1
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Nag Chowdhury S, Anwar MS, Ghosh D. Cluster formation due to repulsive spanning trees in attractively coupled networks. Phys Rev E 2024; 109:044314. [PMID: 38755838 DOI: 10.1103/physreve.109.044314] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2023] [Accepted: 03/22/2024] [Indexed: 05/18/2024]
Abstract
Ensembles of coupled nonlinear oscillators are a popular paradigm and an ideal benchmark for analyzing complex collective behaviors. The onset of cluster synchronization is found to be at the core of various technological and biological processes. The current literature has investigated cluster synchronization by focusing mostly on the case of attractive coupling among the oscillators. However, the case of two coexisting competing interactions is of practical interest due to their relevance in diverse natural settings, including neuronal networks consisting of excitatory and inhibitory neurons, the coevolving social model with voters of opposite opinions, and ecological plant communities with both facilitation and competition, to name a few. In the present article, we investigate the impact of repulsive spanning trees on cluster formation within a connected network of attractively coupled limit-cycle oscillators. We successfully predict which nodes belong to each cluster and the emergent frustration of the connected networks independent of the particular local dynamics at the network nodes. We also determine local asymptotic stability of the cluster states using an approach based on the formulation of a master stability function. We additionally validate the emergence of solitary states and antisynchronization for some specific choices of spanning trees and networks.
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Affiliation(s)
- Sayantan Nag Chowdhury
- Department of Environmental Science and Policy, University of California, Davis, Davis, California 95616, USA
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Md Sayeed Anwar
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
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2
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Elaeva M, Blanter E, Shnirman M, Shapoval A. Asymmetry in the Kuramoto model with nonidentical coupling. Phys Rev E 2023; 107:064201. [PMID: 37464665 DOI: 10.1103/physreve.107.064201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2022] [Accepted: 04/26/2023] [Indexed: 07/20/2023]
Abstract
Synchronization and desynchronization of coupled oscillators appear to be the key property of many physical systems. It is believed that to predict a synchronization (or desynchronization) event, the knowledge on the exact structure of the oscillatory network is required. However, natural sciences often deal with observations where the coupling coefficients are not available. In the present paper we suggest a way to characterize synchronization of two oscillators without the reconstruction of coupling. Our method is based on the Kuramoto chain with three oscillators with constant but nonidentical coupling. We characterize coupling in this chain by two parameters: the coupling strength s and disparity σ. We give an analytical expression of the boundary s_{max} of synchronization occurred when s>s_{max}. We propose asymmetry A of the generalized order parameter induced by the coupling disparity as a new characteristic of the synchronization between two oscillators. For the chain model with three oscillators we present the self-consistent inverse problem. We explore scaling properties of the asymmetry A constructed for the inverse problem. We demonstrate that the asymmetry A in the chain model is maximal when the coupling strength in the model reaches the boundary of synchronization s_{max}. We suggest that the asymmetry A may be derived from the phase difference of any two oscillators if one pretends that they are edges of an abstract chain with three oscillators. Performing such a derivation with the general three-oscillator Kuramoto model, we show that the crossover from the chain to general network of oscillators keeps the interrelation between the asymmetry A and synchronization. Finally, we apply the asymmetry A to describe synchronization of the solar magnetic field proxies and discuss its potential use for the forecast of solar cycle anomalies.
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Affiliation(s)
- M Elaeva
- Department of Higher Mathematics, HSE University, Moscow 109028, Russia
| | - E Blanter
- Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS, Moscow 117997, Russia
| | - M Shnirman
- Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS, Moscow 117997, Russia
| | - A Shapoval
- Department of Mathematics and Computer Science, University of Lodz, Lodz 90-238, Poland and Cybersecurity Center, Universidad Bernardo O'Higgins, Santiago 8370993, Chile
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3
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Ray A, Bröhl T, Mishra A, Ghosh S, Ghosh D, Kapitaniak T, Dana SK, Hens C. Extreme events in a complex network: Interplay between degree distribution and repulsive interaction. CHAOS (WOODBURY, N.Y.) 2022; 32:121103. [PMID: 36587354 DOI: 10.1063/5.0128743] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2022] [Accepted: 11/16/2022] [Indexed: 06/17/2023]
Abstract
The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and, thereby, confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator associated with a node.
