1
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Yawata K, Fukami K, Taira K, Nakao H. Phase autoencoder for limit-cycle oscillators. CHAOS (WOODBURY, N.Y.) 2024; 34:063111. [PMID: 38829787 DOI: 10.1063/5.0205718] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2024] [Accepted: 05/10/2024] [Indexed: 06/05/2024]
Abstract
We present a phase autoencoder that encodes the asymptotic phase of a limit-cycle oscillator, a fundamental quantity characterizing its synchronization dynamics. This autoencoder is trained in such a way that its latent variables directly represent the asymptotic phase of the oscillator. The trained autoencoder can perform two functions without relying on the mathematical model of the oscillator: first, it can evaluate the asymptotic phase and the phase sensitivity function of the oscillator; second, it can reconstruct the oscillator state on the limit cycle in the original space from the phase value as an input. Using several examples of limit-cycle oscillators, we demonstrate that the asymptotic phase and the phase sensitivity function can be estimated only from time-series data by the trained autoencoder. We also present a simple method for globally synchronizing two oscillators as an application of the trained autoencoder.
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Affiliation(s)
- Koichiro Yawata
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Kai Fukami
- Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, California 90095, USA
| | - Kunihiko Taira
- Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, California 90095, USA
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
- Research Center for Autonomous Systems Materialogy, Institute of Innovative Research, Tokyo Institute of Technology, Kanagawa 226-8501, Japan
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2
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Medrano J, Friston K, Zeidman P. Linking fast and slow: The case for generative models. Netw Neurosci 2024; 8:24-43. [PMID: 38562283 PMCID: PMC10861163 DOI: 10.1162/netn_a_00343] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2023] [Accepted: 10/11/2023] [Indexed: 04/04/2024] Open
Abstract
A pervasive challenge in neuroscience is testing whether neuronal connectivity changes over time due to specific causes, such as stimuli, events, or clinical interventions. Recent hardware innovations and falling data storage costs enable longer, more naturalistic neuronal recordings. The implicit opportunity for understanding the self-organised brain calls for new analysis methods that link temporal scales: from the order of milliseconds over which neuronal dynamics evolve, to the order of minutes, days, or even years over which experimental observations unfold. This review article demonstrates how hierarchical generative models and Bayesian inference help to characterise neuronal activity across different time scales. Crucially, these methods go beyond describing statistical associations among observations and enable inference about underlying mechanisms. We offer an overview of fundamental concepts in state-space modeling and suggest a taxonomy for these methods. Additionally, we introduce key mathematical principles that underscore a separation of temporal scales, such as the slaving principle, and review Bayesian methods that are being used to test hypotheses about the brain with multiscale data. We hope that this review will serve as a useful primer for experimental and computational neuroscientists on the state of the art and current directions of travel in the complex systems modelling literature.
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Affiliation(s)
- Johan Medrano
- The Wellcome Centre for Human Neuroimaging, UCL Queen Square Institute of Neurology, London, UK
| | - Karl Friston
- The Wellcome Centre for Human Neuroimaging, UCL Queen Square Institute of Neurology, London, UK
| | - Peter Zeidman
- The Wellcome Centre for Human Neuroimaging, UCL Queen Square Institute of Neurology, London, UK
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3
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Yamaguchi YY, Terada Y. Reconstruction of phase dynamics from macroscopic observations based on linear and nonlinear response theories. Phys Rev E 2024; 109:024217. [PMID: 38491619 DOI: 10.1103/physreve.109.024217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2023] [Accepted: 01/22/2024] [Indexed: 03/18/2024]
Abstract
We propose a method to reconstruct the phase dynamics in rhythmical interacting systems from macroscopic responses to weak inputs by developing linear and nonlinear response theories, which predict the responses in a given system. By solving an inverse problem, the method infers an unknown system: the natural frequency distribution, the coupling function, and the time delay which is inevitable in real systems. In contrast to previous methods, our method requires neither strong invasiveness nor microscopic observations. We demonstrate that the method reconstructs two phase systems from observed responses accurately. The qualitative methodological advantages demonstrated by our quantitative numerical examinations suggest its broad applicability in various fields, including brain systems, which are often observed through macroscopic signals such as electroencephalograms and functional magnetic response imaging.
