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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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Toth K, Wilson D. The influence of synaptic plasticity on critical coupling estimates for neural populations. J Math Biol 2024; 88:39. [PMID: 38441655 DOI: 10.1007/s00285-024-02061-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2023] [Revised: 01/31/2024] [Accepted: 02/04/2024] [Indexed: 03/07/2024]
Abstract
The presence or absence of synaptic plasticity can dramatically influence the collective behavior of populations of coupled neurons. In this work, we consider spike-timing dependent plasticity (STDP) and its resulting influence on phase cohesion in computational models of heterogeneous populations of conductance-based neurons. STDP allows for the influence of individual synapses to change over time, strengthening or weakening depending on the relative timing of the relevant action potentials. Using phase reduction techniques, we derive an upper bound on the critical coupling strength required to retain phase cohesion for a network of synaptically coupled, heterogeneous neurons with STDP. We find that including STDP can significantly alter phase cohesion as compared to a network with static synaptic connections. Analytical results are validated numerically. Our analysis highlights the importance of the relative ordering of action potentials emitted in a population of tonically firing neurons and demonstrates that order switching can degrade the synchronizing influence of coupling when STDP is considered.
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Affiliation(s)
- Kaitlyn Toth
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN, 37966, USA
| | - Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN, 37966, USA.
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3
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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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4
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Mau ETK, Rosenblum M, Pikovsky A. High-order phase reduction for coupled 2D oscillators. CHAOS (WOODBURY, N.Y.) 2023; 33:101101. [PMID: 37831797 DOI: 10.1063/5.0169008] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2023] [Accepted: 09/20/2023] [Indexed: 10/15/2023]
Abstract
Phase reduction is a general approach to describe coupled oscillatory units in terms of their phases, assuming that the amplitudes are enslaved. The coupling should be small for such reduction, but one also expects the reduction to be valid for finite coupling. This paper presents a general framework, allowing us to obtain coupling terms in higher orders of the coupling parameter for generic two-dimensional oscillators and arbitrary coupling terms. The theory is illustrated with an accurate prediction of Arnold's tongue for the van der Pol oscillator exploiting higher-order phase reduction.
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Affiliation(s)
- Erik T K Mau
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany
| | - Michael Rosenblum
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany
| | - Arkady Pikovsky
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany
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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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NPDS toolbox: Neural population (De) synchronization toolbox for MATLAB. Neurocomputing 2022. [DOI: 10.1016/j.neucom.2022.07.060] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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7
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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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8
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Winkler M, Dumont G, Schöll E, Gutkin B. Phase response approaches to neural activity models with distributed delay. BIOLOGICAL CYBERNETICS 2022; 116:191-203. [PMID: 34853889 DOI: 10.1007/s00422-021-00910-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2021] [Accepted: 10/31/2021] [Indexed: 06/13/2023]
Abstract
In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.
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Affiliation(s)
- Marius Winkler
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, 24 rue Lhomond, 75005, Paris, France
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623, Berlin, Germany
| | - Grégory Dumont
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, 24 rue Lhomond, 75005, Paris, France
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623, Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität, Philippstraße 13, 10115, Berlin, Germany
- Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473, Potsdam, Germany
| | - Boris Gutkin
- Group for Neural Theory, LNC INSERM U960, DEC, Ecole Normale Supérieure PSL* University, 24 rue Lhomond, 75005, Paris, France.
- Center for Cognition and Decision Making, Institue for Cognitive Neuroscience, NRU Higher School of Economics, Krivokolenniy sidewalk 3, 101000, Moscow, Russia.
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9
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Toth K, Wilson D. Control of coupled neural oscillations using near-periodic inputs. CHAOS (WOODBURY, N.Y.) 2022; 32:033130. [PMID: 35364826 DOI: 10.1063/5.0076508] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2021] [Accepted: 02/22/2022] [Indexed: 06/14/2023]
Abstract
Deep brain stimulation (DBS) is a commonly used treatment for medication resistant Parkinson's disease and is an emerging treatment for other neurological disorders. More recently, phase-specific adaptive DBS (aDBS), whereby the application of stimulation is locked to a particular phase of tremor, has been proposed as a strategy to improve therapeutic efficacy and decrease side effects. In this work, in the context of these phase-specific aDBS strategies, we investigate the dynamical behavior of large populations of coupled neurons in response to near-periodic stimulation, namely, stimulation that is periodic except for a slowly changing amplitude and phase offset that can be used to coordinate the timing of applied input with a specified phase of model oscillations. Using an adaptive phase-amplitude reduction strategy, we illustrate that for a large population of oscillatory neurons, the temporal evolution of the associated phase distribution in response to near-periodic forcing can be captured using a reduced order model with four state variables. Subsequently, we devise and validate a closed-loop control strategy to disrupt synchronization caused by coupling. Additionally, we identify strategies for implementing the proposed control strategy in situations where underlying model equations are unavailable by estimating the necessary terms of the reduced order equations in real-time from observables.
