1
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Wang Y, Wang L, Fan H, Ma J, Cao H, Wang X. Breathing cluster in complex neuron-astrocyte networks. CHAOS (WOODBURY, N.Y.) 2023; 33:113118. [PMID: 37967261 DOI: 10.1063/5.0146906] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2023] [Accepted: 10/20/2023] [Indexed: 11/17/2023]
Abstract
Brain activities are featured by spatially distributed neural clusters of coherent firings and a spontaneous slow switching of the clusters between the coherent and incoherent states. Evidences from recent in vivo experiments suggest that astrocytes, a type of glial cell regarded previously as providing only structural and metabolic supports to neurons, participate actively in brain functions by regulating the neural firing activities, yet the underlying mechanism remains unknown. Here, introducing astrocyte as a reservoir of the glutamate released from the neuron synapses, we propose the model of the complex neuron-astrocyte network, and investigate the roles of astrocytes in regulating the cluster synchronization behaviors of networked chaotic neurons. It is found that a specific set of neurons on the network are synchronized and form a cluster, while the remaining neurons are kept as desynchronized. Moreover, during the course of network evolution, the cluster is switching between the synchrony and asynchrony states in an intermittent fashion, henceforth the phenomenon of "breathing cluster." By the method of symmetry-based analysis, we conduct a theoretical investigation on the synchronizability of the cluster. It is revealed that the contents of the cluster are determined by the network symmetry, while the breathing of the cluster is attributed to the interplay between the neural network and the astrocyte. The phenomenon of breathing cluster is demonstrated in different network models, including networks with different sizes, nodal dynamics, and coupling functions. The findings shed light on the cellular mechanism of astrocytes in regulating neural activities and give insights into the state-switching of the neocortex.
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Affiliation(s)
- Ya Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Liang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Huawei Fan
- School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
| | - Jun Ma
- Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China
| | - Hui Cao
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
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2
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Ren X, Lei Y, Grebogi C, Baptista MS. The complementary contribution of each order topology into the synchronization of multi-order networks. CHAOS (WOODBURY, N.Y.) 2023; 33:111101. [PMID: 37909900 DOI: 10.1063/5.0177687] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2023] [Accepted: 10/12/2023] [Indexed: 11/03/2023]
Abstract
Higher-order interactions improve our capability to model real-world complex systems ranging from physics and neuroscience to economics and social sciences. There is great interest nowadays in understanding the contribution of higher-order terms to the collective behavior of the network. In this work, we investigate the stability of complete synchronization of complex networks with higher-order structures. We demonstrate that the synchronization level of a network composed of nodes interacting simultaneously via multiple orders is maintained regardless of the intensity of coupling strength across different orders. We articulate that lower-order and higher-order topologies work together complementarily to provide the optimal stable configuration, challenging previous conclusions that higher-order interactions promote the stability of synchronization. Furthermore, we find that simply adding higher-order interactions based on existing connections, as in simple complexes, does not have a significant impact on synchronization. The universal applicability of our work lies in the comprehensive analysis of different network topologies, including hypergraphs and simplicial complexes, and the utilization of appropriate rescaling to assess the impact of higher-order interactions on synchronization stability.
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Affiliation(s)
- Xiaomin Ren
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Youming Lei
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| | - Murilo S Baptista
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
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3
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Kong LW, Lai YC. Short-lived chimera states. CHAOS (WOODBURY, N.Y.) 2023; 33:2894496. [PMID: 37276573 DOI: 10.1063/5.0145573] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2023] [Accepted: 04/21/2023] [Indexed: 06/07/2023]
Abstract
In the classic Kuramoto system of coupled two-dimensional rotators, chimera states characterized by the coexistence of synchronous and asynchronous groups of oscillators are long-lived because the average lifetime of these states increases exponentially with the system size. Recently, it was discovered that, when the rotators in the Kuramoto model are three-dimensional, the chimera states become short-lived in the sense that their lifetime scales with only the logarithm of the dimension-augmenting perturbation. We introduce transverse-stability analysis to understand the short-lived chimera states. In particular, on the unit sphere representing three-dimensional (3D) rotations, the long-lived chimera states in the classic Kuramoto system occur on the equator, to which latitudinal perturbations that make the rotations 3D are transverse. We demonstrate that the largest transverse Lyapunov exponent calculated with respect to these long-lived chimera states is typically positive, making them short-lived. The transverse-stability analysis turns the previous numerical scaling law of the transient lifetime into an exact formula: the "free" proportional constant in the original scaling law can now be precisely determined in terms of the largest transverse Lyapunov exponent. Our analysis reinforces the speculation that in physical systems, chimera states can be short-lived as they are vulnerable to any perturbations that have a component transverse to the invariant subspace in which they live.