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Affiliation(s)
| | - Yu Terada
- Department of Neurobiology, University of California San Diego, La Jolla, California 92093, USA
- Institute for Physics of Intelligence, Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
- Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
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4
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Nicks R, Allen R, Coombes S. Insights into oscillator network dynamics using a phase-isostable framework. CHAOS (WOODBURY, N.Y.) 2024; 34:013141. [PMID: 38271631 DOI: 10.1063/5.0179430] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2023] [Accepted: 12/16/2023] [Indexed: 01/27/2024]
Abstract
Networks of coupled nonlinear oscillators can display a wide range of emergent behaviors under the variation of the strength of the coupling. Network equations for pairs of coupled oscillators where the dynamics of each node is described by the evolution of its phase and slowest decaying isostable coordinate have previously been shown to capture bifurcations and dynamics of the network, which cannot be explained through standard phase reduction. An alternative framework using isostable coordinates to obtain higher-order phase reductions has also demonstrated a similar descriptive ability for two oscillators. In this work, we consider the phase-isostable network equations for an arbitrary but finite number of identical coupled oscillators, obtaining conditions required for the stability of phase-locked states including synchrony. For the mean-field complex Ginzburg-Landau equation where the solutions of the full system are known, we compare the accuracy of the phase-isostable network equations and higher-order phase reductions in capturing bifurcations of phase-locked states. We find the former to be the more accurate and, therefore, employ this to investigate the dynamics of globally linearly coupled networks of Morris-Lecar neuron models (both two and many nodes). We observe qualitative correspondence between results from numerical simulations of the full system and the phase-isostable description demonstrating that in both small and large networks, the phase-isostable framework is able to capture dynamics that the first-order phase description cannot.
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Affiliation(s)
- R Nicks
- School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - R Allen
- School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - S Coombes
- School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
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5
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Mircheski P, Zhu J, Nakao H. Phase-amplitude reduction and optimal phase locking of collectively oscillating networks. CHAOS (WOODBURY, N.Y.) 2023; 33:103111. [PMID: 37831791 DOI: 10.1063/5.0161119] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2023] [Accepted: 09/13/2023] [Indexed: 10/15/2023]
Abstract
We present a phase-amplitude reduction framework for analyzing collective oscillations in networked dynamical systems. The framework, which builds on the phase reduction method, takes into account not only the collective dynamics on the limit cycle but also deviations from it by introducing amplitude variables and using them with the phase variable. The framework allows us to study how networks react to applied inputs or coupling, including their synchronization and phase locking, while capturing the deviations of the network states from the unperturbed dynamics. Numerical simulations are used to demonstrate the effectiveness of the framework for networks composed of FitzHugh-Nagumo elements. The resulting phase-amplitude equations can be used in deriving optimal periodic waveforms or introducing feedback control for achieving fast phase locking while stabilizing the collective oscillations.
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Affiliation(s)
- Petar Mircheski
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Jinjie Zhu
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
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6
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Thümler M, Srinivas SGM, Schröder M, Timme M. Synchrony for Weak Coupling in the Complexified Kuramoto Model. PHYSICAL REVIEW LETTERS 2023; 130:187201. [PMID: 37204897 DOI: 10.1103/physrevlett.130.187201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2022] [Revised: 03/10/2023] [Accepted: 04/04/2023] [Indexed: 05/21/2023]
Abstract
We present the finite-size Kuramoto model analytically continued from real to complex variables and analyze its collective dynamics. For strong coupling, synchrony appears through locked states that constitute attractors, as for the real-variable system. However, synchrony persists in the form of complex locked states for coupling strengths K below the transition K^{(pl)} to classical phase locking. Stable complex locked states indicate a locked subpopulation of zero mean frequency in the real-variable model and their imaginary parts help identifying which units comprise that subpopulation. We uncover a second transition at K^{'}<K^{(pl)} below which complex locked states become linearly unstable yet still exist for arbitrarily small coupling strengths.
