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Khatun AA, Muthanna YA, Punetha N, Jafri HH. Collective dynamics of coupled Lorenz oscillators near the Hopf boundary: Intermittency and chimera states. Phys Rev E 2024; 109:034208. [PMID: 38632727 DOI: 10.1103/physreve.109.034208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2022] [Accepted: 02/05/2024] [Indexed: 04/19/2024]
Abstract
We study collective dynamics of networks of mutually coupled identical Lorenz oscillators near a subcritical Hopf bifurcation. Such systems exhibit induced multistable behavior with interesting spatiotemporal dynamics including synchronization, desynchronization, and chimera states. For analysis, we first consider a ring topology with nearest-neighbor coupling and find that the system may exhibit intermittent behavior due to the complex basin structures and dynamical frustration, where temporal dynamics of the oscillators in the ensemble switches between different attractors. Consequently, different oscillators may show a dynamics that is intermittently synchronized (or desynchronized), giving rise to intermittent chimera states. The behavior of the intermittent laminar phases is characterized by the characteristic time spent in the synchronization manifold, which decays as a power law. Such intermittent dynamics is quite general and is also observed in an ensemble of a large number of oscillators arranged in variety of network topologies including nonlocal, scale-free, random, and small-world networks.
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Affiliation(s)
- Anjuman Ara Khatun
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
- Department of Physics, Indian Institute of Technology Bombay, Mumbai 400 076, India
| | - Yusra Ahmed Muthanna
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
- Physics Department, Taiz University, Taiz 6803, Yemen
| | | | - Haider Hasan Jafri
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
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2
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Venegas-Pineda LG, Jardón-Kojakhmetov H, Cao M. Stable chimera states: A geometric singular perturbation approach. CHAOS (WOODBURY, N.Y.) 2023; 33:113123. [PMID: 37972302 DOI: 10.1063/5.0142122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 10/23/2023] [Indexed: 11/19/2023]
Abstract
Over the past decades, chimera states have attracted considerable attention given their unexpected symmetry-breaking spatiotemporal nature and simultaneously exhibiting synchronous and incoherent behaviors under specific conditions. Despite relevant precursory results of such unforeseen states for diverse physical and topological configurations, there remain structures and mechanisms yet to be unveiled. In this work, using mean-field techniques, we analyze a multilayer network composed of two populations of heterogeneous Kuramoto phase oscillators with coevolutive coupling strengths. Moreover, we employ the geometric singular perturbation theory through the inclusion of a time-scale separation between the dynamics of the network elements and the adaptive coupling strength connecting them, gaining a better insight into the behavior of the system from a fast-slow dynamics perspective. Consequently, we derive the necessary and sufficient condition to produce stable chimera states when considering a coevolutionary intercoupling strength. Additionally, under the aforementioned constraint and with a suitable adaptive law election, it is possible to generate intriguing patterns, such as persistent breathing chimera states. Thereafter, we analyze the geometric properties of the mean-field system with a coevolutionary intracoupling strength and demonstrate the production of stable chimera states. Next, we give arguments for the presence of such patterns in the associated network under specific conditions. Finally, relaxation oscillations and canard cycles, seemingly related to breathing chimeras, are numerically produced under identified conditions due to the geometry of our system.
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Affiliation(s)
- Luis Guillermo Venegas-Pineda
- Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Nijenborgh 9, 9700 AK Groningen, The Netherlands
| | - Hildeberto Jardón-Kojakhmetov
- Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Nijenborgh 9, 9700 AK Groningen, The Netherlands
| | - Ming Cao
- Engineering and Technology Institute Groningen, University of Groningen, Nijenborgh 4, 9700 AE Groningen, The Netherlands
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3
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Gu S, Fang Q, Zhou P, Li N. Chimera states in a large laterally coupled laser array with four different waveguide structures. CHAOS (WOODBURY, N.Y.) 2023; 33:073152. [PMID: 37499249 DOI: 10.1063/5.0151330] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2023] [Accepted: 07/03/2023] [Indexed: 07/29/2023]
Abstract
Chimera states are rich and fascinating phenomena existing in many networks, where the identical oscillators self-organize into spatially separated coexisting domains of coherent and incoherent oscillations. Here, we report these states in the large laterally coupled laser array with four different waveguiding structures, with which a variety of chimera patterns can be revealed. We present the bifurcation diagrams giving birth to them and find that the chimeras exist in the boundary of the steady state and multi-period oscillation solutions, which applies to all the prevalent waveguiding structures considered. We also find that the waveguiding structures play an important role in the chimera states, e.g., the array composed of the index antiguiding with gain-guiding has a wider chimera region compared to other waveguides considered. Additionally, the effects of the crucial parameters including the laser separation ratio, pump rate, frequency detuning, and linewidth enhancement factor on the observed phenomena are discussed. Our analysis shows that the frequency detuning between lasers and the linewidth enhancement factor affects the lifetime and pattern of chimeras. The results could guide the design of laser arrays or introduce more insight into a new understanding of the dynamical behaviors of networks.
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Affiliation(s)
- Shuangquan Gu
- School of Optoelectronic Science and Engineering and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
- Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province and Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, China
| | - Qi Fang
- School of Optoelectronic Science and Engineering and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
- Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province and Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, China
| | - Pei Zhou
- School of Optoelectronic Science and Engineering and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
- Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province and Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, China
| | - Nianqiang Li
- School of Optoelectronic Science and Engineering and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
- Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province and Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, China
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Andrzejak RG, Espinoso A. Chimera states in multiplex networks: Chameleon-like across-layer synchronization. CHAOS (WOODBURY, N.Y.) 2023; 33:2890080. [PMID: 37163994 DOI: 10.1063/5.0146550] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2023] [Accepted: 04/19/2023] [Indexed: 05/12/2023]
Abstract
Different across-layer synchronization types of chimera states in multilayer networks have been discovered recently. We investigate possible relations between them, for example, if the onset of some synchronization type implies the onset of some other type. For this purpose, we use a two-layer network with multiplex inter-layer coupling. Each layer consists of a ring of non-locally coupled phase oscillators. While oscillators in each layer are identical, the layers are made non-identical by introducing mismatches in the oscillators' mean frequencies and phase lag parameters of the intra-layer coupling. We use different metrics to quantify the degree of various across-layer synchronization types. These include phase-locking between individual interacting oscillators, amplitude and phase synchronization between the order parameters of each layer, generalized synchronization between the driver and response layer, and the alignment of the incoherent oscillator groups' position on the two rings. For positive phase lag parameter mismatches, we get a cascaded onset of synchronization upon a gradual increase of the inter-layer coupling strength. For example, the two order parameters show phase synchronization before any of the interacting oscillator pairs does. For negative mismatches, most synchronization types have their onset in a narrow range of the coupling strength. Weaker couplings can destabilize chimera states in the response layer toward an almost fully coherent or fully incoherent motion. Finally, in the absence of a phase lag mismatch, sufficient coupling turns the response dynamics into a replica of the driver dynamics with the phases of all oscillators shifted by a constant lag.