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Affiliation(s)
- Arnob Ray
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Timo Bröhl
- Department of Epileptology, University Hospital Bonn, Venusberg Campus 1, 53127 Bonn, Germany
| | - Arindam Mishra
- Department of Physics, National University of Singapore, Singapore 117551
| | - Subrata Ghosh
- Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology, Gachibowli, Hyderabad 500032, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Tomasz Kapitaniak
- Division of Dynamics, Lodz University of Technology, 90-924 Lodz, Poland
| | - Syamal K Dana
- Department of Mathematics, National Institute of Technology Durgapur, Durgapur 713209, India
| | - Chittaranjan Hens
- Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology, Gachibowli, Hyderabad 500032, India
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4
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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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5
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Aravind M, Sinha S, Parmananda P. Competitive interplay of repulsive coupling and cross-correlated noises in bistable systems. CHAOS (WOODBURY, N.Y.) 2021; 31:061106. [PMID: 34241287 DOI: 10.1063/5.0056173] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2021] [Accepted: 05/25/2021] [Indexed: 06/13/2023]
Abstract
The influence of noise on synchronization has potential impact on physical, chemical, biological, and engineered systems. Research on systems subject to common noise has demonstrated that noise can aid synchronization, as common noise imparts correlations on the sub-systems. In our work, we revisit this idea for a system of bistable dynamical systems, under repulsive coupling, driven by noises with varying degrees of cross correlation. This class of coupling has not been fully explored, and we show that it offers new counter-intuitive emergent behavior. Specifically, we demonstrate that the competitive interplay of noise and coupling gives rise to phenomena ranging from the usual synchronized state to the uncommon anti-synchronized state where the coupled bistable systems are pushed to different wells. Interestingly, this progression from anti-synchronization to synchronization goes through a domain where the system randomly hops between the synchronized and anti-synchronized states. The underlying basis for this striking behavior is that correlated noise preferentially enhances coherence, while the interactions provide an opposing drive to push the states apart. Our results also shed light on the robustness of synchronization obtained in the idealized scenario of perfectly correlated noise, as well as the influence of noise correlation on anti-synchronization. Last, the experimental implementation of our model using bistable electronic circuits, where we were able to sweep a large range of noise strengths and noise correlations in the laboratory realization of this noise-driven coupled system, firmly indicates the robustness and generality of our observations.
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Affiliation(s)
- Manaoj Aravind
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
| | - Sudeshna Sinha
- Indian Institute of Science Education and Research Mohali, Knowledge City, SAS Nagar, Sector 81, Manauli, Punjab P.O. 140306, India
| | - P Parmananda
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
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6
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Crnkić A, Povh J, Jaćimović V, Levnajić Z. Collective dynamics of phase-repulsive oscillators solves graph coloring problem. CHAOS (WOODBURY, N.Y.) 2020; 30:033128. [PMID: 32237769 DOI: 10.1063/1.5127794] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2019] [Accepted: 02/20/2020] [Indexed: 06/11/2023]
Abstract
We show how to couple phase-oscillators on a graph so that collective dynamics "searches" for the coloring of that graph as it relaxes toward the dynamical equilibrium. This translates a combinatorial optimization problem (graph coloring) into a functional optimization problem (finding and evaluating the global minimum of dynamical non-equilibrium potential, done by the natural system's evolution). Using a sample of graphs, we show that our method can serve as a viable alternative to the traditional combinatorial algorithms. Moreover, we show that, with the same computational cost, our method efficiently solves the harder problem of improper coloring of weighed graphs.
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Affiliation(s)
- Aladin Crnkić
- Faculty of Technical Engineering, University of Bihać, Ljubijankićeva, bb., 77000 Bihać, Bosnia and Herzegovina
| | - Janez Povh
- Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva cesta 6, 1000 Ljubljana, Slovenia
| | - Vladimir Jaćimović
- Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put, bb., 81000 Podgorica, Montenegro
| | - Zoran Levnajić
- Complex Systems and Data Science Lab, Faculty of Information Studies in Novo Mesto, Ljubljanska cesta 31A, 8000 Novo Mesto, Slovenia
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7
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Chowdhury SN, Ghosh D, Hens C. Effect of repulsive links on frustration in attractively coupled networks. Phys Rev E 2020; 101:022310. [PMID: 32168719 DOI: 10.1103/physreve.101.022310] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2019] [Accepted: 01/31/2020] [Indexed: 06/10/2023]
Abstract
We investigate the impact of attractive-repulsive interaction in networks of limit cycle oscillators. Mainly we focus on the design principle for generating an antiphase state between adjacent nodes in a complex network. We establish that a partial negative control throughout the branches of a spanning tree inside the positively coupled limit cycle oscillators works efficiently well in comparison with randomly chosen negative links to establish zero frustration (antiphase synchronization) in bipartite graphs. Based on the emergence of zero frustration, we develop a universal 0-π rule to understand the antiphase synchronization in a bipartite graph. Further, this rule is used to construct a nonbipartite graph for a given nonzero frustrated value. We finally show the generality of 0-π rule by implementing it in arbitrary undirected nonbipartite graphs of attractive-repulsively coupled limit cycle oscillators and successfully calculate the nonzero frustration value, which matches with numerical data. The validation of the rule is checked through the bifurcation analysis of small networks. Our work may unveil the underlying mechanism of several synchronization phenomena that exist in a network of oscillators having a mixed type of coupling.
