1
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Chutani M, Tadić B, Gupte N. Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with competing simplex-encoded couplings. Phys Rev E 2021; 104:034206. [PMID: 34654179 DOI: 10.1103/physreve.104.034206] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2021] [Accepted: 08/24/2021] [Indexed: 01/21/2023]
Abstract
Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviors on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here we investigate the phase synchronization (Kuramoto) dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous four-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds d_{s}=4. By numerically solving the set of coupled equations for the phase oscillators associated with the network nodes, we determine the time-averaged system's order parameter to characterize the synchronization level. Our results reveal a variety of synchronization and desynchronization scenarios, including partially synchronized states and nonsymmetrical hysteresis loops, depending on the sign and strength of the pairwise interactions and the geometric frustrations promoted by couplings on triangle faces. For substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronization and the abrupt desynchronization transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronization processes.
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Affiliation(s)
- Malayaja Chutani
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Bosiljka Tadić
- Department of Theoretical Physics, Jožef Stefan Institute, Ljubljana, Slovenia.,Complexity Science Hub Vienna, Vienna, Austria
| | - Neelima Gupte
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India.,Complex Systems and Dynamics Group, Indian Institute of Technology Madras, Chennai 600036, India
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2
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Roy M, Poria S, Hens C. Assortativity-induced explosive synchronization in a complex neuronal network. Phys Rev E 2021; 103:062307. [PMID: 34271687 DOI: 10.1103/physreve.103.062307] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2020] [Accepted: 05/25/2021] [Indexed: 11/07/2022]
Abstract
In this study, we consider a scale-free network of nonidentical Chialvo neurons, coupled through electrical synapses. For sufficiently strong coupling, the system undergoes a transition from completely out of phase synchronized to phase synchronized state. The principal focus of this study is to investigate the effect of the degree of assortativity over the synchronization transition process. It is observed that, depending on assortativity, bistability between two asymptotically stable states allows one to develop a hysteresis loop which transforms the phase transition from second order to first order. An expansion in the area of hysteresis loop is noticeable with increasing degree-degree correlation in the network. Our study also reveals that effective frequencies of nodes simultaneously go through a continuous or sudden transition to the synchronized state with the corresponding phases. Further, we examine the robustness of the results under the effect of network size and average degree, as well as diverse frequency setup. Finally, we investigate the dynamical mechanism in the process of generating explosive synchronization. We observe a significant impact of lower degree nodes behind such phenomena: in a positive assortative network the low degree nodes delay the synchronization transition.
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Affiliation(s)
- Mousumi Roy
- Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata 700009, India
| | - Swarup Poria
- Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata 700009, India
| | - Chittaranjan Hens
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
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3
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Boaretto BRR, Budzinski RC, Prado TL, Lopes SR. Mechanism for explosive synchronization of neural networks. Phys Rev E 2019; 100:052301. [PMID: 31869923 DOI: 10.1103/physreve.100.052301] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Indexed: 06/10/2023]
Abstract
Here we investigate the mechanism for explosive synchronization (ES) of a complex neural network composed of nonidentical neurons and coupled by Newman-Watts small-world matrices. We find a range of nonlocal connection probabilities for which the network displays an abrupt transition to phase synchronization, characterizing ES. The mechanism behind the ES is the following: As the coupling parameter is varied in a network of distinct neurons, ES is likely to occur due to a bistable regime, namely a chaotic nonsynchronized and a regular phase-synchronized state in the phase space. In this case, even small coupling changes make possible a transition between them. The onset of ES occurs via a saddle-node bifurcation of a periodic orbit that leads the network dynamics to display a locally stable phase-synchronized state. The presence of this regime is accompanied by a hysteresis loop on the network dynamics as the coupling parameter is adiabatically increased and decreased. The end of the hysteresis loop is marked by a frontier crisis of the chaotic attractor which also determines the end of the coupling strength interval where ES is possible.
