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Iima M. Optimal external forces of the lock-in phenomena for flow past an inclined plate in uniform flow. Phys Rev E 2024; 109:045102. [PMID: 38755879 DOI: 10.1103/physreve.109.045102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2023] [Accepted: 03/04/2024] [Indexed: 05/18/2024]
Abstract
We theoretically studied the optimal control, frequency lock-in, and phase lock-in phenomena due to the spatially localized periodic forcing in flow past an inclined plate. Although frequency lock-in is evident in many fluid phenomena, especially fluid-structure interactions, not many researchers have investigated it using a theoretical approach based on flow details. We obtained detailed information on the lock-in phenomena to external periodic forcing using phase reduction theory, a mathematical method for extracting the dynamics near the limit cycle. Furthermore, the optimal forces applied to the velocity field were determined under the condition of the minimum forcing energy and maximum lock-in range. The study of uniform periodic forces applied within spatially confined regions led to the conclusion that the effective lock-in position, which includes both the upstream and downstream areas of the plate, depends on the principal frequency of the force. The frequency lock-in range of these forces was analyzed and compared with theoretical predictions.
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Affiliation(s)
- Makoto Iima
- Graduate School of Integrated Life Sciences, Hiroshima University, 1-7-1, Kagamiyama Higashihiroshima, Hiroshima 739-8521, Japan
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2
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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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3
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Bomela W, Singhal B, Li JS. Engineering spatiotemporal patterns: information encoding, processing, and controllability in oscillator ensembles. Biomed Phys Eng Express 2023; 9:045033. [PMID: 37348467 PMCID: PMC10486008 DOI: 10.1088/2057-1976/ace0c9] [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: 12/08/2022] [Revised: 04/21/2023] [Accepted: 06/22/2023] [Indexed: 06/24/2023]
Abstract
The ability to finely manipulate spatiotemporal patterns displayed in neuronal populations is critical for understanding and influencing brain functions, sleep cycles, and neurological pathologies. However, such control tasks are challenged not only by the immense scale but also by the lack of real-time state measurements of neurons in the population, which deteriorates the control performance. In this paper, we formulate the control of dynamic structures in an ensemble of neuron oscillators as a tracking problem and propose a principled control technique for designing optimal stimuli that produce desired spatiotemporal patterns in a network of interacting neurons without requiring feedback information. We further reveal an interesting presentation of information encoding and processing in a neuron ensemble in terms of its controllability property. The performance of the presented technique in creating complex spatiotemporal spiking patterns is demonstrated on neural populations described by mathematically ideal and biophysical models, including the Kuramoto and Hodgkin-Huxley models, as well as real-time experiments on Wein bridge oscillators.
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Affiliation(s)
- Walter Bomela
- Department of Electrical and Systems Engineering, Washington University in St. Louis,
United States of America
| | - Bharat Singhal
- Department of Electrical and Systems Engineering, Washington University in St. Louis,
United States of America
| | - Jr-Shin Li
- Department of Electrical and Systems Engineering, Washington University in St. Louis,
United States of America
- Division of Biology & and Biomedical Sciences, Washington University in St. Louis,
United States of America
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4
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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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5
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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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6
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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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Pyragas K, Fedaravičius AP, Pyragienė T, Tass PA. Entrainment of a network of interacting neurons with minimum stimulating charge. Phys Rev E 2020; 102:012221. [PMID: 32795011 DOI: 10.1103/physreve.102.012221] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2020] [Accepted: 07/07/2020] [Indexed: 11/07/2022]
Abstract
Periodic pulse train stimulation is generically used to study the function of the nervous system and to counteract disease-related neuronal activity, e.g., collective periodic neuronal oscillations. The efficient control of neuronal dynamics without compromising brain tissue is key to research and clinical purposes. We here adapt the minimum charge control theory, recently developed for a single neuron, to a network of interacting neurons exhibiting collective periodic oscillations. We present a general expression for the optimal waveform, which provides an entrainment of a neural network to the stimulation frequency with a minimum absolute value of the stimulating current. As in the case of a single neuron, the optimal waveform is of bang-off-bang type, but its parameters are now determined by the parameters of the effective phase response curve of the entire network, rather than of a single neuron. The theoretical results are confirmed by three specific examples: two small-scale networks of FitzHugh-Nagumo neurons with synaptic and electric couplings, as well as a large-scale network of synaptically coupled quadratic integrate-and-fire neurons.
