchronization when operating on fast time scales relative to the Kuramoto dynamics.

One node driving synchronisation

Abrupt changes of behaviour in complex networks can be triggered by a single node. This work describes the dynamical fundamentals of how the behaviour of one node affects the whole network formed by coupled phase-oscillators with heterogeneous coupling strengths. The synchronisation of phase-oscillators is independent of the distribution of the natural frequencies, weakly depends on the network size, but highly depends on only one key oscillator whose ratio between its natural frequency in a rotating frame and its coupling strength is maximum. This result is based on a novel method to calculate the critical coupling strength with which the phase-oscillators emerge into frequency synchronisation. In addition, we put forward an analytical method to approximately calculate the phase-angles for the synchronous oscillators.

Approximate solution for frequency synchronization in a finite-size Kuramoto model

Scientists have been considering the Kuramoto model to understand the mechanism behind the appearance of collective behavior, such as frequency synchronization (FS) as a paradigm, in real-world networks with a finite number of oscillators. A major current challenge is to obtain an analytical solution for the phase angles. Here, we provide an approximate analytical solution for this problem by deriving a master solution for the finite-size Kuramoto model, with arbitrary finite-variance distribution of the natural frequencies of the oscillators. The master solution embodies all particular solutions of the finite-size Kuramoto model for any frequency distribution and coupling strength larger than the critical one. Furthermore, we present a criterion to determine the stability of the FS solution. This allows one to analytically infer the relationship between the physical parameters and the stable behavior of networks.

Dynamics of globally coupled oscillators: Progress and perspectives

In this paper, we discuss recent progress in research of ensembles of mean field coupled oscillators. Without an ambition to present a comprehensive review, we outline most interesting from our viewpoint results and surprises, as well as interrelations between different approaches.

Measuring predictability of autonomous network transitions into bursting dynamics

Understanding spontaneous transitions between dynamical modes in a network is of significant importance. These transitions may separate pathological and normal functions of the brain. In this paper, we develop a set of measures that, based on spatio-temporal features of network activity, predict autonomous network transitions from asynchronous to synchronous dynamics under various conditions. These metrics quantify spike-timing distributions within a narrow time window as a function of the relative location of the active neurons. We applied these metrics to investigate the properties of these transitions in excitatory-only and excitatory-and-inhibitory networks and elucidate how network topology, noise level, and cellular heterogeneity affect both the reliability and the timeliness of the predictions. The developed measures can be calculated in real time and therefore potentially applied in clinical situations.

Bifurcation behaviors of synchronized regions in logistic map networks with coupling delay

Network synchronized regions play an extremely important role in network synchronization according to the master stability function framework. This paper focuses on network synchronous state stability via studying the effects of nodal dynamics, coupling delay, and coupling way on synchronized regions in Logistic map networks. Theoretical and numerical investigations show that (1) network synchronization is closely associated with its nodal dynamics. Particularly, the synchronized region bifurcation points through which the synchronized region switches from one type to another are in good agreement with those of the uncoupled node system, and chaotic nodal dynamics can greatly impede network synchronization. (2) The coupling delay generally impairs the synchronizability of Logistic map networks, which is also dominated by the parity of delay for some nodal parameters. (3) A simple nonlinear coupling facilitates network synchronization more than the linear one does. The results found in this paper will help to intensify our understanding for the synchronous state stability in discrete-time networks with coupling delay.