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Affiliation(s)
- Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Anaïs Espinoso
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute of Science and Technology, Carrer Baldiri Reixac 10-12, 08028 Barcelona, Catalonia, Spain
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5
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Munyayev VO, Bolotov MI, Smirnov LA, Osipov GV, Belykh I. Cyclops States in Repulsive Kuramoto Networks: The Role of Higher-Order Coupling. PHYSICAL REVIEW LETTERS 2023; 130:107201. [PMID: 36962033 DOI: 10.1103/physrevlett.130.107201] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2022] [Accepted: 02/09/2023] [Indexed: 06/18/2023]
Abstract
Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops' eye. As their mythological counterparts, the cyclops states are giants that dominate the system's phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling's repulsion. Beyond the Kuramoto oscillators, we show that this effect is robustly present in networks of canonical theta neurons with adaptive coupling. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.
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Affiliation(s)
- Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Igor Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
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Provata A. From Turing patterns to chimera states in the 2D Brusselator model. CHAOS (WOODBURY, N.Y.) 2023; 33:033133. [PMID: 37003796 DOI: 10.1063/5.0130539] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2022] [Accepted: 02/23/2023] [Indexed: 06/19/2023]
Abstract
The Brusselator has been used as a prototype model for autocatalytic reactions and, in particular, for the Belousov-Zhabotinsky reaction. When coupled at the diffusive limit, the Brusselator undergoes a Turing bifurcation resulting in the formation of classical Turing patterns, such as spots, stripes, and spirals in two spatial dimensions. In the present study, we use generic nonlocally coupled Brusselators and show that in the limit of the coupling range R→1 (diffusive limit), the classical Turing patterns are recovered, while for intermediate coupling ranges and appropriate parameter values, chimera states are produced. This study demonstrates how the parameters of a typical nonlinear oscillator can be tuned so that the coupled system passes from spatially stable Turing structures to dynamical spatiotemporal chimera states.
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Affiliation(s)
- A Provata
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," 15341 Athens, Greece
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7
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Dogonasheva O, Kasatkin D, Gutkin B, Zakharov D. Multistability and evolution of chimera states in a network of type II Morris-Lecar neurons with asymmetrical nonlocal inhibitory connections. CHAOS (WOODBURY, N.Y.) 2022; 32:101101. [PMID: 36319278 DOI: 10.1063/5.0117845] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2022] [Accepted: 09/12/2022] [Indexed: 06/16/2023]
Abstract
Formation of synchronous activity patterns is an essential property of neuronal networks that has been of central interest to synchronization theory. Chimera states, where both synchronous and asynchronous activities of neurons co-exist in a single network, are particularly poignant examples of such patterns, whose dynamics and multistability may underlie brain function, such as cognitive tasks. However, dynamical mechanisms of coherent state formation in spiking neuronal networks as well as ways to control these states remain unclear. In this paper, we take a step in this direction by considering the evolution of chimera states in a network of class II excitable Morris-Lecar neurons with asymmetrical nonlocal inhibitory connections. Using the adaptive coherence measure, we are able to partition the network parameter space into regions of various collective behaviors (antiphase synchronous clusters, traveling waves, different types of chimera states as well as a spiking death regime) and have shown multistability between the various regimes. We track the evolution of the chimera states as a function of changed key network parameters and found transitions between various types of chimera states. We further find that the network can demonstrate long transients leading to quasi-persistence of activity patterns in the border regions hinting at near-criticality behaviors.
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Affiliation(s)
- O Dogonasheva
- Group of Neural Theory, École Normale Supérieure PSL University, Paris 75005, France
| | - Dmitry Kasatkin
- Department of Nonlinear Dynamics, Institute of Applied Physics RAS, Nizhny Novgorod 603155, Russia
| | - Boris Gutkin
- Group of Neural Theory, École Normale Supérieure PSL University, Paris 75005, France
| | - Denis Zakharov
- Centre for Cognition and Decision Making, National Research University Higher School of Economics, Moscow 101000, Russia
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Botha AE, Ansariara M, Emadi S, Kolahchi MR. Chimera Patterns of Synchrony in a Frustrated Array of Hebb Synapses. Front Comput Neurosci 2022; 16:888019. [PMID: 35814347 PMCID: PMC9260432 DOI: 10.3389/fncom.2022.888019] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2022] [Accepted: 05/30/2022] [Indexed: 11/13/2022] Open
Abstract
The union of the Kuramoto–Sakaguchi model and the Hebb dynamics reproduces the Lisman switch through a bistability in synchronized states. Here, we show that, within certain ranges of the frustration parameter, the chimera pattern can emerge, causing a different, time-evolving, distribution in the Hebbian synaptic strengths. We study the stability range of the chimera as a function of the frustration (phase-lag) parameter. Depending on the range of the frustration, two different types of chimeras can appear spontaneously, i.e., from randomized initial conditions. In the first type, the oscillators in the coherent region rotate, on average, slower than those in the incoherent region; while in the second type, the average rotational frequencies of the two regions are reversed, i.e., the coherent region runs, on average, faster than the incoherent region. We also show that non-stationary behavior at finite N can be controlled by adjusting the natural frequency of a single pacemaker oscillator. By slowly cycling the frequency of the pacemaker, we observe hysteresis in the system. Finally, we discuss how we can have a model for learning and memory.
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Affiliation(s)
- A. E. Botha
- Department of Physics, Science Campus, University of South Africa, Private Bag X6, Johannesburg, South Africa
| | - M. Ansariara
- Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
| | - S. Emadi
- Department of Biological Sciences, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
| | - M. R. Kolahchi
- Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
- *Correspondence: M. R. Kolahchi
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9
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Munyayev VO, Bolotov MI, Smirnov LA, Osipov GV, Belykh IV. Stability of rotatory solitary states in Kuramoto networks with inertia. Phys Rev E 2022; 105:024203. [PMID: 35291064 DOI: 10.1103/physreve.105.024203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2021] [Accepted: 02/03/2022] [Indexed: 06/14/2023]
Abstract
Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and oscillates with a different frequency. Such chimera-type patterns with an incoherent state formed by a single oscillator were observed in various oscillator networks; however, there is still a lack of understanding of how such states can stably appear. Here, we study the stability of solitary states in Kuramoto networks of identical two-dimensional phase oscillators with inertia and a phase-lagged coupling. The presence of inertia can induce rotatory dynamics of the phase difference between the solitary oscillator and the coherent cluster. We derive asymptotic stability conditions for such a solitary state as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (1) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (2) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our stability analysis for the emergence of rotatory chimeras.
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Affiliation(s)
- Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
- Institute of Applied Physics, Russian Academy of Sciences, Ul'yanova Str. 46, Nizhny Novgorod 603950, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Igor V Belykh
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
- Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302, USA
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Clusella P, Pietras B, Montbrió E. Kuramoto model for populations of quadratic integrate-and-fire neurons with chemical and electrical coupling. CHAOS (WOODBURY, N.Y.) 2022; 32:013105. [PMID: 35105122 DOI: 10.1063/5.0075285] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2021] [Accepted: 12/13/2021] [Indexed: 06/14/2023]
Abstract
We derive the Kuramoto model (KM) corresponding to a population of weakly coupled, nearly identical quadratic integrate-and-fire (QIF) neurons with both electrical and chemical coupling. The ratio of chemical to electrical coupling determines the phase lag of the characteristic sine coupling function of the KM and critically determines the synchronization properties of the network. We apply our results to uncover the presence of chimera states in two coupled populations of identical QIF neurons. We find that the presence of both electrical and chemical coupling is a necessary condition for chimera states to exist. Finally, we numerically demonstrate that chimera states gradually disappear as coupling strengths cease to be weak.