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Affiliation(s)
- Kaitlyn Toth
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
| | - Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
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10
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Spiliotis K, Starke J, Franz D, Richter A, Köhling R. Deep brain stimulation for movement disorder treatment: exploring frequency-dependent efficacy in a computational network model. BIOLOGICAL CYBERNETICS 2022; 116:93-116. [PMID: 34894291 PMCID: PMC8866393 DOI: 10.1007/s00422-021-00909-2] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/20/2020] [Accepted: 10/31/2021] [Indexed: 06/14/2023]
Abstract
A large-scale computational model of the basal ganglia network and thalamus is proposed to describe movement disorders and treatment effects of deep brain stimulation (DBS). The model of this complex network considers three areas of the basal ganglia region: the subthalamic nucleus (STN) as target area of DBS, the globus pallidus, both pars externa and pars interna (GPe-GPi), and the thalamus. Parkinsonian conditions are simulated by assuming reduced dopaminergic input and corresponding pronounced inhibitory or disinhibited projections to GPe and GPi. Macroscopic quantities are derived which correlate closely to thalamic responses and hence motor programme fidelity. It can be demonstrated that depending on different levels of striatal projections to the GPe and GPi, the dynamics of these macroscopic quantities (synchronisation index, mean synaptic activity and response efficacy) switch from normal to Parkinsonian conditions. Simulating DBS of the STN affects the dynamics of the entire network, increasing the thalamic activity to levels close to normal, while differing from both normal and Parkinsonian dynamics. Using the mentioned macroscopic quantities, the model proposes optimal DBS frequency ranges above 130 Hz.
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Affiliation(s)
| | - Jens Starke
- Institute of Mathematics, University of Rostock, 18057 Rostock, Germany
| | - Denise Franz
- Oscar-Langendorff-Institute of Physiology, Rostock University Medical Center, Rostock, Germany
| | - Angelika Richter
- Institute of Pharmacology, Pharmacy and Toxicology, Faculty of Veterinary Medicine, University of Leipzig, Leipzig, Germany
| | - Rüdiger Köhling
- Oscar-Langendorff-Institute of Physiology, Rostock University Medical Center, Rostock, Germany
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11
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Hunter I, Norton MM, Chen B, Simonetti C, Moustaka ME, Touboul J, Fraden S. Pattern formation in a four-ring reaction-diffusion network with heterogeneity. Phys Rev E 2022; 105:024310. [PMID: 35291089 DOI: 10.1103/physreve.105.024310] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2021] [Accepted: 01/28/2022] [Indexed: 11/07/2022]
Abstract
In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested in free-running (i.e., non-computer-controlled) systems. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of the theory of dynamical systems with symmeries in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of four microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming homogeneity across the oscillators, theory predicts four categories of stable spatiotemporal phase-locked periodic states and four categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed that three of the four phase-locked states were displaced from their idealized positions and, in the ensemble of measurements, appeared as clusters of different shapes and sizes, and that one of the predicted states was absent. We also observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity in intrinsic frequency. We further elucidate how different patterns of heterogeneity impact each attractor differently through a bifurcation analysis. We show that examining bifurcations along invariant manifolds provides a general framework for developing intuition about how chemical-specific dynamics interact with topology in the presence of heterogeneity that can be applied to other oscillators in other topologies.
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Affiliation(s)
- Ian Hunter
- Brandeis University Physics, Waltham, Massachusetts 02453, USA
| | - Michael M Norton
- Center for Neural Engineering, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Bolun Chen
- Volen National Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02453, USA.,Department of Physics, Boston University, Boston Massachusetts 02215, USA
| | - Chris Simonetti
- Brandeis University Physics, Waltham, Massachusetts 02453, USA
| | | | - Jonathan Touboul
- Volen National Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02453, USA.,Brandeis University Mathematics Department, Waltham, Massachusetts 02453, USA
| | - Seth Fraden
- Brandeis University Physics, Waltham, Massachusetts 02453, USA
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12
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Pérez-Cervera A, Lindner B, Thomas PJ. Isostables for Stochastic Oscillators. PHYSICAL REVIEW LETTERS 2021; 127:254101. [PMID: 35029447 DOI: 10.1103/physrevlett.127.254101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2021] [Revised: 10/18/2021] [Accepted: 11/04/2021] [Indexed: 05/25/2023]
Abstract
Thomas and Lindner [P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.254101], defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.
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Affiliation(s)
- Alberto Pérez-Cervera
- National Research University Higher School of Economics, 109208 Moscow, Russia and Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, Haus 2, 10115 Berlin, Germany and Institute of Physics, Humboldt University at Berlin, Newtonstraße 15, D-12489 Berlin, Germany
| | - Peter J Thomas
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, Ohio 44106, USA
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13
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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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14
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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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15
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Ahmed T, Wilson D. Exploiting circadian memory to hasten recovery from circadian misalignment. CHAOS (WOODBURY, N.Y.) 2021; 31:073130. [PMID: 34340336 DOI: 10.1063/5.0053441] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 06/23/2021] [Indexed: 06/13/2023]
Abstract
Recent years have seen a sustained interest in the development of circadian reentrainment strategies to limit the deleterious effects of jet lag. Due to the dynamical complexity of many circadian models, phase-based model reduction techniques are often an imperative first step in the analysis. However, amplitude coordinates that capture lingering effects (i.e., memory) from past inputs are often neglected. In this work, we focus on these amplitude coordinates using an operational phase and an isostable coordinate framework in the context of the development of jet-lag amelioration strategies. By accounting for the influence of circadian memory, we identify a latent phase shift that can prime one's circadian cycle to reentrain more rapidly to an expected time-zone shift. A subsequent optimal control problem is proposed that balances the trade-off between control effort and the resulting latent phase shift. Data-driven model identification techniques for the inference of necessary reduced order, phase-amplitude-based models are considered in situations where the underlying model equations are unknown, and numerical results are illustrated in both a simple planar model and in a coupled population of circadian oscillators.
