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Luis Ocampo-Espindola J, Singhal B, Li JS, Kiss IZ. Optimal phase-selective entrainment of electrochemical oscillators with different phase response curves. CHAOS (WOODBURY, N.Y.) 2024; 34:073129. [PMID: 38995992 DOI: 10.1063/5.0205480] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2024] [Accepted: 06/25/2024] [Indexed: 07/14/2024]
Abstract
We investigate the entrainment of electrochemical oscillators with different phase response curves (PRCs) using a global signal: the goal is to achieve the desired phase configuration using a minimum-power waveform. Establishing the desired phase relationships in a highly nonlinear networked system exhibiting significant heterogeneities, such as different conditions or parameters for the oscillators, presents a considerable challenge because different units respond differently to the common global entraining signal. In this work, we apply an optimal phase-selective entrainment technique in both a kinetic model and experiments involving electrochemical oscillators in achieving phase synchronized states. We estimate the PRCs of the oscillators at different circuit potentials and external resistance, and entrain pairs and small sets of four oscillators in various phase configurations. We show that for small PRC variations, phase assignment can be achieved using an averaged PRC in the control design. However, when the PRCs are sufficiently different, individual PRCs are needed to entrain the system with the expected phase relationships. The results show that oscillator assemblies with heterogeneous PRCs can be effectively entrained to desired phase configurations in practical settings. These findings open new avenues to applications in biological and engineered oscillator systems where synchronization patterns are essential for system performance.
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Affiliation(s)
| | - Bharat Singhal
- Department of Electrical & Systems Engineering, Washington University in St Louis, St Louis, Missouri 63130, USA
| | - Jr-Shin Li
- Department of Electrical & Systems Engineering, Washington University in St Louis, St Louis, Missouri 63130, USA
| | - István Z Kiss
- Department of Chemistry, Saint Louis University, St. Louis, Missouri 63103, USA
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Gengel E, Kuplik Z, Angel D, Heifetz E. A physics-based model of swarming jellyfish. PLoS One 2023; 18:e0288378. [PMID: 37428796 DOI: 10.1371/journal.pone.0288378] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2022] [Accepted: 06/23/2023] [Indexed: 07/12/2023] Open
Abstract
We propose a model for the structure formation of jellyfish swimming based on active Brownian particles. We address the phenomena of counter-current swimming, avoidance of turbulent flow regions and foraging. We motivate corresponding mechanisms from observations of jellyfish swarming reported in the literature and incorporate them into the generic modelling framework. The model characteristics is tested in three paradigmatic flow environments.
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Affiliation(s)
- Erik Gengel
- Department of Geophysics, Porter School of the Environment and Earth Sciences, Tel Aviv University, Tel Aviv, Israel
| | - Zafrir Kuplik
- The Steinhardt Museum of Natural History, Tel Aviv University, Tel Aviv, Israel
- The Leon Recanati Institute for Maritime Studies, University of Haifa, Mount Carmel, Haifa, Israel
| | - Dror Angel
- The Leon Recanati Institute for Maritime Studies, University of Haifa, Mount Carmel, Haifa, Israel
| | - Eyal Heifetz
- Department of Geophysics, Porter School of the Environment and Earth Sciences, Tel Aviv University, Tel Aviv, Israel
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An extended Hilbert transform method for reconstructing the phase from an oscillatory signal. Sci Rep 2023; 13:3535. [PMID: 36864108 PMCID: PMC9981592 DOI: 10.1038/s41598-023-30405-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2022] [Accepted: 02/22/2023] [Indexed: 03/04/2023] Open
Abstract
Rhythmic activity is ubiquitous in biological systems from the cellular to organism level. Reconstructing the instantaneous phase is the first step in analyzing the essential mechanism leading to a synchronization state from the observed signals. A popular method of phase reconstruction is based on the Hilbert transform, which can only reconstruct the interpretable phase from a limited class of signals, e.g., narrow band signals. To address this issue, we propose an extended Hilbert transform method that accurately reconstructs the phase from various oscillatory signals. The proposed method is developed by analyzing the reconstruction error of the Hilbert transform method with the aid of Bedrosian's theorem. We validate the proposed method using synthetic data and show its systematically improved performance compared with the conventional Hilbert transform method with respect to accurately reconstructing the phase. Finally, we demonstrate that the proposed method is potentially useful for detecting the phase shift in an observed signal. The proposed method is expected to facilitate the study of synchronization phenomena from experimental data.
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Singhal B, Kiss IZ, Li JS. Optimal phase-selective entrainment of heterogeneous oscillator ensembles. SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2023; 22:2180-2205. [PMID: 38835972 PMCID: PMC11149604 DOI: 10.1137/22m1521201] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2024]
Abstract
We develop a framework to design optimal entrainment signals that entrain an ensemble of heterogeneous nonlinear oscillators, described by phase models, at desired phases. We explicitly take into account heterogeneity in both oscillation frequency and the type of oscillators characterized by different Phase Response Curves. The central idea is to leverage the Fourier series representation of periodic functions to decode a phase-selective entrainment task into a quadratic program. We demonstrate our approach using a variety of phase models, where we entrain the oscillators into distinct phase patterns. Also, we show how the generalizability gained from our formulation enables us to meet a wide range of design objectives and constraints, such as minimum-power, fast entrainment, and charge-balanced controls.