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Affiliation(s)
- Pau Clusella
- Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona Biomedical Research Park, 08003 Barcelona, Spain
| | - Bastian Pietras
- Institute of Mathematics, Technical University Berlin, 10623 Berlin, Germany
| | - Ernest Montbrió
- Neuronal Dynamics Group, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
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Goodfellow M, Andrzejak RG, Masoller C, Lehnertz K. What Models and Tools can Contribute to a Better Understanding of Brain Activity? FRONTIERS IN NETWORK PHYSIOLOGY 2022; 2:907995. [PMID: 36926061 PMCID: PMC10013030 DOI: 10.3389/fnetp.2022.907995] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2022] [Accepted: 06/06/2022] [Indexed: 12/18/2022]
Abstract
Despite impressive scientific advances in understanding the structure and function of the human brain, big challenges remain. A deep understanding of healthy and aberrant brain activity at a wide range of temporal and spatial scales is needed. Here we discuss, from an interdisciplinary network perspective, the advancements in physical and mathematical modeling as well as in data analysis techniques that, in our opinion, have potential to further advance our understanding of brain structure and function.
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Affiliation(s)
- Marc Goodfellow
- Living Systems Institute, University of Exeter, Exeter, United Kingdom
| | - Ralph G Andrzejak
- Department of Information and Communication Technologies, University Pompeu Fabra, Barcelona, Spain
| | - Cristina Masoller
- Department of Physics, Universitat Politecnica de Catalunya, Barcelona, Spain
| | - Klaus Lehnertz
- Department of Epileptology, University of Bonn Medical Centre, Bonn, Germany.,Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Bonn, Germany.,Interdisciplinary Center for Complex Systems, University of Bonn, Bonn, Germany
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12
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Laing CR. Interpolating between bumps and chimeras. CHAOS (WOODBURY, N.Y.) 2021; 31:113116. [PMID: 34881576 DOI: 10.1063/5.0070341] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2021] [Accepted: 10/11/2021] [Indexed: 06/13/2023]
Abstract
A "bump" refers to a group of active neurons surrounded by quiescent ones while a "chimera" refers to a pattern in a network in which some oscillators are synchronized while the remainder are asynchronous. Both types of patterns have been studied intensively but are sometimes conflated due to their similar appearance and existence in similar types of networks. Here, we numerically study a hybrid system that linearly interpolates between a network of theta neurons that supports a bump at one extreme and a network of phase oscillators that supports a chimera at the other extreme. Using the Ott/Antonsen ansatz, we derive the equation describing the hybrid network in the limit of an infinite number of oscillators and perform bifurcation analysis on this equation. We find that neither the bump nor chimera persists over the whole range of parameters, and the hybrid system shows a variety of other states such as spatiotemporal chaos, traveling waves, and modulated traveling waves.
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Affiliation(s)
- Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Private Bag 102-904 North Shore Mail Centre, Auckland, New Zealand
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13
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Khatun AA, Jafri HH, Punetha N. Controlling chimera states in chaotic oscillator ensembles through linear augmentation. Phys Rev E 2021; 103:042202. [PMID: 34005985 DOI: 10.1103/physreve.103.042202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2019] [Accepted: 03/07/2021] [Indexed: 11/07/2022]
Abstract
In this work, we show how "chimera states," namely, the dynamical situation when synchronized and desynchronized domains coexist in an oscillator ensemble, can be controlled through a linear augmentation (LA) technique. Specifically, in the networks of coupled chaotic oscillators, we obtain chimera states through induced multistability and demonstrate how LA can be used to control the size and spatial location of the incoherent and coherent populations in the ensemble. We examine basins of attraction of the system to analyze the effects of LA on its multistable behavior and thus on chimera states. Stability of the synchronized dynamics is analyzed through a master stability function. We find that these results are independent of a system's initial conditions and the strategy is applicable to the networks of globally, locally as well as nonlocally coupled oscillators. Our results suggest that LA control can be an effective method to control chimera states and to realize a desired collective dynamics in such ensembles.
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Affiliation(s)
- Anjuman Ara Khatun
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
| | - Haider Hasan Jafri
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
| | - Nirmal Punetha
- Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
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14
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Koulierakis I, Verganelakis DA, Omelchenko I, Zakharova A, Schöll E, Provata A. Structural anomalies in brain networks induce dynamical pacemaker effects. CHAOS (WOODBURY, N.Y.) 2020; 30:113137. [PMID: 33261325 DOI: 10.1063/5.0006207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/29/2020] [Accepted: 10/22/2020] [Indexed: 06/12/2023]
Abstract
Dynamical effects on healthy brains and brains affected by tumor are investigated via numerical simulations. The brains are modeled as multilayer networks consisting of neuronal oscillators whose connectivities are extracted from Magnetic Resonance Imaging (MRI) data. The numerical results demonstrate that the healthy brain presents chimera-like states where regions with high white matter concentrations in the direction connecting the two hemispheres act as the coherent domain, while the rest of the brain presents incoherent oscillations. To the contrary, in brains with destructed structures, traveling waves are produced initiated at the region where the tumor is located. These areas act as the pacemaker of the waves sweeping across the brain. The numerical simulations are performed using two neuronal models: (a) the FitzHugh-Nagumo model and (b) the leaky integrate-and-fire model. Both models give consistent results regarding the chimera-like oscillations in healthy brains and the pacemaker effect in the tumorous brains. These results are considered a starting point for further investigation in the detection of tumors with small sizes before becoming discernible on MRI recordings as well as in tumor development and evolution.
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Affiliation(s)
- I Koulierakis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," 15341 Athens, Greece
| | - D A Verganelakis
- Nuclear Medicine Unit, Oncology Clinic "Marianna V. Vardinoyiannis-ELPIDA," Childrens' Hospital "A. Sofia," 11527 Athens, Greece
| | - I Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - A Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - E Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - A Provata
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," 15341 Athens, Greece
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15
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Lugo H, González-Avella JC, San Miguel M. Local connectivity effects in learning and coordination dynamics in a two-layer network. CHAOS (WOODBURY, N.Y.) 2020; 30:083125. [PMID: 32872817 DOI: 10.1063/5.0006908] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2020] [Accepted: 07/27/2020] [Indexed: 06/11/2023]
Abstract
Anticoordination and chimera states occur in a two-layer model of learning and coordination dynamics in fully connected networks. Learning occurs in the intra-layer networks, while a coordination game is played in the inter-layer network. In this paper, we study the robustness of these states against local effects introduced by the local connectivity of random networks. We identify threshold values for the mean degree of the networks such that below these values, local effects prevent the existence of anticoordination and chimera states found in the fully connected setting. Local effects in the intra-layer learning network are more important than in the inter-layer network in preventing the existence of anticoordination states. The local connectivity of the intra- and inter-layer networks is important to avoid the occurrence of chimera states, but the local effects are stronger in the inter-layer coordination network than in the intra-layer learning network. We also study the effect of local connectivity on the problem of equilibrium selection in the asymmetric coordination game, showing that local effects favor the selection of the Pareto-dominant equilibrium in situations in which the risk-dominant equilibrium is selected in the fully connected network. In this case, again, the important local effects are those associated with the coordination dynamics inter-layer network. Indeed, lower average degree of the network connectivity between layers reduces the probability of achieving the risk-dominant strategy, favoring the Pareto-dominant equilibrium.