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Affiliation(s)
- Talha Ahmed
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Knoxville, Tennessee 37996, USA
| | - Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Knoxville, Tennessee 37996, USA
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16
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Das TS, Wilson D. Data-driven phase-isostable reduction for optimal nonfeedback stabilization of cardiac alternans. Phys Rev E 2021; 103:052203. [PMID: 34134261 DOI: 10.1103/physreve.103.052203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2020] [Accepted: 04/07/2021] [Indexed: 11/07/2022]
Abstract
Phase-isostable reduction is an emerging model reduction strategy that can be used to accurately replicate nonlinear behaviors in systems for which standard phase reduction techniques fail. In this work, we derive relationships between the cycle-to-cycle variance of the reduced isostable coordinates for systems subject to both additive white noise and periodic stimulation. Using this information, we propose a data-driven technique for inferring nonlinear terms of the phase-isostable coordinate reduction framework. We apply the proposed model inference strategy to the biologically motivated problem of eliminating cardiac alternans, an arrhythmia that is widely considered to be a precursor to more deadly cardiac arrhythmias. Using this strategy, by simply measuring a series of action potential durations in response to periodic stimulation, we are able to identify energy-optimal, nonfeedback control inputs to stabilize a period-1, alternans-free solution.
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Affiliation(s)
- Tuhin Subhra Das
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
| | - Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
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17
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Wilson D. Data-driven inference of high-accuracy isostable-based dynamical models in response to external inputs. CHAOS (WOODBURY, N.Y.) 2021; 31:063137. [PMID: 34241295 DOI: 10.1063/5.0042874] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2021] [Accepted: 06/07/2021] [Indexed: 06/13/2023]
Abstract
Isostable reduction is a powerful technique that can be used to characterize behaviors of nonlinear dynamical systems using a basis of slowly decaying eigenfunctions of the Koopman operator. When the underlying dynamical equations are known, previously developed numerical techniques allow for high-order accuracy computation of isostable reduced models. However, in situations where the dynamical equations are unknown, few general techniques are available that provide reliable estimates of the isostable reduced equations, especially in applications where large magnitude inputs are considered. In this work, a purely data-driven inference strategy yielding high-accuracy isostable reduced models is developed for dynamical systems with a fixed point attractor. By analyzing steady-state outputs of nonlinear systems in response to sinusoidal forcing, both isostable response functions and isostable-to-output relationships can be estimated to arbitrary accuracy in an expansion performed in the isostable coordinates. Detailed examples are considered for a population of synaptically coupled neurons and for the one-dimensional Burgers' equation. While linear estimates of the isostable response functions are sufficient to characterize the dynamical behavior when small magnitude inputs are considered, the high-accuracy reduced order model inference strategy proposed here is essential when considering large magnitude inputs.
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Affiliation(s)
- Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Knoxville, Tennessee 37996, USA
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18
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Matchen TD, Moehlis J. Leveraging deep learning to control neural oscillators. BIOLOGICAL CYBERNETICS 2021; 115:219-235. [PMID: 33909165 DOI: 10.1007/s00422-021-00874-w] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2020] [Accepted: 04/10/2021] [Indexed: 06/12/2023]
Abstract
Modulation of the firing times of neural oscillators has long been an important control objective, with applications including Parkinson's disease, Tourette's syndrome, epilepsy, and learning. One common goal for such modulation is desynchronization, wherein two or more oscillators are stimulated to transition from firing in phase with each other to firing out of phase. The optimization of such stimuli has been well studied, but this typically relies on either a reduction of the dimensionality of the system or complete knowledge of the parameters and state of the system. This limits the applicability of results to real problems in neural control. Here, we present a trained artificial neural network capable of accurately estimating the effects of square-wave stimuli on neurons using minimal output information from the neuron. We then apply the results of this network to solve several related control problems in desynchronization, including desynchronizing pairs of neurons and achieving clustered subpopulations of neurons in the presence of coupling and noise.
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Affiliation(s)
- Timothy D Matchen
- Department of Mechanical Engineering, University of California, Santa Barbara, CA, 93106, USA.
| | - Jeff Moehlis
- Department of Mechanical Engineering, Program in Dynamical Neuroscience, University of California, Santa Barbara, CA, 93106, USA
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19
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Wilson D. Degenerate isostable reduction for fixed-point and limit-cycle attractors with defective linearizations. Phys Rev E 2021; 103:022211. [PMID: 33735978 DOI: 10.1103/physreve.103.022211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2020] [Accepted: 02/02/2021] [Indexed: 11/07/2022]
Abstract
Isostable coordinates provide a convenient framework for understanding the transient behavior of dynamical systems with stable attractors. These isostable coordinates are often used to characterize the slowest decaying eigenfunctions of the Koopman operator; by neglecting the rapidly decaying Koopman eigenfunctions a reduced order model can be obtained. Existing work has focused primarily on nondegenerate isostable coordinates, that is, isostable coordinates that are associated with eigenvalues that have identical algebraic and geometric multiplicities. Current isostable reduction methods cannot be applied to characterize the decay associated with a defective eigenvalue. In this work, a degenerate isostable framework is proposed for use when eigenvalues are defective. These degenerate isostable coordinates are investigated in the context of various reduced order modeling frameworks that retain many of the important properties of standard (nondegenerate) isostable reduced modeling strategies. Reduced order modeling examples that require the use of degenerate isostable coordinates are presented with relevance to both circadian physiology and nonlinear fluid flows.