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Affiliation(s)
- Moritz Thümler
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (cfaed) and Institute of Theoretical Physics, Technische Universität Dresden, 01062 Dresden, Germany
| | - Shesha G M Srinivas
- Institute of Physics and Material Sciences, Campus Limpertsberg, Université du Luxembourg 162 A, Avenue de la Faïencerie, Luxembourg
| | - Malte Schröder
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (cfaed) and Institute of Theoretical Physics, Technische Universität Dresden, 01062 Dresden, Germany
| | - Marc Timme
- Chair for Network Dynamics, Institute of Theoretical Physics, Center for Advancing Electronics Dresden (cfaed) and Cluster of Excellence Physics of Life, Technische Universität Dresden, 01062 Dresden, Germany
- Lakeside Labs, Lakeside B04b, 9020 Klagenfurt, Austria
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7
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Kato M, Kori H. Partial synchronization and community switching in phase-oscillator networks and its analysis based on a bidirectional, weighted chain of three oscillators. Phys Rev E 2023; 107:014210. [PMID: 36797893 DOI: 10.1103/physreve.107.014210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2022] [Accepted: 12/22/2022] [Indexed: 01/22/2023]
Abstract
Complex networks often possess communities defined based on network connectivity. When dynamics undergo in a network, one can also consider dynamical communities, i.e., a group of nodes displaying a similar dynamical process. We have investigated both analytically and numerically the development of a dynamical community structure, where the community is referred to as a group of nodes synchronized in frequency, in networks of phase oscillators. We first demonstrate that using a few example networks, the community structure changes when network connectivity or interaction strength is varied. In particular, we found that community switching, i.e., a portion of oscillators change the group to which they synchronize, occurs for a range of parameters. We then propose a three-oscillator model: a bidirectional, weighted chain of three Kuramoto phase oscillators, as a theoretical framework for understanding the community formation and its variation. Our analysis demonstrates that the model shows a variety of partially synchronized patterns: oscillators with similar natural frequencies tend to synchronize for weak coupling, while tightly connected oscillators tend to synchronize for strong coupling. We obtain approximate expressions for the critical coupling strengths by employing a perturbative approach in a weak coupling regime and a geometric approach in strong coupling regimes. Moreover, we elucidate the bifurcation types of transitions between different patterns. Our theory might be useful for understanding the development of partially synchronized patterns in a wider class of complex networks than community structured networks.
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Affiliation(s)
- Masaki Kato
- Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan
| | - Hiroshi Kori
- Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan and Department of Complexity Sciences and Engineering, The University of Tokyo, Kashiwa, Chiba, Japan
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8
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Lehmkuhl S, Fleischer S, Lohmann L, Rosen MS, Chekmenev EY, Adams A, Theis T, Appelt S. RASER MRI: Magnetic resonance images formed spontaneously exploiting cooperative nonlinear interaction. SCIENCE ADVANCES 2022; 8:eabp8483. [PMID: 35857519 PMCID: PMC9278855 DOI: 10.1126/sciadv.abp8483] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2022] [Accepted: 05/27/2022] [Indexed: 05/27/2023]
Abstract
The spatial resolution of magnetic resonance imaging (MRI) is limited by the width of Lorentzian point spread functions associated with the transverse relaxation rate 1/T2*. Here, we show a different contrast mechanism in MRI by establishing RASER (radio-frequency amplification by stimulated emission of radiation) in imaged media. RASER imaging bursts emerge out of noise and without applying radio-frequency pulses when placing spins with sufficient population inversion in a weak magnetic field gradient. Small local differences in initial population inversion density can create stronger image contrast than conventional MRI. This different contrast mechanism is based on the cooperative nonlinear interaction between all slices. On the other hand, the cooperative nonlinear interaction gives rise to imaging artifacts, such as amplitude distortions and side lobes outside of the imaging domain. Contrast mechanism and artifacts are explored experimentally and predicted by simulations on the basis of a proposed RASER MRI theory.
