Criterion for the emergence of explosive synchronization transitions in networks of phase oscillators

Brain network hypersensitivity underlies pain crises in sickle cell disease

Sickle cell disease (SCD) is a genetic disorder causing painful and unpredictable Vaso-occlusive crises (VOCs) through blood vessel blockages. In this study, we propose explosive synchronization (ES) as a novel approach to comprehend the hypersensitivity and occurrence of VOCs in the SCD brain network. We hypothesized that the accumulated disruptions in the brain network induced by SCD might lead to strengthened ES and hypersensitivity. We explored ES's relationship with patient reported outcome measures (PROMs) as well as VOCs by analyzing EEG data from 25 SCD patients and 18 matched controls. SCD patients exhibited lower alpha frequency than controls. SCD patients showed correlation between frequency disassortativity (FDA), an ES condition, and three important PROMs. Furthermore, stronger FDA was observed in SCD patients with a higher frequency of VOCs and EEG recording near VOC. We also conducted computational modeling on SCD brain network to study FDA's role in network sensitivity. Our model demonstrated that a stronger FDA could be linked to increased sensitivity and frequency of VOCs. This study establishes connections between SCD pain and the universal network mechanism, ES, offering a strong theoretical foundation. This understanding will aid predicting VOCs and refining pain management for SCD patients.

Brain network hypersensitivity underlies pain crises in sickle cell disease

Sickle cell disease (SCD) is a genetic disorder causing blood vessel blockages and painful Vaso-occlusive crises (VOCs). VOCs, characterized by severe pain due to blocked blood flow, are recurrent and unpredictable, posing challenges for preventive strategies. In this study we propose explosive synchronization (ES), a phenomenon characterized by abrupt brain network phase transitions, as a novel approach to address this challenge. We hypothesized that the accumulated disruptions in the brain network induced by SCD might lead to strengthened ES and hypersensitivity. We explored ES's relationship with patient reported outcome measures (PROMs) and VOCs by analyzing EEG data from 25 SCD patients and 18 matched controls. SCD patients exhibited significantly lower alpha wave frequency than controls. SCD patients under painful pressure stimulation showed correlation between frequency disassortativity (FDA), an ES condition, and three important PROMs. Furthermore, patients who had a higher frequency of VOCs in the preceding 12 months presented with stronger FDA. The timing of VOC occurrence relative to EEG recordings was significantly associated to FDA. We also conducted computational modeling on SCD brain network to study FDA's role in network sensitivity. Stronger FDA correlated with higher responsivity and complexity in our model. Simulation under noisy environment showed that higher FDA could be linked to increased occurrence frequency of crisis. This study establishes connections between SCD pain and the universal network mechanism, ES, offering a strong theoretical foundation. This understanding will aid predicting VOCs and refining pain management for SCD patients.

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.

Effects of network robustness on explosive synchronization

Current studies have shown that there is a positive correlation between the network assortativity and robustness and that the assortativity also plays an important role in explosive synchronization. In this paper, taking the network robustness as a global property, we investigate its significance as well as the influence of its interaction with the assortativity on explosive synchronization. Our numerical results demonstrate that explosive synchronization is suppressed in extreme situations of both the robustness and assortativity. In addition, through appropriate adjustments of them, a maximum hysteresis area between the forward and backward transitions can be reached. Furthermore, our results might also provide reference for those who are interested in effects of network structure on synchronization, though this problem is still challenging as we show in the discussion.

Explosive death of conjugate coupled Van der Pol oscillators on networks

Explosive death phenomenon has been gradually gaining attention of researchers due to the research boom of explosive synchronization, and it has been observed recently for the identical or nonidentical coupled systems in all-to-all network. In this work, we investigate the emergence of explosive death in networked Van der Pol (VdP) oscillators with conjugate variables coupling. It is demonstrated that the network structures play a crucial role in identifying the types of explosive death behaviors. We also observe that the damping coefficient of the VdP system not only can determine whether the explosive death state is generated but also can adjust the forward transition point. We further show that the backward transition point is independent of the network topologies and the damping coefficient, which is well confirmed by theoretical analysis. Our results reveal the generality of explosive death phenomenon in different network topologies and are propitious to promote a better comprehension for the oscillation quenching behaviors.