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Affiliation(s)
- Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
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20
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Ding Y, Ermentrout B. Traveling waves in non-local pulse-coupled networks. J Math Biol 2021; 82:18. [PMID: 33570663 DOI: 10.1007/s00285-021-01572-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2020] [Revised: 10/29/2020] [Accepted: 01/19/2021] [Indexed: 11/25/2022]
Abstract
Traveling phase waves are commonly observed in recordings of the cerebral cortex and are believed to organize behavior across different areas of the brain. We use this as motivation to analyze a one-dimensional network of phase oscillators that are nonlocally coupled via the phase response curve (PRC) and the Dirac delta function. Existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem. First and second order perturbation theory is applied to get analytic insight and we show that long waves are stable while short waves are unstable. We apply the results to PRCs that come from mitral neurons. We extend the results to smooth pulse-like coupling by reducing the nonlocal equation to a local one and solving the associated boundary value problem.
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Affiliation(s)
- Yujie Ding
- University of Pittsburgh, Pennsylvania, USA
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21
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Wilson D. Analysis of input-induced oscillations using the isostable coordinate framework. CHAOS (WOODBURY, N.Y.) 2021; 31:023131. [PMID: 33653055 DOI: 10.1063/5.0036508] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2020] [Accepted: 01/26/2021] [Indexed: 06/12/2023]
Abstract
Many reduced order modeling techniques for oscillatory dynamical systems are only applicable when the underlying system admits a stable periodic orbit in the absence of input. By contrast, very few reduction frameworks can be applied when the oscillations themselves are induced by coupling or other exogenous inputs. In this work, the behavior of such input-induced oscillations is considered. By leveraging the isostable coordinate framework, a high-accuracy reduced set of equations can be identified and used to predict coupling-induced bifurcations that precipitate stable oscillations. Subsequent analysis is performed to predict the steady state phase-locking relationships. Input-induced oscillations are considered for two classes of coupled dynamical systems. For the first, stable fixed points of systems with parameters near Hopf bifurcations are considered so that the salient dynamical features can be captured using an asymptotic expansion of the isostable coordinate dynamics. For the second, an adaptive phase-amplitude reduction framework is used to analyze input-induced oscillations that emerge in excitable systems. Examples with relevance to circadian and neural physiology are provided that highlight the utility of the proposed techniques.
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Affiliation(s)
- Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
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22
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Kuelbs D, Dunefsky J, Monga B, Moehlis J. Analysis of neural clusters due to deep brain stimulation pulses. BIOLOGICAL CYBERNETICS 2020; 114:589-607. [PMID: 33296013 DOI: 10.1007/s00422-020-00850-w] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/04/2020] [Accepted: 11/20/2020] [Indexed: 06/12/2023]
Abstract
Deep brain stimulation (DBS) is an established method for treating pathological conditions such as Parkinson's disease, dystonia, Tourette syndrome, and essential tremor. While the precise mechanisms which underly the effectiveness of DBS are not fully understood, several theoretical studies of populations of neural oscillators stimulated by periodic pulses have suggested that this may be related to clustering, in which subpopulations of the neurons are synchronized, but the subpopulations are desynchronized with respect to each other. The details of the clustering behavior depend on the frequency and amplitude of the stimulation in a complicated way. In the present study, we investigate how the number of clusters and their stability properties, bifurcations, and basins of attraction can be understood in terms of one-dimensional maps defined on the circle. Moreover, we generalize this analysis to stimuli that consist of pulses with alternating properties, which provide additional degrees of freedom in the design of DBS stimuli. Our results illustrate how the complicated properties of clustering behavior for periodically forced neural oscillator populations can be understood in terms of a much simpler dynamical system.
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Affiliation(s)
| | | | - Bharat Monga
- Department of Mechanical Engineering, University of California, Santa Barbara, CA, 93106, USA
| | - Jeff Moehlis
- Department of Mechanical Engineering, Program in Dynamical Neuroscience, University of California, Santa Barbara, CA, 93106, USA.
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23
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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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24
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Norton MM, Grover P, Hagan MF, Fraden S. Optimal Control of Active Nematics. PHYSICAL REVIEW LETTERS 2020; 125:178005. [PMID: 33156653 DOI: 10.1103/physrevlett.125.178005] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2020] [Accepted: 09/18/2020] [Indexed: 06/11/2023]
Abstract
In this work we present the first systematic framework to sculpt active nematic systems, using optimal control theory and a hydrodynamic model of active nematics. We demonstrate the use of two different control fields, (i) applied vorticity and (ii) activity strength, to shape the dynamics of an extensile active nematic that is confined to a disk. In the absence of control inputs, the system exhibits two attractors, clockwise and counterclockwise circulating states characterized by two co-rotating topological +1/2 defects. We specifically seek spatiotemporal inputs that switch the system from one attractor to the other; we also examine phase-shifting perturbations. We identify control inputs by optimizing a penalty functional with three contributions: total control effort, spatial gradients in the control, and deviations from the desired trajectory. This work demonstrates that optimal control theory can be used to calculate nontrivial inputs capable of restructuring active nematics in a manner that is economical, smooth, and rapid, and therefore will serve as a guide to experimental efforts to control active matter.