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Affiliation(s)
- Kestutis Pyragas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | | | - Tatjana Pyragienė
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Peter A Tass
- Department of Neurosurgery, Stanford University, Stanford, California 94305, USA
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8
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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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9
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Sebek M, Kawamura Y, Nott AM, Kiss IZ. Anti-phase collective synchronization with intrinsic in-phase coupling of two groups of electrochemical oscillators. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190095. [PMID: 31656145 PMCID: PMC6833994 DOI: 10.1098/rsta.2019.0095] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/05/2019] [Indexed: 05/02/2023]
Abstract
The synchronization of two groups of electrochemical oscillators is investigated during the electrodissolution of nickel in sulfuric acid. The oscillations are coupled through combined capacitance and resistance, so that in a single pair of oscillators (nearly) in-phase synchronization is obtained. The internal coupling within each group is relatively strong, but there is a phase difference between the fast and slow oscillators. The external coupling between the two groups is weak. The experiments show that the two groups can exhibit (nearly) anti-phase collective synchronization. Such synchronization occurs only when the external coupling is weak, and the interactions are delayed by the capacitance. When the external coupling is restricted to those between the fast and the slow elements, the anti-phase synchronization is more prominent. The results are interpreted with phase models. The theory predicts that, for anti-phase collective synchronization, there must be a minimum internal phase difference for a given shift in the phase coupling function. This condition is less stringent with external fast-to-slow coupling. The results provide a framework for applications of collective phase synchronization in modular networks where weak coupling between the groups can induce synchronization without rearrangements of the phase dynamics within the groups. 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 Sebek
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St Louis, MO 63103, USA
| | - Yoji Kawamura
- Center for Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, 236-0001 Yokohama, Japan
| | - Ashley M. Nott
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St Louis, MO 63103, USA
| | - István Z. Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St Louis, MO 63103, USA
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10
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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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11
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Li L, Asano F, Tokuda I. High-Speed Sliding Locomotion Generation on Slippery Surface of an Indirectly Controlled Robot With Viscoelastic Body. IEEE Robot Autom Lett 2019. [DOI: 10.1109/lra.2019.2924132] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
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12
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13
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Grines E, Osipov G, Pikovsky A. Describing dynamics of driven multistable oscillators with phase transfer curves. CHAOS (WOODBURY, N.Y.) 2018; 28:106323. [PMID: 30384664 DOI: 10.1063/1.5037290] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2018] [Accepted: 08/06/2018] [Indexed: 06/08/2023]
Abstract
Phase response curve is an important tool in the studies of stable self-sustained oscillations; it describes a phase shift under action of an external perturbation. We consider multistable oscillators with several stable limit cycles. Under a perturbation, transitions from one oscillating mode to another one may occur. We define phase transfer curves to describe the phase shifts at such transitions. This allows for a construction of one-dimensional maps that characterize periodically kicked multistable oscillators. We show that these maps are good approximations of the full dynamics for large periods of forcing.