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Affiliation(s)
- Haydée Lugo
- ICAE and Department of Economic Analysis, Universidad Complutense de Madrid, 28223 Madrid, Spain
| | | | - Maxi San Miguel
- IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain
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16
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Wang Z, Liu Z. A Brief Review of Chimera State in Empirical Brain Networks. Front Physiol 2020; 11:724. [PMID: 32714208 PMCID: PMC7344215 DOI: 10.3389/fphys.2020.00724] [Citation(s) in RCA: 31] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2020] [Accepted: 06/02/2020] [Indexed: 11/24/2022] Open
Abstract
Understanding the human brain and its functions has always been an interesting and challenging problem. Recently, a significant progress on this problem has been achieved on the aspect of chimera state where a coexistence of synchronized and unsynchronized states can be sustained in identical oscillators. This counterintuitive phenomenon is closely related to the unihemispheric sleep in some marine mammals and birds and has recently gotten a hot attention in neural systems, except the previous studies in non-neural systems such as phase oscillators. This review will briefly summarize the main results of chimera state in neuronal systems and pay special attention to the network of cerebral cortex, aiming to accelerate the study of chimera state in brain networks. Some outlooks are also discussed.
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Affiliation(s)
| | - Zonghua Liu
- School of Physics and Electronic Science, East China Normal University, Shanghai, China
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17
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Chandran P, Gopal R, Chandrasekar VK, Athavan N. Chimera-like states induced by additional dynamic nonlocal wirings. CHAOS (WOODBURY, N.Y.) 2020; 30:063106. [PMID: 32611102 DOI: 10.1063/1.5144929] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2020] [Accepted: 05/07/2020] [Indexed: 06/11/2023]
Abstract
We investigate the existence of chimera-like states in a small-world network of chaotically oscillating identical Rössler systems with an addition of randomly switching nonlocal links. By varying the small-world coupling strength, we observe no chimera-like state either in the absence of nonlocal wirings or with static nonlocal wirings. When we give an additional nonlocal wiring to randomly selected nodes and if we allow the random selection of nodes to change with time, we observe the onset of chimera-like states. Upon increasing the number of randomly selected nodes gradually, we find that the incoherent window keeps on shrinking, whereas the chimera-like window widens up. Moreover, the system attains a completely synchronized state comparatively sooner for a lower coupling strength. Also, we show that one can induce chimera-like states by a suitable choice of switching times, coupling strengths, and a number of nonlocal links. We extend the above-mentioned randomized injection of nonlocal wirings for the cases of globally coupled Rössler oscillators and a small-world network of coupled FitzHugh-Nagumo oscillators and obtain similar results.
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Affiliation(s)
- P Chandran
- Department of Physics, H. H. The Rajah's College (affiliated to Bharathidasan University), Pudukkottai 622 001, Tamil Nadu, India
| | - R Gopal
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur 613 401, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur 613 401, Tamil Nadu, India
| | - N Athavan
- Department of Physics, H. H. The Rajah's College (affiliated to Bharathidasan University), Pudukkottai 622 001, Tamil Nadu, India
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18
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Chimera states in hybrid coupled neuron populations. Neural Netw 2020; 126:108-117. [DOI: 10.1016/j.neunet.2020.03.002] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/11/2019] [Revised: 02/03/2020] [Accepted: 03/02/2020] [Indexed: 01/01/2023]
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19
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Drauschke F, Sawicki J, Berner R, Omelchenko I, Schöll E. Effect of topology upon relay synchronization in triplex neuronal networks. CHAOS (WOODBURY, N.Y.) 2020; 30:051104. [PMID: 32491914 DOI: 10.1063/5.0008341] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2020] [Accepted: 04/20/2020] [Indexed: 06/11/2023]
Abstract
Relay synchronization in complex networks is characterized by the synchronization of remote parts of the network due to their interaction via a relay. In multilayer networks, distant layers that are not connected directly can synchronize due to signal propagation via relay layers. In this work, we investigate relay synchronization of partial synchronization patterns like chimera states in three-layer networks of interacting FitzHugh-Nagumo oscillators. We demonstrate that the phenomenon of relay synchronization is robust to topological random inhomogeneities of small-world type in the layer networks. We show that including randomness in the connectivity structure either of the remote network layers or of the relay layer increases the range of interlayer coupling strength where relay synchronization can be observed.
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Affiliation(s)
- Fenja Drauschke
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Rico Berner
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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20
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Kang L, Tian C, Huo S, Liu Z. A two-layered brain network model and its chimera state. Sci Rep 2019; 9:14389. [PMID: 31591418 PMCID: PMC6779761 DOI: 10.1038/s41598-019-50969-5] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2019] [Accepted: 09/23/2019] [Indexed: 01/14/2023] Open
Abstract
Based on the data of cerebral cortex, we present a two-layered brain network model of coupled neurons where the two layers represent the left and right hemispheres of cerebral cortex, respectively, and the links between the two layers represent the inter-couplings through the corpus callosum. By this model we show that abundant patterns of synchronization can be observed, especially the chimera state, depending on the parameters of system such as the coupling strengths and coupling phase. Further, we extend the model to a more general two-layered network to better understand the mechanism of the observed patterns, where each hemisphere of cerebral cortex is replaced by a highly clustered subnetwork. We find that the number of inter-couplings is another key parameter for the emergence of chimera states. Thus, the chimera states come from a matching between the structure parameters such as the number of inter-couplings and clustering coefficient etc and the dynamics parameters such as the intra-, inter-coupling strengths and coupling phase etc. A brief theoretical analysis is provided to explain the borderline of synchronization. These findings may provide helpful clues to understand the mechanism of brain functions.
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Affiliation(s)
- Ling Kang
- 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
- School of Data Science, Tongren University, Tongren, 554300, P.R. China
| | - Siyu Huo
- 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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21
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Höhlein K, Kemeth FP, Krischer K. Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators. Phys Rev E 2019; 100:022217. [PMID: 31574758 DOI: 10.1103/physreve.100.022217] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2019] [Indexed: 01/22/2023]
Abstract
Oscillatory systems with long-range or global coupling offer promising insight into the interplay between high-dimensional (or microscopic) chaotic motion and collective interaction patterns. Within this paper, we use Lyapunov analysis to investigate whether chimera states in globally coupled Stuart-Landau (SL) oscillators exhibit collective degrees of freedom. We compare two types of chimera states, which emerge in SL ensembles with linear and nonlinear global coupling, respectively, the latter introducing a constraint that conserves the oscillation of the mean. Lyapunov spectra reveal that for both chimera states the Lyapunov exponents split into several groups with different convergence properties in the limit of large system size. Furthermore, in both cases the Lyapunov dimension is found to scale extensively and the localization properties of covariant Lypunov vectors manifest the presence of collective Lyapunov modes. Here, however, we find qualitative differences between the two types of chimera states: Whereas the ones in the system under nonlinear global coupling exhibit only slow collective modes corresponding to Lyapunov exponents equal or close to zero, those which experience the linear mean-field coupling exhibit also faster collective modes associated with Lyapunov exponents with large positive or negative values. Furthermore, for the fastest collective mode we showed that it spreads across both synchonous and incoherent oscillators.