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Affiliation(s)
- Sayantan Nag Chowdhury
- 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
| | - Chittaranjan Hens
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108, India
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8
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Johnson JD, Abrams DM. A coupled oscillator model for the origin of bimodality and multimodality. CHAOS (WOODBURY, N.Y.) 2019; 29:073120. [PMID: 31370422 DOI: 10.1063/1.5100289] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2019] [Accepted: 07/08/2019] [Indexed: 06/10/2023]
Abstract
Perhaps because of the elegance of the central limit theorem, it is often assumed that distributions in nature will approach singly-peaked, unimodal shapes reminiscent of the Gaussian normal distribution. However, many systems behave differently, with variables following apparently bimodal or multimodal distributions. Here, we argue that multimodality may emerge naturally as a result of repulsive or inhibitory coupling dynamics, and we show rigorously how it emerges for a broad class of coupling functions in variants of the paradigmatic Kuramoto model.
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Affiliation(s)
- J D Johnson
- Department of Engineering Sciences and Applied Mathematics, McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA
| | - D M Abrams
- Department of Engineering Sciences and Applied Mathematics, McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA
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9
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Kale P, Zalesky A, Gollo LL. Estimating the impact of structural directionality: How reliable are undirected connectomes? Netw Neurosci 2018; 2:259-284. [PMID: 30234180 PMCID: PMC6135560 DOI: 10.1162/netn_a_00040] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2017] [Accepted: 12/19/2017] [Indexed: 11/30/2022] Open
Abstract
Directionality is a fundamental feature of network connections. Most structural brain networks are intrinsically directed because of the nature of chemical synapses, which comprise most neuronal connections. Because of the limitations of noninvasive imaging techniques, the directionality of connections between structurally connected regions of the human brain cannot be confirmed. Hence, connections are represented as undirected, and it is still unknown how this lack of directionality affects brain network topology. Using six directed brain networks from different species and parcellations (cat, mouse, C. elegans, and three macaque networks), we estimate the inaccuracies in network measures (degree, betweenness, clustering coefficient, path length, global efficiency, participation index, and small-worldness) associated with the removal of the directionality of connections. We employ three different methods to render directed brain networks undirected: (a) remove unidirectional connections, (b) add reciprocal connections, and (c) combine equal numbers of removed and added unidirectional connections. We quantify the extent of inaccuracy in network measures introduced through neglecting connection directionality for individual nodes and across the network. We find that the coarse division between core and peripheral nodes remains accurate for undirected networks. However, hub nodes differ considerably when directionality is neglected. Comparing the different methods to generate undirected networks from directed ones, we generally find that the addition of reciprocal connections (false positives) causes larger errors in graph-theoretic measures than the removal of the same number of directed connections (false negatives). These findings suggest that directionality plays an essential role in shaping brain networks and highlight some limitations of undirected connectomes.