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Affiliation(s)
- Ling-Wei Kong
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
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4
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Letellier C, Olsen LF, Mangiarotti S. Chaos: From theory to applications for the 80th birthday of Otto E. Rössler. CHAOS (WOODBURY, N.Y.) 2021; 31:060402. [PMID: 34241288 DOI: 10.1063/5.0058332] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2021] [Accepted: 06/02/2021] [Indexed: 06/13/2023]
Affiliation(s)
- Christophe Letellier
- Normandie Université-CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France
| | - Lars F Olsen
- Institute of Biochemistry and Molecular Biology, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
| | - Sylvain Mangiarotti
- Centre d'Études Spatiales de la Biosphère, OMP-Cesbio/UPS-CNRS-CNES-IRD-INRAe, Observatoire Midi-Pyrénées, 18 Avenue Édouard Belin, 31401 Toulouse, France
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5
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Chatterjee AK, Kundu A, Kulkarni M. Spatiotemporal spread of perturbations in a driven dissipative Duffing chain: An out-of-time-ordered correlator approach. Phys Rev E 2020; 102:052103. [PMID: 33327101 DOI: 10.1103/physreve.102.052103] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2020] [Accepted: 08/03/2020] [Indexed: 11/07/2022]
Abstract
Out-of-time-ordered correlators (OTOCs) have been extensively used as a major tool for exploring quantum chaos, and recently there has been a classical analog. Studies have been limited to closed systems. In this work, we probe an open classical many-body system, more specifically, a spatially extended driven dissipative chain of coupled Duffing oscillators using the classical OTOC to investigate the spread and growth (decay) of an initially localized perturbation in the chain. Correspondingly, we find three distinct types of dynamical behavior: the sustained chaos, transient chaos, and nonchaotic region, as clearly exhibited by different geometrical shapes in the OTOC heat map. To quantify such differences, we look at instantaneous speed (IS), finite-time Lyapunov exponents (FTLEs), and velocity-dependent Lyapunov exponents (VDLEs) extracted from OTOCs. Introduction of these quantities turns out to be instrumental in diagnosing and demarcating different regimes of dynamical behavior. To gain control over open nonlinear systems, it is important to look at the variation of these quantities with respect to parameters. As we tune drive, dissipation, and coupling, FTLEs and IS exhibit transition between sustained chaos and nonchaotic regimes with intermediate transient chaos regimes and highly intermittent sustained chaos points. In the limit of zero nonlinearity, we present exact analytical results for the driven dissipative harmonic system, and we find that our analytical results can very well describe the nonchaotic regime as well as the late-time behavior in the transient regime of the Duffing chain. We believe that this analysis is an important step forward towards understanding nonlinear dynamics, chaos, and spatiotemporal spread of perturbations in many-particle open systems.
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Affiliation(s)
- Amit Kumar Chatterjee
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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6
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Dos Santos V, Borges FS, Iarosz KC, Caldas IL, Szezech JD, Viana RL, Baptista MS, Batista AM. Basin of attraction for chimera states in a network of Rössler oscillators. CHAOS (WOODBURY, N.Y.) 2020; 30:083115. [PMID: 32872816 DOI: 10.1063/5.0014013] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2020] [Accepted: 07/20/2020] [Indexed: 06/11/2023]
Abstract
Chimera states are spatiotemporal patterns in which coherent and incoherent dynamics coexist simultaneously. These patterns were observed in both locally and nonlocally coupled oscillators. We study the existence of chimera states in networks of coupled Rössler oscillators. The Rössler oscillator can exhibit periodic or chaotic behavior depending on the control parameters. In this work, we show that the existence of coherent, incoherent, and chimera states depends not only on the coupling strength, but also on the initial state of the network. The initial states can belong to complex basins of attraction that are not homogeneously distributed. Due to this fact, we characterize the basins by means of the uncertainty exponent and basin stability. In our simulations, we find basin boundaries with smooth, fractal, and riddled structures.