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Affiliation(s)
- Sören Lehmkuhl
- Institute of Microstructure Technology, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany
- Department of Chemistry, North Carolina State University, Raleigh, NC 27606, USA
| | - Simon Fleischer
- Institute of Technical and Macromolecular Chemistry, RWTH Aachen University, 52056 Aachen, Germany
| | - Lars Lohmann
- Institute of Technical and Macromolecular Chemistry, RWTH Aachen University, 52056 Aachen, Germany
| | - Matthew S. Rosen
- Massachusetts General Hospital, A. A. Martinos Center for Biomedical Imaging, Boston, MA 02129, USA
- Department of Physics, Harvard University, Cambridge, MA 02138, USA
| | - Eduard Y. Chekmenev
- Department of Chemistry, Integrative Biosciences (IBio), Karmanos Cancer Institute (KCI), Wayne State University, Detroit, MI 48202, USA
- Russian Academy of Sciences, Leninskiy Prospekt 14, Moscow 119991, Russia
| | - Alina Adams
- Institute of Technical and Macromolecular Chemistry, RWTH Aachen University, 52056 Aachen, Germany
| | - Thomas Theis
- Department of Chemistry, North Carolina State University, Raleigh, NC 27606, USA
- Department of Physics, North Carolina State University, Raleigh, NC 27695, USA
- Joint Department of Biomedical Engineering, University of North Carolina at Chapel Hill and North Carolina State University, Raleigh, NC 27695, USA
| | - Stephan Appelt
- Institute of Technical and Macromolecular Chemistry, RWTH Aachen University, 52056 Aachen, Germany
- Central Institute for Engineering, Electronics and Analytics – Electronic Systems (ZEA-2), Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany
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9
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Namura N, Takata S, Yamaguchi K, Kobayashi R, Nakao H. Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data. Phys Rev E 2022; 106:014204. [PMID: 35974495 DOI: 10.1103/physreve.106.014204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2022] [Accepted: 06/13/2022] [Indexed: 06/15/2023]
Abstract
We propose a method for estimating the asymptotic phase and amplitude functions of limit-cycle oscillators using observed time series data without prior knowledge of their dynamical equations. The estimation is performed by polynomial regression and can be solved as a convex optimization problem. The validity of the proposed method is numerically illustrated by using two-dimensional limit-cycle oscillators as examples. As an application, we demonstrate data-driven fast entrainment with amplitude suppression using the optimal periodic input derived from the estimated phase and amplitude functions.
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Affiliation(s)
- Norihisa Namura
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Shohei Takata
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Katsunori Yamaguchi
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Ryota Kobayashi
- Graduate School of Frontier Sciences, The University of Tokyo, Chiba 277-8561, Japan; Mathematics and Informatics Center, The University of Tokyo, Tokyo 113-8656, Japan; and JST, PRESTO, Saitama 332-0012, Japan
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
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10
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Kato Y, Nakao H. A definition of the asymptotic phase for quantum nonlinear oscillators from the Koopman operator viewpoint. CHAOS (WOODBURY, N.Y.) 2022; 32:063133. [PMID: 35778147 DOI: 10.1063/5.0088559] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2022] [Accepted: 05/24/2022] [Indexed: 06/15/2023]
Abstract
We propose a definition of the asymptotic phase for quantum nonlinear oscillators from the viewpoint of the Koopman operator theory. The asymptotic phase is a fundamental quantity for the analysis of classical limit-cycle oscillators, but it has not been defined explicitly for quantum nonlinear oscillators. In this study, we define the asymptotic phase for quantum oscillatory systems by using the eigenoperator of the backward Liouville operator associated with the fundamental oscillation frequency. By using the quantum van der Pol oscillator with a Kerr effect as an example, we illustrate that the proposed asymptotic phase appropriately yields isochronous phase values in both semiclassical and strong quantum regimes.
