1
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Ocampo-Espindola JL, Nikhil KL, Li JS, Herzog ED, Kiss IZ. Synchronization, clustering, and weak chimeras in a densely coupled transcription-based oscillator model for split circadian rhythms. CHAOS (WOODBURY, N.Y.) 2023; 33:083105. [PMID: 37535024 PMCID: PMC10403273 DOI: 10.1063/5.0156135] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2023] [Accepted: 07/08/2023] [Indexed: 08/04/2023]
Abstract
The synchronization dynamics for the circadian gene expression in the suprachiasmatic nucleus is investigated using a transcriptional circadian clock gene oscillator model. With global coupling in constant dark (DD) conditions, the model exhibits a one-cluster phase synchronized state, in dim light (dim LL), bistability between one- and two-cluster states and in bright LL, a two-cluster state. The two-cluster phase synchronized state, where some oscillator pairs synchronize in-phase, and some anti-phase, can explain the splitting of the circadian clock, i.e., generation of two bouts of daily activities with certain species, e.g., with hamsters. The one- and two-cluster states can be reached by transferring the animal from DD or bright LL to dim LL, i.e., the circadian synchrony has a memory effect. The stability of the one- and two-cluster states was interpreted analytically by extracting phase models from the ordinary differential equation models. In a modular network with two strongly coupled oscillator populations with weak intragroup coupling, with appropriate initial conditions, one group is synchronized to the one-cluster state and the other group to the two-cluster state, resulting in a weak-chimera state. Computational modeling suggests that the daily rhythms in sleep-wake depend on light intensity acting on bilateral networks of suprachiasmatic nucleus (SCN) oscillators. Addition of a network heterogeneity (coupling between the left and right SCN) allowed the system to exhibit chimera states. The simulations can guide experiments in the circadian rhythm research to explore the effect of light intensity on the complexities of circadian desynchronization.
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Affiliation(s)
| | - K. L. Nikhil
- Department of Biology, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130-4899, USA
| | - Jr-Shin Li
- Department of Electrical and Systems Engineering, Washington University in St Louis, 1 Brookings Drive, St. Louis, Missouri 63130, USA
| | - Erik D. Herzog
- Department of Biology, Washington University in St. Louis, One Brookings Drive, St. Louis, Missouri 63130-4899, USA
| | - István Z. Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Ave., St. Louis, Missouri 63103, USA
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2
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Galinsky VL, Frank LR. Neuronal avalanches: Sandpiles of self-organized criticality or critical dynamics of brain waves? FRONTIERS OF PHYSICS 2023; 18:45301. [PMID: 37008280 PMCID: PMC10062440 DOI: 10.1007/s11467-023-1273-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2022] [Accepted: 02/23/2023] [Indexed: 06/19/2023]
Abstract
Analytical expressions for scaling of brain wave spectra derived from the general nonlinear wave Hamiltonian form show excellent agreement with experimental "neuronal avalanche" data. The theory of the weakly evanescent nonlinear brain wave dynamics [Phys. Rev. Research 2, 023061 (2020); J. Cognitive Neurosci. 32, 2178 (2020)] reveals the underlying collective processes hidden behind the phenomenological statistical description of the neuronal avalanches and connects together the whole range of brain activity states, from oscillatory wave-like modes, to neuronal avalanches, to incoherent spiking, showing that the neuronal avalanches are just the manifestation of the different nonlinear side of wave processes abundant in cortical tissue. In a more broad way these results show that a system of wave modes interacting through all possible combinations of the third order nonlinear terms described by a general wave Hamiltonian necessarily produces anharmonic wave modes with temporal and spatial scaling properties that follow scale free power laws. To the best of our knowledge this has never been reported in the physical literature and may be applicable to many physical systems that involve wave processes and not just to neuronal avalanches.
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Affiliation(s)
- Vitaly L. Galinsky
- Center for Scientific Computation in Imaging, University of California at San Diego, La Jolla, CA 92037-0854, USA
| | - Lawrence R. Frank
- Center for Scientific Computation in Imaging, University of California at San Diego, La Jolla, CA 92037-0854, USA
- Center for Functional MRI, University of California at San Diego, La Jolla, CA 92037-0677, USA
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3
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Berner R, Lu A, Sokolov IM. Synchronization transitions in Kuramoto networks with higher-mode interaction. CHAOS (WOODBURY, N.Y.) 2023; 33:073138. [PMID: 37463093 DOI: 10.1063/5.0151038] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2023] [Accepted: 06/21/2023] [Indexed: 07/20/2023]
Abstract
Synchronization is an omnipresent collective phenomenon in nature and technology, whose understanding is still elusive for real-world systems in particular. We study the synchronization transition in a phase oscillator system with two nonvanishing Fourier-modes in the interaction function, hence going beyond the Kuramoto paradigm. We show that the transition scenarios crucially depend on the interplay of the two coupling modes. We describe the multistability induced by the presence of a second coupling mode. By extending the collective coordinate approach, we describe the emergence of various states observed in the transition from incoherence to coherence. Remarkably, our analysis suggests that, in essence, the two-mode coupling gives rise to states characterized by two independent but interacting groups of oscillators. We believe that these findings will stimulate future research on dynamical systems, including complex interaction functions beyond the Kuramoto-type.