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Affiliation(s)
- Michael M Norton
- Center for Neural Engineering, Department of Engineering Science and Materials, Pennsylvania State University, University Park, Pennsylvania 16801, USA and Physics Department, Brandeis University, Waltham, Massachusetts 02453, USA
| | - Piyush Grover
- Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA
| | - Michael F Hagan
- Physics Department, Brandeis University, Waltham, Massachusetts 02453, USA
| | - Seth Fraden
- Physics Department, Brandeis University, Waltham, Massachusetts 02453, USA
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25
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Pérez-Cervera A, M-Seara T, Huguet G. Global phase-amplitude description of oscillatory dynamics via the parameterization method. CHAOS (WOODBURY, N.Y.) 2020; 30:083117. [PMID: 32872842 DOI: 10.1063/5.0010149] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2020] [Accepted: 07/13/2020] [Indexed: 05/25/2023]
Abstract
In this paper, we use the parameterization method to provide a complete description of the dynamics of an n-dimensional oscillator beyond the classical phase reduction. The parameterization method allows us, via efficient algorithms, to obtain a parameterization of the attracting invariant manifold of the limit cycle in terms of the phase-amplitude variables. The method has several advantages. It provides analytically a Fourier-Taylor expansion of the parameterization up to any order, as well as a simplification of the dynamics that allows for a numerical globalization of the manifolds. Thus, one can obtain the local and global isochrons and isostables, including the slow attracting manifold, up to high accuracy, which offer a geometrical portrait of the oscillatory dynamics. Furthermore, it provides straightforwardly the infinitesimal phase and amplitude response functions, that is, the extended infinitesimal phase and amplitude response curves, which monitor the phase and amplitude shifts beyond the asymptotic state. Thus, the methodology presented yields an accurate description of the phase dynamics for perturbations not restricted to the limit cycle but to its attracting invariant manifold. Finally, we explore some strategies to reduce the dimension of the dynamics, including the reduction of the dynamics to the slow stable submanifold. We illustrate our methods by applying them to different three-dimensional single neuron and neural population models in neuroscience.
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Affiliation(s)
- Alberto Pérez-Cervera
- Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avda. Diagonal 647, 08028 Barcelona, Spain
| | - Tere M-Seara
- Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avda. Diagonal 647, 08028 Barcelona, Spain
| | - Gemma Huguet
- Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avda. Diagonal 647, 08028 Barcelona, Spain
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26
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Bick C, Goodfellow M, Laing CR, Martens EA. Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:9. [PMID: 32462281 PMCID: PMC7253574 DOI: 10.1186/s13408-020-00086-9] [Citation(s) in RCA: 91] [Impact Index Per Article: 22.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 05/07/2020] [Indexed: 05/03/2023]
Abstract
Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.
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Affiliation(s)
- Christian Bick
- Centre for Systems, Dynamics, and Control, University of Exeter, Exeter, UK.
- Department of Mathematics, University of Exeter, Exeter, UK.
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK.
- Mathematical Institute, University of Oxford, Oxford, UK.
- Institute for Advanced Study, Technische Universität München, Garching, Germany.
| | - Marc Goodfellow
- Department of Mathematics, University of Exeter, Exeter, UK
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK
- Living Systems Institute, University of Exeter, Exeter, UK
- Wellcome Trust Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter, UK
| | - Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark.
- Department of Biomedical Science, University of Copenhagen, Copenhagen N, Denmark.
- Centre for Translational Neuroscience, University of Copenhagen, Copenhagen N, Denmark.
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27
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Optimal open-loop desynchronization of neural oscillator populations. J Math Biol 2020; 81:25-64. [PMID: 32418056 DOI: 10.1007/s00285-020-01501-1] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2019] [Revised: 02/28/2020] [Indexed: 10/24/2022]
Abstract
Deep brain stimulation (DBS) is an increasingly used medical treatment for various neurological disorders. While its mechanisms are not fully understood, experimental evidence suggests that through application of periodic electrical stimulation DBS may act to desynchronize pathologically synchronized populations of neurons resulting desirable changes to a larger brain circuit. However, the underlying mathematical mechanisms by which periodic stimulation can engender desynchronization in a coupled population of neurons is not well understood. In this work, a reduced phase-amplitude reduction framework is used to characterize the desynchronizing influence of periodic stimulation on a population of coupled oscillators. Subsequently, optimal control theory allows for the design of periodic, open-loop stimuli with the capacity to destabilize completely synchronized solutions while simultaneously stabilizing rotating block solutions. This framework exploits system nonlinearities in order to strategically modify unstable Floquet exponents. In the limit of weak neural coupling, it is shown that this method only requires information about the phase response curves of the individual neurons. The effects of noise and heterogeneity are also considered and numerical results are presented. This framework could ultimately be used to inform the design of more efficient deep brain stimulation waveforms for the treatment of neurological disease.