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Affiliation(s)
- B R R Boaretto
- Departamento de Física, Universidade Federal do Paraná, 81531-980, Curitiba, Paraná, Brazil
| | - R C Budzinski
- Departamento de Física, Universidade Federal do Paraná, 81531-980, Curitiba, Paraná, Brazil
| | - T L Prado
- Departamento de Física, Universidade Federal do Paraná, 81531-980, Curitiba, Paraná, Brazil
| | - S R Lopes
- Departamento de Física, Universidade Federal do Paraná, 81531-980, Curitiba, Paraná, Brazil
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4
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Liu Y, Kurths J. Effects of network robustness on explosive synchronization. Phys Rev E 2019; 100:012312. [PMID: 31499821 DOI: 10.1103/physreve.100.012312] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2019] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- Yang Liu
- Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany.,Department of Computer Science, Technische Universität Berlin, 10587 Berlin, Germany
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany.,Department of Physics, Humboldt University Berlin, 12489 Berlin, Germany.,Saratov State University, 410012 Saratov, Russia
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5
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Cao L, Tian C, Wang Z, Zhang X, Liu Z. Influence of stochastic perturbations on the cluster explosive synchronization of second-order Kuramoto oscillators on networks. Phys Rev E 2018; 97:022220. [PMID: 29548119 DOI: 10.1103/physreve.97.022220] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2017] [Indexed: 06/08/2023]
Abstract
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.
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Affiliation(s)
- Liang Cao
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China
| | - Changhai Tian
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China
| | - Zhenhua Wang
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China
| | - Xiyun Zhang
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China
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6
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Lee U, Kim M, Lee K, Kaplan CM, Clauw DJ, Kim S, Mashour GA, Harris RE. Functional Brain Network Mechanism of Hypersensitivity in Chronic Pain. Sci Rep 2018; 8:243. [PMID: 29321621 PMCID: PMC5762762 DOI: 10.1038/s41598-017-18657-4] [Citation(s) in RCA: 32] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2017] [Accepted: 12/14/2017] [Indexed: 11/15/2022] Open
Abstract
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.
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Affiliation(s)
- UnCheol Lee
- Department of Anesthesiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA.,Center for Consciousness Science, University of Michigan Medical School, Domino's Farms, P.O. Box 385, Ann Arbor, MI, 48105, USA
| | - Minkyung Kim
- Department of Anesthesiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA.,Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, South Korea
| | - KyoungEun Lee
- Department of Anesthesiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA
| | - Chelsea M Kaplan
- Neuroscience Graduate Program, University of Michigan, Ann Arbor, MI, USA
| | - Daniel J Clauw
- Department of Anesthesiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA.,Chronic Pain and Fatigue Research Center, University of Michigan, Ann Arbor, MI, 48105, USA
| | - Seunghwan Kim
- Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, South Korea
| | - George A Mashour
- Department of Anesthesiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA. .,Center for Consciousness Science, University of Michigan Medical School, Domino's Farms, P.O. Box 385, Ann Arbor, MI, 48105, USA. .,Neuroscience Graduate Program, University of Michigan, Ann Arbor, MI, USA.
| | - Richard E Harris
- Department of Anesthesiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA. .,Neuroscience Graduate Program, University of Michigan, Ann Arbor, MI, USA. .,Chronic Pain and Fatigue Research Center, University of Michigan, Ann Arbor, MI, 48105, USA.
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7
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Kim M, Kim S, Mashour GA, Lee U. Relationship of Topology, Multiscale Phase Synchronization, and State Transitions in Human Brain Networks. Front Comput Neurosci 2017; 11:55. [PMID: 28713258 PMCID: PMC5492767 DOI: 10.3389/fncom.2017.00055] [Citation(s) in RCA: 35] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2017] [Accepted: 06/07/2017] [Indexed: 12/29/2022] Open
Abstract
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.