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Affiliation(s)
- Rico Berner
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Annie Lu
- Department of Mathematics, Washington State University, Pullman, Washington 99164-3113, USA
| | - Igor M Sokolov
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
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4
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Bergoin R, Torcini A, Deco G, Quoy M, Zamora-López G. Inhibitory neurons control the consolidation of neural assemblies via adaptation to selective stimuli. Sci Rep 2023; 13:6949. [PMID: 37117236 PMCID: PMC10147639 DOI: 10.1038/s41598-023-34165-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2022] [Accepted: 04/25/2023] [Indexed: 04/30/2023] Open
Abstract
Brain circuits display modular architecture at different scales of organization. Such neural assemblies are typically associated to functional specialization but the mechanisms leading to their emergence and consolidation still remain elusive. In this paper we investigate the role of inhibition in structuring new neural assemblies driven by the entrainment to various inputs. In particular, we focus on the role of partially synchronized dynamics for the creation and maintenance of structural modules in neural circuits by considering a network of excitatory and inhibitory [Formula: see text]-neurons with plastic Hebbian synapses. The learning process consists of an entrainment to temporally alternating stimuli that are applied to separate regions of the network. This entrainment leads to the emergence of modular structures. Contrary to common practice in artificial neural networks-where the acquired weights are typically frozen after the learning session-we allow for synaptic adaptation even after the learning phase. We find that the presence of inhibitory neurons in the network is crucial for the emergence and the post-learning consolidation of the modular structures. Indeed networks made of purely excitatory neurons or of neurons not respecting Dale's principle are unable to form or to maintain the modular architecture induced by the stimuli. We also demonstrate that the number of inhibitory neurons in the network is directly related to the maximal number of neural assemblies that can be consolidated, supporting the idea that inhibition has a direct impact on the memory capacity of the neural network.
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Affiliation(s)
- Raphaël Bergoin
- ETIS, UMR 8051, ENSEA, CY Cergy Paris Université, CNRS, 6 Av. du Ponceau, 95000, Cergy-Pontoise, France.
- Center for Brain and Cognition, Department of Information and Communications Technologies, Pompeu Fabra University, Carrer Ramón Trias i Fargas 25-27, 08005, Barcelona, Spain.
| | - Alessandro Torcini
- Laboratoire de Physique Théorique et Modélisation, UMR 8089, CY Cergy Paris Université, CNRS, 2 Av. Adolphe Chauvin, 95032, Cergy-Pontoise, France
| | - Gustavo Deco
- Center for Brain and Cognition, Department of Information and Communications Technologies, Pompeu Fabra University, Carrer Ramón Trias i Fargas 25-27, 08005, Barcelona, Spain
- Instituciò Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluis Companys 23, 08010, Barcelona, Spain
| | - Mathias Quoy
- ETIS, UMR 8051, ENSEA, CY Cergy Paris Université, CNRS, 6 Av. du Ponceau, 95000, Cergy-Pontoise, France
- IPAL, CNRS, 1 Fusionopolis Way #21-01 Connexis (South Tower), Singapore, 138632, Singapore
| | - Gorka Zamora-López
- Center for Brain and Cognition, Department of Information and Communications Technologies, Pompeu Fabra University, Carrer Ramón Trias i Fargas 25-27, 08005, Barcelona, Spain
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5
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Abstract
Analytical expressions for scaling of brain wave spectra derived from the general non-linear wave Hamiltonian form show excellent agreement with experimental "neuronal avalanche" data. The theory of the weakly evanescent non-linear brain wave dynamics reveals the underlying collective processes hidden behind the phenomenological statistical description of the neuronal avalanches and connects together the whole range of brain activity states, from oscillatory wave-like modes, to neuronal avalanches, to incoherent spiking, showing that the neuronal avalanches are just the manifestation of the different non-linear side of wave processes abundant in cortical tissue. In a more broad way these results show that a system of wave modes interacting through all possible combinations of the third order non-linear terms described by a general wave Hamiltonian necessarily produces anharmonic wave modes with temporal and spatial scaling properties that follow scale free power laws. To the best of our knowledge this has never been reported in the physical literature and may be applicable to many physical systems that involve wave processes and not just to neuronal avalanches.
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Affiliation(s)
- Vitaly L. Galinsky
- Center for Scientific Computation in Imaging, University of California, San Diego, San Diego, CA, United States
| | - Lawrence R. Frank
- Center for Scientific Computation in Imaging, University of California, San Diego, San Diego, CA, United States
- Center for Functional MRI, University of California, San Diego, San Diego, CA, United States
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6
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Berner R, Sawicki J, Thiele M, Löser T, Schöll E. Critical Parameters in Dynamic Network Modeling of Sepsis. FRONTIERS IN NETWORK PHYSIOLOGY 2022; 2:904480. [PMID: 36926088 PMCID: PMC10012967 DOI: 10.3389/fnetp.2022.904480] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2022] [Accepted: 05/03/2022] [Indexed: 11/13/2022]
Abstract
In this work, we propose a dynamical systems perspective on the modeling of sepsis and its organ-damaging consequences. We develop a functional two-layer network model for sepsis based upon the interaction of parenchymal cells and immune cells via cytokines, and the coevolutionary dynamics of parenchymal, immune cells, and cytokines. By means of the simple paradigmatic model of phase oscillators in a two-layer system, we analyze the emergence of organ threatening interactions between the dysregulated immune system and the parenchyma. We demonstrate that the complex cellular cooperation between parenchyma and stroma (immune layer) either in the physiological or in the pathological case can be related to dynamical patterns of the network. In this way we explain sepsis by the dysregulation of the healthy homeostatic state (frequency synchronized) leading to a pathological state (desynchronized or multifrequency cluster) in the parenchyma. We provide insight into the complex stabilizing and destabilizing interplay of parenchyma and stroma by determining critical interaction parameters. The coupled dynamics of parenchymal cells (metabolism) and nonspecific immune cells (response of the innate immune system) is represented by nodes of a duplex layer. Cytokine interaction is modeled by adaptive coupling weights between nodes representing immune cells (with fast adaptation timescale) and parenchymal cells (slow adaptation timescale), and between pairs of parenchymal and immune cells in the duplex network (fixed bidirectional coupling). The proposed model allows for a functional description of organ dysfunction in sepsis and the recurrence risk in a plausible pathophysiological context.