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Affiliation(s)
- Evgeny Grines
- Department of Control Theory and Dynamics of Systems, Lobachevsky State University of Nizhny Novgorod, 23, Prospekt Gagarina, Nizhny Novgorod 603950, Russia
| | - Grigory Osipov
- Department of Control Theory and Dynamics of Systems, Lobachevsky State University of Nizhny Novgorod, 23, Prospekt Gagarina, Nizhny Novgorod 603950, Russia
| | - Arkady Pikovsky
- Department of Control Theory and Dynamics of Systems, Lobachevsky State University of Nizhny Novgorod, 23, Prospekt Gagarina, Nizhny Novgorod 603950, Russia
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14
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15
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Bomela WB, Dasanayake IS, Li JS, Chen Y, Kiss IZ. Optimal Phase-to-Phase Control of Chemical Oscillations. Ind Eng Chem Res 2018. [DOI: 10.1021/acs.iecr.8b00505] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/10/2023]
Affiliation(s)
- Walter B. Bomela
- Washington University in St. Louis, St. Louis, Missouri 63130, United States
| | - Isuru S. Dasanayake
- Washington University in St. Louis, St. Louis, Missouri 63130, United States
| | - Jr-Shin Li
- Washington University in St. Louis, St. Louis, Missouri 63130, United States
| | - Yifei Chen
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, United States
| | - István Z. Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, United States
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16
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Tiana-Alsina J, Quintero-Quiroz C, Panozzo M, Torrent MC, Masoller C. Experimental study of modulation waveforms for entraining the spikes emitted by a semiconductor laser with optical feedback. OPTICS EXPRESS 2018; 26:9298-9309. [PMID: 29715883 DOI: 10.1364/oe.26.009298] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2018] [Accepted: 03/26/2018] [Indexed: 06/08/2023]
Abstract
The entrainment phenomenon, by which an oscillator adjusts its natural rhythm to an external periodic signal, has been observed in many natural systems. Recently, attention has focused on which are the optimal conditions for achieving entrainment. Here we use a semiconductor laser with optical feedback, operating in the low-frequency fluctuations (LFFs) regime, as a testbed for a controlled entrainment experiment. In the LFF regime the laser intensity displays abrupt spikes, which can be entrained to a weak periodic signal that directly modulates the laser pump current. We compare the performance of three modulation waveforms for producing 1:1 locking (one spike is emitted in each modulation cycle), as well as higher order locking regimes. We characterize the parameter regions where high-quality locking occurs, and those where the laser emits spikes which are not entrained to the external signal. The role of the modulation amplitude and frequency, and the role of the dc value of the laser pump current (that controls the natural spike frequency) in the entrainment quality are discussed.
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Nakao H, Yasui S, Ota M, Arai K, Kawamura Y. Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations. CHAOS (WOODBURY, N.Y.) 2018; 28:045103. [PMID: 31906627 DOI: 10.1063/1.5009669] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
A general phase reduction method for a network of coupled dynamical elements exhibiting collective oscillations, which is applicable to arbitrary networks of heterogeneous dynamical elements, is developed. A set of coupled adjoint equations for phase sensitivity functions, which characterize the phase response of the collective oscillation to small perturbations applied to individual elements, is derived. Using the phase sensitivity functions, collective oscillation of the network under weak perturbation can be described approximately by a one-dimensional phase equation. As an example, mutual synchronization between a pair of collectively oscillating networks of excitable and oscillatory FitzHugh-Nagumo elements with random coupling is studied.
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Affiliation(s)
- Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Sho Yasui
- Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Masashi Ota
- Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Kensuke Arai
- Department of Statistics and Mathematics, Boston University, Boston, Massachusetts 02215, USA
| | - Yoji Kawamura
- Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan
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18
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Kawamura Y, Shirasaka S, Yanagita T, Nakao H. Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems. Phys Rev E 2018; 96:012224. [PMID: 29347085 DOI: 10.1103/physreve.96.012224] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2017] [Indexed: 11/07/2022]
Abstract
Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case in which the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.
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Affiliation(s)
- Yoji Kawamura
- Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan
| | - Sho Shirasaka
- Research Center for Advanced Science and Technology, University of Tokyo, Tokyo 153-8904, Japan
| | - Tatsuo Yanagita
- Osaka Electro-Communication University, Neyagawa 572-8530, Japan
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan.,Department of Mechanical Engineering, University of California, Santa Barbara, California 93106, USA
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Shirasaka S, Watanabe N, Kawamura Y, Nakao H. Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling. Phys Rev E 2017; 96:012223. [PMID: 29347076 DOI: 10.1103/physreve.96.012223] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2017] [Indexed: 12/22/2022]
Abstract
We consider optimization of the linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, we derive the coupling matrix between different components of the oscillator states that maximizes the linear stability of the synchronized state under given constraints on the overall coupling intensity and the stationary phase difference. The improvement in the linear stability is illustrated by using several types of limit-cycle oscillators as examples.
