1
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Arai T, Ota K, Funato T, Tsuchiya K, Aoyagi T, Aoi S. Interlimb coordination is not strictly controlled during walking. Commun Biol 2024; 7:1152. [PMID: 39304734 DOI: 10.1038/s42003-024-06843-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2023] [Accepted: 09/04/2024] [Indexed: 09/22/2024] Open
Abstract
In human walking, the left and right legs move alternately, half a stride out of phase with each other. Although various parameters, such as stride frequency and length, vary with walking speed, the antiphase relationship remains unchanged. In contrast, during walking in left-right asymmetric situations, the relative phase shifts from the antiphase condition to compensate for the asymmetry. Interlimb coordination is important for adaptive walking and we expect that interlimb coordination is strictly controlled during walking. However, the control mechanism remains unclear. In the present study, we derived a quantity that models the control of interlimb coordination during walking using two coupled oscillators based on the phase reduction theory and Bayesian inference method. The results were not what we expected. Specifically, we found that the relative phase is not actively controlled until the deviation from the antiphase condition exceeds a certain threshold. In other words, the control of interlimb coordination has a dead zone like that in the case of the steering wheel of an automobile. It is conjectured that such forgoing of control enhances energy efficiency and maneuverability. Our discovery of the dead zone in the control of interlimb coordination provides useful insight for understanding gait control in humans.
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Affiliation(s)
- Takahiro Arai
- Center for Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama, 236-0001, Japan
| | - Kaiichiro Ota
- Cybozu, Inc., 2-7-1 Nihombashi, Chuo-ku, Tokyo, 103-6027, Japan
| | - Tetsuro Funato
- Department of Mechanical Engineering and Intelligent Systems, Graduate School of Informatics and Engineering, The University of Electro-Communications, 1-5-1 Choufugaoka, Choufu, Tokyo, 182-8585, Japan
| | - Kazuo Tsuchiya
- Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto daigaku-Katsura, Nishikyo-ku, Kyoto, 615-8540, Japan
| | - Toshio Aoyagi
- Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan
| | - Shinya Aoi
- Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, The University of Osaka, 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japan.
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2
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Lancia L. Instantaneous phase of rhythmic behaviour under volitional control. Hum Mov Sci 2024; 96:103249. [PMID: 39047306 DOI: 10.1016/j.humov.2024.103249] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/26/2023] [Revised: 06/18/2024] [Accepted: 06/21/2024] [Indexed: 07/27/2024]
Abstract
The phase of signals representing cyclic behavioural patterns provides valuable information for understanding the mechanisms driving the observed behaviours. Methods usually adopted to estimate the phase, which are based on projecting the signal onto the complex plane, have strict requirements on its frequency content, which limits their application. To overcome these limitations, input signals can be processed using band-pass filters or decomposition techniques. In this paper, we briefly review these approaches and propose a new one. Our approach is based on the principles of Empirical Mode Decomposition (EMD), but unlike EMD, it does not aim to decompose the input signal. This avoids the many problems that can occur when extracting a signal's components one by one. The proposed approach estimates the phase of experimental signals that have one main oscillatory component modulated by slower activity and perturbed by weak, sparse, or random activity at faster time scales. We illustrate how our approach works by estimating the phase dynamics of synthetic signals and real-world signals representing knee angles during flexion/extension activity, heel height during gait, and the activity of different organs involved in speech production.
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Affiliation(s)
- Leonardo Lancia
- Laboratoire Parole et Langage, Aix-Marseille Université / CNRS, 5 av. Pasteur, 13100 Aix-en-Provence, France.
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3
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Xia Y, Hu Z, Wei D, Chen K, Peng Y, Yang M. Biomimetic Synchronization in Biciliated Robots. PHYSICAL REVIEW LETTERS 2024; 133:048302. [PMID: 39121428 DOI: 10.1103/physrevlett.133.048302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2023] [Revised: 04/15/2024] [Accepted: 06/10/2024] [Indexed: 08/11/2024]
Abstract
Direct mechanical coupling is known to be critical for establishing synchronization among cilia. However, the actual role of the connections is still elusive-partly because controlled experiments in living samples are challenging. Here, we employ an artificial ciliary system to address this issue. Two cilia are formed by chains of self-propelling robots and anchored to a shared base so that they are purely mechanically coupled. The system mimics biological ciliary beating but allows fine control over the beating dynamics. With different schemes of mechanical coupling, artificial cilia exhibit rich motility patterns. Particularly, their synchronous beating display two distinct modes-analogous to those observed in C. reinhardtii, the biciliated model organism for studying synchronization. Close examination suggests that the system evolves towards the most dissipative mode. Using this guideline in both simulations and experiments, we are able to direct the system into a desired state by altering the modes' respective dissipation. Our results have significant implications in understanding the synchronization of cilia.