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Affiliation(s)
- Rico Berner
- Institut für Physik, Humboldt-Universität zu Berlin, Berlin, Germany.,Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany.,Potsdam Institute for Climate Impact Research, Potsdam, Germany.,Fachhochschule Nordwestschweiz FHNW, Basel, Switzerland
| | - Max Thiele
- Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany
| | | | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany.,Potsdam Institute for Climate Impact Research, Potsdam, Germany.,Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, Berlin, Germany
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7
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Kusters GLA, Tito NB, Storm C, van der Schoot P. Controlling permeation in electrically deforming liquid crystal network films: A dynamical Landau theory. Phys Rev E 2021; 104:054701. [PMID: 34942727 DOI: 10.1103/physreve.104.054701] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2021] [Accepted: 11/02/2021] [Indexed: 11/07/2022]
Abstract
Liquid crystal networks exploit the coupling between the responsivity of liquid crystalline mesogens, e.g., to electric fields, and the (visco)elastic properties of a polymer network. Because of this, these materials have been put forward for a wide array of applications, including responsive surfaces such as artificial skins and membranes. For such applications, the desired functional response must generally be realized under strict geometrical constraints, such as provided by supported thin films. To model such settings, we present a dynamical, spatially heterogeneous Landau-type theory for electrically actuated liquid crystal network films. We find that the response of the liquid crystal network permeates the film from top to bottom, and illustrate how this affects the timescale associated with macroscopic deformation. Finally, by linking our model parameters to experimental quantities, we suggest that the permeation rate can be controlled by varying the aspect ratio of the mesogens and their degree of orientational order when crosslinked into the polymer network, for which we predict a single optimum. Our results contribute specifically to the rational design of future applications involving transport or on-demand release of molecular cargo in liquid crystal network films.
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Affiliation(s)
- Guido L A Kusters
- Department of Applied Physics, Eindhoven University of Technology, The Netherlands
| | | | - Cornelis Storm
- Department of Applied Physics, Eindhoven University of Technology, The Netherlands.,Institute for Complex Molecular Systems, Eindhoven University of Technology, The Netherlands
| | - Paul van der Schoot
- Department of Applied Physics, Eindhoven University of Technology, The Netherlands
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8
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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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9
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Yoneda R, Harada K, Yamaguchi YY. Critical exponents in coupled phase-oscillator models on small-world networks. Phys Rev E 2021; 102:062212. [PMID: 33465963 DOI: 10.1103/physreve.102.062212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2020] [Accepted: 11/23/2020] [Indexed: 11/07/2022]
Abstract
A coupled phase-oscillator model consists of phase oscillators, each of which has the natural frequency obeying a probability distribution and couples with other oscillators through a given periodic coupling function. This type of model is widely studied since it describes the synchronization transition, which emerges between the nonsynchronized state and partially synchronized states. The synchronization transition is characterized by several critical exponents, and we focus on the critical exponent defined by coupling strength dependence of the order parameter for revealing universality classes. In a typical interaction represented by the perfect graph, an infinite number of universality classes is yielded by dependency on the natural frequency distribution and the coupling function. Since the synchronization transition is also observed in a model on a small-world network, whose number of links is proportional to the number of oscillators, a natural question is whether the infinite number of universality classes remains in small-world networks irrespective of the order of links. Our numerical results suggest that the number of universality classes is reduced to one and the critical exponent is shared in the considered models having coupling functions up to second harmonics with unimodal and symmetric natural frequency distributions.
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Affiliation(s)
- Ryosuke Yoneda
- Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
| | - Kenji Harada
- Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
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10
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Zhang M, Beetle C, Kelso JAS, Tognoli E. Connecting empirical phenomena and theoretical models of biological coordination across scales. J R Soc Interface 2019; 16:20190360. [PMID: 31409241 PMCID: PMC6731488 DOI: 10.1098/rsif.2019.0360] [Citation(s) in RCA: 25] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2019] [Accepted: 07/15/2019] [Indexed: 11/18/2022] Open
Abstract
Coordination in living systems-from cells to people-must be understood at multiple levels of description. Analyses and modelling of empirically observed patterns of biological coordination often focus either on ensemble-level statistics in large-scale systems with many components, or on detailed dynamics in small-scale systems with few components. The two approaches have proceeded largely independent of each other. To bridge this gap between levels and scales, we have recently conducted a human experiment of mid-scale social coordination specifically designed to reveal coordination at multiple levels (ensemble, subgroups and dyads) simultaneously. Based on this experiment, the present work shows that, surprisingly, a single system of equations captures key observations at all relevant levels. It also connects empirically validated models of large- and small-scale biological coordination-the Kuramoto and extended Haken-Kelso-Bunz (HKB) models-and the hallmark phenomena that each is known to capture. For example, it exhibits both multistability and metastability observed in small-scale empirical research (via the second-order coupling and symmetry breaking in extended HKB) and the growth of biological complexity as a function of scale (via the scalability of the Kuramoto model). Only by incorporating both of these features simultaneously can we reproduce the essential coordination behaviour observed in our experiment.