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Affiliation(s)
- Sho Shirasaka
- Research Center for Advanced Science and Technology, University of Tokyo, Tokyo 153-8904, Japan
| | - Nobuhiro Watanabe
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Yoji Kawamura
- Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan.,Department of Mechanical Engineering, University of California, Santa Barbara, California 93106, USA
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Shirasaka S, Kurebayashi W, Nakao H. Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems. CHAOS (WOODBURY, N.Y.) 2017; 27:023119. [PMID: 28249399 DOI: 10.1063/1.4977195] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Phase reduction framework for limit-cycling systems based on isochrons has been used as a powerful tool for analyzing the rhythmic phenomena. Recently, the notion of isostables, which complements the isochrons by characterizing amplitudes of the system state, i.e., deviations from the limit-cycle attractor, has been introduced to describe the transient dynamics around the limit cycle [Wilson and Moehlis, Phys. Rev. E 94, 052213 (2016)]. In this study, we introduce a framework for a reduced phase-amplitude description of transient dynamics of stable limit-cycling systems. In contrast to the preceding study, the isostables are treated in a fully consistent way with the Koopman operator analysis, which enables us to avoid discontinuities of the isostables and to apply the framework to system states far from the limit cycle. We also propose a new, convenient bi-orthogonalization method to obtain the response functions of the amplitudes, which can be interpreted as an extension of the adjoint covariant Lyapunov vector to transient dynamics in limit-cycling systems. We illustrate the utility of the proposed reduction framework by estimating the optimal injection timing of external input that efficiently suppresses deviations of the system state from the limit cycle in a model of a biochemical oscillator.
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Affiliation(s)
- Sho Shirasaka
- Graduate School of Information Science and Engineering, Tokyo Institute of Technology, O-okayama 2-12-1, Meguro, Tokyo 152-8552, Japan
| | - Wataru Kurebayashi
- Faculty of Software and Information Technology, Aomori University, Kobata 2-3-1, Aomori, Aomori 030-0943, Japan
| | - Hiroya Nakao
- School of Engineering, Tokyo Institute of Technology, O-okayama 2-12-1, Meguro, Tokyo 152-8552, Japan
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Shirasaka S, Kurebayashi W, Nakao H. Phase reduction theory for hybrid nonlinear oscillators. Phys Rev E 2017; 95:012212. [PMID: 28208342 DOI: 10.1103/physreve.95.012212] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2016] [Indexed: 11/07/2022]
Abstract
Hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena. However, the phase reduction theory, a fundamental framework for analyzing the synchronization of limit-cycle oscillations in rhythmic systems, has mostly been restricted to smooth dynamical systems. Here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis, control, and optimization of nonlinear oscillators whose smooth models are unavailable or intractable. On the basis of the generalized theory, we analyze injection locking of hybrid limit-cycle oscillators by periodic forcing and reveal their characteristic synchronization properties, such as ultrafast and robust entrainment to the periodic forcing and logarithmic scaling at the synchronization transition. We also illustrate the theory by analyzing the synchronization dynamics of a simple physical model of biped locomotion.
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Affiliation(s)
- Sho Shirasaka
- Graduate School of Information Science and Engineering, Tokyo Institute of Technology, O-okayama 2-12-1, Meguro, Tokyo 152-8552, Japan
| | - Wataru Kurebayashi
- Faculty of Software and Information Technology, Aomori University, Kobata 2-3-1, Aomori, Aomori 030-0943, Japan
| | - Hiroya Nakao
- School of Engineering, Tokyo Institute of Technology, O-okayama 2-12-1, Meguro, Tokyo 152-8552, Japan
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22
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Novičenko V, Ratas I. Optimal waveform for the entrainment of oscillators perturbed by an amplitude-modulated high-frequency force. Phys Rev E 2017; 94:062213. [PMID: 28085462 DOI: 10.1103/physreve.94.062213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2016] [Indexed: 11/07/2022]
Abstract
We analyze limit cycle oscillators under perturbation constructed as a product of two signals, namely, an envelope with a period close to natural period of an oscillator and a high-frequency carrier signal. A theory for obtaining an envelope waveform that achieves the maximal frequency interval of entrained oscillators is presented. The optimization problem for fixed power and maximal allowed amplitude is solved by employing the phase reduction method and the Pontryagin's maximum principle. We have shown that the optimal envelope waveform is a bang-bang-type solution. Also, we have found "inversion" symmetry that relates two signals with different powers but the same interval of entrained frequencies. The theoretical results are confirmed numerically on FitzHugh-Nagumo oscillators.
