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León I, Pazó D. Efficient moment-based approach to the simulation of infinitely many heterogeneous phase oscillators. CHAOS (WOODBURY, N.Y.) 2022; 32:063124. [PMID: 35778114 DOI: 10.1063/5.0093001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2022] [Accepted: 05/27/2022] [Indexed: 06/15/2023]
Abstract
The dynamics of ensembles of phase oscillators are usually described considering their infinite-size limit. In practice, however, this limit is fully accessible only if the Ott-Antonsen theory can be applied, and the heterogeneity is distributed following a rational function. In this work, we demonstrate the usefulness of a moment-based scheme to reproduce the dynamics of infinitely many oscillators. Our analysis is particularized for Gaussian heterogeneities, leading to a Fourier-Hermite decomposition of the oscillator density. The Fourier-Hermite moments obey a set of hierarchical ordinary differential equations. As a preliminary experiment, the effects of truncating the moment system and implementing different closures are tested in the analytically solvable Kuramoto model. The moment-based approach proves to be much more efficient than the direct simulation of a large oscillator ensemble. The convenience of the moment-based approach is exploited in two illustrative examples: (i) the Kuramoto model with bimodal frequency distribution, and (ii) the "enlarged Kuramoto model" (endowed with nonpairwise interactions). In both systems, we obtain new results inaccessible through direct numerical integration of populations.
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Affiliation(s)
- Iván León
- Instituto de Física de Cantabria (IFCA), Universidad de Cantabria-CSIC, Avda. Los Castros, s/n, 39005 Santander, Spain
| | - Diego Pazó
- Instituto de Física de Cantabria (IFCA), Universidad de Cantabria-CSIC, Avda. Los Castros, s/n, 39005 Santander, Spain
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León I, Pazó D. Enlarged Kuramoto model: Secondary instability and transition to collective chaos. Phys Rev E 2022; 105:L042201. [PMID: 35590592 DOI: 10.1103/physreve.105.l042201] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2021] [Accepted: 03/21/2022] [Indexed: 06/15/2023]
Abstract
The emergence of collective synchrony from an incoherent state is a phenomenon essentially described by the Kuramoto model. This canonical model was derived perturbatively, by applying phase reduction to an ensemble of heterogeneous, globally coupled Stuart-Landau oscillators. This derivation neglects nonlinearities in the coupling constant. We show here that a comprehensive analysis requires extending the Kuramoto model up to quadratic order. This "enlarged Kuramoto model" comprises three-body (nonpairwise) interactions, which induce strikingly complex phenomenology at certain parameter values. As the coupling is increased, a secondary instability renders the synchronized state unstable, and subsequent bifurcations lead to collective chaos. An efficient numerical study of the thermodynamic limit, valid for Gaussian heterogeneity, is carried out by means of a Fourier-Hermite decomposition of the oscillator density.
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Affiliation(s)
- Iván León
- Instituto de Física de Cantabria (IFCA), Universidad de Cantabria-CSIC, 39005 Santander, Spain
| | - Diego Pazó
- Instituto de Física de Cantabria (IFCA), Universidad de Cantabria-CSIC, 39005 Santander, Spain
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Cencetti G, Clusella P, Fanelli D. Pattern invariance for reaction-diffusion systems on complex networks. Sci Rep 2018; 8:16226. [PMID: 30385860 PMCID: PMC6212431 DOI: 10.1038/s41598-018-34372-0] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2018] [Accepted: 10/11/2018] [Indexed: 11/09/2022] Open
Abstract
Given a reaction-diffusion system interacting via a complex network, we propose two different techniques to modify the network topology while preserving its dynamical behaviour. In the region of parameters where the homogeneous solution gets spontaneously destabilized, perturbations grow along the unstable directions made available across the networks of connections, yielding irregular spatio-temporal patterns. We exploit the spectral properties of the Laplacian operator associated to the graph in order to modify its topology, while preserving the unstable manifold of the underlying equilibrium. The new network is isodynamic to the former, meaning that it reproduces the dynamical response (pattern) to a perturbation, as displayed by the original system. The first method acts directly on the eigenmodes, thus resulting in a general redistribution of link weights which, in some cases, can completely change the structure of the original network. The second method uses localization properties of the eigenvectors to identify and randomize a subnetwork that is mostly embedded only into the stable manifold. We test both techniques on different network topologies using the Ginzburg-Landau system as a reference model. Whereas the correlation between patterns on isodynamic networks generated via the first recipe is larger, the second method allows for a finer control at the level of single nodes. This work opens up a new perspective on the multiple possibilities for identifying the family of discrete supports that instigate equivalent dynamical responses on a multispecies reaction-diffusion system.