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Affiliation(s)
| | | | - Da Wei
- Beijing National Laboratory for Condensed Matter Physics and Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Ke Chen
- Beijing National Laboratory for Condensed Matter Physics and Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
| | | | - Mingcheng Yang
- Beijing National Laboratory for Condensed Matter Physics and Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
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4
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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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5
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Wei D, Quaranta G, Aubin-Tam ME, Tam DSW. The younger flagellum sets the beat for Chlamydomonas reinhardtii. eLife 2024; 13:e86102. [PMID: 38752724 PMCID: PMC11098555 DOI: 10.7554/elife.86102] [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: 01/11/2023] [Accepted: 04/03/2024] [Indexed: 05/18/2024] Open
Abstract
Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by fine-tuning the coordination. The model organism for studying flagellate motility, Chlamydomonas reinhardtii, employs synchronous, breaststroke-like flagellar beating to swim, and it modulates the beating amplitudes differentially to steer. This strategy hinges on both inherent flagellar asymmetries (e.g. different response to chemical messengers) and such asymmetries being effectively coordinated in the synchronous beating. In C. reinhardtii, the synchrony of beating is known to be supported by a mechanical connection between flagella; however, how flagellar asymmetries persist in the synchrony remains elusive. For example, it has been speculated for decades that one flagellum leads the beating, as its dynamic properties (i.e. frequency, waveform, etc.) appear to be copied by the other one. In this study, we combine experiments, computations, and modeling efforts to elucidate the roles played by each flagellum in synchronous beating. With a non-invasive technique to selectively load each flagellum, we show that the coordinated beating essentially only responds to load exerted on the cis flagellum; and that such asymmetry in response derives from a unilateral coupling between the two flagella. Our results highlight a distinct role for each flagellum in coordination and have implication for biflagellates' tactic behaviors.
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Affiliation(s)
- Da Wei
- Department of Bionanoscience, Delft University of TechnologyDelftNetherlands
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of SciencesBeijingChina
| | - Greta Quaranta
- Laboratory for Aero and Hydrodynamics, Delft University of TechnologyDelftNetherlands
| | - Marie-Eve Aubin-Tam
- Department of Bionanoscience, Delft University of TechnologyDelftNetherlands
| | - Daniel SW Tam
- Laboratory for Aero and Hydrodynamics, Delft University of TechnologyDelftNetherlands
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6
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Yamaguchi YY, Terada Y. Reconstruction of phase dynamics from macroscopic observations based on linear and nonlinear response theories. Phys Rev E 2024; 109:024217. [PMID: 38491619 DOI: 10.1103/physreve.109.024217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2023] [Accepted: 01/22/2024] [Indexed: 03/18/2024]
Abstract
We propose a method to reconstruct the phase dynamics in rhythmical interacting systems from macroscopic responses to weak inputs by developing linear and nonlinear response theories, which predict the responses in a given system. By solving an inverse problem, the method infers an unknown system: the natural frequency distribution, the coupling function, and the time delay which is inevitable in real systems. In contrast to previous methods, our method requires neither strong invasiveness nor microscopic observations. We demonstrate that the method reconstructs two phase systems from observed responses accurately. The qualitative methodological advantages demonstrated by our quantitative numerical examinations suggest its broad applicability in various fields, including brain systems, which are often observed through macroscopic signals such as electroencephalograms and functional magnetic response imaging.
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Affiliation(s)
| | - Yu Terada
- Department of Neurobiology, University of California San Diego, La Jolla, California 92093, USA
- Institute for Physics of Intelligence, Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
- Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
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Rosenblum M, Pikovsky A. Inferring connectivity of an oscillatory network via the phase dynamics reconstruction. FRONTIERS IN NETWORK PHYSIOLOGY 2023; 3:1298228. [PMID: 38073862 PMCID: PMC10704096 DOI: 10.3389/fnetp.2023.1298228] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2023] [Accepted: 11/13/2023] [Indexed: 06/10/2024]
Abstract
We review an approach for reconstructing oscillatory networks' undirected and directed connectivity from data. The technique relies on inferring the phase dynamics model. The central assumption is that we observe the outputs of all network nodes. We distinguish between two cases. In the first one, the observed signals represent smooth oscillations, while in the second one, the data are pulse-like and can be viewed as point processes. For the first case, we discuss estimating the true phase from a scalar signal, exploiting the protophase-to-phase transformation. With the phases at hand, pairwise and triplet synchronization indices can characterize the undirected connectivity. Next, we demonstrate how to infer the general form of the coupling functions for two or three oscillators and how to use these functions to quantify the directional links. We proceed with a different treatment of networks with more than three nodes. We discuss the difference between the structural and effective phase connectivity that emerges due to high-order terms in the coupling functions. For the second case of point-process data, we use the instants of spikes to infer the phase dynamics model in the Winfree form directly. This way, we obtain the network's coupling matrix in the first approximation in the coupling strength.