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Affiliation(s)
- Viktor Novičenko
- Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio Avenue 3, LT-10222 Vilnius, Lithuania
| | - Irmantas Ratas
- Center for Physical Sciences and Technology, Saulėtekio Avenue 3, LT-10222 Vilnius, Lithuania
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23
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Kawamura Y, Nakao H. Optimization of noise-induced synchronization of oscillator networks. Phys Rev E 2016; 94:032201. [PMID: 27739705 DOI: 10.1103/physreve.94.032201] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2016] [Indexed: 11/07/2022]
Abstract
We investigate common-noise-induced synchronization between two identical networks of coupled phase oscillators exhibiting fully locked collective oscillations. Using the collective phase description method for fully locked oscillators, we demonstrate that two noninteracting networks of coupled phase oscillators can exhibit in-phase synchronization between the networks when driven by weak common noise. We derive the Lyapunov exponent characterizing the relaxation time for synchronization and develop a method of obtaining the optimal input pattern of common noise to achieve fast synchronization. We illustrate the theory using three representative networks with heterogeneous, global, and local coupling. The theoretical results are validated by direct numerical simulations.
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Affiliation(s)
- Yoji Kawamura
- Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan and Research and Development Center for Marine Biosciences, Japan Agency for Marine-Earth Science and Technology, Yokosuka 237-0061, Japan
| | - Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
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24
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Kashima K. Noise Response Data Reveal Novel Controllability Gramian for Nonlinear Network Dynamics. Sci Rep 2016; 6:27300. [PMID: 27264780 PMCID: PMC4893695 DOI: 10.1038/srep27300] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2015] [Accepted: 05/16/2016] [Indexed: 12/01/2022] Open
Abstract
Control of nonlinear large-scale dynamical networks, e.g., collective behavior of agents interacting via a scale-free connection topology, is a central problem in many scientific and engineering fields. For the linear version of this problem, the so-called controllability Gramian has played an important role to quantify how effectively the dynamical states are reachable by a suitable driving input. In this paper, we first extend the notion of the controllability Gramian to nonlinear dynamics in terms of the Gibbs distribution. Next, we show that, when the networks are open to environmental noise, the newly defined Gramian is equal to the covariance matrix associated with randomly excited, but uncontrolled, dynamical state trajectories. This fact theoretically justifies a simple Monte Carlo simulation that can extract effectively controllable subdynamics in nonlinear complex networks. In addition, the result provides a novel insight into the relationship between controllability and statistical mechanics.
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Affiliation(s)
- Kenji Kashima
- Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
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25
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Pikovsky A. Maximizing Coherence of Oscillations by External Locking. PHYSICAL REVIEW LETTERS 2015; 115:070602. [PMID: 26317705 DOI: 10.1103/physrevlett.115.070602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2015] [Indexed: 06/04/2023]
Abstract
We study how coherence of noisy oscillations can be optimally enhanced by external locking. Based on the condition of minimizing the phase diffusion constant, we find the optimal forcing explicitly in the limits of small and large noise, in dependence of the phase sensitivity of the oscillator. We show analytically that the form of the optimal force bifurcates with the noise intensity; this is confirmed by the analysis of an optimal locking forcing for an experimentally obtained phase sensitivity of a neural cell. In the limit of small noise, the results are compared with purely deterministic conditions of optimal locking.