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Affiliation(s)
- Minkyung Kim
- Department of Physics, Pohang University of Science and TechnologyPohang, South Korea.,Center for Consciousness Science, University of Michigan Medical SchoolAnn Arbor, MI, United States
| | - Seunghwan Kim
- Department of Physics, Pohang University of Science and TechnologyPohang, South Korea
| | - George A Mashour
- Center for Consciousness Science, University of Michigan Medical SchoolAnn Arbor, MI, United States.,Department of Anesthesiology, University of Michigan Medical SchoolAnn Arbor, MI, United States
| | - UnCheol Lee
- Center for Consciousness Science, University of Michigan Medical SchoolAnn Arbor, MI, United States.,Department of Anesthesiology, University of Michigan Medical SchoolAnn Arbor, MI, United States
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8
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Bi H, Hu X, Boccaletti S, Wang X, Zou Y, Liu Z, Guan S. Coexistence of Quantized, Time Dependent, Clusters in Globally Coupled Oscillators. PHYSICAL REVIEW LETTERS 2016; 117:204101. [PMID: 27886476 DOI: 10.1103/physrevlett.117.204101] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2015] [Indexed: 06/06/2023]
Abstract
We report on a novel collective state, occurring in globally coupled nonidentical oscillators in the proximity of the point where the transition from the system's incoherent to coherent phase converts from explosive to continuous. In such a state, the oscillators form quantized clusters, where neither their phases nor their instantaneous frequencies are locked. The oscillators' instantaneous speeds are different within the clusters, but they form a characteristic cusped pattern and, more importantly, they behave periodically in time so that their average values are the same. Given its intrinsic specular nature with respect to the recently introduced Chimera states, the phase is termed the Bellerophon state. We provide an analytical and numerical description of Bellerophon states, and furnish practical hints on how to seek them in a variety of experimental and natural systems.
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Affiliation(s)
- Hongjie Bi
- Department of Physics, East China Normal University, Shanghai 200241, China
| | - Xin Hu
- Suzhou Institute of Nano-Tech and Nano-Bionics, Chinese Academy of Sciences, Suzhou 215123, China
| | - S Boccaletti
- CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy
- The Embassy of Italy in Tel Aviv, 25 Hamered Street, 68125 Tel Aviv, Israel
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xian 710062, China
| | - Yong Zou
- Department of Physics, East China Normal University, Shanghai 200241, China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai 200241, China
| | - Shuguang Guan
- Department of Physics, East China Normal University, Shanghai 200241, China
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9
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Danziger MM, Moskalenko OI, Kurkin SA, Zhang X, Havlin S, Boccaletti S. Explosive synchronization coexists with classical synchronization in the Kuramoto model. CHAOS (WOODBURY, N.Y.) 2016; 26:065307. [PMID: 27369869 DOI: 10.1063/1.4953345] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
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.
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Affiliation(s)
| | - Olga I Moskalenko
- Faculty of Nonlinear Processes, Saratov State University, Astrakhanskaya, 83, Saratov 410012, Russia
| | - Semen A Kurkin
- Faculty of Nonlinear Processes, Saratov State University, Astrakhanskaya, 83, Saratov 410012, Russia
| | - Xiyun Zhang
- Department of Physics, East China Normal University, Shanghai 200062, China
| | - Shlomo Havlin
- Department of Physics, Bar-Ilan University, Ramat Gan, Israel
| | - Stefano Boccaletti
- CNR-Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence, Italy
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10
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Synchronization of phase oscillators with frequency-weighted coupling. Sci Rep 2016; 6:21926. [PMID: 26903110 PMCID: PMC4763290 DOI: 10.1038/srep21926] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2015] [Accepted: 02/02/2016] [Indexed: 11/08/2022] Open
Abstract
Recently, the first-order synchronization transition has been studied in systems of coupled phase oscillators. In this paper, we propose a framework to investigate the synchronization in the frequency-weighted Kuramoto model with all-to-all couplings. A rigorous mean-field analysis is implemented to predict the possible steady states. Furthermore, a detailed linear stability analysis proves that the incoherent state is only neutrally stable below the synchronization threshold. Nevertheless, interestingly, the amplitude of the order parameter decays exponentially (at least for short time) in this regime, resembling the Landau damping effect in plasma physics. Moreover, the explicit expression for the critical coupling strength is determined by both the mean-field method and linear operator theory. The mechanism of bifurcation for the incoherent state near the critical point is further revealed by the amplitude expansion theory, which shows that the oscillating standing wave state could also occur in this model for certain frequency distributions. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.