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Affiliation(s)
- Michael Rosenblum
- Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany
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8
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Lukarski D, Petkoski S, Ji P, Stankovski T. Delta-alpha cross-frequency coupling for different brain regions. CHAOS (WOODBURY, N.Y.) 2023; 33:103126. [PMID: 37844293 DOI: 10.1063/5.0157979] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2023] [Accepted: 09/26/2023] [Indexed: 10/18/2023]
Abstract
Neural interactions occur on different levels and scales. It is of particular importance to understand how they are distributed among different neuroanatomical and physiological relevant brain regions. We investigated neural cross-frequency couplings between different brain regions according to the Desikan-Killiany brain parcellation. The adaptive dynamic Bayesian inference method was applied to EEG measurements of healthy resting subjects in order to reconstruct the coupling functions. It was found that even after averaging over all subjects, the mean coupling function showed a characteristic waveform, confirming the direct influence of the delta-phase on the alpha-phase dynamics in certain brain regions and that the shape of the coupling function changes for different regions. While the averaged coupling function within a region was of similar form, the region-averaged coupling function was averaged out, which implies that there is a common dependence within separate regions across the subjects. It was also found that for certain regions the influence of delta on alpha oscillations is more pronounced and that oscillations that influence other are more evenly distributed across brain regions than the influenced oscillations. When presenting the information on brain lobes, it was shown that the influence of delta emanating from the brain as a whole is greatest on the alpha oscillations of the cingulate frontal lobe, and at the same time the influence of delta from the cingulate parietal brain lobe is greatest on the alpha oscillations of the whole brain.
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Affiliation(s)
- Dushko Lukarski
- Faculty of Medicine, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia
- University Clinic for Radiotherapy and Oncology, 1000 Skopje, Macedonia
| | - Spase Petkoski
- Aix Marseille Univ, INSERM, Inst Neurosci Syst (INS), 13005 Marseille, France
| | - Peng Ji
- Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, 200433 Shanghai, China
| | - Tomislav Stankovski
- Faculty of Medicine, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia
- Department of Physics, Lancaster University, LA1 4YB Lancaster, United Kingdom
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9
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Barba-Franco JJ, Gallegos A, Jaimes-Reátegui R, Muñoz-Maciel J, Pisarchik AN. Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators. CHAOS (WOODBURY, N.Y.) 2023; 33:073126. [PMID: 37433655 DOI: 10.1063/5.0141054] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2023] [Accepted: 06/23/2023] [Indexed: 07/13/2023]
Abstract
We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.
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Affiliation(s)
- J J Barba-Franco
- Departamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144, Colonia Paseos de la Monta na, 47460 Lagos de Moreno, Jalisco, Mexico
| | - A Gallegos
- Departamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144, Colonia Paseos de la Monta na, 47460 Lagos de Moreno, Jalisco, Mexico
| | - R Jaimes-Reátegui
- Departamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144, Colonia Paseos de la Monta na, 47460 Lagos de Moreno, Jalisco, Mexico
| | - J Muñoz-Maciel
- Departamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144, Colonia Paseos de la Monta na, 47460 Lagos de Moreno, Jalisco, Mexico
| | - A N Pisarchik
- Center for Biomedical Technology, Universidad Politécnica de Madrid, Campus de Montegancedo, Pozuelo de Alarcón, 28223 Madrid, Spain
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Manasova D, Stankovski T. Neural Cross-Frequency Coupling Functions in Sleep. Neuroscience 2023:S0306-4522(23)00227-0. [PMID: 37225051 DOI: 10.1016/j.neuroscience.2023.05.016] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2022] [Revised: 04/27/2023] [Accepted: 05/16/2023] [Indexed: 05/26/2023]
Abstract
The human brain presents a heavily connected complex system. From a relatively fixed anatomy, it can enable a vast repertoire of functions. One important brain function is the process of natural sleep, which alters consciousness and voluntary muscle activity. On neural level, these alterations are accompanied by changes of the brain connectivity. In order to reveal the changes of connectivity associated with sleep, we present a methodological framework for reconstruction and assessment of functional interaction mechanisms. By analyzing EEG (electroencephalogram) recordings from human whole night sleep, first, we applied a time-frequency wavelet transform to study the existence and strength of brainwave oscillations. Then we applied a dynamical Bayesian inference on the phase dynamics in the presence of noise. With this method we reconstructed the cross-frequency coupling functions, which revealed the mechanism of how the interactions occur and manifest. We focus our analysis on the delta-alpha coupling function and observe how this cross-frequency coupling changes during the different sleep stages. The results demonstrated that the delta-alpha coupling function was increasing gradually from Awake to NREM3 (non-rapid eye movement), but only during NREM2 and NREM3 deep sleep it was significant in respect of surrogate data testing. The analysis on the spatially distributed connections showed that this significance is strong only for within the single electrode region and in the front-to-back direction. The presented methodological framework is for the whole-night sleep recordings, but it also carries general implications for other global neural states.
