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Bollt E, Fish J, Kumar A, Roque Dos Santos E, Laurienti PJ. Fractal basins as a mechanism for the nimble brain. Sci Rep 2023; 13:20860. [PMID: 38012212 PMCID: PMC10682042 DOI: 10.1038/s41598-023-45664-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2023] [Accepted: 10/22/2023] [Indexed: 11/29/2023] Open
Abstract
An interesting feature of the brain is its ability to respond to disparate sensory signals from the environment in unique ways depending on the environmental context or current brain state. In dynamical systems, this is an example of multi-stability, the ability to switch between multiple stable states corresponding to specific patterns of brain activity/connectivity. In this article, we describe chimera states, which are patterns consisting of mixed synchrony and incoherence, in a brain-inspired dynamical systems model composed of a network with weak individual interactions and chaotic/periodic local dynamics. We illustrate the mechanism using synthetic time series interacting on a realistic anatomical brain network derived from human diffusion tensor imaging. We introduce the so-called vector pattern state (VPS) as an efficient way of identifying chimera states and mapping basin structures. Clustering similar VPSs for different initial conditions, we show that coexisting attractors of such states reveal intricately "mingled" fractal basin boundaries that are immediately reachable. This could explain the nimble brain's ability to rapidly switch patterns between coexisting attractors.
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Affiliation(s)
- Erik Bollt
- Department of Electrical and Computer Engineering, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA.
- Clarkson Center for Complex Systems Science, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA.
| | - Jeremie Fish
- Department of Electrical and Computer Engineering, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA
- Clarkson Center for Complex Systems Science, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA
| | - Anil Kumar
- Department of Electrical and Computer Engineering, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA
- Clarkson Center for Complex Systems Science, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA
| | - Edmilson Roque Dos Santos
- Department of Electrical and Computer Engineering, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA
- Clarkson Center for Complex Systems Science, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA
- Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Av. Trab. São Carlense, 400, São Carlos, SP, 13566-590, Brazil
| | - Paul J Laurienti
- Department of Radiology, Wake Forest University School of Medicine, 475 Vine Street, Winston-Salem, NC, 27101, USA
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Mishra A, Saha S, Dana SK. Chimeras in globally coupled oscillators: A review. CHAOS (WOODBURY, N.Y.) 2023; 33:092101. [PMID: 37703474 DOI: 10.1063/5.0143872] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 08/21/2023] [Indexed: 09/15/2023]
Abstract
The surprising phenomenon of chimera in an ensemble of identical oscillators is no more strange behavior of network dynamics and reality. By this time, this symmetry breaking self-organized collective dynamics has been established in many networks, a ring of non-locally coupled oscillators, globally coupled networks, a three-dimensional network, and multi-layer networks. A variety of coupling and dynamical models in addition to the phase oscillators has been used for a successful observation of chimera patterns. Experimental verification has also been done using metronomes, pendula, chemical, and opto-electronic systems. The phenomenon has also been shown to appear in small networks, and hence, it is not size-dependent. We present here a brief review of the origin of chimera patterns restricting our discussions to networks of globally coupled identical oscillators only. The history of chimeras in globally coupled oscillators is older than what has been reported in nonlocally coupled phase oscillators much later. We elaborate the story of the origin of chimeras in globally coupled oscillators in a chronological order, within our limitations, and with brief descriptions of the significant contributions, including our personal experiences. We first introduce chimeras in non-locally coupled and other network configurations, in general, and then discuss about globally coupled networks in more detail.
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Affiliation(s)
- Arindam Mishra
- Department of Physics, National University of Singapore, Singapore 117551
| | - Suman Saha
- Cognitive Brain Dynamics Laboratory, National Brain Research Centre, Gurugram 122051, India
| | - Syamal K Dana
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
- Division of Dynamics, Lodz University of Technology, 90-924 Lodz, Poland
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Lee S, Krischer K. Heteroclinic switching between chimeras in a ring of six oscillator populations. CHAOS (WOODBURY, N.Y.) 2023; 33:2894497. [PMID: 37276574 DOI: 10.1063/5.0147228] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2023] [Accepted: 05/15/2023] [Indexed: 06/07/2023]
Abstract
In a network of coupled oscillators, a symmetry-broken dynamical state characterized by the coexistence of coherent and incoherent parts can spontaneously form. It is known as a chimera state. We study chimera states in a network consisting of six populations of identical Kuramoto-Sakaguchi phase oscillators. The populations are arranged in a ring, and oscillators belonging to one population are uniformly coupled to all oscillators within the same population and to those in the two neighboring populations. This topology supports the existence of different configurations of coherent and incoherent populations along the ring, but all of them are linearly unstable in most of the parameter space. Yet, chimera dynamics is observed from random initial conditions in a wide parameter range, characterized by one incoherent and five synchronized populations. These observable states are connected to the formation of a heteroclinic cycle between symmetric variants of saddle chimeras, which gives rise to a switching dynamics. We analyze the dynamical and spectral properties of the chimeras in the thermodynamic limit using the Ott-Antonsen ansatz and in finite-sized systems employing Watanabe-Strogatz reduction. For a heterogeneous frequency distribution, a small heterogeneity renders a heteroclinic switching dynamics asymptotically attracting. However, for a large heterogeneity, the heteroclinic orbit does not survive; instead, it is replaced by a variety of attracting chimera states.