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11
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Kim M, Mashour GA, Moraes SB, Vanini G, Tarnal V, Janke E, Hudetz AG, Lee U. Functional and Topological Conditions for Explosive Synchronization Develop in Human Brain Networks with the Onset of Anesthetic-Induced Unconsciousness. Front Comput Neurosci 2016; 10:1. [PMID: 26834616 PMCID: PMC4720783 DOI: 10.3389/fncom.2016.00001] [Citation(s) in RCA: 37] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2015] [Accepted: 01/04/2016] [Indexed: 11/18/2022] Open
Abstract
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.
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Affiliation(s)
- Minkyung Kim
- Department of Anesthesiology, University of Michigan Medical SchoolAnn Arbor, MI, USA; Center for Consciousness Science, University of Michigan Medical SchoolAnn Arbor, MI, USA; Department of Physics, Pohang University of Science and TechnologyPohang, South Korea
| | - George A Mashour
- Department of Anesthesiology, University of Michigan Medical SchoolAnn Arbor, MI, USA; Center for Consciousness Science, University of Michigan Medical SchoolAnn Arbor, MI, USA; Neuroscience Graduate Program, University of Michigan Medical SchoolAnn Arbor, MI, USA
| | - Stefanie-Blain Moraes
- Department of Anesthesiology, University of Michigan Medical School Ann Arbor, MI, USA
| | - Giancarlo Vanini
- Department of Anesthesiology, University of Michigan Medical School Ann Arbor, MI, USA
| | - Vijay Tarnal
- Department of Anesthesiology, University of Michigan Medical School Ann Arbor, MI, USA
| | - Ellen Janke
- Department of Anesthesiology, University of Michigan Medical School Ann Arbor, MI, USA
| | - Anthony G Hudetz
- Department of Anesthesiology, University of Michigan Medical SchoolAnn Arbor, MI, USA; Center for Consciousness Science, University of Michigan Medical SchoolAnn Arbor, MI, USA; Neuroscience Graduate Program, University of Michigan Medical SchoolAnn Arbor, MI, USA
| | - Uncheol Lee
- Department of Anesthesiology, University of Michigan Medical SchoolAnn Arbor, MI, USA; Center for Consciousness Science, University of Michigan Medical SchoolAnn Arbor, MI, USA
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12
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Navas A, Villacorta-Atienza JA, Leyva I, Almendral JA, Sendiña-Nadal I, Boccaletti S. Effective centrality and explosive synchronization in complex networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062820. [PMID: 26764757 DOI: 10.1103/physreve.92.062820] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2015] [Indexed: 06/05/2023]
Abstract
Synchronization of networked oscillators is known to depend fundamentally on the interplay between the dynamics of the graph's units and the microscopic arrangement of the network's structure. We here propose an effective network whose topological properties reflect the interplay between the topology and dynamics of the original network. On that basis, we are able to introduce the effective centrality, a measure that quantifies the role and importance of each network's node in the synchronization process. In particular, in the context of explosive synchronization, we use such a measure to assess the propensity of a graph to sustain an irreversible transition to synchronization. We furthermore discuss a strategy to induce the explosive behavior in a generic network, by acting only upon a fraction of its nodes.