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Affiliation(s)
- Dragana Manasova
- Sorbonne Université, Institut du Cerveau-Paris Brain Institute-ICM, Inserm, CNRS, APHP, Hôpital de la Pitié Salpêtrière, Paris, France; Université Paris Cité, Paris, France
| | - Tomislav Stankovski
- Faculty of Medicine, Ss Cyril and Methodius University, Skopje 1000, North Macedonia; Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom.
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11
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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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12
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Yoon H. Age-dependent cardiorespiratory directional coupling in wake-resting state. Physiol Meas 2022; 43. [PMID: 36575156 DOI: 10.1088/1361-6579/acaa1b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2022] [Accepted: 12/08/2022] [Indexed: 12/13/2022]
Abstract
Objective.Cooperation in the cardiorespiratory system helps maintain internal stability. Various types of system interactions have been investigated; however, the characteristics of the interactions have mostly been studied using data collected in well-defined physiological states, such as sleep. Furthermore, most analyses provided general information about the interaction, making it difficult to quantify how the systems influenced one another.Approach.Cardiorespiratory directional coupling was investigated in different age groups (20 young and 19 elderly subjects) in a wake-resting state. The directionality index (DI) was calculated using instantaneous phases from the heartbeat interval and respiratory signal to provide information about the strength and direction of interaction between the systems. Statistical analysis was performed between the groups on the DI and independent measures of directionality (ncr: influence from cardiac system to respiratory system, and ncc: influence from the respiratory system to the cardiac system).Main results.The values of DI were -0.52 and -0.17 in the young and elderly groups, respectively (p< 0.001). Furthermore, the values of ncrand nccwere found to be significantly different between the groups (p< 0.001), respectively.Significance.Changes in both directions between the systems influence different aspects of cardiorespiratory coupling between the groups. This observation could be linked to different levels of autonomic modulation associated with ageing. Our approach could aid in quantitatively tracking and comprehending how systems interact in response to physiological and environmental changes. It could also be used to understand how abnormal interaction characteristics influence physiological system dysfunctions and disorders.
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Affiliation(s)
- Heenam Yoon
- Department of Human-Centered Artificial Intelligence, Sangmyung University, Seoul 03016, Republic of Korea
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13
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Wang X, Sipahi R, Porfiri M. Spatiotemporal patterns of firearm acquisition in the United States in different presidential terms. CHAOS (WOODBURY, N.Y.) 2022; 32:073115. [PMID: 35907731 DOI: 10.1063/5.0096773] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2022] [Accepted: 05/17/2022] [Indexed: 06/15/2023]
Abstract
This study develops mathematical tools and approaches to investigate spatiotemporal patterns of firearm acquisition in the U.S. complemented by hypothesis testing and statistical analysis. First, state-level and nation-level instant background check (BC) data are employed as proxy of firearm acquisition corresponding to 1999-2021. The relative-phase time-series of BC in each U.S. state is recovered and utilized to calculate the time-series of the U.S. states' synchronization degree. We reveal that U.S. states present a high-level degree of synchronization except in 2010-2011 and after 2018. Comparing these results with respect to a sitting U.S. president provides additional information: specifically, any two presidential terms are characterized by statistically different synchronization degrees except G. W. Bush's first term and B. H. Obama's second term. Next, to detail variations of BC, short-time Fourier transform, dimensionality reduction techniques, and diffusion maps are implemented within a time-frequency representation. Firearm acquisition in the high frequency band is described by a low-dimensional embedding, in the form of a plane with two embedding coordinates. Data points on the embedding plane identify separate clusters that signify state transitions in the original BC data with respect to different time windows. Through this analysis, we reveal that the frequency content of the BC data has a time-dependent characteristic. By comparing the diffusion map at hand with respect to a presidential term, we find that at least one of the embedding coordinates presents statistically significant variations between any two presidential terms except B. H. Obama's first term and D. J. Trump's pre-COVID term. The results point at a possible interplay between firearm acquisition in the U.S. and a presidential term.