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Affiliation(s)
- Mengsen Zhang
- Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, FL, USA
| | | | - J. A. Scott Kelso
- Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, FL, USA
- Intelligent Systems Research Centre, Ulster University, Derry∼Londonderry, Northern Ireland
| | - Emmanuelle Tognoli
- Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, FL, USA
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11
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Saw VL, Chung NN, Quek WL, Pang YEI, Chew LY. Bus bunching as a synchronisation phenomenon. Sci Rep 2019; 9:6887. [PMID: 31053731 PMCID: PMC6499774 DOI: 10.1038/s41598-019-43310-7] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2019] [Accepted: 04/18/2019] [Indexed: 11/09/2022] Open
Abstract
Bus bunching is a perennial phenomenon that not only diminishes the efficiency of a bus system, but also prevents transit authorities from keeping buses on schedule. We present a physical theory of buses serving a loop of bus stops as a ring of coupled self-oscillators, analogous to the Kuramoto model. Sustained bunching is a repercussion of the process of phase synchronisation whereby the phases of the oscillators are locked to each other. This emerges when demand exceeds a critical threshold. Buses also bunch at low demand, albeit temporarily, due to frequency detuning arising from different human drivers’ distinct natural speeds. We calculate the critical transition when complete phase locking (full synchronisation) occurs for the bus system, and posit the critical transition to completely no phase locking (zero synchronisation). The intermediate regime is the phase where clusters of partially phase locked buses exist. Intriguingly, these theoretical results are in close correspondence to real buses in a university’s shuttle bus system.
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Affiliation(s)
- Vee-Liem Saw
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore.,Data Science and Artificial Intelligence Research Centre, Nanyang Technological University, Block N4 #02a-32, Nanyang Avenue, Singapore, 639798, Singapore
| | - Ning Ning Chung
- Complexity Institute, Nanyang Technological University, 61 Nanyang Drive, Singapore, 637335, Singapore
| | - Wei Liang Quek
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore
| | - Yi En Ian Pang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore
| | - Lock Yue Chew
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore. .,Data Science and Artificial Intelligence Research Centre, Nanyang Technological University, Block N4 #02a-32, Nanyang Avenue, Singapore, 639798, Singapore. .,Complexity Institute, Nanyang Technological University, 61 Nanyang Drive, Singapore, 637335, Singapore.
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12
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Abstract
We investigate the dynamics of a population of identical biomolecules mimicked as electric dipoles with random orientations and positions in space and oscillating with their intrinsic frequencies. The biomolecules, beyond being coupled among themselves via the dipolar interaction, are also driven by a common external energy supply. A collective mode emerges by decreasing the average distance among the molecules as testified by the emergence of a clear peak in the power spectrum of the total dipole moment. This is due to a coherent vibration of the most part of the molecules at a frequency definitely larger than their own frequencies corresponding to a partial cluster synchronization of the biomolecules. These results can be verified experimentally via spectroscopic investigations of the strength of the intermolecular electrodynamic interactions, thus being able to test the possible biological relevance of the observed macroscopic mode.
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13
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Ogawa S, Yamaguchi YY. Landau-like theory for universality of critical exponents in quasistationary states of isolated mean-field systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062108. [PMID: 26172662 DOI: 10.1103/physreve.91.062108] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2014] [Indexed: 06/04/2023]
Abstract
An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two nonclassical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends the universality class of the nonclassical exponents to spatially periodic one-dimensional systems and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum.
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Affiliation(s)
- Shun Ogawa
- Aix Marseille Université, Université de Toulon, CNRS, Centre de Physique Théorique UMR7332, 13288 Marseille, France
| | - Yoshiyuki Y Yamaguchi
- Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, 606-8501, Kyoto, Japan
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14
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Dodla R, Wilson CJ. Interaction function of oscillating coupled neurons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042704. [PMID: 24229210 PMCID: PMC3928969 DOI: 10.1103/physreve.88.042704] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2013] [Revised: 08/28/2013] [Indexed: 06/02/2023]
Abstract
Large scale simulations of electrically coupled neuronal oscillators often employ the phase coupled oscillator paradigm to understand and predict network behavior. We study the nature of the interaction between such coupled oscillators using weakly coupled oscillator theory. By employing piecewise linear approximations for phase response curves and voltage time courses and parametrizing their shapes, we compute the interaction function for all such possible shapes and express it in terms of discrete Fourier modes. We find that reasonably good approximation is achieved with four Fourier modes that comprise of both sine and cosine terms.