Influence of stochastic perturbations on the cluster explosive synchronization of second-order Kuramoto oscillators on networks

Explosive synchronization in networked second-order Kuramoto oscillators has been well studied recently and it is revealed that the synchronization process is featured by cluster explosive synchronization. However, little attention has been paid to the influence of noise or perturbation. We here study this problem and discuss the influences of noise and perturbation. For the former, we interestingly find that noise has significant influence on the cluster explosive synchronization of those nodes with smaller degrees, i.e., their synchronization will change from the first-order to second-order transition and the critical points for both the forward and backward synchronization depend on the strength of noise. Especially, when the strength of noise is in an optimal range, a synchronization of the nodes with smaller degrees will be induced in the region of coupling strength where they do not display synchronization in the absence of noise. For the latter, we find that the effect of perturbation is similar to that of noise when its duration W is small. However, the perturbation will induce a change from cluster explosive synchronization to explosive synchronization when W is large. Furthermore, a brief theory is provided to explain the influence of perturbations on the critical points.

Functional Brain Network Mechanism of Hypersensitivity in Chronic Pain

Fibromyalgia (FM) is a chronic widespread pain condition characterized by augmented multi-modal sensory sensitivity. Although the mechanisms underlying this sensitivity are thought to involve an imbalance in excitatory and inhibitory activity throughout the brain, the underlying neural network properties associated with hypersensitivity to pain stimuli are largely unknown. In network science, explosive synchronization (ES) was introduced as a mechanism of hypersensitivity in diverse biological and physical systems that display explosive and global propagations with small perturbations. We hypothesized that ES may also be a mechanism of the hypersensitivity in FM brains. To test this hypothesis, we analyzed resting state electroencephalogram (EEG) of 10 FM patients. First, we examined theoretically well-known ES conditions within functional brain networks reconstructed from EEG, then tested whether a brain network model with ES conditions identified in the EEG data is sensitive to an external perturbation. We demonstrate for the first time that the FM brain displays characteristics of ES conditions, and that these factors significantly correlate with chronic pain intensity. The simulation data support the conclusion that networks with ES conditions are more sensitive to perturbation compared to non-ES network. The model and empirical data analysis provide convergent evidence that ES may be a network mechanism of FM hypersensitivity.

Self-similarity in explosive synchronization of complex networks

We report that explosive synchronization of networked oscillators (a process through which the transition to coherence occurs without intermediate stages but is rather characterized by a sudden and abrupt jump from the network's asynchronous to synchronous motion) is related to self-similarity of synchronous clusters of different size. Self-similarity is revealed by destructing the network synchronous state during the backward transition and observed with the decrease of the coupling strength between the nodes of the network. As illustrative examples, networks of Kuramoto oscillators with different topologies of links have been considered. For each one of such topologies, the destruction of the synchronous state goes step by step with self-similar configurations of interacting oscillators. At the critical point, the invariance of the phase distribution in the synchronized cluster with respect to the cluster size is reported.

Relationship of Topology, Multiscale Phase Synchronization, and State Transitions in Human Brain Networks

How the brain reconstitutes consciousness and cognition after a major perturbation like general anesthesia is an important question with significant neuroscientific and clinical implications. Recent empirical studies in animals and humans suggest that the recovery of consciousness after anesthesia is not random but ordered. Emergence patterns have been classified as progressive and abrupt transitions from anesthesia to consciousness, with associated differences in duration and electroencephalogram (EEG) properties. We hypothesized that the progressive and abrupt emergence patterns from the unconscious state are associated with, respectively, continuous and discontinuous synchronization transitions in functional brain networks. The discontinuous transition is explainable with the concept of explosive synchronization, which has been studied almost exclusively in network science. We used the Kuramato model, a simple oscillatory network model, to simulate progressive and abrupt transitions in anatomical human brain networks acquired from diffusion tensor imaging (DTI) of 82 brain regions. To facilitate explosive synchronization, distinct frequencies for hub nodes with a large frequency disassortativity (i.e., higher frequency nodes linking with lower frequency nodes, or vice versa) were applied to the brain network. In this simulation study, we demonstrated that both progressive and abrupt transitions follow distinct synchronization processes at the individual node, cluster, and global network levels. The characteristic synchronization patterns of brain regions that are “progressive and earlier” or “abrupt but delayed” account for previously reported behavioral responses of gradual and abrupt emergence from the unconscious state. The characteristic network synchronization processes observed at different scales provide new insights into how regional brain functions are reconstituted during progressive and abrupt emergence from the unconscious state. This theoretical approach also offers a principled explanation of how the brain reconstitutes consciousness and cognitive functions after physiologic (sleep), pharmacologic (anesthesia), and pathologic (coma) perturbations.