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Affiliation(s)
- A Navas
- Center for Biomedical Technology, Univ. Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - J A Villacorta-Atienza
- Department of Applied Mathematics, Facultad de Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain
| | - I Leyva
- Center for Biomedical Technology, Univ. Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
- Complex Systems Group & GISC, Univ. Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
| | - J A Almendral
- Center for Biomedical Technology, Univ. Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
- Complex Systems Group & GISC, Univ. Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
| | - I Sendiña-Nadal
- Center for Biomedical Technology, Univ. Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
- Complex Systems Group & GISC, Univ. Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
| | - S Boccaletti
- CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy
- Embassy of Italy in Israel, Trade Tower, 25 Hamered St., 68125 Tel Aviv, Israel
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13
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Qu J, Wang SJ, Jusup M, Wang Z. Effects of random rewiring on the degree correlation of scale-free networks. Sci Rep 2015; 5:15450. [PMID: 26482005 PMCID: PMC4611853 DOI: 10.1038/srep15450] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2015] [Accepted: 09/08/2015] [Indexed: 01/01/2023] Open
Abstract
Random rewiring is used to generate null networks for the purpose of analyzing the topological properties of scale-free networks, yet the effects of random rewiring on the degree correlation are subject to contradicting interpretations in the literature. We comprehensively analyze the degree correlation of randomly rewired scale-free networks and show that random rewiring increases disassortativity by reducing the average degree of the nearest neighbors of high-degree nodes. The effect can be captured by the measures of the degree correlation that consider all links in the network, but not by analogous measures that consider only links between degree peers, hence the potential for contradicting interpretations. We furthermore find that random and directional rewiring affect the topology of a scale-free network differently, even if the degree correlation of the rewired networks is the same. Consequently, the network dynamics is changed, which is proven here by means of the biased random walk.
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Affiliation(s)
- Jing Qu
- School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710119, China
| | - Sheng-Jun Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710119, China
| | - Marko Jusup
- Faculty of Sciences, Kyushu University, Fukuoka 819-0395, Japan
| | - Zhen Wang
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Fukuoka 816-8580, Japan
- School of Automation, Northwestern Polytechnical University, Xi’an 710072, China
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14
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Explosive or Continuous: Incoherent state determines the route to synchronization. Sci Rep 2015; 5:12039. [PMID: 26160578 PMCID: PMC4498219 DOI: 10.1038/srep12039] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2015] [Accepted: 06/15/2015] [Indexed: 11/12/2022] Open
Abstract
Abrupt and continuous spontaneous emergence of collective synchronization of coupled oscillators have attracted much attention. In this paper, we propose a dynamical ensemble order parameter equation that enables us to grasp the essential low-dimensional dynamical mechanism of synchronization in networks of coupled oscillators. Different solutions of the dynamical ensemble order parameter equation build correspondences with diverse collective states, and different bifurcations reveal various transitions among these collective states. The structural relationship between the incoherent state and the synchronous state leads to different routes of transitions to synchronization, either continuous or discontinuous. The explosive synchronization is determined by the bistable state where the measure of each state and the critical points are obtained analytically by using the dynamical ensemble order parameter equation. Our method and results hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, provide an effective approach in understanding the routes to synchronization in more general complex networks.
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Zhou W, Chen L, Bi H, Hu X, Liu Z, Guan S. Explosive synchronization with asymmetric frequency distribution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012812. [PMID: 26274230 DOI: 10.1103/physreve.92.012812] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/12/2015] [Indexed: 06/04/2023]
Abstract
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.
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Affiliation(s)
- Wenchang Zhou
- Department of Physics, East China Normal University, Shanghai, 200241, China
| | - Lumin Chen
- Department of Physics, East China Normal University, Shanghai, 200241, China
- Management School, The University of Sheffield, Western Bank, Sheffield S10 2TN, United Kingdom
| | - Hongjie Bi
- Department of Physics, East China Normal University, Shanghai, 200241, China
| | - Xin Hu
- Department of Physics, East China Normal University, Shanghai, 200241, China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai, 200241, China
- State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Shuguang Guan
- Department of Physics, East China Normal University, Shanghai, 200241, China
- State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
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Vlasov V, Zou Y, Pereira T. Explosive synchronization is discontinuous. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012904. [PMID: 26274244 DOI: 10.1103/physreve.92.012904] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2015] [Indexed: 06/04/2023]
Abstract
Spontaneous explosive is an abrupt transition to collective behavior taking place in heterogeneous networks when the frequencies of the nodes are positively correlated with the node degree. This explosive transition was conjectured to be discontinuous. Indeed, numerical investigations reveal a hysteresis behavior associated with the transition. Here, we analyze explosive synchronization in star graphs. We show that in the thermodynamic limit the transition to (and out of) collective behavior is indeed discontinuous. The discontinuous nature of the transition is related to the nonlinear behavior of the order parameter, which in the thermodynamic limit exhibits multiple fixed points. Moreover, we unravel the hysteresis behavior in terms of the graph parameters. Our numerical results show that finite-size graphs are well described by our predictions.