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Affiliation(s)
- Ramana Dodla
- Department of Biology, University of Texas at San Antonio, San Antonio, Texas 78249, USA
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15
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Nishikawa I, Tanaka G, Aihara K. Nonstandard scaling law of fluctuations in finite-size systems of globally coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:024102. [PMID: 24032967 DOI: 10.1103/physreve.88.024102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2013] [Indexed: 06/02/2023]
Abstract
Universal scaling laws form one of the central issues in physics. A nonstandard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we found that a statistical quantity related to fluctuations follows a nonstandard scaling law with respect to the system size in a synchronized state of globally coupled nonidentical phase oscillators [I. Nishikawa et al., Chaos 22, 013133 (2012)]. However, it is still unclear how widely this nonstandard scaling law is observed. In the present paper, we discuss the conditions required for the unusual scaling law in globally coupled oscillator systems and validate the conditions by numerical simulations of several different models.
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Affiliation(s)
- Isao Nishikawa
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
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16
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Temirbayev AA, Nalibayev YD, Zhanabaev ZZ, Ponomarenko VI, Rosenblum M. Autonomous and forced dynamics of oscillator ensembles with global nonlinear coupling: an experimental study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062917. [PMID: 23848758 DOI: 10.1103/physreve.87.062917] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2013] [Revised: 05/14/2013] [Indexed: 06/02/2023]
Abstract
We perform experiments with 72 electronic limit-cycle oscillators, globally coupled via a linear or nonlinear feedback loop. While in the linear case we observe a standard Kuramoto-like synchronization transition, in the nonlinear case, with increase of the coupling strength, we first observe a transition to full synchrony and then a desynchronization transition to a quasiperiodic state. However, in this state the ensemble remains coherent so that the amplitude of the mean field is nonzero, but the frequency of the mean field is larger than frequencies of all oscillators. Next, we analyze effects of common periodic forcing of the linearly or nonlinearly coupled ensemble and demonstrate regimes when the mean field is entrained by the force whereas the oscillators are not.
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Affiliation(s)
- Amirkhan A Temirbayev
- Physical-Technical Department, al-Farabi Kazakh National University, al-Farabi Avenue 71, 050040 Almaty, Kazakhstan
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Skardal PS, Restrepo JG. Hierarchical synchrony of phase oscillators in modular networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:016208. [PMID: 22400644 DOI: 10.1103/physreve.85.016208] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2011] [Indexed: 05/18/2023]
Abstract
We study synchronization of sinusoidally coupled phase oscillators on networks with modular structure and a large number of oscillators in each community. Of particular interest is the hierarchy of local and global synchrony, i.e., synchrony within and between communities, respectively. Using the recent ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)], we find that the degree of local synchrony can be determined from a set of coupled low-dimensional equations. If the number of communities in the network is large, a low-dimensional description of global synchrony can be also found. Using these results, we study bifurcations between different types of synchrony. We find that, depending on the relative strength of local and global coupling, the transition to synchrony in the network can be mediated by local or global effects.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA.
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18
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Skardal PS, Ott E, Restrepo JG. Cluster synchrony in systems of coupled phase oscillators with higher-order coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:036208. [PMID: 22060476 DOI: 10.1103/physreve.84.036208] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2011] [Indexed: 05/31/2023]
Abstract
We study the phenomenon of cluster synchrony that occurs in ensembles of coupled phase oscillators when higher-order modes dominate the coupling between oscillators. For the first time, we develop a complete analytic description of the dynamics in the limit of a large number of oscillators and use it to quantify the degree of cluster synchrony, cluster asymmetry, and switching. We use a variation of the recent dimensionality-reduction technique of Ott and Antonsen [Chaos 18, 037113 (2008)] and find an analytic description of the degree of cluster synchrony valid on a globally attracting manifold. Shaped by this manifold, there is an infinite family of steady-state distributions of oscillators, resulting in a high degree of multistability in the cluster asymmetry. We also show how through external forcing the degree of asymmetry can be controlled, and suggest that systems displaying cluster synchrony can be used to encode and store data.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Applied Mathematics, University of Colorado at Boulder, Colorado 80309, USA.
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19
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Uchida N. Many-body theory of synchronization by long-range interactions. PHYSICAL REVIEW LETTERS 2011; 106:064101. [PMID: 21405469 DOI: 10.1103/physrevlett.106.064101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2010] [Indexed: 05/30/2023]
Abstract
Synchronization of coupled oscillators on a d-dimensional lattice with the power-law coupling G(r) = g0/rα and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of ϵ = α/d-1. For α ≤ d, the system exhibits a sharp synchronization transition as described by the conventional mean-field theory. For α > d, the transition is smeared by the quenched disorder, and the macroscopic order parameter ψ decays slowly with g0 as |ψ| ∝ g(0)(2).
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Affiliation(s)
- Nariya Uchida
- Department of Physics, Tohoku University, Sendai 980-8578, Japan
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20
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Ghosh A, Roy D, Jirsa VK. Simple model for bursting dynamics of neurons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:041930. [PMID: 19905365 DOI: 10.1103/physreve.80.041930] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2009] [Revised: 09/26/2009] [Indexed: 05/28/2023]
Abstract
Neuronal cells in isolation or as an assembly exhibit bursting behavior on two different time scales. We introduce a simple one-dimensional model which requires only one phase variable to describe the phenomenon of parabolic bursting. The analysis in the continuum limit reveals that for any unimodal distribution of frequencies, the qualitative properties of the full and the reduced model are identical. Further, we derive analytically an exact low-dimensional description of a globally coupled network of bursting oscillators for our model. Study of the stability for this low-dimensional model reveals different dynamical signatures in the parameter space. We demonstrate that the structure of the parameter space remains independent of the number of spikes per burst.