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Affiliation(s)
- Arkady Pikovsky
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany and Department of Control Theory, Nizhni Novgorod State University, Gagarin Avenue 23, 606950 Nizhni Novgorod, Russia
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26
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Pyragas K, Novičenko V. Phase reduction of a limit cycle oscillator perturbed by a strong amplitude-modulated high-frequency force. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012910. [PMID: 26274250 DOI: 10.1103/physreve.92.012910] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2015] [Indexed: 06/04/2023]
Abstract
The phase reduction method for a limit cycle oscillator subjected to a strong amplitude-modulated high-frequency force is developed. An equation for the phase dynamics is derived by introducing a new, effective phase response curve. We show that if the effective phase response curve is everywhere positive (negative), then an entrainment of the oscillator to an envelope frequency is possible only when this frequency is higher (lower) than the natural frequency of the oscillator. Also, by using the Pontryagin maximum principle, we have derived an optimal waveform of the perturbation that ensures an entrainment of the oscillator with minimal power. The theoretical results are demonstrated with the Stuart-Landau oscillator and model neurons.
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Affiliation(s)
- Kestutis Pyragas
- Center for Physical Sciences and Technology, A. Goštauto 11, LT-01108 Vilnius, Lithuania
| | - Viktor Novičenko
- Center for Physical Sciences and Technology, A. Goštauto 11, LT-01108 Vilnius, Lithuania
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27
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Wilson D, Holt AB, Netoff TI, Moehlis J. Optimal entrainment of heterogeneous noisy neurons. Front Neurosci 2015; 9:192. [PMID: 26074762 PMCID: PMC4448041 DOI: 10.3389/fnins.2015.00192] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2015] [Accepted: 05/15/2015] [Indexed: 11/13/2022] Open
Abstract
We develop a methodology to design a stimulus optimized to entrain nonlinear, noisy limit cycle oscillators with uncertain properties. Conditions are derived which guarantee that the stimulus will entrain the oscillators despite these uncertainties. Using these conditions, we develop an energy optimal control strategy to design an efficient entraining stimulus and apply it to numerical models of noisy phase oscillators and to in vitro hippocampal neurons. In both instances, the optimal stimuli outperform other similar but suboptimal entraining stimuli. Because this control strategy explicitly accounts for both noise and inherent uncertainty of model parameters, it could have experimental relevance to neural circuits where robust spike timing plays an important role.
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Affiliation(s)
- Dan Wilson
- Department of Mechanical Engineering, University of California, Santa Barbara Santa Barbara, CA, USA
| | - Abbey B Holt
- Graduate Program in Neuroscience, University of Minnesota Minneapolis, MN, USA
| | - Theoden I Netoff
- Graduate Program in Neuroscience, University of Minnesota Minneapolis, MN, USA ; Department of Biomedical Engineering, University of Minnesota Minneapolis, MN, USA
| | - Jeff Moehlis
- Department of Mechanical Engineering, University of California, Santa Barbara Santa Barbara, CA, USA
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28
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Serkh K, Forger DB. Optimal schedules of light exposure for rapidly correcting circadian misalignment. PLoS Comput Biol 2014; 10:e1003523. [PMID: 24722195 PMCID: PMC3983044 DOI: 10.1371/journal.pcbi.1003523] [Citation(s) in RCA: 57] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/23/2013] [Accepted: 01/29/2014] [Indexed: 11/24/2022] Open
Abstract
Jet lag arises from a misalignment of circadian biological timing with the timing of human activity, and is caused by rapid transmeridian travel. Jet lag's symptoms, such as depressed cognitive alertness, also arise from work and social schedules misaligned with the timing of the circadian clock. Using experimentally validated mathematical models, we develop a new methodology to find mathematically optimal schedules of light exposure and avoidance for rapidly re-entraining the human circadian system. In simulations, our schedules are found to significantly outperform other recently proposed schedules. Moreover, our schedules appear to be significantly more robust to both noise in light and to inter-individual variations in endogenous circadian period than other proposed schedules. By comparing the optimal schedules for thousands of different situations, and by using general mathematical arguments, we are also able to translate our findings into general principles of optimal circadian re-entrainment. These principles include: 1) a class of schedules where circadian amplitude is only slightly perturbed, optimal for dim light and for small shifts 2) another class of schedules where shifting occurs along the shortest path in phase-space, optimal for bright light and for large shifts 3) the determination that short light pulses are less effective than sustained light if the goal is to re-entrain quickly, and 4) the determination that length of daytime should be significantly shorter when delaying the clock than when advancing it. When our body's internal timekeeping system becomes misaligned with the time of day in the outside world, many negative effects can be felt, including decreased performance, improper sleep, and jet lag. When misalignment is prolonged, it can also lead to serious medical conditions, including cancer, cardiovascular disease, and possibly even late-onset diabetes. Rapid readjustment of our internal daily (circadian) clock by properly timed exposure to light, which is the strongest signal to our internal circadian clock, is therefore important to the large proportion of the population which suffers from misalignment, including transmeridian travelers, shift workers, and individuals with circadian disorders. Here we develop a methodology to determine schedules of light exposure which may shift the human circadian clock in the minimum time. By calculating thousands of schedules, we show how the human circadian pacemaker is predicted to be capable of shifting much more rapidly than previously thought, simply by adjusting the timing of the beginning and end of each day. Schedules are summarized into general principles of optimal shifting, which can be applied without knowledge of the schedules themselves.