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Affiliation(s)
- Penelope Kale
- QIMR Berghofer Medical Research Institute, Australia
- University of Queensland, Australia
| | - Andrew Zalesky
- Melbourne Neuropsychiatry Centre and Department of Biomedical Engineering, University of Melbourne, Australia
| | - Leonardo L. Gollo
- QIMR Berghofer Medical Research Institute, Australia
- University of Queensland, Australia
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10
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Yan Y, Xu J, Wiercigroch M. Basins of attraction of the bistable region of time-delayed cutting dynamics. Phys Rev E 2017; 96:032205. [PMID: 29347033 DOI: 10.1103/physreve.96.032205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2017] [Indexed: 06/07/2023]
Abstract
This paper investigates the effects of bistability in a nonsmooth time-delayed dynamical system, which is often manifested in science and engineering. Previous studies on cutting dynamics have demonstrated persistent coexistence of chatter and chatter-free responses in a bistable region located in the linearly stable zone. As there is no widely accepted definition of basins of attraction for time-delayed systems, bistable regions are coined as unsafe zones (UZs). Hence, we have attempted to define the basins of attraction and stability basins for a typical delayed system to get insight into the bistability in systems with time delays. Special attention was paid to the influences of delayed initial conditions, starting points, and states at time zero on the long-term dynamics of time-delayed systems. By using this concept, it has been confirmed that the chatter is prone to occur when the waviness frequency in the workpiece surface coincides with the effective natural frequency of the cutting process. Further investigations unveil a thin "boundary layer" inside the UZ in the immediate vicinity of the stability boundary, in which we observe an extremely fast growth of the chatter basin stability. The results reveal that the system is more stable when the initial cutting depth is smaller. The physics of the tool deflection at the instant of the tool-workpiece engagement is used to evaluate the cutting safety, and the safe level could be zero when the geometry of tool engagement is unfavorable. Finally, the basins of attraction are used to quench the chatter by a single strike, where the resultant "islands" offer an opportunity to suppress the chatter even when the cutting is very close to the stability boundary.
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Affiliation(s)
- Yao Yan
- School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Jian Xu
- School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
| | - Marian Wiercigroch
- Centre for Applied Dynamics Research, School of Engineering, King's College, University of Aberdeen, Aberdeen, AB24 3UE Scotland, UK
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11
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Emergence of local synchronization in neuronal networks with adaptive couplings. PLoS One 2017; 12:e0178975. [PMID: 28575125 PMCID: PMC5456398 DOI: 10.1371/journal.pone.0178975] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2016] [Accepted: 05/22/2017] [Indexed: 11/26/2022] Open
Abstract
Local synchronization, both prolonged and transient, of oscillatory neuronal behavior in cortical networks plays a fundamental role in many aspects of perception and cognition. Here we study networks of Hindmarsh-Rose neurons with a new type of adaptive coupling, and show that these networks naturally produce both permanent and transient synchronization of local clusters of neurons. These deterministic systems exhibit complex dynamics with 1/fη power spectra, which appears to be a consequence of a novel form of self-organized criticality.
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12
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Sathiyadevi K, Karthiga S, Chandrasekar VK, Senthilkumar DV, Lakshmanan M. Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings. Phys Rev E 2017; 95:042301. [PMID: 28505842 DOI: 10.1103/physreve.95.042301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2016] [Indexed: 06/07/2023]
Abstract
Spontaneous symmetry breaking is an important phenomenon observed in various fields including physics and biology. In this connection, we here show that the trade-off between attractive and repulsive couplings can induce spontaneous symmetry breaking in a homogeneous system of coupled oscillators. With a simple model of a system of two coupled Stuart-Landau oscillators, we demonstrate how the tendency of attractive coupling in inducing in-phase synchronized (IPS) oscillations and the tendency of repulsive coupling in inducing out-of-phase synchronized oscillations compete with each other and give rise to symmetry breaking oscillatory states and interesting multistabilities. Further, we provide explicit expressions for synchronized and antisynchronized oscillatory states as well as the so called oscillation death (OD) state and study their stability. If the Hopf bifurcation parameter (λ) is greater than the natural frequency (ω) of the system, the attractive coupling favors the emergence of an antisymmetric OD state via a Hopf bifurcation whereas the repulsive coupling favors the emergence of a similar state through a saddle-node bifurcation. We show that an increase in the repulsive coupling not only destabilizes the IPS state but also facilitates the reentrance of the IPS state.
