1
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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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2
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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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3
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Effects of Synaptic Pruning on Phase Synchronization in Chimera States of Neural Network. APPLIED SCIENCES-BASEL 2022. [DOI: 10.3390/app12041942] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
This research explores the effect of synaptic pruning on a ring-shaped neural network of non-locally coupled FitzHugh–Nagumo (FHN) oscillators. The neurons in the pruned region synchronize with each other, and they repel the coherent domain of the chimera states. Furthermore, the width of the pruned region decides the precision and efficiency of the control effect on the position of coherent domains. This phenomenon gives a systematic comprehension of the relation between pruning and synchronization in neural networks from a new aspect that has never been addressed. An explanation of this mechanism is also given.
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4
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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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5
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Andrzejak RG. Chimeras confined by fractal boundaries in the complex plane. CHAOS (WOODBURY, N.Y.) 2021; 31:053104. [PMID: 34240923 DOI: 10.1063/5.0049631] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2021] [Accepted: 04/16/2021] [Indexed: 06/13/2023]
Abstract
Complex-valued quadratic maps either converge to fixed points, enter into periodic cycles, show aperiodic behavior, or diverge to infinity. Which of these scenarios takes place depends on the map's complex-valued parameter c and the initial conditions. The Mandelbrot set is defined by the set of c values for which the map remains bounded when initiated at the origin of the complex plane. In this study, we analyze the dynamics of a coupled network of two pairs of two quadratic maps in dependence on the parameter c. Across the four maps, c is kept the same whereby the maps are identical. In analogy to the behavior of individual maps, the network iterates either diverge to infinity or remain bounded. The bounded solutions settle into different stable states, including full synchronization and desynchronization of all maps. Furthermore, symmetric partially synchronized states of within-pair synchronization and across-pair synchronization as well as a symmetry broken chimera state are found. The boundaries between bounded and divergent solutions in the domain of c are fractals showing a rich variety of intriguingly esthetic patterns. Moreover, the set of bounded solutions is divided into countless subsets throughout all length scales in the complex plane. Each individual subset contains only one state of synchronization and is enclosed within fractal boundaries by c values leading to divergence.
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Affiliation(s)
- Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Catalonia, Spain
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6
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Controlling the Chimera Form in the Leaky Integrate-and-Fire Model. ADVANCES IN EXPERIMENTAL MEDICINE AND BIOLOGY 2021; 1338:247-258. [DOI: 10.1007/978-3-030-78775-2_30] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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7
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Bolotov MI, Smirnov LA, Osipov GV, Pikovsky A. Locking and regularization of chimeras by periodic forcing. Phys Rev E 2020; 102:042218. [PMID: 33212667 DOI: 10.1103/physreve.102.042218] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2019] [Accepted: 09/28/2020] [Indexed: 11/07/2022]
Abstract
We study how a chimera state in a one-dimensional medium of nonlocally coupled oscillators responds to a homogeneous in space periodic in time external force. On a macroscopic level, where a chimera can be considered as an oscillating object, forcing leads to entrainment of the chimera's basic frequency inside an Arnold tongue. On a mesoscopic level, where a chimera can be viewed as an inhomogeneous, stationary, or nonstationary pattern, strong forcing can lead to regularization of an unstationary chimera. On a microscopic level of the dynamics of individual oscillators, forcing outside of the Arnold tongue leads to a multiplateau state with nontrivial locking properties.