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Affiliation(s)
- Yuzuru Kato
- Department of Complex and Intelligent Systems, Future University Hakodate, Hokkaido 041-8655, Japan
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
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11
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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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12
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Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory. MATHEMATICS 2021. [DOI: 10.3390/math9182188] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
The asymptotic phase is a fundamental quantity for the analysis of deterministic limit-cycle oscillators, and generalized definitions of the asymptotic phase for stochastic oscillators have also been proposed. In this article, we show that the asymptotic phase and also amplitude can be defined for classical and semiclassical stochastic oscillators in a natural and unified manner by using the eigenfunctions of the Koopman operator of the system. We show that the proposed definition gives appropriate values of the phase and amplitude for strongly stochastic limit-cycle oscillators, excitable systems undergoing noise-induced oscillations, and also for quantum limit-cycle oscillators in the semiclassical regime.
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13
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Takata S, Kato Y, Nakao H. Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory. CHAOS (WOODBURY, N.Y.) 2021; 31:093124. [PMID: 34598448 DOI: 10.1063/5.0054603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2021] [Accepted: 08/30/2021] [Indexed: 06/13/2023]
Abstract
Optimal entrainment of limit-cycle oscillators by strong periodic inputs is studied on the basis of the phase-amplitude reduction and Floquet theory. Two methods for deriving the input waveforms that keep the system state close to the original limit cycle are proposed, which enable the use of strong inputs for entrainment. The first amplitude-feedback method uses feedback control to suppress deviations of the system state from the limit cycle, while the second amplitude-penalty method seeks an input waveform that does not excite large deviations from the limit cycle in the feedforward framework. Optimal entrainment of the van der Pol and Willamowski-Rössler oscillators with real or complex Floquet exponents is analyzed as examples. It is demonstrated that the proposed methods can achieve considerably faster entrainment and provide wider entrainment ranges than the conventional method that relies only on phase reduction.
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Affiliation(s)
- Shohei Takata
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Yuzuru Kato
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
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14
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Nakao H, Yamaguchi K, Katayama S, Yanagita T. Sparse optimization of mutual synchronization in collectively oscillating networks. CHAOS (WOODBURY, N.Y.) 2021; 31:063113. [PMID: 34241311 DOI: 10.1063/5.0049091] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2021] [Accepted: 05/14/2021] [Indexed: 06/13/2023]
Abstract
We consider a pair of collectively oscillating networks of dynamical elements and optimize their internetwork coupling for efficient mutual synchronization based on the phase reduction theory developed by Nakao et al. [Chaos 28, 045103 (2018)]. The dynamical equations describing a pair of weakly coupled networks are reduced to a pair of coupled phase equations, and the linear stability of the synchronized state between the networks is represented as a function of the internetwork coupling matrix. We seek the optimal coupling by minimizing the Frobenius and L1 norms of the internetwork coupling matrix for the prescribed linear stability of the synchronized state. Depending on the norm, either a dense or sparse internetwork coupling yielding efficient mutual synchronization of the networks is obtained. In particular, a sparse yet resilient internetwork coupling is obtained by L1-norm optimization with additional constraints on the individual connection weights.