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Affiliation(s)
- Vagner Dos Santos
- Program of Post-graduation in Science, State University of Ponta Grossa, Ponta Grossa, Paraná 84030-900, Brazil
| | - Fernando S Borges
- Center for Mathematics, Computation and Cognition, Federal University of ABC, São Bernardo do Campo, São Paulo 09606-045, Brazil
| | - Kelly C Iarosz
- Institute of Physics, University of São Paulo, São Paulo 05508-900, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, São Paulo 05508-900, Brazil
| | - J D Szezech
- Program of Post-graduation in Science, State University of Ponta Grossa, Ponta Grossa, Paraná 84030-900, Brazil
| | - Ricardo L Viana
- Department of Physics, Federal University of Paraná, Curitiba, Paraná 80060-000, Brazil
| | - Murilo S Baptista
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, AB24 3UE Aberdeen, Scotland, United Kingdom
| | - Antonio M Batista
- Program of Post-graduation in Science, State University of Ponta Grossa, Ponta Grossa, Paraná 84030-900, Brazil
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7
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Santos MS, Protachevicz PR, Iarosz KC, Caldas IL, Viana RL, Borges FS, Ren HP, Szezech JD, Batista AM, Grebogi C. Spike-burst chimera states in an adaptive exponential integrate-and-fire neuronal network. CHAOS (WOODBURY, N.Y.) 2019; 29:043106. [PMID: 31042937 DOI: 10.1063/1.5087129] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2018] [Accepted: 03/20/2019] [Indexed: 06/09/2023]
Abstract
Chimera states are spatiotemporal patterns in which coherence and incoherence coexist. We observe the coexistence of synchronous (coherent) and desynchronous (incoherent) domains in a neuronal network. The network is composed of coupled adaptive exponential integrate-and-fire neurons that are connected by means of chemical synapses. In our neuronal network, the chimera states exhibit spatial structures both with spike and burst activities. Furthermore, those desynchronized domains not only have either spike or burst activity, but we show that the structures switch between spikes and bursts as the time evolves. Moreover, we verify the existence of multicluster chimera states.
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Affiliation(s)
- Moises S Santos
- Department of Physics, Federal University of Paraná, 80060-000 Curitiba, PR, Brazil
| | - Paulo R Protachevicz
- Graduate in Science Program-Physics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Kelly C Iarosz
- Institute of Physics, University of São Paulo, 05508-900 São Paulo, SP, Brazil
| | - Iberê L Caldas
- Institute of Physics, University of São Paulo, 05508-900 São Paulo, SP, Brazil
| | - Ricardo L Viana
- Department of Physics, Federal University of Paraná, 80060-000 Curitiba, PR, Brazil
| | - Fernando S Borges
- Center for Mathematics Computation and Cognition, Federal University of ABC, 09606-045 São Bernardo do Campo, SP, Brazil
| | - Hai-Peng Ren
- Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xian University of Technology, Xi'an 710048, People's Republic of China
| | - José D Szezech
- Graduate in Science Program-Physics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Antonio M Batista
- Graduate in Science Program-Physics, State University of Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
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8
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Yao N, Huang ZG, Ren HP, Grebogi C, Lai YC. Self-adaptation of chimera states. Phys Rev E 2019; 99:010201. [PMID: 30780345 DOI: 10.1103/physreve.99.010201] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2018] [Indexed: 06/09/2023]
Abstract
Chimera states in spatiotemporal dynamical systems have been investigated in physical, chemical, and biological systems, and have been shown to be robust against random perturbations. How do chimera states achieve their robustness? We uncover a self-adaptation behavior by which, upon a spatially localized perturbation, the coherent component of the chimera state spontaneously drifts to an optimal location as far away from the perturbation as possible, exposing only its incoherent component to the perturbation to minimize the disturbance. A systematic numerical analysis of the evolution of the spatiotemporal pattern of the chimera state towards the optimal stable state reveals an exponential relaxation process independent of the spatial location of the perturbation, implying that its effects can be modeled as restoring and damping forces in a mechanical system and enabling the articulation of a phenomenological model. Not only is the model able to reproduce the numerical results, it can also predict the trajectory of drifting. Our finding is striking as it reveals that, inherently, chimera states possess a kind of "intelligence" in achieving robustness through self-adaptation. The behavior can be exploited for the controlled generation of chimera states with their coherent component placed in any desired spatial region of the system.