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Affiliation(s)
- Maxim I Bolotov
- Department of Control Theory, Research and Education Mathematical Center "Mathematics for Future Technologies," Nizhny Novgorod State University, Gagarin Av. 23, 603950, Nizhny Novgorod, Russia
| | - Lev A Smirnov
- Department of Control Theory, Research and Education Mathematical Center "Mathematics for Future Technologies," Nizhny Novgorod State University, Gagarin Av. 23, 603950, Nizhny Novgorod, Russia.,Institute of Applied Physics of the Russian Academy of Sciences, Ul'yanov Str. 46, 603950, Nizhny Novgorod, Russia
| | - Grigory V Osipov
- Department of Control Theory, Research and Education Mathematical Center "Mathematics for Future Technologies," Nizhny Novgorod State University, Gagarin Av. 23, 603950, Nizhny Novgorod, Russia
| | - Arkady Pikovsky
- Department of Control Theory, Research and Education Mathematical Center "Mathematics for Future Technologies," Nizhny Novgorod State University, Gagarin Av. 23, 603950, Nizhny Novgorod, Russia.,Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24-25, 14476 Potsdam, Germany.,National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia
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8
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Ruzzene G, Omelchenko I, Sawicki J, Zakharova A, Schöll E, Andrzejak RG. Remote pacemaker control of chimera states in multilayer networks of neurons. Phys Rev E 2020; 102:052216. [PMID: 33327161 DOI: 10.1103/physreve.102.052216] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 10/30/2020] [Indexed: 06/12/2023]
Abstract
Networks of coupled nonlinear oscillators allow for the formation of nontrivial partially synchronized spatiotemporal patterns, such as chimera states, in which there are coexisting coherent (synchronized) and incoherent (desynchronized) domains. These complementary domains form spontaneously, and it is impossible to predict where the synchronized group will be positioned within the network. Therefore, possible ways to control the spatial position of the coherent and incoherent groups forming the chimera states are of high current interest. In this work we investigate how to control chimera patterns in multiplex networks of FitzHugh-Nagumo neurons, and in particular we want to prove that it is possible to remotely control chimera states exploiting the multiplex structure. We introduce a pacemaker oscillator within the network: this is an oscillator that does not receive input from the rest of the network but is sending out information to its neighbors. The pacemakers can be positioned in one or both layers. Their presence breaks the spatial symmetry of the layer in which they are introduced and allows us to control the position of the incoherent domain. We demonstrate how the remote control is possible for both uni- and bidirectional coupling between the layers. Furthermore we show which are the limitations of our control mechanisms when it is generalized from single-layer to multilayer networks.
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Affiliation(s)
- Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Iryna Omelchenko
- 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
| | - Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
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9
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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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10
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Lilienkamp T, Parlitz U. Susceptibility of transient chimera states. Phys Rev E 2020; 102:032219. [PMID: 33075925 DOI: 10.1103/physreve.102.032219] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2019] [Accepted: 08/26/2020] [Indexed: 11/07/2022]
Abstract
Chaotic dynamics of a dynamical system is not necessarily persistent. If there is (without any active intervention from outside) a transition towards a (possibly nonchaotic) attractor, this phenomenon is called transient chaos, which can be observed in a variety of systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. Also, chimera states, which show coherent and incoherent dynamics in spatially distinct regions of the system, are often chaotic transients. In many practical cases, the control of the chaotic dynamics (either the termination or the preservation of the chaotic dynamics) is desired. Although the self-termination typically occurs quite abruptly and can so far in general not be properly predicted, previous studies showed that in many systems a 'terminal transient phase" (TTP) prior to the self-termination existed, where the system was less susceptible against small but finite perturbations in different directions in state space. In this study, we show that, in the specific case of chimera states, these susceptible directions can be related to the structure of the chimera, which we divide into the coherent part, the incoherent part and the boundary in between. That means, in practice, if self-termination is close we can identify the direction of perturbation which is likely to maintain the chaotic dynamics (the chimera state). This finding improves the general understanding of the state space structure during the TTP, and could contribute also to practical applications like future control strategies of epileptic seizures which have been recently related to the collapse of chimera states.
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Affiliation(s)
- Thomas Lilienkamp
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.,German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.,German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany.,Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
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11
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Rybalova E, Bukh A, Strelkova G, Anishchenko V. Spiral and target wave chimeras in a 2D lattice of map-based neuron models. CHAOS (WOODBURY, N.Y.) 2019; 29:101104. [PMID: 31675811 DOI: 10.1063/1.5126178] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2019] [Accepted: 09/23/2019] [Indexed: 06/10/2023]
Abstract
We study the dynamics of a two-dimensional lattice of nonlocally coupled-map-based neuron models represented by Rulkov maps. It is firstly shown that this discrete-time neural network can exhibit spiral and target waves and corresponding chimera states when the control parameters (the coupling strength and the coupling radius) are varied. It is demonstrated that one-core, multicore, and ring-shaped core spiral chimeras can be realized in the network. We also reveal a novel type of chimera structure-a target wave chimera. We explore the transition from spiral wave chimeras to target wave structures when varying the coupling parameters. We report for the first time that the spiral wave regime can be suppressed by applying noise excitations, and the subsequent transition to the target wave mode occurs.
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Affiliation(s)
- E Rybalova
- Department of Physics, Saratov National Research State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - A Bukh
- Department of Physics, Saratov National Research State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - G Strelkova
- Department of Physics, Saratov National Research State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - V Anishchenko
- Department of Physics, Saratov National Research State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
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