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Affiliation(s)
- Giulia Cencetti
- Università degli Studi di Firenze, Dipartimento di Ingegneria dell'Informazione, Florence, Italy.
- Università degli Studi di Firenze, Dipartimento di Fisica e Astronomia and CSDC, Florence, Italy.
- INFN Sezione di Firenze, Florence, Italy.
| | - Pau Clusella
- Università degli Studi di Firenze, Dipartimento di Fisica e Astronomia and CSDC, Florence, Italy
- Institute for Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen, UK
| | - Duccio Fanelli
- Università degli Studi di Firenze, Dipartimento di Fisica e Astronomia and CSDC, Florence, Italy
- INFN Sezione di Firenze, Florence, Italy
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Hannay KM, Forger DB, Booth V. Macroscopic models for networks of coupled biological oscillators. SCIENCE ADVANCES 2018; 4:e1701047. [PMID: 30083596 PMCID: PMC6070363 DOI: 10.1126/sciadv.1701047] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2017] [Accepted: 06/20/2018] [Indexed: 05/20/2023]
Abstract
The study of synchronization of coupled biological oscillators is fundamental to many areas of biology including neuroscience, cardiac dynamics, and circadian rhythms. Mathematical models of these systems may involve hundreds of variables in thousands of individual cells resulting in an extremely high-dimensional description of the system. This often contrasts with the low-dimensional dynamics exhibited on the collective or macroscopic scale for these systems. We introduce a macroscopic reduction for networks of coupled oscillators motivated by an elegant structure we find in experimental measurements of circadian protein expression and several mathematical models for coupled biological oscillators. The observed structure in the collective amplitude of the oscillator population differs from the well-known Ott-Antonsen ansatz, but its emergence can be characterized through a simple argument depending only on general phase-locking behavior in coupled oscillator systems. We further demonstrate its emergence in networks of noisy heterogeneous oscillators with complex network connectivity. Applying this structure, we derive low-dimensional macroscopic models for oscillator population activity. This approach allows for the incorporation of cellular-level experimental data into the macroscopic model whose parameters and variables can then be directly associated with tissue- or organism-level properties, thereby elucidating the core properties driving the collective behavior of the system.
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Affiliation(s)
- Kevin M. Hannay
- Department of Mathematics, Schreiner University, Kerrville, TX 78028, USA
- Corresponding author.
| | - Daniel B. Forger
- Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
- Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI 48109, USA
| | - Victoria Booth
- Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
- Department of Anesthesiology, University of Michigan, Ann Arbor, MI 48109, USA
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Daido H. Boundary in the dynamic phase of globally coupled oscillatory and excitable units. Phys Rev E 2017; 96:012210. [PMID: 29347184 DOI: 10.1103/physreve.96.012210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2017] [Indexed: 06/07/2023]
Abstract
There is a crucial boundary between dynamic phase 1 and dynamic phase 2 of globally coupled oscillatory and excitable units, where the mean field is constant and oscillates in the former and the latter, respectively. This boundary is theoretically derived here for a large population of dynamical units, each having only a phase variable, where it is assumed that both the coupling strength and the distribution width of bifurcation parameters are equally small. This theory, which is applicable only if all or most of the units are intrinsically oscillatory, is confirmed to agree with simulation results for two different distribution densities.