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Affiliation(s)
- Nan Yao
- Department of Applied Physics, Xi'an University of Technology, Xi'an 710048, China
| | - Zi-Gang Huang
- The Key Laboratory of Biomedical Information Engineering of Ministry of Education, National Engineering Research Center of Health Care and Medical Devices, The Key Laboratory of Neuro-informatics & Rehabilitation Engineering of Ministry of Civil Affairs, and Institute of Health and Rehabilitation Science, School of Life Science and Technology, Xi'an Jiaotong University, Xi'an 710049, China
| | - Hai-Peng Ren
- Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xi'an University of Technology, Xi'an 710048, China
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
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9
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Santos V, Szezech JD, Batista AM, Iarosz KC, Baptista MS, Ren HP, Grebogi C, Viana RL, Caldas IL, Maistrenko YL, Kurths J. Riddling: Chimera's dilemma. CHAOS (WOODBURY, N.Y.) 2018; 28:081105. [PMID: 30180624 DOI: 10.1063/1.5048595] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2018] [Accepted: 08/07/2018] [Indexed: 06/08/2023]
Abstract
We investigate the basin of attraction properties and its boundaries for chimera states in a circulant network of Hénon maps. It is known that coexisting basins of attraction lead to a hysteretic behaviour in the diagrams of the density of states as a function of a varying parameter. Chimera states, for which coherent and incoherent domains occur simultaneously, emerge as a consequence of the coexistence of basin of attractions for each state. Consequently, the distribution of chimera states can remain invariant by a parameter change, and it can also suffer subtle changes when one of the basins ceases to exist. A similar phenomenon is observed when perturbations are applied in the initial conditions. By means of the uncertainty exponent, we characterise the basin boundaries between the coherent and chimera states, and between the incoherent and chimera states. This way, we show that the density of chimera states can be not only moderately sensitive but also highly sensitive to initial conditions. This chimera's dilemma is a consequence of the fractal and riddled nature of the basin boundaries.
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Affiliation(s)
- V Santos
- Graduate in Science Program, State University of Ponta Grossa, Ponta Grossa, Paraná 84030-900, Brazil
| | - J D Szezech
- Graduate in Science Program, State University of Ponta Grossa, Ponta Grossa, Paraná 84030-900, Brazil
| | - A M Batista
- Graduate in Science Program, State University of Ponta Grossa, Ponta Grossa, Paraná 84030-900, Brazil
| | - K C Iarosz
- Potsdam Institute for Climate Impact Research, Potsdam, Brandenburg 14473, Germany
| | - M S Baptista
- Institute for Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom
| | - H P Ren
- Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xian University of Technology, Xi'an 710048, People's Republic of China
| | - C Grebogi
- Institute for Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom
| | - R L Viana
- Department of Physics, Federal University of Paraná, Curitiba, Paraná 80060-000, Brazil
| | - I L Caldas
- Institute of Physics, University of São Paulo, São Paulo, São Paulo 05508-900, Brazil
| | - Y L Maistrenko
- Potsdam Institute for Climate Impact Research, Potsdam, Brandenburg 14473, Germany
| | - J Kurths
- Potsdam Institute for Climate Impact Research, Potsdam, Brandenburg 14473, Germany
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10
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Clerc MG, Coulibaly S, Ferré MA, Rojas RG. Chimera states in a Duffing oscillators chain coupled to nearest neighbors. CHAOS (WOODBURY, N.Y.) 2018; 28:083126. [PMID: 30180634 DOI: 10.1063/1.5025038] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2018] [Accepted: 07/31/2018] [Indexed: 06/08/2023]
Abstract
Coupled nonlinear oscillators can present complex spatiotemporal behaviors. Here, we report the coexistence of coherent and incoherent domains, called chimera states, in an array of identical Duffing oscillators coupled to their nearest neighbors. The chimera states show a significant variation of amplitude in the desynchronized domain. These intriguing states are observed in the bistability region between a homogeneous state and a spatiotemporal chaotic one. These dynamical behaviors are characterized by their Lyapunov spectra and their global phase coherence order parameter. The local coupling between oscillators prevents one domain from invading the other one. Depending on initial conditions, a family of chimera states appear, organized in a snaking-like diagram.