Explosive synchronization coexists with classical synchronization in the Kuramoto model

Explosive synchronization has recently been reported in a system of adaptively coupled Kuramoto oscillators, without any conditions on the frequency or degree of the nodes. Here, we find that, in fact, the explosive phase coexists with the standard phase of the Kuramoto oscillators. We determine this by extending the mean-field theory of adaptively coupled oscillators with full coupling to the case with partial coupling of a fraction f. This analysis shows that a metastable region exists for all finite values of f > 0, and therefore explosive synchronization is expected for any perturbation of adaptively coupling added to the standard Kuramoto model. We verify this theory with GPU-accelerated simulations on very large networks (N ∼ 10(6)) and find that, in fact, an explosive transition with hysteresis is observed for all finite couplings. By demonstrating that explosive transitions coexist with standard transitions in the limit of f → 0, we show that this behavior is far more likely to occur naturally than was previously believed.

Functional and Topological Conditions for Explosive Synchronization Develop in Human Brain Networks with the Onset of Anesthetic-Induced Unconsciousness

Sleep, anesthesia, and coma share a number of neural features but the recovery profiles are radically different. To understand the mechanisms of reversibility of unconsciousness at the network level, we studied the conditions for gradual and abrupt transitions in conscious and anesthetized states. We hypothesized that the conditions for explosive synchronization (ES) in human brain networks would be present in the anesthetized brain just over the threshold of unconsciousness. To test this hypothesis, functional brain networks were constructed from multi-channel electroencephalogram (EEG) recordings in seven healthy subjects across conscious, unconscious, and recovery states. We analyzed four variables that are involved in facilitating ES in generic, non-biological networks: (1) correlation between node degree and frequency, (2) disassortativity (i.e., the tendency of highly-connected nodes to link with less-connected nodes, or vice versa), (3) frequency difference of coupled nodes, and (4) an inequality relationship between local and global network properties, which is referred to as the suppressive rule. We observed that the four network conditions for ES were satisfied in the unconscious state. Conditions for ES in the human brain suggest a potential mechanism for rapid recovery from the lightly-anesthetized state. This study demonstrates for the first time that the network conditions for ES, formerly shown in generic networks only, are present in empirically-derived functional brain networks. Further investigations with deep anesthesia, sleep, and coma could provide insight into the underlying causes of variability in recovery profiles of these unconscious states.

Explosive synchronization with asymmetric frequency distribution

In this work, we study the synchronization in a generalized Kuramoto model with frequency-weighted coupling. In particular, we focus on the situations in which the frequency distributions of oscillators are asymmetric. For typical unimodal frequency distributions, such as Lorentzian, Gaussian, triangle, and even special Rayleigh, we find that the synchronization transition in the model generally converts from the first order to the second order as the central frequency shifts toward positive direction. We characterize two interesting coherent states in the system: In the former, two phase-locking clusters are formed, rotating with the same frequency. They correspond to those oscillators with relatively high frequencies while the oscillators with relatively small frequencies are not entrained. In the latter, two phase-locking clusters rotate with different frequencies, leading to the oscillation of the order parameter. We further conduct theoretical analysis to reveal the relation between the asymmetric frequency distribution and the conversion of synchronization type, and justify the coherent states observed in the system.

Effects of assortative mixing in the second-order Kuramoto model

In this paper we analyze the second-order Kuramoto model in the presence of a positive correlation between the heterogeneity of the connections and the natural frequencies in scale-free networks. We numerically show that discontinuous transitions emerge not just in disassortative but also in strongly assortative networks, in contrast with the first-order model. We also find that the effect of assortativity on network synchronization can be compensated by adjusting the phase damping. Our results show that it is possible to control collective behavior of damped Kuramoto oscillators by tuning the network structure or by adjusting the dissipation related to the phases' movement.

Effects of degree correlations on the explosive synchronization of scale-free networks

We study the organization of finite-size, large ensembles of phase oscillators networking via scale-free topologies in the presence of a positive correlation between the oscillators' natural frequencies and the network's degrees. Under those circumstances, abrupt transitions to synchronization are known to occur in growing scale-free networks, while the transition has a completely different nature for static random configurations preserving the same structure-dynamics correlation. We show that the further presence of degree-degree correlations in the network structure has important consequences on the nature of the phase transition characterizing the passage from the phase-incoherent to the phase-coherent network state. While high levels of positive and negative mixing consistently induce a second-order phase transition, moderate values of assortative mixing, such as those ubiquitously characterizing social networks in the real world, greatly enhance the irreversible nature of explosive synchronization in scale-free networks. The latter effect corresponds to a maximization of the area and of the width of the hysteretic loop that differentiates the forward and backward transitions to synchronization.