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Affiliation(s)
- Xu Wang
- Department of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts 02115, USA
| | - Rifat Sipahi
- Department of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts 02115, USA
| | - Maurizio Porfiri
- Center for Urban Science and Progress, Tandon School of Engineering, New York University, Brooklyn, New York 11201, USA
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14
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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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15
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Sorrentino P, Ambrosanio M, Rucco R, Cabral J, Gollo LL, Breakspear M, Baselice F. Detection of Cross-Frequency Coupling Between Brain Areas: An Extension of Phase Linearity Measurement. Front Neurosci 2022; 16:846623. [PMID: 35546895 PMCID: PMC9083011 DOI: 10.3389/fnins.2022.846623] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/31/2021] [Accepted: 02/21/2022] [Indexed: 11/25/2022] Open
Abstract
The current paper proposes a method to estimate phase to phase cross-frequency coupling between brain areas, applied to broadband signals, without any a priori hypothesis about the frequency of the synchronized components. N:m synchronization is the only form of cross-frequency synchronization that allows the exchange of information at the time resolution of the faster signal, hence likely to play a fundamental role in large-scale coordination of brain activity. The proposed method, named cross-frequency phase linearity measurement (CF-PLM), builds and expands upon the phase linearity measurement, an iso-frequency connectivity metrics previously published by our group. The main idea lies in using the shape of the interferometric spectrum of the two analyzed signals in order to estimate the strength of cross-frequency coupling. We first provide a theoretical explanation of the metrics. Then, we test the proposed metric on simulated data from coupled oscillators synchronized in iso- and cross-frequency (using both Rössler and Kuramoto oscillator models), and subsequently apply it on real data from brain activity. Results show that the method is useful to estimate n:m synchronization, based solely on the phase of the signals (independently of the amplitude), and no a-priori hypothesis is available about the expected frequencies.
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Affiliation(s)
- Pierpaolo Sorrentino
- Systems Neuroscience Institute, Marseille, France.,Hermitage Capodimonte Hospital, Naples, Italy
| | | | | | - Joana Cabral
- Life and Health Sciences Research Institute (ICVS), University of Minho, Braga, Portugal.,Department of Psychiatry, University of Oxford, Oxford, United Kingdom
| | - Leonardo L Gollo
- Turner Institute for Brain and Mental Health, Monash University, Melbourne, VIC, Australia.,QIMR Berghofer Medical Research Institute, Brisbane, QLD, Australia
| | - Michael Breakspear
- QIMR Berghofer Medical Research Institute, Brisbane, QLD, Australia.,Hunter Medical Research Institute, Newcastle, NSW, Australia
| | - Fabio Baselice
- Egineering Department, University of Naples Parthenope, Naples, Italy
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16
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Tönjes R, Kori H. Phase and frequency linear response theory for hyperbolic chaotic oscillators. CHAOS (WOODBURY, N.Y.) 2022; 32:043124. [PMID: 35489838 DOI: 10.1063/5.0064519] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2021] [Accepted: 04/04/2022] [Indexed: 06/14/2023]
Abstract
We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes in frequency are experimentally accessible, giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response.
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Affiliation(s)
- Ralf Tönjes
- Institute of Physics and Astronomy, Potsdam University, 14476 Potsdam-Golm, Germany
| | - Hiroshi Kori
- Department of Complexity Sciences and Engineering, University of Tokyo, Kashiwa, 277-8561 Chiba, Japan
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17
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Espinoso A, Andrzejak RG. Phase irregularity: A conceptually simple and efficient approach to characterize electroencephalographic recordings from epilepsy patients. Phys Rev E 2022; 105:034212. [PMID: 35428047 DOI: 10.1103/physreve.105.034212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2021] [Accepted: 03/06/2022] [Indexed: 06/14/2023]
Abstract
The severe neurological disorder epilepsy affects almost 1% of the world population. For patients who suffer from pharmacoresistant focal-onset epilepsy, electroencephalographic (EEG) recordings are essential for the localization of the brain area where seizures start. Apart from the visual inspection of the recordings, quantitative EEG signal analysis techniques proved to be useful for this purpose. Among other features, regularity versus irregularity and phase coherence versus phase independence allowed characterizing brain dynamics from the measured EEG signals. Can phase irregularities also characterize brain dynamics? To address this question, we use the univariate coefficient of phase velocity variation, defined as the ratio of phase velocity standard deviation and the mean phase velocity. Beyond that, as a bivariate measure we use the classical mean phase coherence to quantify the degree of phase locking. All phase-based measures are combined with surrogates to test null hypotheses about the dynamics underlying the signals. In the first part of our analysis, we use the Rössler model system to study our approach under controlled conditions. In the second part, we use the Bern-Barcelona EEG database which consists of focal and nonfocal signals extracted from seizure-free recordings. Focal signals are recorded from brain areas where the first seizure EEG signal changes can be detected, and nonfocal signals are recorded from areas that are not involved in the seizure at its onset. Our results show that focal signals have less phase variability and more phase coherence than nonfocal signals. Once combined with surrogates, the mean phase velocity proved to have the highest discriminative power between focal and nonfocal signals. In conclusion, conceptually simple and easy to compute phase-based measures can help to detect features induced by epilepsy from EEG signals. This holds not only for the classical mean phase coherence but even more so for univariate measures of phase irregularity.