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Affiliation(s)
- Kirill Serkh
- Department of Applied Mathematics, Graduate School of Arts & Sciences, Yale University, New Haven, Connecticut, United States of America
- Department of Mathematics, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Daniel B. Forger
- Department of Mathematics, University of Michigan, Ann Arbor, Michigan, United States of America
- Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, Michigan, United States of America
- * E-mail: .
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29
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Kawamura Y, Nakao H. Noise-induced synchronization of oscillatory convection and its optimization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012912. [PMID: 24580302 DOI: 10.1103/physreve.89.012912] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2013] [Indexed: 06/03/2023]
Abstract
We investigate common-noise-induced phase synchronization between uncoupled identical Hele-Shaw cells exhibiting oscillatory convection. Using the phase description method for oscillatory convection, we demonstrate that the uncoupled systems of oscillatory Hele-Shaw convection can exhibit in-phase synchronization when driven by weak common noise. We derive the Lyapunov exponent determining the relaxation time for the synchronization, and develop a method for obtaining the optimal spatial pattern of the common noise to achieve synchronization. The theoretical results are confirmed by direct numerical simulations.
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Affiliation(s)
- Yoji Kawamura
- Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan
| | - Hiroya Nakao
- Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo 152-8552, Japan
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30
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Hasegawa Y, Arita M. Circadian clocks optimally adapt to sunlight for reliable synchronization. J R Soc Interface 2013; 11:20131018. [PMID: 24352677 PMCID: PMC3899870 DOI: 10.1098/rsif.2013.1018] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022] Open
Abstract
Circadian oscillation provides selection advantages through synchronization to the daylight cycle. However, a reliable clock must be designed through two conflicting properties: entrainability to synchronize internal time with periodic stimuli such as sunlight, and regularity to oscillate with a precise period. These two aspects do not easily coexist, because better entrainability favours higher sensitivity which may sacrifice regularity. To investigate conditions for satisfying the two properties, we analytically calculated the optimal phase-response curve with a variational method. Our results indicate an existence of a dead zone, i.e. a time period during which input stimuli neither advance nor delay the clock. A dead zone appears only when input stimuli obey the time course of actual solar radiation, but a simple sine curve cannot yield a dead zone. Our calculation demonstrates that every circadian clock with a dead zone is optimally adapted to the daylight cycle.
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Affiliation(s)
- Yoshihiko Hasegawa
- Department of Biophysics and Biochemistry, Graduate School of Science, The University of Tokyo, , Tokyo 113-0033, Japan
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31
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Zlotnik A, Chen Y, Kiss IZ, Tanaka HA, Li JS. Optimal waveform for fast entrainment of weakly forced nonlinear oscillators. PHYSICAL REVIEW LETTERS 2013; 111:024102. [PMID: 23889405 DOI: 10.1103/physrevlett.111.024102] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2013] [Revised: 04/02/2013] [Indexed: 05/18/2023]
Abstract
For many biological and engineered systems, a central function or design goal is to abbreviate the time required to synchronize a rhythmic process to an external forcing signal. We present a theory for deriving the input that effectively minimizes the average transient time required to entrain a phase model, which enables a practical technique for constructing fast entrainment waveforms for general nonlinear oscillators. This result is verified in numerical simulations using the Hodgkin-Huxley neuron model, and in experiments on an oscillatory electrochemical system.