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Affiliation(s)
- Vladimir Vlasov
- Institute of Physics, Potsdam University, 14469 Potsdam, Germany
| | - Yong Zou
- Department of Physics, East China Normal University, Shanghai 200062, China
| | - Tiago Pereira
- Department of Mathematics, Imperial College London, London SW72AZ, United Kingdom
- Institute of Mathematical and Computer Sciences, Universidade de São Paulo, São Paulo, Brazil
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Kumar P, Verma DK, Parmananda P, Boccaletti S. Experimental evidence of explosive synchronization in mercury beating-heart oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062909. [PMID: 26172774 DOI: 10.1103/physreve.91.062909] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2015] [Indexed: 06/04/2023]
Abstract
We report experimental evidence of explosive synchronization in coupled chemo-mechanical systems, namely in mercury beating-heart (MBH) oscillators. Connecting four MBH oscillators in a star network configuration and setting natural frequencies of each oscillator in proportion to the number of its links, a gradual increase of the coupling strength results in an abrupt and irreversible (first-order-like) transition from the system's unordered to ordered phase. On its turn, such a transition indicates the emergence of a bistable regime wherein coexisting states can be experimentally revealed. Finally, we prove how such a regime allows an experimental implementation of magneticlike states of synchronization, by the use of an external signal.
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Affiliation(s)
- Pawan Kumar
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai-400 076, India
| | - Dinesh Kumar Verma
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai-400 076, India
| | - P Parmananda
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai-400 076, India
| | - S Boccaletti
- CNR-Institute of Complex Systems, Via Madonna del Prato, 10, 50019 Sesto Fiorentino (FI), Italy
- Embassy of Italy in Israel, Trade Tower, 25 Hamered St., 68125 Tel Aviv, Israel
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Peron TKD, Ji P, Rodrigues FA, Kurths J. Effects of assortative mixing in the second-order Kuramoto model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052805. [PMID: 26066210 DOI: 10.1103/physreve.91.052805] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2014] [Indexed: 06/04/2023]
Abstract
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.
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Affiliation(s)
- Thomas K Dm Peron
- Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, São Paulo, Brazil
| | - Peng Ji
- Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany
- Department of Physics, Humboldt University, 12489 Berlin, Germany
| | - Francisco A Rodrigues
- Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970 São Carlos, São Paulo, Brazil
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany
- Department of Physics, Humboldt University, 12489 Berlin, Germany
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
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Sendiña-Nadal I, Leyva I, Navas A, Villacorta-Atienza JA, Almendral JA, Wang Z, Boccaletti S. Effects of degree correlations on the explosive synchronization of scale-free networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032811. [PMID: 25871161 DOI: 10.1103/physreve.91.032811] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2014] [Indexed: 06/04/2023]
Abstract
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.
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Affiliation(s)
- I Sendiña-Nadal
- Complex Systems Group, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - I Leyva
- Complex Systems Group, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - A Navas
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - J A Villacorta-Atienza
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - J A Almendral
- Complex Systems Group, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - Z Wang
- Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
- Center for Nonlinear Studies, Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Hong Kong) and Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
| | - S Boccaletti
- CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy
- Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv, Israel
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Chen Y, Cao Z, Wang S, Hu G. Self-organized correlations lead to explosive synchronization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022810. [PMID: 25768555 DOI: 10.1103/physreve.91.022810] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2014] [Indexed: 06/04/2023]
Abstract
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.
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Affiliation(s)
- Yang Chen
- School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Zhoujian Cao
- Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Shihong Wang
- School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Gang Hu
- Department of Physics, Beijing Normal University, Beijing 100875, China
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Zhang X, Boccaletti S, Guan S, Liu Z. Explosive synchronization in adaptive and multilayer networks. PHYSICAL REVIEW LETTERS 2015; 114:038701. [PMID: 25659026 DOI: 10.1103/physrevlett.114.038701] [Citation(s) in RCA: 129] [Impact Index Per Article: 14.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2014] [Indexed: 05/23/2023]
Abstract
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.