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Affiliation(s)
- M G Clerc
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla, 487-3 Santiago, Chile
| | - S Coulibaly
- Université de Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - M A Ferré
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla, 487-3 Santiago, Chile
| | - R G Rojas
- Intituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla, 4059 Valparaíso, Chile
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11
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Hizanidis J, Kouvaris NE, Gorka ZL, Díaz-Guilera A, Antonopoulos CG. Chimera-like States in Modular Neural Networks. Sci Rep 2016; 6:19845. [PMID: 26796971 PMCID: PMC4726386 DOI: 10.1038/srep19845] [Citation(s) in RCA: 65] [Impact Index Per Article: 8.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2015] [Accepted: 12/18/2015] [Indexed: 11/17/2022] Open
Abstract
Chimera states, namely the coexistence of coherent and incoherent behavior, were previously analyzed in complex networks. However, they have not been extensively studied in modular networks. Here, we consider a neural network inspired by the connectome of the C. elegans soil worm, organized into six interconnected communities, where neurons obey chaotic bursting dynamics. Neurons are assumed to be connected with electrical synapses within their communities and with chemical synapses across them. As our numerical simulations reveal, the coaction of these two types of coupling can shape the dynamics in such a way that chimera-like states can happen. They consist of a fraction of synchronized neurons which belong to the larger communities, and a fraction of desynchronized neurons which are part of smaller communities. In addition to the Kuramoto order parameter ρ, we also employ other measures of coherence, such as the chimera-like χ and metastability λ indices, which quantify the degree of synchronization among communities and along time, respectively. We perform the same analysis for networks that share common features with the C. elegans neural network. Similar results suggest that under certain assumptions, chimera-like states are prominent phenomena in modular networks, and might provide insight for the behavior of more complex modular networks.
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Affiliation(s)
- Johanne Hizanidis
- Crete Center for Quantum Complexity and Nanotechnology, Physics Department, University of Crete, 71003 Heraklion, Greece
- National Center for Scientific Research “Demokritos”, 15310 Athens, Greece
| | - Nikos E. Kouvaris
- Department of Physics, University of Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
| | - Zamora-López Gorka
- Center for Brain and Cognition, Universitat Pompeu Fabra, Barcelona, Spain
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
| | - Albert Díaz-Guilera
- Department of Physics, University of Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
| | - Chris G. Antonopoulos
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, CO4 3SQ Colchester, UK
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12
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Abstract
A remarkable phenomenon in spatiotemporal dynamical systems is chimera state, where the structurally and dynamically identical oscillators in a coupled networked system spontaneously break into two groups, one exhibiting coherent motion and another incoherent. This phenomenon was typically studied in the setting of non-local coupling configurations. We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift. We find the emergence of multicluster chimera states. Remarkably, as a parameter characterizing the amount of link removal is increased, chimera states of distinct numbers of clusters emerge and persist in different parameter regions. We develop a phenomenological theory, based on enhanced or reduced interactions among oscillators in different spatial groups, to explain why chimera states of certain numbers of clusters occur in certain parameter regions. The theoretical prediction agrees well with numerics.
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13
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Schmidt L, Krischer K. Clustering as a prerequisite for chimera states in globally coupled systems. PHYSICAL REVIEW LETTERS 2015; 114:034101. [PMID: 25658999 DOI: 10.1103/physrevlett.114.034101] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2014] [Indexed: 06/04/2023]
Abstract
The coexistence of coherently and incoherently oscillating parts in a system of identical oscillators with symmetrical coupling, i.e., a chimera state, is even observable with uniform global coupling. We address the question of the prerequisites for these states to occur in globally coupled systems. By analyzing two different types of chimera states found for nonlinear global coupling, we show that a clustering mechanism to split the ensemble into two groups is needed as a first step. In fact, the chimera states inherit properties from the cluster states in which they originate. Remarkably, they can exist in parameter space between cluster and chaotic states, as well as between cluster and synchronized states.