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Affiliation(s)
- K Sathiyadevi
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamil Nadu, India
| | - S Karthiga
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamil Nadu, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695 016, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
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13
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Thakur B, Sharma D, Sen A, Johnston GL. Collective dynamics of time-delay-coupled phase oscillators in a frustrated geometry. Phys Rev E 2017; 95:012204. [PMID: 28208395 DOI: 10.1103/physreve.95.012204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2016] [Indexed: 06/06/2023]
Abstract
We study the effect of time delay on the dynamics of a system of repulsively coupled nonlinear oscillators that are configured as a geometrically frustrated network. In the absence of time delay, frustrated systems are known to possess a high degree of multistability among a large number of coexisting collective states except for the fully synchronized state that is normally obtained for attractively coupled systems. Time delay in the coupling is found to remove this constraint and to lead to such a synchronized ground state over a range of parameter values. A quantitative study of the variation of frustration in a system with the amount of time delay has been made and a universal scaling behavior is found. The variation in frustration as a function of the product of time delay and the collective frequency of the system is seen to lie on a characteristic curve that is common for all natural frequencies of the identical oscillators and coupling strengths. Thus time delay can be used as a tuning parameter to control the amount of frustration in a system and thereby influence its collective behavior. Our results can be of potential use in a host of practical applications in physical and biological systems in which frustrated configurations and time delay are known to coexist.
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Affiliation(s)
- Bhumika Thakur
- Institute for Plasma Research, HBNI, Bhat, Gandhinagar 382428, India
| | - Devendra Sharma
- Institute for Plasma Research, HBNI, Bhat, Gandhinagar 382428, India
| | - Abhijit Sen
- Institute for Plasma Research, HBNI, Bhat, Gandhinagar 382428, India
| | - George L Johnston
- EduTron Corporation, 5 Cox Road, Winchester, Massachusetts 01890, USA
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14
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Esfahani ZG, Gollo LL, Valizadeh A. Stimulus-dependent synchronization in delayed-coupled neuronal networks. Sci Rep 2016; 6:23471. [PMID: 27001428 PMCID: PMC4802300 DOI: 10.1038/srep23471] [Citation(s) in RCA: 36] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2015] [Accepted: 03/07/2016] [Indexed: 02/04/2023] Open
Abstract
Time delay is a general feature of all interactions. Although the effects of delayed interaction are often neglected when the intrinsic dynamics is much slower than the coupling delay, they can be crucial otherwise. We show that delayed coupled neuronal networks support transitions between synchronous and asynchronous states when the level of input to the network changes. The level of input determines the oscillation period of neurons and hence whether time-delayed connections are synchronizing or desynchronizing. We find that synchronizing connections lead to synchronous dynamics, whereas desynchronizing connections lead to out-of-phase oscillations in network motifs and to frustrated states with asynchronous dynamics in large networks. Since the impact of a neuronal network to downstream neurons increases when spikes are synchronous, networks with delayed connections can serve as gatekeeper layers mediating the firing transfer to other regions. This mechanism can regulate the opening and closing of communicating channels between cortical layers on demand.
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Affiliation(s)
- Zahra G Esfahani
- Department of physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
| | - Leonardo L Gollo
- Systems Neuroscience Group, QIMR Berghofer Medical Research Institute, Brisbane, Queensland, Australia
| | - Alireza Valizadeh
- Department of physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran.,School of Cognitive Sciences, IPM, Niavaran, Tehran, Iran
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15
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Astakhov S, Gulai A, Fujiwara N, Kurths J. The role of asymmetrical and repulsive coupling in the dynamics of two coupled van der Pol oscillators. CHAOS (WOODBURY, N.Y.) 2016; 26:023102. [PMID: 26931583 DOI: 10.1063/1.4940967] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2015] [Accepted: 01/14/2016] [Indexed: 06/05/2023]
Abstract
A system of two asymmetrically coupled van der Pol oscillators has been studied. We show that the introduction of a small asymmetry in coupling leads to the appearance of a "wideband synchronization channel" in the bifurcational structure of the parameter space. An increase of asymmetry and transition to repulsive interaction leads to the formation of multistability. As the result, the tip of the Arnold's tongue widens due to the formation of folds defined by saddle-node bifurcation curves for the limit cycles on the torus.