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Affiliation(s)
- Hiroaki Daido
- Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan
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Clusella P, Politi A. Noise-induced stabilization of collective dynamics. Phys Rev E 2017; 95:062221. [PMID: 28709323 DOI: 10.1103/physreve.95.062221] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2017] [Indexed: 06/07/2023]
Abstract
We illustrate a counterintuitive effect of an additive stochastic force, which acts independently on each element of an ensemble of globally coupled oscillators. We show that a very small white noise not only broadens the clusters, wherever they are induced by the deterministic forces, but can also stabilize a linearly unstable collective periodic regime: self-consistent partial synchrony. With the help of microscopic simulations we are able to identify two noise-induced bifurcations. A macroscopic analysis, based on a perturbative solution of the associated nonlinear Fokker-Planck equation, confirms the numerical studies and allows determining the eigenvalues of the stability problem. We finally argue about the generality of the phenomenon.
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Affiliation(s)
- Pau Clusella
- Institute for Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
- Dipartimento di Fisica, Università di Firenze, I-50019 Sesto Fiorentino, Italy
| | - Antonio Politi
- Institute for Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
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Pikovsky A, Rosenblum M. Dynamics of globally coupled oscillators: Progress and perspectives. CHAOS (WOODBURY, N.Y.) 2015; 25:097616. [PMID: 26428569 DOI: 10.1063/1.4922971] [Citation(s) in RCA: 94] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
In this paper, we discuss recent progress in research of ensembles of mean field coupled oscillators. Without an ambition to present a comprehensive review, we outline most interesting from our viewpoint results and surprises, as well as interrelations between different approaches.
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Affiliation(s)
- Arkady Pikovsky
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
| | - Michael Rosenblum
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
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Hannay KM, Booth V, Forger DB. Collective phase response curves for heterogeneous coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022923. [PMID: 26382491 DOI: 10.1103/physreve.92.022923] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2015] [Indexed: 06/05/2023]
Abstract
Phase response curves (PRCs) have become an indispensable tool in understanding the entrainment and synchronization of biological oscillators. However, biological oscillators are often found in large coupled heterogeneous systems and the variable of physiological importance is the collective rhythm resulting from an aggregation of the individual oscillations. To study this phenomena we consider phase resetting of the collective rhythm for large ensembles of globally coupled Sakaguchi-Kuramoto oscillators. Making use of Ott-Antonsen theory we derive an asymptotically valid analytic formula for the collective PRC. A result of this analysis is a characteristic scaling for the change in the amplitude and entrainment points for the collective PRC compared to the individual oscillator PRC. We support the analytical findings with numerical evidence and demonstrate the applicability of the theory to large ensembles of coupled neuronal oscillators.
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Affiliation(s)
- Kevin M Hannay
- Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - Victoria Booth
- Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA
- Department of Anesthesiology, University of Michigan, Ann Arbor, Michigan 48109, USA
| | - Daniel B Forger
- Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA
- Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, Michigan 48109, USA
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Komarov M, Pikovsky A. Intercommunity resonances in multifrequency ensembles of coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012906. [PMID: 26274246 DOI: 10.1103/physreve.92.012906] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2015] [Indexed: 06/04/2023]
Abstract
We generalize the Kuramoto model of globally coupled oscillators to multifrequency communities. A situation when mean frequencies of two subpopulations are close to the resonance 2:1 is considered in detail. We construct uniformly rotating solutions describing synchronization inside communities and between them. Remarkably, cross coupling across the frequencies can promote synchrony even when ensembles are separately asynchronous. We also show that the transition to synchrony due to the cross coupling is accompanied by a huge multiplicity of distinct synchronous solutions, which is directly related to a multibranch entrainment. On the other hand, for synchronous populations, the cross-frequency coupling can destroy phase locking and lead to chaos of mean fields.