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Affiliation(s)
- Bharat Singhal
- Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, Missouri 63130, USA
| | - István Z Kiss
- Department of Chemistry, Saint Louis University, St. Louis, Missouri 63103, USA
| | - Jr-Shin Li
- Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, Missouri 63130, USA
- Division of Biology & and Biomedical Sciences, Washington University in St. Louis, St. Louis, Missouri 63130, USA
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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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Ji H, Fouad AD, Teng S, Liu A, Alvarez-Illera P, Yao B, Li Z, Fang-Yen C. Phase response analyses support a relaxation oscillator model of locomotor rhythm generation in Caenorhabditis elegans. eLife 2021; 10:e69905. [PMID: 34569934 PMCID: PMC8560089 DOI: 10.7554/elife.69905] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2021] [Accepted: 09/24/2021] [Indexed: 01/25/2023] Open
Abstract
Neural circuits coordinate with muscles and sensory feedback to generate motor behaviors appropriate to an animal's environment. In C. elegans, the mechanisms by which the motor circuit generates undulations and modulates them based on the environment are largely unclear. We quantitatively analyzed C. elegans locomotion during free movement and during transient optogenetic muscle inhibition. Undulatory movements were highly asymmetrical with respect to the duration of bending and unbending during each cycle. Phase response curves induced by brief optogenetic inhibition of head muscles showed gradual increases and rapid decreases as a function of phase at which the perturbation was applied. A relaxation oscillator model based on proprioceptive thresholds that switch the active muscle moment was developed and is shown to quantitatively agree with data from free movement, phase responses, and previous results for gait adaptation to mechanical loadings. Our results suggest a neuromuscular mechanism underlying C. elegans motor pattern generation within a compact circuit.
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Affiliation(s)
- Hongfei Ji
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Anthony D Fouad
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Shelly Teng
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Alice Liu
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Pilar Alvarez-Illera
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Bowen Yao
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Zihao Li
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
| | - Christopher Fang-Yen
- Department of Bioengineering, School of Engineering and Applied Science, University of PennsylvaniaPhiladelphiaUnited States
- Department of Neuroscience, Perelman School of Medicine, University of PennsylvaniaPhiladelphiaUnited States
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Tokuda IT, Levnajic Z, Ishimura K. A practical method for estimating coupling functions in complex dynamical systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190015. [PMID: 31656141 PMCID: PMC6833996 DOI: 10.1098/rsta.2019.0015] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/02/2019] [Indexed: 06/10/2023]
Abstract
A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101. (doi:10.1103/PhysRevLett.99.064101)) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community. 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)
- Isao T. Tokuda
- Department of Mechanical Engineering, Ritsumeikan University, Kusatsu, Japan
| | - Zoran Levnajic
- Complex Systems and Data Science Lab, Faculty of Information Studies in Novo Mesto, Novo Mesto, Slovenia
| | - Kazuyoshi Ishimura
- Department of Mechanical Engineering, Ritsumeikan University, Kusatsu, Japan
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Cestnik R, Abel M. Inferring the dynamics of oscillatory systems using recurrent neural networks. CHAOS (WOODBURY, N.Y.) 2019; 29:063128. [PMID: 31266337 DOI: 10.1063/1.5096918] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2019] [Accepted: 06/06/2019] [Indexed: 06/09/2023]
Abstract
We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit.
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Affiliation(s)
- Rok Cestnik
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
| | - Markus Abel
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
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Vanag VK. "Cognitive" modes in small networks of almost identical chemical oscillators with pulsatile inhibitory coupling. CHAOS (WOODBURY, N.Y.) 2019; 29:033106. [PMID: 30927858 DOI: 10.1063/1.5063322] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2018] [Accepted: 02/11/2019] [Indexed: 06/09/2023]
Abstract
The Lavrova-Vanag (LV) model of the periodical Belousov-Zhabotinsky (BZ) reaction has been investigated at pulsed self-perturbations, when a sharp spike of the BZ reaction induces a short inhibitory pulse that perturbs the BZ reaction after some time τ since each spike. The dynamics of this BZ system is strongly dependent on the amplitude Cinh of the perturbing pulses. At Cinh > Ccr, a new pseudo-steady state (SS) emerges far away from the limit cycle of the unperturbed BZ oscillator. The perturbed BZ system spends rather long time in the vicinity of this pseudo-SS, which serves as a trap for phase trajectories. As a result, the dynamics of the BZ system changes qualitatively. We observe new modes with packed spikes separated by either long "silent" dynamics or small-amplitude oscillations around pseudo-SS, depending on Cinh. Networks of two or three LV-BZ oscillators with strong pulsatile coupling and self-inhibition are able to generate so-called "cognitive" modes, which are very sensitive to small changes in Cinh. We demonstrate how the coupling between the BZ oscillators in these networks should be organized to find "cognitive" modes.
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Affiliation(s)
- Vladimir K Vanag
- Centre for Nonlinear Chemistry, Immanuel Kant Baltic Federal University, 14 A. Nevskogo Str., Kaliningrad 236041, Russia
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