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Affiliation(s)
- Xiyun Zhang
- Department of Physics, East China Normal University, Shanghai 200062, China
| | - Stefano Boccaletti
- CNR-Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence, Italy and The Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv, Israel
| | - Shuguang Guan
- Department of Physics, East China Normal University, Shanghai 200062, China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai 200062, China
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Exact solution for first-order synchronization transition in a generalized Kuramoto model. Sci Rep 2014; 4:7262. [PMID: 25434404 PMCID: PMC4248286 DOI: 10.1038/srep07262] [Citation(s) in RCA: 53] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2014] [Accepted: 11/11/2014] [Indexed: 11/08/2022] Open
Abstract
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.
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Zhang X, Zou Y, Boccaletti S, Liu Z. Explosive synchronization as a process of explosive percolation in dynamical phase space. Sci Rep 2014; 4:5200. [PMID: 24903808 PMCID: PMC4650870 DOI: 10.1038/srep05200] [Citation(s) in RCA: 54] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2014] [Accepted: 05/16/2014] [Indexed: 11/09/2022] Open
Abstract
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.
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Affiliation(s)
- Xiyun Zhang
- Department of Physics, East China Normal University, Shanghai, 200062, China
| | - Yong Zou
- Department of Physics, East China Normal University, Shanghai, 200062, China
| | - S Boccaletti
- CNR- Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence, Italy
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai, 200062, China
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Skardal PS, Arenas A. Disorder induces explosive synchronization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062811. [PMID: 25019837 DOI: 10.1103/physreve.89.062811] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2014] [Indexed: 06/03/2023]
Abstract
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.
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Affiliation(s)
- Per Sebastian Skardal
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Alex Arenas
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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Zou Y, Pereira T, Small M, Liu Z, Kurths J. Basin of attraction determines hysteresis in explosive synchronization. PHYSICAL REVIEW LETTERS 2014; 112:114102. [PMID: 24702374 DOI: 10.1103/physrevlett.112.114102] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2014] [Indexed: 06/03/2023]
Abstract
Spontaneous explosive emergent behavior takes place in heterogeneous networks when the frequencies of the nodes are positively correlated to the node degree. A central feature of such explosive transitions is a hysteretic behavior at the transition to synchronization. We unravel the underlying mechanisms and show that the dynamical origin of the hysteresis is a change of basin of attraction of the synchronization state. Our findings hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, reveal how microscopic network parameters such as node degree and frequency affect the global network properties and can be used for network design and control.
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Affiliation(s)
- Yong Zou
- Department of Physics, East China Normal University, Shanghai 200062, China and Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany
| | - Tiago Pereira
- Department of Mathematics, Imperial College London, London SW72AZ, United Kingdom and London Mathematical Laboratory, London WC2N 6DF, United Kingdom
| | - Michael Small
- School of Mathematics and Statistics, University of Western Australia, Crawley, Western Australia 6009, Australia
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai 200062, China
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany and Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB243UE, United Kingdom
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Zhu L, Tian L, Shi D. Criterion for the emergence of explosive synchronization transitions in networks of phase oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042921. [PMID: 24229263 DOI: 10.1103/physreve.88.042921] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2013] [Indexed: 06/02/2023]
Abstract
The emergence of explosive synchronization transitions in networks of phase oscillators recently has become one of the most interesting topics. It is widely believed that the large frequency mismatch of a pair of oscillators (also known as disassortativity in frequency) is a direct cause of an explosive synchronization. It is found that, besides the disassortativity in frequency, the disassortativity in node degree also shows up in connection with the first-order synchronization transition. In this paper, we simulate the Kuramoto model on top of a family of networks with different degree-degree and frequency-frequency correlation patterns. Results show that only when the degrees and natural frequencies of the network's nodes are both disassortative can an explosive synchronization occur.
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Affiliation(s)
- Liuhua Zhu
- College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
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