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Affiliation(s)
- Lennart Schmidt
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, James-Franck-Str. 1, D-85748 Garching, Germany and Institute for Advanced Study - Technische Universität München, Lichtenbergstr. 2a, D-85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, James-Franck-Str. 1, D-85748 Garching, Germany
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14
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Yao N, Huang ZG, Lai YC, Zheng ZG. Robustness of chimera states in complex dynamical systems. Sci Rep 2013; 3:3522. [PMID: 24343533 PMCID: PMC3865463 DOI: 10.1038/srep03522] [Citation(s) in RCA: 45] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2013] [Accepted: 11/29/2013] [Indexed: 11/20/2022] Open
Abstract
The remarkable phenomenon of chimera state in systems of non-locally coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interests. In such a state, different groups of oscillators can exhibit characteristically distinct types of dynamical behaviors, in spite of identity of the oscillators. But how robust are chimera states against random perturbations to the structure of the underlying network? We address this fundamental issue by studying the effects of random removal of links on the probability for chimera states. Using direct numerical calculations and two independent theoretical approaches, we find that the likelihood of chimera state decreases with the probability of random-link removal. A striking finding is that, even when a large number of links are removed so that chimera states are deemed not possible, in the state space there are generally both coherent and incoherent regions. The regime of chimera state is a particular case in which the oscillators in the coherent region happen to be synchronized or phase-locked.
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Affiliation(s)
- Nan Yao
- 1] Department of Physics and the Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China [2] School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA
| | - Zi-Gang Huang
- 1] School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA [2] Lanzhou University, and Institute of Modern Physics of CAS, Lanzhou 730000, China
| | - Ying-Cheng Lai
- 1] School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA [2] Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
| | - Zhi-Gang Zheng
- Department of Physics and the Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China
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Chembo Kouomou Y, Woafo P. Transitions from spatiotemporal chaos to cluster and complete synchronization states in a shift-invariant set of coupled nonlinear oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:046205. [PMID: 12786458 DOI: 10.1103/physreve.67.046205] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2002] [Revised: 12/02/2002] [Indexed: 05/24/2023]
Abstract
We study the spatiotemporal dynamics of a ring of diffusely coupled single-well Duffing oscillators. The transitions from spatiotemporal chaos to cluster and complete synchronization states are particularly investigated, as well as the Hopf bifurcations to instability. It is found that the underlying mechanism of these transitions relies on the motion of the representative points corresponding to the system's nondegenerated spatial transverse Fourier modes in the parametric Strutt diagram. A scaling law is used to demonstrate that the compact interval of the scalar coupling parameter values leading to cluster synchronization broadens in a square-power-like fashion as the number of oscillators is increased. The analytical approach is confirmed by numerical simulations.
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Affiliation(s)
- Y Chembo Kouomou
- Laboratoire de Mécanique, Faculté des Sciences, Université de Yaoundé I, Boîte Postale 812 Yaoundé, Cameroon
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Kouomou YC, Woafo P. Generalized correlated states in a ring of coupled nonlinear oscillators with a local injection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:066201. [PMID: 12513378 DOI: 10.1103/physreve.66.066201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2002] [Indexed: 05/24/2023]
Abstract
In this paper, we study the spatiotemporal dynamics of a ring of diffusely coupled nonlinear oscillators. Floquet theory is used to investigate the various dynamical states of the ring, as well as the Hopf bifurcations between them. A local injection scheme is applied to synchronize the ring with an external master oscillator. The shift-invariance symmetry is thereby broken, leading to the emergence of generalized correlated states. The transition boundaries from these correlated states to spatiotemporal chaos and complete synchronization are also derived. Numerical simulations are performed to support the analytic approach.
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Affiliation(s)
- Y Chembo Kouomou
- Laboratoire de Mécanique, Faculté des Sciences, Université de Yaoundé I, Boîte Postale 812, Cameroon
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Venkataramani SC, Ott E. Pattern selection in extended periodically forced systems: a continuum coupled map approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:046202. [PMID: 11308925 DOI: 10.1103/physreve.63.046202] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2000] [Indexed: 05/23/2023]
Abstract
We propose that a useful approach to the modeling of periodically forced extended systems is through continuum coupled map (CCM) models. CCM models are discrete time, continuous space models, mapping a continuous spatially varying field xi(n)(x) from time n to time n+1. The efficacy of CCM models is illustrated by an application to experiments of Umbanhowar, Melo, and Swinney [Nature 382, 793 (1996)] on vertically vibrated granular layers. Using a simple CCM model incorporating temporal period doubling and spatial patterning at a preferred length scale, we obtain results that bear remarkable similarities to the experimental observations. The fact that the model does not make use of physics specific to granular layers suggests that similar phenomena may be observed in other (nongranular) periodically forced, strongly dissipative systems. We also present a framework for the analysis of pattern selection in CCM models using a truncated modal expansion. Through the analysis, we predict scaling laws of various quantities, and these laws may be verifiable experimentally.