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Affiliation(s)
- Kevin Höhlein
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, D-85748 Garching, Germany
| | - Felix P Kemeth
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, D-85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, D-85748 Garching, Germany
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22
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Kaminker V, Wackerbauer R. Alternating activity patterns and a chimeralike state in a network of globally coupled excitable Morris-Lecar neurons. CHAOS (WOODBURY, N.Y.) 2019; 29:053121. [PMID: 31154794 DOI: 10.1063/1.5093483] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2019] [Accepted: 04/30/2019] [Indexed: 06/09/2023]
Abstract
Spatiotemporal chaos collapses to either a rest state or a propagating pulse in a ring network of diffusively coupled, excitable Morris-Lecar neurons. Adding global varying synaptic coupling to the ring network reveals complex transient behavior. Spatiotemporal chaos collapses into a transient pulse that reinitiates spatiotemporal chaos to allow sequential pattern switching until a collapse to the rest state. A domain of irregular neuron activity coexists with a domain of inactive neurons forming a transient chimeralike state. Transient spatial localization of the chimeralike state is observed for stronger synapses.
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Affiliation(s)
- Vitaliy Kaminker
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
| | - Renate Wackerbauer
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
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23
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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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24
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Omelchenko I, Zakharova A. Intriguing coexistence of synchrony and asynchrony in the brain: Comment on "Chimera states in neuronal networks: A review" by Soumen Majhi, Bidesh K. Bera, Dibakar Ghosh, Matjaẑ Perc. Phys Life Rev 2019; 28:134-136. [PMID: 30910384 DOI: 10.1016/j.plrev.2019.02.013] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2019] [Accepted: 02/27/2019] [Indexed: 11/17/2022]
Affiliation(s)
- Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.
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25
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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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26
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Cano AV, Cosenza MG. Asymmetric cluster and chimera dynamics in globally coupled systems. CHAOS (WOODBURY, N.Y.) 2018; 28:113119. [PMID: 30501202 DOI: 10.1063/1.5043398] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2018] [Accepted: 11/06/2018] [Indexed: 06/09/2023]
Abstract
We investigate the emergence of chimera and cluster states possessing asymmetric dynamics in globally coupled systems, where the trajectories of oscillators belonging to different subpopulations exhibit different dynamical properties. In an asymmetric chimera state, the trajectory of an element in the synchronized subset is stationary or periodic, while that of an oscillator in the desynchronized subset is chaotic. In an asymmetric cluster state, the periods of the trajectories of elements belonging to different clusters are different. We consider a network of globally coupled chaotic maps as a simple model for the occurrence of such asymmetric states in spatiotemporal systems. We employ the analogy between a single map subject to a constant drive and the effective local dynamics in the globally coupled map system to elucidate the mechanisms for the emergence of asymmetric chimera and cluster states in the latter system. By obtaining the dynamical responses of the driven map, we establish a condition for the equivalence of the dynamics of the driven map and that of the system of globally coupled maps. This condition is applied to predict parameter values and subset partitions for the formation of asymmetric cluster and chimera states in the globally coupled system.
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Affiliation(s)
- A V Cano
- Grupo de Caos y Sistemas Complejos, Centro de Física Fundamental, Universidad de Los Andes, Mérida, Venezuela
| | - M G Cosenza
- Grupo de Caos y Sistemas Complejos, Centro de Física Fundamental, Universidad de Los Andes, Mérida, Venezuela
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27
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Majhi S, Ghosh D. Alternating chimeras in networks of ephaptically coupled bursting neurons. CHAOS (WOODBURY, N.Y.) 2018; 28:083113. [PMID: 30180636 DOI: 10.1063/1.5022612] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2018] [Accepted: 07/26/2018] [Indexed: 06/08/2023]
Abstract
The distinctive phenomenon of the chimera state has been explored in neuronal systems under a variety of different network topologies during the last decade. Nevertheless, in all the works, the neurons are presumed to interact with each other directly with the help of synapses only. But, the influence of ephaptic coupling, particularly magnetic flux across the membrane, is mostly unexplored and should essentially be dealt with during the emergence of collective electrical activities and propagation of signals among the neurons in a network. Through this article, we report the development of an emerging dynamical state, namely, the alternating chimera, in a network of identical neuronal systems induced by an external electromagnetic field. Owing to this interaction scenario, the nonlinear neuronal oscillators are coupled indirectly via electromagnetic induction with magnetic flux, through which neurons communicate in spite of the absence of physical connections among them. The evolution of each neuron, here, is described by the three-dimensional Hindmarsh-Rose dynamics. We demonstrate that the presence of such non-locally and globally interacting external environments induces a stationary alternating chimera pattern in the ensemble of neurons, whereas in the local coupling limit, the network exhibits a transient chimera state whenever the local dynamics of the neurons is of the chaotic square-wave bursting type. For periodic square-wave bursting of the neurons, a similar qualitative phenomenon has been witnessed with the exception of the disappearance of cluster states for non-local and global interactions. Besides these observations, we advance our work while providing confirmation of the findings for neuronal ensembles exhibiting plateau bursting dynamics and also put forward the fact that the plateau pattern actually favors the alternating chimera more than others. These results may deliver better interpretations for different aspects of synchronization appearing in a network of neurons through field coupling that also relaxes the prerequisite of synaptic connectivity for realizing the chimera state in neuronal networks.
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Affiliation(s)
- Soumen Majhi
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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28
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Kasimatis T, Hizanidis J, Provata A. Three-dimensional chimera patterns in networks of spiking neuron oscillators. Phys Rev E 2018; 97:052213. [PMID: 29906870 DOI: 10.1103/physreve.97.052213] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2018] [Indexed: 06/08/2023]
Abstract
We study the stable spatiotemporal patterns that arise in a three-dimensional (3D) network of neuron oscillators, whose dynamics is described by the leaky integrate-and-fire (LIF) model. More specifically, we investigate the form of the chimera states induced by a 3D coupling matrix with nonlocal topology. The observed patterns are in many cases direct generalizations of the corresponding two-dimensional (2D) patterns, e.g., spheres, layers, and cylinder grids. We also find cylindrical and "cross-layered" chimeras that do not have an equivalent in 2D systems. Quantitative measures are calculated, such as the ratio of synchronized and unsynchronized neurons as a function of the coupling range, the mean phase velocities, and the distribution of neurons in mean phase velocities. Based on these measures, the chimeras are categorized in two families. The first family of patterns is observed for weaker coupling and exhibits higher mean phase velocities for the unsynchronized areas of the network. The opposite holds for the second family, where the unsynchronized areas have lower mean phase velocities. The various measures demonstrate discontinuities, indicating criticality as the parameters cross from the first family of patterns to the second.
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Affiliation(s)
- T Kasimatis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," 15310 Athens, Greece
- School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Athens, Greece
| | - J Hizanidis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," 15310 Athens, Greece
- Department of Physics, University of Crete, 71003 Heraklion, Greece
- National University of Science and Technology MISiS, Leninsky Prospect 4, Moscow, 119049, Russia
| | - A Provata
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," 15310 Athens, Greece
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29
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Chouzouris T, Omelchenko I, Zakharova A, Hlinka J, Jiruska P, Schöll E. Chimera states in brain networks: Empirical neural vs. modular fractal connectivity. CHAOS (WOODBURY, N.Y.) 2018; 28:045112. [PMID: 31906648 DOI: 10.1063/1.5009812] [Citation(s) in RCA: 46] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Abstract
Complex spatiotemporal patterns, called chimera states, consist of coexisting coherent and incoherent domains and can be observed in networks of coupled oscillators. The interplay of synchrony and asynchrony in complex brain networks is an important aspect in studies of both the brain function and disease. We analyse the collective dynamics of FitzHugh-Nagumo neurons in complex networks motivated by its potential application to epileptology and epilepsy surgery. We compare two topologies: an empirical structural neural connectivity derived from diffusion-weighted magnetic resonance imaging and a mathematically constructed network with modular fractal connectivity. We analyse the properties of chimeras and partially synchronized states and obtain regions of their stability in the parameter planes. Furthermore, we qualitatively simulate the dynamics of epileptic seizures and study the influence of the removal of nodes on the network synchronizability, which can be useful for applications to epileptic surgery.