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Affiliation(s)
- Sergey Astakhov
- Information Security of Automated Systems Department, Yuri Gagarin State Technical University of Saratov, Politekhnitcheskaya st. 77, Saratov 410054, Russia
| | - Artem Gulai
- Radioelectronics and Telecommunications Department, Yuri Gagarin State Technical University of Saratov, Politekhnitcheskaya st. 77, Saratov 410054, Russia
| | - Naoya Fujiwara
- Center for Spatial Information Science, The University of Tokyo, 277-8568 Chiba, Japan
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany
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Fukunaga T, Imasaka T, Ito A, Sugitani Y, Konishi K, Hara N. Dynamical behavior and peak power reduction in a pair of energy storage oscillators coupled by delayed power price. Phys Rev E 2016; 93:022220. [PMID: 26986344 DOI: 10.1103/physreve.93.022220] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2015] [Indexed: 11/07/2022]
Abstract
This paper investigates dynamics of a management system for controlling a pair of energy storages. The system involves the following two characteristics: each storage behaves in a manner that reduces the number of charge noncharge cycles and begins to be charged when the price of power is lower than a particular price threshold. The price is proportional to the past total power flow from a power grid to all storages. A peak of the total power flow occurs when these storages are charged simultaneously. From the viewpoint of nonlinear dynamics, the energy storages can be considered as relaxation oscillators coupled by a delay connection. Our analytical results suggest that the peak can be reduced by inducing an antiphase synchronization in coupled oscillators. We confirm these analytical results through numerical simulations. In addition, we numerically investigate the dynamical behavior in 10 storages and find that time delay in the connection is important in reducing the peak.
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Affiliation(s)
- Tomohiro Fukunaga
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531 Japan
| | - Tomoaki Imasaka
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531 Japan
| | - Akira Ito
- Engineering R&D Center, DENSO CORPORATION, 1-1 Showa-cho, Kariya, Aichi 448-8661 Japan
| | - Yoshiki Sugitani
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531 Japan
| | - Keiji Konishi
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531 Japan
| | - Naoyuki Hara
- Department of Electrical and Information Systems, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531 Japan
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Zaks MA, Tomov P. Onset of time dependence in ensembles of excitable elements with global repulsive coupling. Phys Rev E 2016; 93:020201. [PMID: 26986274 DOI: 10.1103/physreve.93.020201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2015] [Indexed: 06/05/2023]
Abstract
We consider the effect of global repulsive coupling on an ensemble of identical excitable elements. An increase of the coupling strength destabilizes the synchronous equilibrium and replaces it with many attracting oscillatory states, created in the transcritical heteroclinic bifurcation. The period of oscillations is inversely proportional to the distance from the critical parameter value. If the elements interact with the global field via the first Fourier harmonics of their phases, the stable equilibrium is in one step replaced by the attracting continuum of periodic motions.
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Affiliation(s)
- Michael A Zaks
- Institute of Physics and Astronomy, Potsdam University, 14465 Potsdam, Germany
| | - Petar Tomov
- Institute of Physics, Humboldt University at Berlin, 12489 Berlin, Germany
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18
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Gollo LL, Breakspear M. The frustrated brain: from dynamics on motifs to communities and networks. Philos Trans R Soc Lond B Biol Sci 2015; 369:rstb.2013.0532. [PMID: 25180310 DOI: 10.1098/rstb.2013.0532] [Citation(s) in RCA: 46] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023] Open
Abstract
Cognitive function depends on an adaptive balance between flexible dynamics and integrative processes in distributed cortical networks. Patterns of zero-lag synchrony likely underpin numerous perceptual and cognitive functions. Synchronization fulfils integration by reducing entropy, while adaptive function mandates that a broad variety of stable states be readily accessible. Here, we elucidate two complementary influences on patterns of zero-lag synchrony that derive from basic properties of brain networks. First, mutually coupled pairs of neuronal subsystems-resonance pairs-promote stable zero-lag synchrony among the small motifs in which they are embedded, and whose effects can propagate along connected chains. Second, frustrated closed-loop motifs disrupt synchronous dynamics, enabling metastable configurations of zero-lag synchrony to coexist. We document these two complementary influences in small motifs and illustrate how these effects underpin stable versus metastable phase-synchronization patterns in prototypical modular networks and in large-scale cortical networks of the macaque (CoCoMac). We find that the variability of synchronization patterns depends on the inter-node time delay, increases with the network size and is maximized for intermediate coupling strengths. We hypothesize that the dialectic influences of resonance versus frustration may form a dynamic substrate for flexible neuronal integration, an essential platform across diverse cognitive processes.