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Affiliation(s)
- S C Venkataramani
- Department of Mathematics, University of Chicago, Chicago, IL 60637, USA.
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Woafo P, Chedjou JC, Fotsin HB. Dynamics of a system consisting of a van der Pol oscillator coupled to a Duffing oscillator. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:5929-5934. [PMID: 9965808 DOI: 10.1103/physreve.54.5929] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lai YC. Unpredictability of the asymptotic attractors in phase-coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:2902-2908. [PMID: 9962966 DOI: 10.1103/physreve.51.2902] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kozlowski J, Parlitz U, Lauterborn W. Bifurcation analysis of two coupled periodically driven Duffing oscillators. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:1861-1867. [PMID: 9962844 DOI: 10.1103/physreve.51.1861] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lai YC, Winslow RL. Fractal basin boundaries in coupled map lattices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:3470-3473. [PMID: 9962397 DOI: 10.1103/physreve.50.3470] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lai YC, Winslow RL. Riddled parameter space in spatiotemporal chaotic dynamical systems. PHYSICAL REVIEW LETTERS 1994; 72:1640-1643. [PMID: 10055663 DOI: 10.1103/physrevlett.72.1640] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Shinbrot T, Ottino JM. Geometric method to create coherent structures in chaotic flows. PHYSICAL REVIEW LETTERS 1993; 71:843-846. [PMID: 10055382 DOI: 10.1103/physrevlett.71.843] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Abraham FF. Spatiotemporal dynamics of an intrinsically chaotic field. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:1625-1628. [PMID: 9960185 DOI: 10.1103/physreve.47.1625] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Müller B, Trayanov A. Deterministic chaos in non-Abelian lattice gauge theory. PHYSICAL REVIEW LETTERS 1992; 68:3387-3390. [PMID: 10045691 DOI: 10.1103/physrevlett.68.3387] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Moon FC. Experimental measurement of chaotic attractors in solid mechanics(a)). CHAOS (WOODBURY, N.Y.) 1991; 1:31-41. [PMID: 12779894 DOI: 10.1063/1.165815] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
In this paper a review is given of experimental techniques in chaotic dynamics of solid mechanical systems based on modern ideas of nonlinear dynamics. These methods include Poincare maps, double Poincare sections, symbol dynamics, and fractal dimension. The physical problems discussed include nonlinear elastic beams, forced motion of a string, flow-induced vibration of a rod, forced motions of a magnetic pendulum, and rigid body dynamics of a magnet and high-temperature superconductor.
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Moon FC, Holmes W, Khoury P. Symbol dynamic maps of spatial-temporal chaotic vibrations in a string of impact oscillators. CHAOS (WOODBURY, N.Y.) 1991; 1:65-68. [PMID: 12779897 DOI: 10.1063/1.165818] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Spatially complex, temporally chaotic dynamics of N-coupled impact oscillators connected by a string are studied experimentally using a discrete measure of the motion for each of the masses. For N=8, a binary assignment of symbols, corresponding to whether or not the masses impact an amplitude constraint, is used to code the spatial pattern as a binary number and to store its change in time in a computer. A spatial pattern return map is then used to observe the change in spatial patterns with time. Bifurcations in spatial impact patterns are observed in this experiment. An entropy measure is also used to characterize the dynamics. Numerical simulation shows behavior similar to the experimental system.
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Affiliation(s)
- F. C. Moon
- Cornell University, Ithaca, New York 14853
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Fabiny L, Wiesenfeld K. Clustering behavior of oscillator arrays. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:2640-2648. [PMID: 9905329 DOI: 10.1103/physreva.43.2640] [Citation(s) in RCA: 30] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Jaroszewicz A, Sukiennicki A. Spatiotemporal dynamics in a chain of damped spins. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:3554-3561. [PMID: 9997671 DOI: 10.1103/physrevb.43.3554] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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