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Affiliation(s)
- Teresa Chouzouris
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Jaroslav Hlinka
- Institute of Computer Science, Czech Academy of Sciences, Pod Vodarenskou vezi 2, 18207 Prague, Czech Republic
| | - Premysl Jiruska
- Institute of Physiology, Czech Academy of Sciences, Videnska 1083, 14220 Prague, Czech Republic
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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30
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Sathiyadevi K, Chandrasekar VK, Senthilkumar DV, Lakshmanan M. Distinct collective states due to trade-off between attractive and repulsive couplings. Phys Rev E 2018; 97:032207. [PMID: 29776099 DOI: 10.1103/physreve.97.032207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2018] [Indexed: 06/08/2023]
Abstract
We investigate the effect of repulsive coupling together with an attractive coupling in a network of nonlocally coupled oscillators. To understand the complex interaction between these two couplings we introduce a control parameter in the repulsive coupling which plays a crucial role in inducing distinct complex collective patterns. In particular, we show the emergence of various cluster chimera death states through a dynamically distinct transition route, namely the oscillatory cluster state and coherent oscillation death state as a function of the repulsive coupling in the presence of the attractive coupling. In the oscillatory cluster state, the oscillators in the network are grouped into two distinct dynamical states of homogeneous and inhomogeneous oscillatory states. Further, the network of coupled oscillators follow the same transition route in the entire coupling range. Depending upon distinct coupling ranges, the system displays different number of clusters in the death state and oscillatory state. We also observe that the number of coherent domains in the oscillatory cluster state exponentially decreases with increase in coupling range and obeys a power-law decay. Additionally, we show analytical stability for observed solitary state, synchronized state, and incoherent oscillation death state.
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Affiliation(s)
- K Sathiyadevi
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695016, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli-620 024, Tamil Nadu, India
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31
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Ratas I, Pyragas K. Symmetry breaking in two interacting populations of quadratic integrate-and-fire neurons. Phys Rev E 2018; 96:042212. [PMID: 29347512 DOI: 10.1103/physreve.96.042212] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2017] [Indexed: 11/07/2022]
Abstract
We analyze the dynamics of two coupled identical populations of quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The populations are heterogeneous; they include both inherently spiking and excitable neurons. The coupling within and between the populations is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rates and the mean membrane potentials in both populations. The reduced equations are exact in the infinite-size limit. The bifurcation analysis of the equations reveals a rich variety of nonsymmetric patterns, including a splay state, antiphase periodic oscillations, chimera-like states, and chaotic oscillations as well as bistabilities between various states. The validity of the reduced equations is confirmed by direct numerical simulations of the finite-size networks.
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Affiliation(s)
- Irmantas Ratas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Kestutis Pyragas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
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32
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Omelchenko I, Omel'chenko OE, Zakharova A, Schöll E. Optimal design of tweezer control for chimera states. Phys Rev E 2018; 97:012216. [PMID: 29448470 DOI: 10.1103/physreve.97.012216] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2017] [Indexed: 06/08/2023]
Abstract
Chimera states are complex spatio-temporal patterns which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyze the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHugh-Nagumo oscillators, and determine the ranges of optimal control parameters. We demonstrate that the tweezer control scheme allows for stabilization of chimera states with different shapes, and can be used as an instrument for controlling the coherent domains size, as well as the maximum average frequency difference of the oscillators.
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Affiliation(s)
- Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | | | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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33
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Kotwal T, Jiang X, Abrams DM. Connecting the Kuramoto Model and the Chimera State. PHYSICAL REVIEW LETTERS 2017; 119:264101. [PMID: 29328734 DOI: 10.1103/physrevlett.119.264101] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2017] [Indexed: 05/14/2023]
Abstract
Since its discovery in 2002, the chimera state has frequently been described as a counterintuitive, puzzling phenomenon. The Kuramoto model, in contrast, has become a celebrated paradigm useful for understanding a range of phenomena related to phase transitions, synchronization, and network effects. Here we show that the chimera state can be understood as emerging naturally through a symmetry-breaking bifurcation from the Kuramoto model's partially synchronized state. Our analysis sheds light on recent observations of chimera states in laser arrays, chemical oscillators, and mechanical pendula.
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Affiliation(s)
- Tejas Kotwal
- Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India
| | - Xin Jiang
- LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
| | - Daniel M Abrams
- Department of Engineering Sciences and Applied Mathematics; Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
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34
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Tian C, Bi H, Zhang X, Guan S, Liu Z. Asymmetric couplings enhance the transition from chimera state to synchronization. Phys Rev E 2017; 96:052209. [PMID: 29347748 DOI: 10.1103/physreve.96.052209] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2017] [Indexed: 06/07/2023]
Abstract
Chimera state has been well studied recently, but little attention has been paid to its transition to synchronization. We study this topic here by considering two groups of adaptively coupled Kuramoto oscillators. By searching the final states of different initial conditions, we find that the system can easily show a chimera state with robustness to initial conditions, in contrast to the sensitive dependence of chimera state on initial conditions in previous studies. Further, we show that, in the case of symmetric couplings, the behaviors of the two groups are always complementary to each other, i.e., robustness of chimera state, except a small basin of synchronization. Interestingly, we reveal that the basin of synchronization will be significantly increased when either the coupling of inner groups or that of intergroups are asymmetric. This transition from the attractor of chimera state to the attractor of synchronization is closely related to both the phase delay and the asymmetric degree of coupling strengths, resulting in a diversity of attractor's patterns. A theory based on the Ott-Antonsen ansatz is given to explain the numerical simulations. This finding may be meaningful for the control of competition between two attractors in biological systems, such as the cardiac rhythm and ventricular fibrillation, etc.
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Affiliation(s)
- Changhai Tian
- Department of Physics, East China Normal University, Shanghai 200062, People's Republic of China
- School of Data Science, Tongren University, Tongren 554300, People's Republic of China
| | - Hongjie Bi
- Department of Physics, East China Normal University, Shanghai 200062, People's Republic of China
| | - Xiyun Zhang
- Department of Physics, East China Normal University, Shanghai 200062, People's Republic of China
| | - Shuguang Guan
- Department of Physics, East China Normal University, Shanghai 200062, People's Republic of China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai 200062, People's Republic of China
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35
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Dai Q, Liu D, Cheng H, Li H, Yang J. Two-frequency chimera state in a ring of nonlocally coupled Brusselators. PLoS One 2017; 12:e0187067. [PMID: 29077732 PMCID: PMC5659769 DOI: 10.1371/journal.pone.0187067] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2017] [Accepted: 10/12/2017] [Indexed: 11/19/2022] Open
Abstract
Chimera states, which consist of coexisting domains of spatially coherent and incoherent dynamics, have been intensively investigated in the past decade. In this work, we report a special chimera state, 2-frequency chimera state, in one-dimensional ring of nonlocally coupled Brusselators. In a 2-frequency chimera state, there exist two types of coherent domains and oscillators in different types of coherent domains have different mean phase velocities. We present the stability diagram of 2-frequency chimera state and study the transition between the 2-frequency chimera state and an ordinary 2-cluster chimera state.