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Affiliation(s)
- Leonardo L Gollo
- Systems Neuroscience Group, QIMR Berghofer Medical Research Institute, Brisbane, Queensland, Australia
| | - Michael Breakspear
- Systems Neuroscience Group, QIMR Berghofer Medical Research Institute, Brisbane, Queensland, Australia School of Psychiatry, University of New South Wales and The Black Dog Institute, Sydney, New South Wales, Australia The Royal Brisbane and Women's Hospital, Brisbane, Queensland, Australia
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19
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Astakhov S, Fujiwara N, Gulay A, Tsukamoto N, Kurths J. Hopf bifurcation and multistability in a system of phase oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032908. [PMID: 24125326 DOI: 10.1103/physreve.88.032908] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2012] [Revised: 04/26/2013] [Indexed: 06/02/2023]
Abstract
We study the phase reduction of two coupled van der Pol oscillators with asymmetric repulsive coupling under an external harmonic force. We show that the system of two phase oscillators undergoes a Hopf bifurcation and possesses multistability on a 2π-periodic phase plane. We describe the bifurcation mechanisms of formation of multistability in the phase-reduced system and show that the Andronov-Hopf bifurcation in the phase-reduced system is not an artifact of the reduction approach but, indeed, has its prototype in the nonreduced system. The bifurcational mechanisms presented in the paper enable one to describe synchronization effects in a wide class of interacting systems with repulsive coupling e.g., genetic oscillators.
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Affiliation(s)
- Sergey Astakhov
- Physics Department, Chair of Radiophysics and Nonlinear Dynamics, Saratov State University, 410012 Saratov, Russia
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20
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Zhang X, Ruan Z, Liu Z. An efficient approach to suppress the negative role of contrarian oscillators in synchronization. CHAOS (WOODBURY, N.Y.) 2013; 23:033135. [PMID: 24089971 DOI: 10.1063/1.4821426] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
It has been found that contrarian oscillators usually take a negative role in the collective behaviors formed by conformist oscillators. However, experiments revealed that it is also possible to achieve a strong coherence even when there are contrarians in the system such as neuron networks with both excitable and inhibitory neurons. To understand the underlying mechanism of this abnormal phenomenon, we here consider a complex network of coupled Kuramoto oscillators with mixed positive and negative couplings and present an efficient approach, i.e., tit-for-tat strategy, to suppress the negative role of contrarian oscillators in synchronization and thus increase the order parameter of synchronization. Two classes of contrarian oscillators are numerically studied and a brief theoretical analysis is provided to explain the numerical results.
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Affiliation(s)
- Xiyun Zhang
- Department of Physics, East China Normal University, Shanghai 200062, People's Republic of China
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21
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Ren Q, Long Q, Zhao J. Symmetry and symmetry breaking in a Kuramoto model induced on a Möbius strip. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022811. [PMID: 23496572 DOI: 10.1103/physreve.87.022811] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2012] [Indexed: 06/01/2023]
Abstract
The concept of bounded confidence in social dynamics is introduced into the Kuramoto model. Then a model with projective symmetry is naturally induced by a principal Z(2) bundle over a circle. A Möbius strip is constructed as the manifold of interactions such that two "sides" of a local section correspond to antipodal couplings with opposite signs. We show analytically that symmetric polarization of synchronized states (conformist vs contrarian) can emerge from the collective behavior of homogeneously coupled oscillators. In the continuum limit, a generalized asymmetric model reverts to the symmetric case under uniform initial condition, whereas for networks of finite size, it exhibits richer dynamics or the ordinary behavior of the classic model via symmetry breaking, depending on the relevant parameters values. The symmetry and symmetry breaking of the synchronization state are analogous to the concepts of equality and majority of opinions in sociophysical models of opinion formation. The synchronized clusters can be one (consensus), two (polarization), or more (fragmentation), as opinion clusters.
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Affiliation(s)
- Quansheng Ren
- School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
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22
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Levnajić Z. Evolutionary design of non-frustrated networks of phase-repulsive oscillators. Sci Rep 2012; 2:967. [PMID: 23243494 PMCID: PMC3522069 DOI: 10.1038/srep00967] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2012] [Accepted: 11/21/2012] [Indexed: 11/17/2022] Open
Abstract
Evolutionary optimisation algorithm is employed to design networks of phase-repulsive oscillators that achieve an anti-phase synchronised state. By introducing the link frustration, the evolutionary process is implemented by rewiring the links with probability proportional to their frustration, until the final network displaying a unique non-frustrated dynamical state is reached. Resulting networks are bipartite and with zero clustering. In addition, the designed non-frustrated anti-phase synchronised networks display a clear topological scale. This contrasts usually studied cases of networks with phase-attractive dynamics, whose performance towards full synchronisation is typically enhanced by the presence of a topological hierarchy.
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Affiliation(s)
- Zoran Levnajić
- Faculty of information studies in Novo mesto, Novo mesto, Slovenia.
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