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28
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Duchet B, Weerasinghe G, Cagnan H, Brown P, Bick C, Bogacz R. Phase-dependence of response curves to deep brain stimulation and their relationship: from essential tremor patient data to a Wilson-Cowan model. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:4. [PMID: 32232686 PMCID: PMC7105566 DOI: 10.1186/s13408-020-00081-0] [Citation(s) in RCA: 16] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2019] [Accepted: 03/12/2020] [Indexed: 05/15/2023]
Abstract
Essential tremor manifests predominantly as a tremor of the upper limbs. One therapy option is high-frequency deep brain stimulation, which continuously delivers electrical stimulation to the ventral intermediate nucleus of the thalamus at about 130 Hz. Constant stimulation can lead to side effects, it is therefore desirable to find ways to stimulate less while maintaining clinical efficacy. One strategy, phase-locked deep brain stimulation, consists of stimulating according to the phase of the tremor. To advance methods to optimise deep brain stimulation while providing insights into tremor circuits, we ask the question: can the effects of phase-locked stimulation be accounted for by a canonical Wilson-Cowan model? We first analyse patient data, and identify in half of the datasets significant dependence of the effects of stimulation on the phase at which stimulation is provided. The full nonlinear Wilson-Cowan model is fitted to datasets identified as statistically significant, and we show that in each case the model can fit to the dynamics of patient tremor as well as to the phase response curve. The vast majority of top fits are stable foci. The model provides satisfactory prediction of how patient tremor will react to phase-locked stimulation by predicting patient amplitude response curves although they were not explicitly fitted. We also approximate response curves of the significant datasets by providing analytical results for the linearisation of a stable focus model, a simplification of the Wilson-Cowan model in the stable focus regime. We report that the nonlinear Wilson-Cowan model is able to describe response to stimulation more precisely than the linearisation.
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Affiliation(s)
- Benoit Duchet
- Nuffield Department of Clinical Neuroscience, University of Oxford, Oxford, UK
- MRC Brain Network Dynamics Unit, University of Oxford, Oxford, UK
| | - Gihan Weerasinghe
- Nuffield Department of Clinical Neuroscience, University of Oxford, Oxford, UK
- MRC Brain Network Dynamics Unit, University of Oxford, Oxford, UK
| | - Hayriye Cagnan
- Nuffield Department of Clinical Neuroscience, University of Oxford, Oxford, UK
- MRC Brain Network Dynamics Unit, University of Oxford, Oxford, UK
- Wellcome Centre for Human Neuroimaging, UCL Institute of Neurology, London, UK
| | - Peter Brown
- Nuffield Department of Clinical Neuroscience, University of Oxford, Oxford, UK
- MRC Brain Network Dynamics Unit, University of Oxford, Oxford, UK
| | - Christian Bick
- Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford, UK
- Centre for Systems, Dynamics, and Control and Department of Mathematics, University of Exeter, Exeter, UK
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK
| | - Rafal Bogacz
- Nuffield Department of Clinical Neuroscience, University of Oxford, Oxford, UK
- MRC Brain Network Dynamics Unit, University of Oxford, Oxford, UK
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29
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Wilson D. Phase-amplitude reduction far beyond the weakly perturbed paradigm. Phys Rev E 2020; 101:022220. [PMID: 32168672 DOI: 10.1103/physreve.101.022220] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2019] [Accepted: 01/27/2020] [Indexed: 06/10/2023]
Abstract
While phase reduction is a well-established technique for the analysis of perturbed limit cycle oscillators, practical application requires perturbations to be sufficiently weak thereby limiting its utility in many situations. Here, a general strategy is developed for constructing a set of phase-amplitude reduced equations that is valid to arbitrary orders of accuracy in the amplitude coordinates. This reduction framework can be used to investigate the behavior of oscillatory dynamical systems far beyond the weakly perturbed paradigm. Additionally, a patchwork phase-amplitude reduction method is suggested that is useful when exceedingly large magnitude perturbations are considered. This patchwork method incorporates the high-accuracy phase-amplitude reductions of multiple nearby periodic orbits that result from modifications to nominal parameters. The proposed method of high-accuracy phase-amplitude reduction can be readily implemented numerically and examples are provided where reductions are computed up to fourteenth order accuracy.
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Affiliation(s)
- Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
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30
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Kato Y, Nakao H. Semiclassical optimization of entrainment stability and phase coherence in weakly forced quantum limit-cycle oscillators. Phys Rev E 2020; 101:012210. [PMID: 32069673 DOI: 10.1103/physreve.101.012210] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2019] [Indexed: 06/10/2023]
Abstract
Optimal entrainment of a quantum nonlinear oscillator to a periodically modulated weak harmonic drive is studied in the semiclassical regime. By using the semiclassical phase-reduction theory recently developed for quantum nonlinear oscillators [Y. Kato, N. Yamamoto, and H. Nakao, Phys. Rev. Res. 1, 033012 (2019)10.1103/PhysRevResearch.1.033012], two types of optimization problems, one for the stability and the other for the phase coherence of the entrained state, are considered. The optimal waveforms of the periodic amplitude modulation can be derived by applying the classical optimization methods to the semiclassical phase equation that approximately describes the quantum limit-cycle dynamics. Using a quantum van der Pol oscillator with squeezing and Kerr effects as an example, the performance of optimization is numerically analyzed. It is shown that the optimized waveform for the entrainment stability yields faster entrainment to the driving signal than the case with a simple sinusoidal waveform, while that for the phase coherence yields little improvement from the sinusoidal case. These results are explained from the properties of the phase sensitivity function.