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Affiliation(s)
- Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Katsunori Yamaguchi
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Shingo Katayama
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Tatsuo Yanagita
- Department of Engineering Science, Osaka Electro-Communication University, Neyagawa, Osaka 572-8530, Japan
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15
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Abstract
We investigate oscillations in coupled systems. The methodology is based on the Hopf bifurcation theorem and a condition extended from the Routh–Hurwitz criterion. Such a condition leads to locating the bifurcation values of the parameters. With such an approach, we analyze a single-cell system modeling the minimal genetic negative feedback loop and the coupled-cell system composed by these single-cell systems. We study the oscillatory properties for these systems and compare these properties between the model with Hill-type repression and the one with protein-sequestration-based repression. As the parameters move from the Hopf bifurcation value for single cells to the one for coupled cells, we compute the eigenvalues of the linearized systems to obtain the magnitude of the collective frequency when the periodic solution of the coupled-cell system is generated. Extending from this information on the parameter values, we further compute and compare the collective frequency for the coupled-cell system and the average frequency of the decoupled individual cells. To compare these scenarios with other biological oscillators, we perform parallel analysis and computations on a segmentation clock model.
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16
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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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17
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León I, Pazó D. Quasi phase reduction of all-to-all strongly coupled λ-ω oscillators near incoherent states. Phys Rev E 2020; 102:042203. [PMID: 33212714 DOI: 10.1103/physreve.102.042203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2020] [Accepted: 09/10/2020] [Indexed: 11/07/2022]
Abstract
The dynamics of an ensemble of N weakly coupled limit-cycle oscillators can be captured by their N phases using standard phase reduction techniques. However, it is a phenomenological fact that all-to-all strongly coupled limit-cycle oscillators may behave as "quasiphase oscillators," evidencing the need of novel reduction strategies. We introduce, here, quasi phase reduction (QPR), a scheme suited for identical oscillators with polar symmetry (λ-ω systems). By applying QPR, we achieve a reduction to N+2 degrees of freedom: N phase oscillators interacting through one independent complex variable. This "quasi phase model" is asymptotically valid in the neighborhood of incoherent states, irrespective of the coupling strength. The effectiveness of QPR is illustrated in a particular case, an ensemble of Stuart-Landau oscillators, obtaining exact stability boundaries of uniform and nonuniform incoherent states for a variety of couplings. An extension of QPR beyond the neighborhood of incoherence is also explored. Finally, a general QPR model with N+2M degrees of freedom is obtained for coupling through the first M harmonics.
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Affiliation(s)
- Iván León
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
| | - Diego Pazó
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
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Nicolaou ZG, Sebek M, Kiss IZ, Motter AE. Coherent Dynamics Enhanced by Uncorrelated Noise. PHYSICAL REVIEW LETTERS 2020; 125:094101. [PMID: 32915595 DOI: 10.1103/physrevlett.125.094101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2020] [Revised: 06/17/2020] [Accepted: 07/21/2020] [Indexed: 06/11/2023]
Abstract
Synchronization is a widespread phenomenon observed in physical, biological, and social networks, which persists even under the influence of strong noise. Previous research on oscillators subject to common noise has shown that noise can actually facilitate synchronization, as correlations in the dynamics can be inherited from the noise itself. However, in many spatially distributed networks, such as the mammalian circadian system, the noise that different oscillators experience can be effectively uncorrelated. Here, we show that uncorrelated noise can in fact enhance synchronization when the oscillators are coupled. Strikingly, our analysis also shows that uncorrelated noise can be more effective than common noise in enhancing synchronization. We first establish these results theoretically for phase and phase-amplitude oscillators subject to either or both additive and multiplicative noise. We then confirm the predictions through experiments on coupled electrochemical oscillators. Our findings suggest that uncorrelated noise can promote rather than inhibit coherence in natural systems and that the same effect can be harnessed in engineered systems.