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Affiliation(s)
- Seungjae Lee
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
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Liu W, Xie J, Liu H, Xiao J. Heterogeneity induced splay state of amplitude envelope in globally coupled oscillators. CHAOS (WOODBURY, N.Y.) 2022; 32:123117. [PMID: 36587328 DOI: 10.1063/5.0130753] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2022] [Accepted: 11/11/2022] [Indexed: 06/17/2023]
Abstract
Splay states of the amplitude envelope are stably observed as a heterogenous node is introduced into the globally coupled identical oscillators with repulsive coupling. With the increment of the frequency mismatches between the heterogenous nodes and the rest identical globally coupled oscillators, the formal stable splay state based on the time series becomes unstable, while a splay state based on the new-born amplitude envelopes of time series is stably observed among the rest identical oscillators. The characteristics of the splay state based on the amplitude envelope are numerically and theoretically presented for different parameters of the coupling strength ϵ and the frequency mismatches Δω for small coupling strength and large frequency mismatches. We expect that all these results could reveal the generality of splay states in coupled nonidentical oscillators and help to understand the rich dynamics of amplitude envelopes in multidisciplinary fields.
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Affiliation(s)
- Weiqing Liu
- School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
| | - Jiangnan Xie
- School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
| | - Hanchang Liu
- School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
| | - Jinghua Xiao
- School of Science, Beijing University of Posts and Communications, Beijing 100876, China
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Ebrahimzadeh P, Schiek M, Maistrenko Y. Mixed-mode chimera states in pendula networks. CHAOS (WOODBURY, N.Y.) 2022; 32:103118. [PMID: 36319296 DOI: 10.1063/5.0103071] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Accepted: 09/21/2022] [Indexed: 06/16/2023]
Abstract
We report the emergence of peculiar chimera states in networks of identical pendula with global phase-lagged coupling. The states reported include both rotating and quiescent modes, i.e., with non-zero and zero average frequencies. This kind of mixed-mode chimeras may be interpreted as images of bump states known in neuroscience in the context of modeling the working memory. We illustrate this striking phenomenon for a network of N = 100 coupled pendula, followed by a detailed description of the minimal non-trivial case of N = 3. Parameter regions for five characteristic types of the system behavior are identified, which consist of two mixed-mode chimeras with one and two rotating pendula, classical weak chimera with all three pendula rotating, synchronous rotation, and quiescent state. The network dynamics is multistable: up to four of the states can coexist in the system phase state as demonstrated through the basins of attraction. The analysis suggests that the robust mixed-mode chimera states can generically describe the complex dynamics of diverse pendula-like systems widespread in nature.
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Affiliation(s)
- P Ebrahimzadeh
- ZEA-2: Electronics Systems, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany
| | - M Schiek
- ZEA-2: Electronics Systems, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany
| | - Y Maistrenko
- ZEA-2: Electronics Systems, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany
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Kachhara S, Ambika G. Frequency chimera state induced by differing dynamical timescales. Phys Rev E 2021; 104:064214. [PMID: 35030851 DOI: 10.1103/physreve.104.064214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2021] [Accepted: 12/04/2021] [Indexed: 06/14/2023]
Abstract
We report the occurrence of a self-emerging frequency chimera state in spatially extended systems of coupled oscillators, where the coherence and incoherence are defined with respect to the emergent frequency of the oscillations. This is generated by the local coupling among nonlinear oscillators evolving under differing dynamical timescales starting from random initial conditions. We show how they self-organize to structured patterns with spatial domains of coherence that are in frequency synchronization, coexisting with domains that are incoherent in frequencies. Our study has relevance in understanding such patterns observed in real-world systems like neuronal systems, power grids, social and ecological networks, where differing dynamical timescales is natural and realistic among the interacting systems.
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Affiliation(s)
- Sneha Kachhara
- Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India
| | - G Ambika
- Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India
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Jaros P, Levchenko R, Kapitaniak T, Maistrenko Y. Chimera states for directed networks. CHAOS (WOODBURY, N.Y.) 2021; 31:103111. [PMID: 34717326 DOI: 10.1063/5.0059765] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2021] [Accepted: 09/28/2021] [Indexed: 06/13/2023]
Abstract
We demonstrate that chimera behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny chimera islands arise in the parameter space. They are surrounded by developed chaotic switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, as we show for a hundred oscillators (cyclic century), the islands merge into a single chimera continent, which incorporates the world of chimeras of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as the coupling range decreases.
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Affiliation(s)
- Patrycja Jaros
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | | | - Tomasz Kapitaniak
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Yuri Maistrenko
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
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Berner R, Yanchuk S, Maistrenko Y, Schöll E. Generalized splay states in phase oscillator networks. CHAOS (WOODBURY, N.Y.) 2021; 31:073128. [PMID: 34340340 DOI: 10.1063/5.0056664] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2021] [Accepted: 06/24/2021] [Indexed: 06/13/2023]
Abstract
Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we introduce generalized m-splay states constituting a special subclass of phase-locked states with vanishing mth order parameter. Such states typically manifest incoherent dynamics, and they often create high-dimensional families of solutions (splay manifolds). For a general class of phase oscillator networks, we provide explicit linear stability conditions for splay states and exemplify our results with the well-known Kuramoto-Sakaguchi model. Importantly, our stability conditions are expressed in terms of just a few observables such as the order parameter or the trace of the Jacobian. As a result, these conditions are simple and applicable to networks of arbitrary size. We generalize our findings to phase oscillators with inertia and adaptively coupled phase oscillator models.
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Affiliation(s)
- Rico Berner
- Institute of Theoretical Physics, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Serhiy Yanchuk
- Institute of Mathematics, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
| | - Yuri Maistrenko
- Forschungszentrum Jülich GmbH, Wilhelm-Johnen-Straße, 52428 Jülich, Germany
| | - Eckehard Schöll
- Institute of Theoretical Physics, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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