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Affiliation(s)
- Qionglin Dai
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
| | - Danna Liu
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
| | - Hongyan Cheng
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
| | - Haihong Li
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
| | - Junzhong Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
- * E-mail:
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36
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Andrzejak RG, Ruzzene G, Malvestio I. Generalized synchronization between chimera states. CHAOS (WOODBURY, N.Y.) 2017; 27:053114. [PMID: 28576111 DOI: 10.1063/1.4983841] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Networks of coupled oscillators in chimera states are characterized by an intriguing interplay of synchronous and asynchronous motion. While chimera states were initially discovered in mathematical model systems, there is growing experimental and conceptual evidence that they manifest themselves also in natural and man-made networks. In real-world systems, however, synchronization and desynchronization are not only important within individual networks but also across different interacting networks. It is therefore essential to investigate if chimera states can be synchronized across networks. To address this open problem, we use the classical setting of ring networks of non-locally coupled identical phase oscillators. We apply diffusive drive-response couplings between pairs of such networks that individually show chimera states when there is no coupling between them. The drive and response networks are either identical or they differ by a variable mismatch in their phase lag parameters. In both cases, already for weak couplings, the coherent domain of the response network aligns its position to the one of the driver networks. For identical networks, a sufficiently strong coupling leads to identical synchronization between the drive and response. For non-identical networks, we use the auxiliary system approach to demonstrate that generalized synchronization is established instead. In this case, the response network continues to show a chimera dynamics which however remains distinct from the one of the driver. Hence, segregated synchronized and desynchronized domains in individual networks congregate in generalized synchronization across networks.
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Affiliation(s)
- Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Catalonia, Spain
| | - Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Catalonia, Spain
| | - Irene Malvestio
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Catalonia, Spain
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37
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Ujjwal SR, Punetha N, Prasad A, Ramaswamy R. Emergence of chimeras through induced multistability. Phys Rev E 2017; 95:032203. [PMID: 28415241 DOI: 10.1103/physreve.95.032203] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2016] [Indexed: 11/07/2022]
Abstract
Chimeras, namely coexisting desynchronous and synchronized dynamics, are formed in an ensemble of identically coupled identical chaotic oscillators when the coupling induces multiple stable attractors, and further when the basins of the different attractors are intertwined in a complex manner. When there is coupling-induced multistability, an ensemble of identical chaotic oscillators-with global coupling, or also under the influence of common noise or an external drive (chaotic, periodic, or quasiperiodic)-inevitably exhibits chimeric behavior. Induced multistability in the system leads to the formation of distinct subpopulations, one or more of which support synchronized dynamics, while in others the motion is asynchronous or incoherent. We study the mechanism for the emergence of such chimeric states, and we discuss the generality of our results.
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Affiliation(s)
- Sangeeta Rani Ujjwal
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Nirmal Punetha
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, D-01187 Dresden, Germany
| | - Awadhesh Prasad
- Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
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38
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Schmidt A, Kasimatis T, Hizanidis J, Provata A, Hövel P. Chimera patterns in two-dimensional networks of coupled neurons. Phys Rev E 2017; 95:032224. [PMID: 28415206 DOI: 10.1103/physreve.95.032224] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2016] [Indexed: 06/07/2023]
Abstract
We discuss synchronization patterns in networks of FitzHugh-Nagumo and leaky integrate-and-fire oscillators coupled in a two-dimensional toroidal geometry. A common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have demonstrated that both models when coupled nonlocally in one-dimensional ring networks produce chimera states for a large range of parameter values. In this study, we give evidence of a plethora of two-dimensional chimera patterns of various shapes, including spots, rings, stripes, and grids, observed in both models, as well as additional patterns found mainly in the FitzHugh-Nagumo system. Both systems exhibit multistability: For the same parameter values, different initial conditions give rise to different dynamical states. Transitions occur between various patterns when the parameters (coupling range, coupling strength, refractory period, and coupling phase) are varied. Many patterns observed in the two models follow similar rules. For example, the diameter of the rings grows linearly with the coupling radius.
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Affiliation(s)
- Alexander Schmidt
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Theodoros Kasimatis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos", 15310 Athens, Greece
- School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Athens, Greece
| | - Johanne Hizanidis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos", 15310 Athens, Greece
- Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of Crete, 71003 Heraklion, Greece
| | - Astero Provata
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos", 15310 Athens, Greece
| | - Philipp Hövel
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany
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39
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Cano AV, Cosenza MG. Chimeras and clusters in networks of hyperbolic chaotic oscillators. Phys Rev E 2017; 95:030202. [PMID: 28415379 DOI: 10.1103/physreve.95.030202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2017] [Indexed: 06/07/2023]
Abstract
We show that chimera states, where differentiated subsets of synchronized and desynchronized dynamical elements coexist, can emerge in networks of hyperbolic chaotic oscillators subject to global interactions. As local dynamics we employ Lozi maps, which possess hyperbolic chaotic attractors. We consider a globally coupled system of these maps and use two statistical quantities to describe its collective behavior: the average fraction of elements belonging to clusters and the average standard deviation of state variables. Chimera states, clusters, complete synchronization, and incoherence are thus characterized on the space of parameters of the system. We find that chimera states are related to the formation of clusters in the system. In addition, we show that chimera states arise for a sufficiently long range of interactions in nonlocally coupled networks of these maps. Our results reveal that, under some circumstances, hyperbolicity does not impede the formation of chimera states in networks of coupled chaotic systems, as it had been previously hypothesized.
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Affiliation(s)
- A V Cano
- Grupo de Caos y Sistemas Complejos, Centro de Física Fundamental, Universidad de Los Andes, Mérida 5101, Mérida, Venezuela
| | - M G Cosenza
- Grupo de Caos y Sistemas Complejos, Centro de Física Fundamental, Universidad de Los Andes, Mérida 5101, Mérida, Venezuela
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40
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Kalle P, Sawicki J, Zakharova A, Schöll E. Chimera states and the interplay between initial conditions and non-local coupling. CHAOS (WOODBURY, N.Y.) 2017; 27:033110. [PMID: 28364760 DOI: 10.1063/1.4977866] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We study chimera states in a network of non-locally coupled Stuart-Landau oscillators. We investigate the impact of initial conditions in combination with non-local coupling. Based on an analytical argument, we show how the coupling phase and the coupling strength are linked to the occurrence of chimera states, flipped profiles of the mean phase velocity, and the transition from a phase- to an amplitude-mediated chimera state.