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Affiliation(s)
- 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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31
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Wilson D. A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems. CHAOS (WOODBURY, N.Y.) 2020; 30:013121. [PMID: 32013514 DOI: 10.1063/1.5126122] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2019] [Accepted: 12/24/2019] [Indexed: 06/10/2023]
Abstract
Phase-amplitude reduction is of growing interest as a strategy for the reduction and analysis of oscillatory dynamical systems. Augmentation of the widely studied phase reduction with amplitude coordinates can be used to characterize transient behavior in directions transverse to a limit cycle to give a richer description of the dynamical behavior. Various definitions for amplitude coordinates have been suggested, but none are particularly well suited for implementation in experimental systems where output recordings are readily available but the underlying equations are typically unknown. In this work, a reduction framework is developed for inferring a phase-amplitude reduced model using only the observed model output from an arbitrarily high-dimensional system. This framework employs a proper orthogonal reduction strategy to identify important features of the transient decay of solutions to the limit cycle. These features are explicitly related to previously developed phase and isostable coordinates and used to define so-called data-driven phase and isostable coordinates that are valid in the entire basin of attraction of a limit cycle. The utility of this reduction strategy is illustrated in examples related to neural physiology and is used to implement an optimal control strategy that would otherwise be computationally intractable. The proposed data-driven phase and isostable coordinate system and associated reduced modeling framework represent a useful tool for the study of nonlinear dynamical systems in situations where the underlying dynamical equations are unknown and in particularly high-dimensional or complicated numerical systems for which standard phase-amplitude reduction techniques are not computationally feasible.
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Affiliation(s)
- Dan Wilson
- Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA
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32
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Kuramoto Y, Nakao H. On the concept of dynamical reduction: the case of coupled oscillators. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190041. [PMID: 31656146 PMCID: PMC6834004 DOI: 10.1098/rsta.2019.0041] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/02/2019] [Indexed: 05/26/2023]
Abstract
An overview is given on two representative methods of dynamical reduction known as centre-manifold reduction and phase reduction. These theories are presented in a somewhat more unified fashion than the theories in the past. The target systems of reduction are coupled limit-cycle oscillators. Particular emphasis is placed on the remarkable structural similarity existing between these theories. While the two basic principles, i.e. (i) reduction of dynamical degrees of freedom and (ii) transformation of reduced evolution equation to a canonical form, are shared commonly by reduction methods in general, it is shown how these principles are incorporated into the above two reduction theories in a coherent manner. Regarding the phase reduction, a new formulation of perturbative expansion is presented for discrete populations of oscillators. The style of description is intended to be so informal that one may digest, without being bothered with technicalities, what has been done after all under the word reduction. 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)
- Yoshiki Kuramoto
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan(emeritus)
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
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33
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Rosenblum M, Pikovsky A. Nonlinear phase coupling functions: a numerical study. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190093. [PMID: 31656143 PMCID: PMC6833997 DOI: 10.1098/rsta.2019.0093] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/04/2023]
Abstract
Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator. 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)
- Michael Rosenblum
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
- Control Theory Department, Institute of Information Technologies, Mathematics and Mechanics, Lobachevsky University Nizhny Novgorod, Nizhny Novgorod, Russia
- e-mail:
| | - Arkady Pikovsky
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
- Control Theory Department, Institute of Information Technologies, Mathematics and Mechanics, Lobachevsky University Nizhny Novgorod, Nizhny Novgorod, Russia
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Norton MM, Tompkins N, Blanc B, Cambria MC, Held J, Fraden S. Dynamics of Reaction-Diffusion Oscillators in Star and other Networks with Cyclic Symmetries Exhibiting Multiple Clusters. PHYSICAL REVIEW LETTERS 2019; 123:148301. [PMID: 31702219 DOI: 10.1103/physrevlett.123.148301] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2019] [Revised: 09/09/2019] [Indexed: 06/10/2023]
Abstract
We experimentally and theoretically investigate the dynamics of inhibitory coupled self-driven oscillators on a star network in which a single central hub node is connected to k peripheral arm nodes. The system consists of water-in-oil Belousov-Zhabotinsky ∼100 μm emulsion drops contained in storage wells etched in silicon wafers. We observed three dynamical attractors by varying the number of arms in the star graph and the coupling strength: (i) unlocked, uncorrelated phase shifts between all oscillators; (ii) locked, arm hubs synchronized in phase with a k-dependent phase shift between the arm and central hub; and (iii) center silent, a central hub stopped oscillating and the arm hubs oscillated without synchrony. We compare experiment to theory. For case (ii), we identified a logarithmic dependence of the phase shift on star degree, and were able to discriminate between contributions to the phase shift arising from star topology and oscillator chemistry.
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Affiliation(s)
- Michael M Norton
- Physics Department, Brandeis University, Waltham, Massachusetts 02453, USA
| | - Nathan Tompkins
- Physics Department, Brandeis University, Waltham, Massachusetts 02453, USA
- Physics Department, Wabash College, Crawfordsville, Indiana 47933, USA
| | - Baptiste Blanc
- Physics Department, Brandeis University, Waltham, Massachusetts 02453, USA
| | | | - Jesse Held
- Physics Department, Brandeis University, Waltham, Massachusetts 02453, USA
| | - Seth Fraden
- Physics Department, Brandeis University, Waltham, Massachusetts 02453, USA
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Watanabe N, Kato Y, Shirasaka S, Nakao H. Optimization of linear and nonlinear interaction schemes for stable synchronization of weakly coupled limit-cycle oscillators. Phys Rev E 2019; 100:042205. [PMID: 31770949 DOI: 10.1103/physreve.100.042205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/09/2019] [Indexed: 06/10/2023]
Abstract
Optimization of mutual synchronization between a pair of limit-cycle oscillators with weak symmetric coupling is considered in the framework of the phase-reduction theory. By generalizing our previous study [S. Shirasaka, N. Watanabe, Y. Kawamura, and H. Nakao, Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling, Phys. Rev. E 96, 012223 (2017)2470-004510.1103/PhysRevE.96.012223] on the optimization of cross-diffusion coupling matrices between the oscillators, we consider optimization of mutual coupling signals to maximize the linear stability of the synchronized state, which are functionals of the past time sequences of the oscillator states. For the case of linear coupling, optimization of the delay time and linear filtering of coupling signals are considered. For the case of nonlinear coupling, general drive-response coupling is considered and the optimal response and driving functions are derived. The theoretical results are illustrated by numerical simulations.