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Affiliation(s)
- Anandamohan Ghosh
- Theoretical Neuroscience Group, Institut des Sciences du Mouvement, UMR 6233, CNRS and Université de la Méditerranée, 163 Avenue de Luminy, 13288 Marseille, France.
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21
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Ostborn P. Renormalization of oscillator lattices with disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:051114. [PMID: 19518423 DOI: 10.1103/physreve.79.051114] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/2008] [Revised: 04/08/2009] [Indexed: 05/27/2023]
Abstract
A real-space renormalization transformation is constructed for lattices of nonidentical oscillators with dynamics of the general form dvarphi_{k}/dt=omega_{k}+g summation operator_{l}f_{lk}(varphi_{l},varphi_{k}) . The transformation acts on ensembles of such lattices. Critical properties corresponding to a second-order phase transition toward macroscopic synchronization are deduced. The analysis is potentially exact but relies in part on unproven assumptions. Numerically, second-order phase transitions with the predicted properties are observed as g increases in two structurally different two-dimensional oscillator models. One model has smooth coupling f_{lk}(varphi_{l},varphi_{k})=phi(varphi_{l}-varphi_{k}) , where phi(x) is nonodd. The other model is pulse coupled, with f_{lk}(varphi_{l},varphi_{k})=delta(varphi_{l})phi(varphi_{k}) . Lower bounds for the critical dimensions for different types of coupling are obtained. For nonodd coupling, macroscopic synchronization cannot be ruled out for any dimension D> or =1 , whereas in the case of odd coupling, the well-known result that it can be ruled out for D<3 is regained.
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Affiliation(s)
- Per Ostborn
- Division of Mathematical Physics, Lund University, S-221 00 Lund, Sweden
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22
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Childs LM, Strogatz SH. Stability diagram for the forced Kuramoto model. CHAOS (WOODBURY, N.Y.) 2008; 18:043128. [PMID: 19123638 DOI: 10.1063/1.3049136] [Citation(s) in RCA: 91] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We analyze the periodically forced Kuramoto model. This system consists of an infinite population of phase oscillators with random intrinsic frequencies, global sinusoidal coupling, and external sinusoidal forcing. It represents an idealization of many phenomena in physics, chemistry, and biology in which mutual synchronization competes with forced synchronization. In other words, the oscillators in the population try to synchronize with one another while also trying to lock onto an external drive. Previous work on the forced Kuramoto model uncovered two main types of attractors, called forced entrainment and mutual entrainment, but the details of the bifurcations between them were unclear. Here we present a complete bifurcation analysis of the model for a special case in which the infinite-dimensional dynamics collapse to a two-dimensional system. Exact results are obtained for the locations of Hopf, saddle-node, and Takens-Bogdanov bifurcations. The resulting stability diagram bears a striking resemblance to that for the weakly nonlinear forced van der Pol oscillator.
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Affiliation(s)
- Lauren M Childs
- Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA.
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23
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McCoy BJ, Madras G. CLUSTER KINETICS OF PHASE TRANSITIONS: APPLICATIONS TO INNOVATIVE TECHNOLOGIES. CHEM ENG COMMUN 2008. [DOI: 10.1080/00986440802290094] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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24
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De Smet F, Aeyels D. Resonances and entrainment breakup in Kuramoto models with multimodal frequency densities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:066212. [PMID: 18643355 DOI: 10.1103/physreve.77.066212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2007] [Revised: 05/05/2008] [Indexed: 05/26/2023]
Abstract
We characterize some intriguing aspects of the entrainment behavior of coupled oscillators by means of a perturbation analysis of the partially synchronized solution of the classical Kuramoto-Sakaguchi model. The analysis reveals that partial entrainment may disappear with increasing coupling strength. It also predicts the occurrence of resonances: partial entrainment is induced in oscillators with natural frequencies in specific intervals not corresponding to high oscillator densities. The results are illustrated by simulations.
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Affiliation(s)
- Filip De Smet
- SYSTeMS Research Group, Department of Electrical Energy, Systems and Automation, Ghent University, Technologiepark Zwijnaarde 914, 9052 Zwijnaarde, Belgium.
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25
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Wood K, Van den Broeck C, Kawai R, Lindenberg K. Continuous and discontinuous phase transitions and partial synchronization in stochastic three-state oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041132. [PMID: 17994961 DOI: 10.1103/physreve.76.041132] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2007] [Indexed: 05/25/2023]
Abstract
We investigate both continuous (second-order) and discontinuous (first-order) transitions to macroscopic synchronization within a single class of discrete, stochastic (globally) phase-coupled oscillators. We provide analytical and numerical evidence that the continuity of the transition depends on the coupling coefficients and, in some nonuniform populations, on the degree of quenched disorder. Hence, in a relatively simple setting this class of models exhibits the qualitative behaviors characteristic of a variety of considerably more complicated models. In addition, we study the microscopic basis of synchronization above threshold and detail the counterintuitive subtleties relating measurements of time-averaged frequencies and mean-field oscillations. Most notably, we observe a state of suprathreshold partial synchronization in which time-averaged frequency measurements from individual oscillators do not correspond to the frequency of macroscopic oscillations observed in the population.