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Affiliation(s)
- Anatoly Zlotnik
- Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, Missouri 63130, USA.
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32
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Granada AE, Bordyugov G, Kramer A, Herzel H. Human chronotypes from a theoretical perspective. PLoS One 2013; 8:e59464. [PMID: 23544070 PMCID: PMC3609763 DOI: 10.1371/journal.pone.0059464] [Citation(s) in RCA: 76] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2012] [Accepted: 02/14/2013] [Indexed: 11/18/2022] Open
Abstract
The endogenous circadian timing system has evolved to synchronize an organism to periodically recurring environmental conditions. Those external time cues are called Zeitgebers. When entrained by a Zeitgeber, the intrinsic oscillator adopts a fixed phase relation to the Zeitgeber. Here, we systematically study how the phase of entrainment depends on clock and Zeitgeber properties. We combine numerical simulations of amplitude-phase models with predictions from analytically tractable models. In this way we derive relations between the phase of entrainment to the mismatch between the endogenous and Zeitgeber period, the Zeitgeber strength, and the range of entrainment. A core result is the “180° rule” asserting that the phase varies over a range of about 180° within the entrainment range. The 180° rule implies that clocks with a narrow entrainment range (“strong oscillators”) exhibit quite flexible entrainment phases. We argue that this high sensitivity of the entrainment phase contributes to the wide range of human chronotypes.
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Affiliation(s)
- Adrián E. Granada
- Institute for Theoretical Biology, Humboldt University, Berlin, Germany
| | - Grigory Bordyugov
- Institute for Theoretical Biology, Humboldt University, Berlin, Germany
| | - Achim Kramer
- Laboratory of Chronobiology, Charité Universitätsmedizin, Berlin, Germany
| | - Hanspeter Herzel
- Institute for Theoretical Biology, Humboldt University, Berlin, Germany
- * E-mail:
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33
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Abstract
In this paper, we derive the minimum-energy periodic control that entrains an ensemble of structurally similar neural oscillators to a desired frequency. The state-space representation of a nominal oscillator is reduced to a phase model by computing its limit cycle and phase response curve, from which the optimal control is derived by using formal averaging and the calculus of variations. We focus on the case of a 1:1 entrainment ratio and suggest a simple numerical method for approximating the optimal controls. The method is applied to asymptotically control the spiking frequency of neural oscillators modeled using the Hodgkin-Huxley equations. Simulations are used to illustrate the optimality of entrainment controls derived using phase models when applied to the original state-space system, which is crucial for using phase models in control synthesis for practical applications. This work addresses a fundamental problem in the field of neural dynamics and provides a theoretical contribution to the optimal frequency control of uncertain oscillating systems.
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Affiliation(s)
- Anatoly Zlotnik
- Department of Electrical and Systems Engineering, Washington University, Saint Louis, MO 63130, USA.
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34
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Dasanayake I, Li JS. Optimal design of minimum-power stimuli for phase models of neuron oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061916. [PMID: 21797412 DOI: 10.1103/physreve.83.061916] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2010] [Revised: 02/15/2011] [Indexed: 05/31/2023]
Abstract
In this paper, we study optimal control problems of spiking neurons whose dynamics are described by a phase model. We design minimum-power current stimuli (controls) that lead to targeted spiking times. In particular, we consider bounded control amplitude and characterize the range of possible spiking times determined by the bound, which can be chosen sufficiently small within the range where the phase model is valid. We show that for a given bound the corresponding feasible spiking times are optimally achieved by piecewise continuous controls. We present analytic expressions with numerical simulations of the minimum-power stimuli for several phase models. We demonstrate the applicability of our method with an experimentally determined phase response curve.
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Affiliation(s)
- Isuru Dasanayake
- Washington University in St. Louis, St. Louis, Missouri 63130, USA.
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