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Affiliation(s)
- Peter Kalle
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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41
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Kouvaris NE, Requejo RJ, Hizanidis J, Díaz-Guilera A. Chimera states in a network-organized public goods game with destructive agents. CHAOS (WOODBURY, N.Y.) 2016; 26:123108. [PMID: 28039967 DOI: 10.1063/1.4971974] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We found that a network-organized metapopulation of cooperators, defectors, and destructive agents playing the public goods game with mutations can collectively reach global synchronization or chimera states. Global synchronization is accompanied by a collective periodic burst of cooperation, whereas chimera states reflect the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation. Numerical simulations have shown that the system's dynamics switches between these two steady states through a first order transition. Depending on the parameters determining the dynamical and topological properties, chimera states with different numbers of coherent and incoherent clusters are observed. Our results present the first systematic study of chimera states and their characterization in the context of evolutionary game theory. This provides a valuable insight into the details of their occurrence, extending the relevance of such states to natural and social systems.
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Affiliation(s)
- Nikos E Kouvaris
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
| | - Rubén J Requejo
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
| | - Johanne Hizanidis
- Crete Center for Quantum Complexity and Nanotechnology, Physics Department, University of Crete, 71003 Heraklion, Greece
| | - Albert Díaz-Guilera
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
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42
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Belykh IV, Brister BN, Belykh VN. Bistability of patterns of synchrony in Kuramoto oscillators with inertia. CHAOS (WOODBURY, N.Y.) 2016; 26:094822. [PMID: 27781476 DOI: 10.1063/1.4961435] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern. We derive analytical conditions for the co-existence of stable two-cluster patterns with constant and oscillating phase shifts. We demonstrate that the dynamics, that governs the bistability of the phase shifts, is described by a driven pendulum equation. We also discuss the implications of our stability results to the stability of chimeras.
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Affiliation(s)
- Igor V Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, 30 Pryor Street, Atlanta, Georgia 30303, USA
| | - Barrett N Brister
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, 30 Pryor Street, Atlanta, Georgia 30303, USA
| | - Vladimir N Belykh
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23, Gagarin Ave., 603950 Nizhny Novgorod, Russia
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43
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Ulonska S, Omelchenko I, Zakharova A, Schöll E. Chimera states in networks of Van der Pol oscillators with hierarchical connectivities. CHAOS (WOODBURY, N.Y.) 2016; 26:094825. [PMID: 27781460 DOI: 10.1063/1.4962913] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We analyse chimera states in ring networks of Van der Pol oscillators with hierarchical coupling topology. We investigate the stepwise transition from a nonlocal to a hierarchical topology and propose the network clustering coefficient as a measure to establish a link between the existence of chimera states and the compactness of the initial base pattern of a hierarchical topology; we show that a large clustering coefficient promotes the occurrence of chimeras. Depending on the level of hierarchy and base pattern, we obtain chimera states with different numbers of incoherent domains. We investigate the chimera regimes as a function of coupling strength and nonlinearity parameter of the individual oscillators. The analysis of a network with larger base pattern resulting in larger clustering coefficient reveals two different types of chimera states and highlights the increasing role of amplitude dynamics.
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Affiliation(s)
- Stefan Ulonska
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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44
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Omelchenko I, Omel'chenko OE, Zakharova A, Wolfrum M, Schöll E. Tweezers for Chimeras in Small Networks. PHYSICAL REVIEW LETTERS 2016; 116:114101. [PMID: 27035303 DOI: 10.1103/physrevlett.116.114101] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2015] [Indexed: 05/26/2023]
Abstract
We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generally difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized.
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Affiliation(s)
- Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | | | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | | | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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45
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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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46
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Abstract
We study the existence of chimera states in pulse-coupled networks of bursting Hindmarsh-Rose neurons with nonlocal, global, and local (nearest neighbor) couplings. Through a linear stability analysis, we discuss the behavior of the stability function in the incoherent (i.e., disorder), coherent, chimera, and multichimera states. Surprisingly, we find that chimera and multichimera states occur even using local nearest neighbor interaction in a network of identical bursting neurons alone. This is in contrast with the existence of chimera states in populations of nonlocally or globally coupled oscillators. A chemical synaptic coupling function is used which plays a key role in the emergence of chimera states in bursting neurons. The existence of chimera, multichimera, coherent, and disordered states is confirmed by means of the recently introduced statistical measures and mean phase velocity.
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Affiliation(s)
- Bidesh K Bera
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli-620024, India
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47
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Panaggio MJ, Abrams DM, Ashwin P, Laing CR. Chimera states in networks of phase oscillators: The case of two small populations. Phys Rev E 2016; 93:012218. [PMID: 26871084 DOI: 10.1103/physreve.93.012218] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2015] [Indexed: 05/01/2023]
Abstract
Chimera states are dynamical patterns in networks of coupled oscillators in which regions of synchronous and asynchronous oscillation coexist. Although these states are typically observed in large ensembles of oscillators and analyzed in the continuum limit, chimeras may also occur in systems with finite (and small) numbers of oscillators. Focusing on networks of 2N phase oscillators that are organized in two groups, we find that chimera states, corresponding to attracting periodic orbits, appear with as few as two oscillators per group and demonstrate that for N>2 the bifurcations that create them are analogous to those observed in the continuum limit. These findings suggest that chimeras, which bear striking similarities to dynamical patterns in nature, are observable and robust in small networks that are relevant to a variety of real-world systems.
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Affiliation(s)
- Mark J Panaggio
- Mathematics Department, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803, USA
| | - Daniel M Abrams
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
- Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA
| | - Peter Ashwin
- Centre for Systems, Dynamics and Control, Harrison Building, Exeter EX4 4QF, United Kingdom
| | - Carlo R Laing
- Institute of Natural and Mathematical Sciences, Massey University, Private Bag 102-904 NSMC, Auckland, New Zealand
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48
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Ujjwal SR, Punetha N, Ramaswamy R. Phase oscillators in modular networks: The effect of nonlocal coupling. Phys Rev E 2016; 93:012207. [PMID: 26871073 DOI: 10.1103/physreve.93.012207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2015] [Indexed: 06/05/2023]
Abstract
We study the dynamics of nonlocally coupled phase oscillators in a modular network. The interactions include a phase lag, α. Depending on the various parameters the system exhibits a number of different dynamical states. In addition to global synchrony there can also be modular synchrony when each module can synchronize separately to a different frequency. There can also be multicluster frequency chimeras, namely coherent domains consisting of modules that are separately synchronized to different frequencies, coexisting with modules within which the dynamics is desynchronized. We apply the Ott-Antonsen ansatz in order to reduce the effective dimensionality and thereby carry out a detailed analysis of the different dynamical states.
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Affiliation(s)
- Sangeeta Rani Ujjwal
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India
| | - Nirmal Punetha
- Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India
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49
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Bastidas VM, Omelchenko I, Zakharova A, Schöll E, Brandes T. Quantum signatures of chimera states. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062924. [PMID: 26764791 DOI: 10.1103/physreve.92.062924] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2015] [Indexed: 06/05/2023]
Abstract
Chimera states are complex spatiotemporal patterns in networks of identical oscillators, characterized by the coexistence of synchronized and desynchronized dynamics. Here we propose to extend the phenomenon of chimera states to the quantum regime, and uncover intriguing quantum signatures of these states. We calculate the quantum fluctuations about semiclassical trajectories and demonstrate that chimera states in the quantum regime can be characterized by bosonic squeezing, weighted quantum correlations, and measures of mutual information. Our findings reveal the relation of chimera states to quantum information theory, and give promising directions for experimental realization of chimera states in quantum systems.
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Affiliation(s)
- V M Bastidas
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - I Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - A Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - E Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - T Brandes
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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50
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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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