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Affiliation(s)
- Kevin Wood
- Department of Chemistry and Biochemistry and Institute for Nonlinear Science, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093-0340, USA
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26
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Shalfeev VD, Ivanchenko MV, Forti GL. Chaotic spatial bifurcation by complex coupling. CHAOS (WOODBURY, N.Y.) 2007; 17:023103. [PMID: 17614657 DOI: 10.1063/1.2671871] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
A spatial bifurcation (a transition from stationary to oscillatory regime) in a chain of unidirectionally coupled phase systems is studied. It is shown that complication of coupling terms can make this bifurcation spatially chaotic in contrast to the previously observed "regular" and "predictable" type. It is demonstrated that the found type of spatial bifurcation corresponds to a smooth (predictable) manifold in the parameter space, while its spatial location gets actually unpredictable being governed by regularities of chaotic behavior. We infer that complex collective dynamics may arise in networks with plain architecture and simple dynamics of individual elements if nontrivial coupling is realized.
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Affiliation(s)
- Vladimir D Shalfeev
- Department of Radiophysics, Nizhny Novgorod State University, 23 Gagarin Avenue, 603950 Nizhny Novgorod, Russia
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27
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Giannuzzi F, Marinazzo D, Nardulli G, Pellicoro M, Stramaglia S. Phase diagram of a generalized Winfree model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:051104. [PMID: 17677019 DOI: 10.1103/physreve.75.051104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2007] [Indexed: 05/16/2023]
Abstract
We study the phase diagram of a generalized Winfree model. The modification is such that the coupling depends on the fraction of synchronized oscillators, a situation which has been noted in some experiments on coupled Josephson junctions and mechanical systems. We let the global coupling k be a function of the Kuramoto order parameter r through an exponent z such that z=1 corresponds to the standard Winfree model, z<1 strengthens the coupling at low r (low amount of synchronization), and at z>1 , the coupling is weakened for low r . Using both analytical and numerical approaches, we find that z controls the size of the incoherent phase region and that one may make the incoherent behavior less typical by choosing z<1 . We also find that the original Winfree model is a rather special case; indeed, the partial locked behavior disappears for z>1 . At fixed k and varying gamma , the stability boundary of the locked phase corresponds to a transition that is continuous for z<1 and first order for z>1 . This change in the nature of the transition is in accordance with a previous study of a similarly modified Kuramoto model.
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Affiliation(s)
- F Giannuzzi
- Dipartimento Interateneo di Fisica, Università di Bari, Bari, Italy.
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28
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Kawamura Y. Chimera Ising walls in forced nonlocally coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:056204. [PMID: 17677145 DOI: 10.1103/physreve.75.056204] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2006] [Indexed: 05/16/2023]
Abstract
Nonlocally coupled oscillator systems can exhibit an exotic spatiotemporal structure called a chimera, where the system splits into two groups of oscillators with sharp boundaries, one of which is phase locked and the other phase randomized. Two examples of chimera states are known: the first one appears in a ring of phase oscillators, and the second is associated with two-dimensional rotating spiral waves. In this paper, we report yet another example of the chimera state that is associated with the so-called Ising walls in one-dimensional spatially extended systems. This chimera state is exhibited by a nonlocally coupled complex Ginzburg-Landau equation with external forcing.
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Affiliation(s)
- Yoji Kawamura
- Department of Physics, Graduate School of Sciences, Kyoto University, Kyoto 606-8502, Japan.
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29
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Quinn DD, Rand RH, Strogatz SH. Singular unlocking transition in the Winfree model of coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:036218. [PMID: 17500780 DOI: 10.1103/physreve.75.036218] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2006] [Indexed: 05/15/2023]
Abstract
The Winfree model consists of a population of globally coupled phase oscillators with randomly distributed natural frequencies. As the coupling strength and the spread of natural frequencies are varied, the various stable states of the model can undergo bifurcations, nearly all of which have been characterized previously. The one exception is the unlocking transition, in which the frequency-locked state disappears abruptly as the spread of natural frequencies exceeds a critical width. Viewed as a function of the coupling strength, this critical width defines a bifurcation curve in parameter space. For the special case where the frequency distribution is uniform, earlier work had uncovered a puzzling singularity in this bifurcation curve. Here we seek to understand what causes the singularity. Using the Poincaré-Lindstedt method of perturbation theory, we analyze the locked state and its associated unlocking transition, first for an arbitrary distribution of natural frequencies, and then for discrete systems of N oscillators. We confirm that the bifurcation curve becomes singular for a continuum uniform distribution, yet find that it remains well behaved for any finite N , suggesting that the continuum limit is responsible for the singularity.
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Affiliation(s)
- D Dane Quinn
- Department of Mechanical Engineering, The University of Akron, Akron, Ohio 44325-3903, USA.
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30
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Miyazaki J, Kinoshita S. Method for determining a coupling function in coupled oscillators with application to Belousov-Zhabotinsky oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:056209. [PMID: 17279986 DOI: 10.1103/physreve.74.056209] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2006] [Indexed: 05/13/2023]
Abstract
A coupling function that describes the interaction between self-sustained oscillators in a phase equation is derived and applied experimentally to Belousov-Zhabotinsky (BZ) oscillators. It is demonstrated that the synchronous behavior of coupled BZ reactors is explained extremely well in terms of the coupling function thus obtained. This method does not require comprehensive knowledge of either the oscillation mechanism or the interaction among the oscillators, both of these being often difficult to elucidate in an actual system. These facts enable us to accurately analyze the weakly coupled entrainment phenomenon through the direct measurement of the coupling function.