Self-organized correlations lead to explosive synchronization

Very recently, a first-order phase transition, named explosive synchronization (ES), has attracted great attention due to its remarkable novelty in theory and significant impact in applications. However, so far, all observations of ES have been associated with various correlation constraints on system parameters, which restrict its generality and applications. Here we consider heterogeneous networks around Hopf bifurcation point described by chemical reaction-diffusion systems and also by their reduced order parameter versions, the complex Ginzburg-Landau equations, and demonstrate that explosive synchronization can appear as an emergent feature of oscillatory networks, and the restrictions on specific parameter correlations used so far for ES can be lifted entirely. Theoretical analyses and numerical simulations show with a perfect agreement that explosive synchronization can appear in networks with nodes having identical natural frequencies, and necessary correlation conditions for ES can be realized in a self-organized manner by network evolution.

Explosive synchronization in adaptive and multilayer networks

At this time, explosive synchronization (ES) of networked oscillators is thought of as being rooted in the setting of specific microscopic correlation features between the natural frequencies of the oscillators and their effective coupling strengths. We show that ES is, in fact, far more general and can occur in adaptive and multilayer networks in the absence of such correlation properties. We first report evidence of ES for single-layer networks where a fraction f of the nodes have links adaptively controlled by a local order parameter, and we then extend the study to a variety of two-layer networks with a fraction f of their nodes coupled with each other by means of dependency links. In the latter case, we give evidence of ES regardless of the differences in the frequency distribution, in the topology of connections between the layers, or both. Finally, we provide a rigorous, analytical treatment to properly ground all of the observed scenarios and to advance the understanding of the actual mechanisms at the basis of ES in real-world systems.

Exact solution for first-order synchronization transition in a generalized Kuramoto model

First-order, or discontinuous, synchronization transition, i.e. an abrupt and irreversible phase transition with hysteresis to the synchronized state of coupled oscillators, has attracted much attention along the past years. We here report the analytical solution of a generalized Kuramoto model, and derive a series of exact results for the first-order synchronization transition, including i) the exact, generic, solutions for the critical coupling strengths for both the forward and backward transitions, ii) the closed form of the forward transition point and the linear stability analysis for the incoherent state (for a Lorentzian frequency distribution), and iii) the closed forms for both the stable and unstable coherent states (and their stabilities) for the backward transition. Our results, together with elucidating the first-order nature of the transition, provide insights on the mechanisms at the basis of such a synchronization phenomenon.

Explosive synchronization as a process of explosive percolation in dynamical phase space

Explosive synchronization and explosive percolation are currently two independent phenomena occurring in complex networks, where the former takes place in dynamical phase space while the latter in configuration space. It has been revealed that the mechanism of EP can be explained by the Achlioptas process, where the formation of a giant component is controlled by a suppressive rule. We here introduce an equivalent suppressive rule for ES. Before the critical point of ES, the suppressive rule induces the presence of multiple, small sized, synchronized clusters, while inducing the abrupt formation of a giant cluster of synchronized oscillators at the critical coupling strength. We also show how the explosive character of ES degrades into a second-order phase transition when the suppressive rule is broken. These results suggest that our suppressive rule can be considered as a dynamical counterpart of the Achlioptas process, indicating that ES and EP can be unified into a same framework.

Disorder induces explosive synchronization

We study explosive synchronization, a phenomenon characterized by first-order phase transitions between incoherent and synchronized states in networks of coupled oscillators. While explosive synchronization has been the subject of many recent studies, in each case strong conditions on the heterogeneity of the network, its link weights, or its initial construction are imposed to engineer a first-order phase transition. This raises the question of how robust explosive synchronization is in view of more realistic structural and dynamical properties. Here we show that explosive synchronization can be induced in mildly heterogeneous networks by the addition of quenched disorder to the oscillators' frequencies, demonstrating that it is not only robust to, but moreover promoted by, this natural mechanism. We support these findings with numerical and analytical results, presenting simulations of a real neural network as well as a self-consistency theory used to study synthetic networks.