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Affiliation(s)
- Zachary G Nicolaou
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
| | - Michael Sebek
- Department of Chemistry, Saint Louis University, St. Louis, Missouri 63103, USA
- Network Science Institute, Northeastern University, Boston, Massachusetts 02115, USA
| | - István Z Kiss
- Department of Chemistry, Saint Louis University, St. Louis, Missouri 63103, USA
| | - Adilson E Motter
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
- Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA
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Tyulkina IV, Goldobin DS, Klimenko LS, Poperechny IS, Raikher YL. Collective in-plane magnetization in a two-dimensional XY macrospin system within the framework of generalized Ott-Antonsen theory. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190259. [PMID: 32279627 PMCID: PMC7202769 DOI: 10.1098/rsta.2019.0259] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
The problem of magnetic transitions between the low-temperature (macrospin ordered) phases in two-dimensional XY arrays is addressed. The system is modelled as a plane structure of identical single-domain particles arranged in a square lattice and coupled by the magnetic dipole-dipole interaction; all the particles possess a strong easy-plane magnetic anisotropy. The basic state of the system in the considered temperature range is an antiferromagnetic (AF) stripe structure, where the macrospins (particle magnetic moments) are still involved in thermofluctuational motion: the superparamagnetic blocking Tb temperature is lower than that (Taf) of the AF transition. The description is based on the stochastic equations governing the dynamics of individual magnetic moments, where the interparticle interaction is added in the mean-field approximation. With the technique of a generalized Ott-Antonsen theory, the dynamics equations for the order parameters (including the macroscopic magnetization and the AF order parameter) and the partition function of the system are rigorously obtained and analysed. We show that inside the temperature interval of existence of the AF phase, a static external field tilted to the plane of the array is able to induce first-order phase transitions from AF to ferromagnetic state; the phase diagrams displaying stable and metastable regions of the system are presented. This article is part of the theme issue 'Patterns in soft and biological matters'.
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Affiliation(s)
- Irina V. Tyulkina
- Institute of Continuous Media Mechanics, UB RAS, Academician Korolev Street 1, Perm 614068, Russia
| | - Denis S. Goldobin
- Institute of Continuous Media Mechanics, UB RAS, Academician Korolev Street 1, Perm 614068, Russia
- Department of Theoretical Physics, Perm State University, Bukirev Street 15, Perm 614990, Russia
| | - Lyudmila S. Klimenko
- Institute of Continuous Media Mechanics, UB RAS, Academician Korolev Street 1, Perm 614068, Russia
- Department of Theoretical Physics, Perm State University, Bukirev Street 15, Perm 614990, Russia
| | - Igor S. Poperechny
- Institute of Continuous Media Mechanics, UB RAS, Academician Korolev Street 1, Perm 614068, Russia
| | - Yuriy L. Raikher
- Institute of Continuous Media Mechanics, UB RAS, Academician Korolev Street 1, Perm 614068, Russia
- e-mail:
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Stankovski T, Pereira T, McClintock PVE, Stefanovska A. Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190039. [PMID: 31656134 PMCID: PMC6834002 DOI: 10.1098/rsta.2019.0039] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 08/13/2019] [Indexed: 06/10/2023]
Abstract
Dynamical systems are widespread, with examples in physics, chemistry, biology, population dynamics, communications, climatology and social science. They are rarely isolated but generally interact with each other. These interactions can be characterized by coupling functions-which contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how each interaction occurs. Coupling functions can be used, not only to understand, but also to control and predict the outcome of the interactions. This theme issue assembles ground-breaking work on coupling functions by leading scientists. After overviewing the field and describing recent advances in the theory, it discusses novel methods for the detection and reconstruction of coupling functions from measured data. It then presents applications in chemistry, neuroscience, cardio-respiratory physiology, climate, electrical engineering and social science. Taken together, the collection summarizes earlier work on coupling functions, reviews recent developments, presents the state of the art, and looks forward to guide the future evolution of the field. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.
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Affiliation(s)
- Tomislav Stankovski
- Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
- Faculty of Medicine, Ss Cyril and Methodius University, Skopje 1000, Macedonia
| | - Tiago Pereira
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
- Institute of Mathematical and Computer Sciences, University of Sao Paulo, Sao Carlos 13566-590, Brazil
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