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Affiliation(s)
- Anaïs Espinoso
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain and Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute of Science and Technology, Carrer Baldiri Reixac 10-12, 08028 Barcelona, Catalonia, Spain
| | - Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
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18
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Calabrese C, Bardy BG, De Lellis P, di Bernardo M. Modeling Frequency Reduction in Human Groups Performing a Joint Oscillatory Task. Front Psychol 2022; 12:753758. [PMID: 35058838 PMCID: PMC8765722 DOI: 10.3389/fpsyg.2021.753758] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2021] [Accepted: 12/07/2021] [Indexed: 11/13/2022] Open
Abstract
In human groups performing oscillatory tasks, it has been observed that the frequency of participants' oscillations reduces when compared to that acquired in solo. This experimental observation is not captured by the standard Kuramoto oscillators, often employed to model human synchronization. In this work, we aim at capturing this observed phenomenon by proposing three alternative modifications of the standard Kuramoto model that are based on three different biologically-relevant hypotheses underlying group synchronization. The three models are tuned, validated and compared against experiments on a group synchronization task, which is a multi-agent extension of the so-called mirror game.
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Affiliation(s)
- Carmela Calabrese
- Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy.,EuroMov Digital Health in Motion, University of Montpellier IMT Mines Ales, Montpellier, France
| | - Benoît G Bardy
- EuroMov Digital Health in Motion, University of Montpellier IMT Mines Ales, Montpellier, France
| | - Pietro De Lellis
- Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy
| | - Mario di Bernardo
- Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy
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19
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Abstract
Experiments that compare rhythmic properties across different genetic alterations and entrainment conditions underlie some of the most important breakthroughs in circadian biology. A robust estimation of the rhythmic properties of the circadian signals goes hand in hand with these discoveries. Widely applied traditional signal analysis methods such as fitting cosine functions or Fourier transformations rely on the assumption that oscillation periods do not change over time. However, novel high-resolution recording techniques have shown that, most commonly, circadian signals exhibit time-dependent changes of periods and amplitudes which cannot be captured with the traditional approaches. In this chapter we introduce a method to determine time-dependent properties of oscillatory signals, using the novel open-source Python-based Biological Oscillations Analysis Toolkit (pyBOAT). We show with examples how to detect rhythms, compute and interpret high-resolution time-dependent spectral results, analyze the main oscillatory component, and to subsequently determine these main components' time-dependent instantaneous period, amplitude, and phase. We introduce step-by-step how such an analysis can be done by means of the easy-to-use point-and-click graphical user interface (GUI) provided by pyBOAT or executed within a Python programming environment. Concepts are explained using simulated signals as well as experimentally obtained time series.
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Affiliation(s)
- Christoph Schmal
- Institute for Theoretical Biology, Humboldt Universität zu Berlin, Berlin, Germany.