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Affiliation(s)
- Maxim Komarov
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str 24, D-14476, Potsdam, Germany
- Department of Cell Biology and Neuroscience, University of California Riverside, 900 University Ave. Riverside, California 92521, USA
- Department of Control Theory, Nizhni Novgorod State University, Gagarin Av. 23, 606950, Nizhni Novgorod, Russia
| | - Arkady Pikovsky
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str 24, D-14476, Potsdam, Germany
- Department of Control Theory, Nizhni Novgorod State University, Gagarin Av. 23, 606950, Nizhni Novgorod, Russia
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Rosenblum M, Pikovsky A. Two types of quasiperiodic partial synchrony in oscillator ensembles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012919. [PMID: 26274259 DOI: 10.1103/physreve.92.012919] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2015] [Indexed: 05/21/2023]
Abstract
We analyze quasiperiodic partially synchronous states in an ensemble of Stuart-Landau oscillators with global nonlinear coupling. We reveal two types of such dynamics: in the first case the time-averaged frequencies of oscillators and of the mean field differ, while in the second case they are equal, but the motion of oscillators is additionally modulated. We describe transitions from the synchronous state to both types of quasiperiodic dynamics, and a transition between two different quasiperiodic states. We present an example of a bifurcation diagram, where we show the borderlines for all these transitions, as well as domain of bistability.
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Affiliation(s)
- Michael Rosenblum
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, D-14476 Potsdam-Golm, Germany
| | - Arkady Pikovsky
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, D-14476 Potsdam-Golm, Germany
- Department of Control Theory, Nizhni Novgorod State University, Gagarin Avenue 23, 606950 Nizhni Novgorod, Russia
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Politi A, Rosenblum M. Equivalence of phase-oscillator and integrate-and-fire models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042916. [PMID: 25974571 DOI: 10.1103/physreve.91.042916] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2014] [Indexed: 06/04/2023]
Abstract
A quantitative comparison of various classes of oscillators (integrate-and-fire, Winfree, and Kuramoto-Daido type) is performed in the weak-coupling limit for a fully connected network of identical units. An almost perfect agreement is found, with only tiny differences among the models. We also show that the regime of self-consistent partial synchronization is rather general and can be observed for arbitrarily small coupling strength in any model class. As a byproduct of our study, we are able to show that an integrate-and-fire model with a generic pulse shape can be always transformed into a similar model with δ pulses and a suitable phase response curve.
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Affiliation(s)
- Antonio Politi
- Institute for Complex Systems and Mathematical Biology and SUPA, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| | - Michael Rosenblum
- Department of Physics and Astronomy, University of Potsdam, Karl-Libknecht-Strasse 24/25, 14476 Potsdam, Germany
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Maistrenko Y, Penkovsky B, Rosenblum M. Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:060901. [PMID: 25019710 DOI: 10.1103/physreve.89.060901] [Citation(s) in RCA: 51] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2013] [Indexed: 05/20/2023]
Abstract
We discuss the desynchronization transition in networks of globally coupled identical oscillators with attractive and repulsive interactions. We show that, if attractive and repulsive groups act in antiphase or close to that, a solitary state emerges with a single repulsive oscillator split up from the others fully synchronized. With further increase of the repulsing strength, the synchronized cluster becomes fuzzy and the dynamics is given by a variety of stationary states with zero common forcing. Intriguingly, solitary states represent the natural link between coherence and incoherence. The phenomenon is described analytically for phase oscillators with sine coupling and demonstrated numerically for more general amplitude models.
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Affiliation(s)
- Yuri Maistrenko
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany and Institute of Mathematics and National Centre for Medical and Biotechnical Research, National Academy of Sciences of Ukraine, Kiev, Ukraine
| | - Bogdan Penkovsky
- National University of "Kyiv-Mohyla Academy," Kiev, Ukraine and FEMTO-ST/Optics Department, University of Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France
| | - Michael Rosenblum
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany
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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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Daido H. Generic scaling at the onset of macroscopic mutual entrainment in limit-cycle oscillators with uniform all-to-all coupling. PHYSICAL REVIEW LETTERS 1994; 73:760-763. [PMID: 10057530 DOI: 10.1103/physrevlett.73.760] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/15/2023]
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