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Affiliation(s)
- J Miyazaki
- Graduate School of Frontier Biosciences, Osaka University, Suita 565-0871, Japan.
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31
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Miyazaki J, Kinoshita S. Determination of a coupling function in multicoupled oscillators. PHYSICAL REVIEW LETTERS 2006; 96:194101. [PMID: 16803103 DOI: 10.1103/physrevlett.96.194101] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2005] [Indexed: 05/03/2023]
Abstract
A new method to determine a coupling function in a phase model is theoretically derived for coupled self-sustained oscillators and applied to Belousov-Zhabotinsky (BZ) oscillators. The synchronous behavior of two coupled BZ reactors is explained extremely well in terms of the coupling function thus obtained. This method is expected to be applicable to weakly coupled multioscillator systems, in which mutual coupling among nearly identical oscillators occurs in a similar manner. The importance of higher-order harmonic terms involved in the coupling function is also discussed.
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Affiliation(s)
- Jun Miyazaki
- Graduate School of Frontier Biosciences, Osaka University, Suita, Japan
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32
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Moon SJ, Ghanem R, Kevrekidis IG. Coarse graining the dynamics of coupled oscillators. PHYSICAL REVIEW LETTERS 2006; 96:144101. [PMID: 16712078 DOI: 10.1103/physrevlett.96.144101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2005] [Indexed: 05/09/2023]
Abstract
We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of nonidentical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics.
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Affiliation(s)
- Sung Joon Moon
- Department of Chemical Engineering & Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, NJ 08544, USA
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33
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Tsimring LS, Rulkov NF, Larsen ML, Gabbay M. Repulsive synchronization in an array of phase oscillators. PHYSICAL REVIEW LETTERS 2005; 95:014101. [PMID: 16090619 DOI: 10.1103/physrevlett.95.014101] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2005] [Indexed: 05/03/2023]
Abstract
We study the dynamics of a repulsively coupled array of phase oscillators. For an array of globally coupled identical oscillators, repulsive coupling results in a family of synchronized regimes characterized by zero mean field. If the number of oscillators is sufficiently large, phase locking among oscillators is destroyed, independently of the coupling strength, when the oscillators' natural frequencies are not the same. In locally coupled networks, however, phase locking occurs even for nonidentical oscillators when the coupling strength is sufficiently strong.
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Affiliation(s)
- L S Tsimring
- Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402, USA
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34
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Zaks MA, Neiman AB, Feistel S, Schimansky-Geier L. Noise-controlled oscillations and their bifurcations in coupled phase oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:066206. [PMID: 14754296 DOI: 10.1103/physreve.68.066206] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2003] [Indexed: 05/24/2023]
Abstract
We derive in Gaussian approximation dynamical equations for the first two cumulants of the mean field fluctuations in a system of globally coupled stochastic phase oscillators. In these equations the intensity of noise serves as an explicit control parameter. Its variation generates transitions between three dynamical regimes: (i) stationary, (ii) rotatory and (iii) locally oscillatory (breathing). The latter regime has previously not been reported in studies of globally coupled noisy phase oscillators. Our detailed bifurcation analysis is supported by numerical simulations of an ensemble of coupled stochastic phase oscillators. Similar regimes are also found in simulations of globally coupled stochastic FitzHugh-Nagumo elements.
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Affiliation(s)
- M A Zaks
- Institute of Physics, Humboldt-University of Berlin, Newtonstrasse 15, 12489 Berlin, Germany
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35
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Ariaratnam JT, Strogatz SH. Phase diagram for the Winfree model of coupled nonlinear oscillators. PHYSICAL REVIEW LETTERS 2001; 86:4278-81. [PMID: 11328154 DOI: 10.1103/physrevlett.86.4278] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2000] [Indexed: 05/20/2023]
Abstract
In 1967 Winfree proposed a mean-field model for the spontaneous synchronization of chorusing crickets, flashing fireflies, circadian pacemaker cells, or other large populations of biological oscillators. Here we give the first bifurcation analysis of the model, for a tractable special case. The system displays rich collective dynamics as a function of the coupling strength and the spread of natural frequencies. Besides incoherence, frequency locking, and oscillator death, there exist hybrid solutions that combine two or more of these states. We present the phase diagram and derive several of the stability boundaries analytically.
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Affiliation(s)
- J T Ariaratnam
- Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
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36
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Daido H. Multi-branch entrainment and multi-peaked order-functions in a phase model of limit-cycle oscillators with uniform all-to-all coupling. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/5/002] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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37
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38
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Daido H. Multibranch Entrainment and Scaling in Large Populations of Coupled Oscillators. PHYSICAL REVIEW LETTERS 1996; 77:1406-1409. [PMID: 10063068 DOI: 10.1103/physrevlett.77.1406] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
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39
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Wiesenfeld K, Colet P, Strogatz SH. Synchronization transitions in a disordered Josephson series array. PHYSICAL REVIEW LETTERS 1996; 76:404-407. [PMID: 10061448 DOI: 10.1103/physrevlett.76.404] [Citation(s) in RCA: 84] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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40
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Crawford JD. Scaling and singularities in the entrainment of globally coupled oscillators. PHYSICAL REVIEW LETTERS 1995; 74:4341-4344. [PMID: 10058476 DOI: 10.1103/physrevlett.74.4341] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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