| | - Gregor Mönke
- European Molecular Biology Laboratory, Heidelberg, Germany
| | - Adrián E Granada
- Charité Comprehensive Cancer Center, Charité Universitätsmedizin Berlin, Berlin, Germany
- German Cancer Consortium (DKTK), German Cancer Research Center (DKFZ), Heidelberg, Germany
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20
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Lukarski D, Stavrov D, Stankovski T. Variability of cardiorespiratory interactions under different breathing patterns. Biomed Signal Process Control 2022. [DOI: 10.1016/j.bspc.2021.103152] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022]
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21
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Mau ETK, Rosenblum M. Optimizing charge-balanced pulse stimulation for desynchronization. CHAOS (WOODBURY, N.Y.) 2022; 32:013103. [PMID: 35105136 DOI: 10.1063/5.0070036] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/03/2021] [Accepted: 12/09/2021] [Indexed: 06/14/2023]
Abstract
Collective synchronization in a large population of self-sustained units appears both in natural and engineered systems. Sometimes this effect is in demand, while in some cases, it is undesirable, which calls for control techniques. In this paper, we focus on pulsatile control, with the goal to either increase or decrease the level of synchrony. We quantify this level by the entropy of the phase distribution. Motivated by possible applications in neuroscience, we consider pulses of a realistic shape. Exploiting the noisy Kuramoto-Winfree model, we search for the optimal pulse profile and the optimal stimulation phase. For this purpose, we derive an expression for the change of the phase distribution entropy due to the stimulus. We relate this change to the properties of individual units characterized by generally different natural frequencies and phase response curves and the population's state. We verify the general result by analyzing a two-frequency population model and demonstrating a good agreement of the theory and numerical simulations.
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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
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22
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Lombardi M, Liuzza D, Bernardo MD. Using Learning to Control Artificial Avatars in Human Motor Coordination Tasks. IEEE T ROBOT 2021. [DOI: 10.1109/tro.2021.3073771] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
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23
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Spontaneous emergence of leadership patterns drives synchronization in complex human networks. Sci Rep 2021; 11:18379. [PMID: 34526559 PMCID: PMC8443630 DOI: 10.1038/s41598-021-97656-y] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2021] [Accepted: 08/04/2021] [Indexed: 02/08/2023] Open
Abstract
Synchronization of human networks is fundamental in many aspects of human endeavour. Recently, much research effort has been spent on analyzing how motor coordination emerges in human groups (from rocking chairs to violin players) and how it is affected by coupling structure and strength. Here we uncover the spontaneous emergence of leadership (based on physical signaling during group interaction) as a crucial factor steering the occurrence of synchronization in complex human networks where individuals perform a joint motor task. In two experiments engaging participants in an arm movement synchronization task, in the physical world as well as in the digital world, we found that specific patterns of leadership emerged and increased synchronization performance. Precisely, three patterns were found, involving a subtle interaction between phase of the motion and amount of influence. Such patterns were independent of the presence or absence of physical interaction, and persisted across manipulated spatial configurations. Our results shed light on the mechanisms that drive coordination and leadership in human groups, and are consequential for the design of interactions with artificial agents, avatars or robots, where social roles can be determinant for a successful interaction.
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24
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Lombardi M, Liuzza D, di Bernardo M. Dynamic Input Deep Learning Control of Artificial Avatars in a Multi-Agent Joint Motor Task. Front Robot AI 2021; 8:665301. [PMID: 34434967 PMCID: PMC8381333 DOI: 10.3389/frobt.2021.665301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2021] [Accepted: 07/20/2021] [Indexed: 11/13/2022] Open
Abstract
In many real-word scenarios, humans and robots are required to coordinate their movements in joint tasks to fulfil a common goal. While several examples regarding dyadic human robot interaction exist in the current literature, multi-agent scenarios in which one or more artificial agents need to interact with many humans are still seldom investigated. In this paper we address the problem of synthesizing an autonomous artificial agent to perform a paradigmatic oscillatory joint task in human ensembles while exhibiting some desired human kinematic features. We propose an architecture based on deep reinforcement learning which is flexible enough to make the artificial agent interact with human groups of different sizes. As a paradigmatic coordination task we consider a multi-agent version of the mirror game, an oscillatory motor task largely used in the literature to study human motor coordination.
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Affiliation(s)
- Maria Lombardi
- Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom.,Department of Electrical Engineering and Information Technology, University of Naples Federico II, Naples, Italy
| | - Davide Liuzza
- ENEA Fusion and Nuclear Safety Department, Frascati, Italy
| | - Mario di Bernardo
- Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom.,Scuola Superiore Meridionale, University of Naples Federico II, Naples, Italy
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25
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Smirnov DA. Phase-dynamic causalities within dynamical effects framework. CHAOS (WOODBURY, N.Y.) 2021; 31:073127. [PMID: 34340361 DOI: 10.1063/5.0055586] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2021] [Accepted: 06/23/2021] [Indexed: 06/13/2023]
Abstract
This work investigates numerics of several widely known phase-dynamic quantifiers of directional (causal) couplings between oscillatory systems: transfer entropy (TE), differential quantifier, and squared-coefficients quantifier based on an evolution map. The study is performed on the system of two stochastic Kuramoto oscillators within the framework of dynamical causal effects. The quantifiers are related to each other and to an asymptotic effect of the coupling on phase diffusion. Several novel findings are listed as follows: (i) for a non-synchronous regime and high enough noise levels, the TE rate multiplied by a certain characteristic time (called here reduced TE) equals twice an asymptotic effect of a directional coupling on phase diffusion; (ii) "information flow" expressed by the TE rate unboundedly rises with the coupling coefficient even in the domain of effective synchronization; (iii) in any effective synchronization regime, the reduced TE is equal to 1/8 n.u. in each direction for equal coupling coefficients and equal noise intensities, and it is in general a simple function of the ratio of noise intensities and the ratio of coupling coefficients.