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Affiliation(s)
- Nobuhiro Watanabe
- 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
| | - Sho Shirasaka
- Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
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León I, Pazó D. Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation. Phys Rev E 2019; 100:012211. [PMID: 31499758 DOI: 10.1103/physreve.100.012211] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2019] [Indexed: 05/01/2023]
Abstract
Phase reduction is a powerful technique that makes possible to describe the dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase. For ensembles of oscillators, a classical example of phase reduction is the derivation of the Kuramoto model from the mean-field complex Ginzburg-Landau equation (MF-CGLE). Still, the Kuramoto model is a first-order phase approximation that displays either full synchronization or incoherence, but none of the nontrivial dynamics of the MF-CGLE. This fact calls for an expansion beyond the first order in the coupling constant. We develop an isochron-based scheme to obtain the second-order phase approximation, which reproduces the weak-coupling dynamics of the MF-CGLE. The practicality of our method is evidenced by extending the calculation up to third order. Each new term of the power-series expansion contributes with additional higher-order multibody (i.e., nonpairwise) interactions. This points to intricate multibody phase interactions as the source of pure collective chaos in the MF-CGLE at moderate coupling.
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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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Monga B, Moehlis J. Optimal phase control of biological oscillators using augmented phase reduction. BIOLOGICAL CYBERNETICS 2019; 113:161-178. [PMID: 29959510 DOI: 10.1007/s00422-018-0764-z] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2018] [Accepted: 06/15/2018] [Indexed: 06/08/2023]
Abstract
We develop a novel optimal control algorithm to change the phase of an oscillator using a minimum energy input, which also minimizes the oscillator's transversal distance to the uncontrolled periodic orbit. Our algorithm uses a two-dimensional reduction technique based on both isochrons and isostables. We develop a novel method to eliminate cardiac alternans by connecting our control algorithm with the underlying physiological problem. We also describe how the devised algorithm can be used for spike timing control which can potentially help with motor symptoms of essential and parkinsonian tremor, and aid in treating jet lag. To demonstrate the advantages of this algorithm, we compare it with a previously proposed optimal control algorithm based on standard phase reduction for the Hopf bifurcation normal form, and models for cardiac pacemaker cells, thalamic neurons, and circadian gene regulation cycle in the suprachiasmatic nucleus. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier is close to unity; in such cases, the previously proposed control algorithm fails.
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Affiliation(s)
- Bharat Monga
- Department of Mechanical Engineering, University of California, Santa Barbara, CA, 93106, USA.
| | - Jeff Moehlis
- Department of Mechanical Engineering, University of California, Santa Barbara, CA, 93106, USA
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Thomas PJ, Olufsen M, Sepulchre R, Iglesias PA, Ijspeert A, Srinivasan M. Control theory in biology and medicine : Introduction to the special issue. BIOLOGICAL CYBERNETICS 2019; 113:1-6. [PMID: 30701314 DOI: 10.1007/s00422-018-00791-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/06/2023]
Abstract
From September-December 2017, the Mathematical Biosciences Institute at Ohio State University hosted a series of workshops on control theory in biology and medicine, including workshops on control and modulation of neuronal and motor systems, control of cellular and molecular systems, control of disease / personalized medicine across heterogeneous populations, and sensorimotor control of animals and robots. This special issue presents tutorials and research articles by several of the participants in the MBI workshops.
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Affiliation(s)
- Peter J Thomas
- Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, Ohio, USA.
| | - Mette Olufsen
- Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA
| | | | - Pablo A Iglesias
- Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland, USA
| | - Auke Ijspeert
- Institute of Bioengineering, EPFL, Lausanne, Switzerland
| | - Manoj Srinivasan
- Department of Mechanical and Aerospace Engineering, Ohio State University, Columbus, Ohio, USA
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Rosenblum M, Pikovsky A. Numerical phase reduction beyond the first order approximation. CHAOS (WOODBURY, N.Y.) 2019; 29:011105. [PMID: 30709152 DOI: 10.1063/1.5079617] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2018] [Accepted: 12/05/2018] [Indexed: 05/21/2023]
Abstract
We develop a numerical approach to reconstruct the phase dynamics of driven or coupled self-sustained oscillators. Employing a simple algorithm for computation of the phase of a perturbed system, we construct numerically the equation for the evolution of the phase. Our simulations demonstrate that the description of the dynamics solely by phase variables can be valid for rather strong coupling strengths and large deviations from the limit cycle. Coupling functions depend crucially on the coupling and are generally non-decomposable in phase response and forcing terms. We also discuss the limitations of the approach.
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Affiliation(s)
- Michael Rosenblum
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
| | - Arkady Pikovsky
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
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