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Affiliation(s)
- Dmitry A Smirnov
- Saratov Branch, Kotelnikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, 38 Zelyonaya Street, Saratov 410019, Russia
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26
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Coupling between Blood Pressure and Subarachnoid Space Width Oscillations during Slow Breathing. ENTROPY 2021; 23:e23010113. [PMID: 33467769 PMCID: PMC7830105 DOI: 10.3390/e23010113] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/28/2020] [Revised: 12/29/2020] [Accepted: 01/12/2021] [Indexed: 12/14/2022]
Abstract
The precise mechanisms connecting the cardiovascular system and the cerebrospinal fluid (CSF) are not well understood in detail. This paper investigates the couplings between the cardiac and respiratory components, as extracted from blood pressure (BP) signals and oscillations of the subarachnoid space width (SAS), collected during slow ventilation and ventilation against inspiration resistance. The experiment was performed on a group of 20 healthy volunteers (12 females and 8 males; BMI =22.1±3.2 kg/m2; age 25.3±7.9 years). We analysed the recorded signals with a wavelet transform. For the first time, a method based on dynamical Bayesian inference was used to detect the effective phase connectivity and the underlying coupling functions between the SAS and BP signals. There are several new findings. Slow breathing with or without resistance increases the strength of the coupling between the respiratory and cardiac components of both measured signals. We also observed increases in the strength of the coupling between the respiratory component of the BP and the cardiac component of the SAS and vice versa. Slow breathing synchronises the SAS oscillations, between the brain hemispheres. It also diminishes the similarity of the coupling between all analysed pairs of oscillators, while inspiratory resistance partially reverses this phenomenon. BP–SAS and SAS–BP interactions may reflect changes in the overall biomechanical characteristics of the brain.
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27
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Lehnertz K, Bröhl T, Rings T. The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges. Front Physiol 2020; 11:598694. [PMID: 33408639 PMCID: PMC7779628 DOI: 10.3389/fphys.2020.598694] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2020] [Accepted: 11/30/2020] [Indexed: 12/30/2022] Open
Abstract
The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.
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Affiliation(s)
- Klaus Lehnertz
- Department of Epileptology, University of Bonn Medical Centre, Bonn, Germany
- Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Bonn, Germany
- Interdisciplinary Center for Complex Systems, University of Bonn, Bonn, Germany
| | - Timo Bröhl
- Department of Epileptology, University of Bonn Medical Centre, Bonn, Germany
- Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Bonn, Germany
| | - Thorsten Rings
- Department of Epileptology, University of Bonn Medical Centre, Bonn, Germany
- Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Bonn, Germany
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28
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Skazkina VV, Karavaev AS, Borovkova EI, Simonyan MA, Prokhorov MD, Ponomarenko VI, Dubinkina ES, Bezruchko BP, Gridnev VI, Kiselev AR. Uncovering Interaction Between The Loops Of Autonomic Regulation Of Blood Circulation From Long Time Series. RUSSIAN OPEN MEDICAL JOURNAL 2020. [DOI: 10.15275/rusomj.2020.0403] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022] Open
Abstract
The purpose of this work is to study the interaction between the autonomic regulatory loops of blood circulation from long time series. Methods ― We simultaneously recorded four-hour signals of electrocardiogram and photoplethysmogram from the ear and finger of ten healthy adults. We determined the intervals of phase synchronization of the studied regulatory loops and analyzed the dependence of their length on the recording time. The deviations of the total percentage of phase synchronization (index S) from its mean value were estimated in moving non-overlapping windows. Results ― For studied signals we found no significant correlation between the length of synchronization epoch and the time of its beginning. A sharp increase in the deviation of the index S from its mean was shown at the end of the experiment. Conclusion ― The increase in the deviation from the mean at the end of our records is most likely associated more with psychosomatic influences than with hormonal regulation or immobilization stress.
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Affiliation(s)
| | | | | | | | - Mikhail D. Prokhorov
- Saratov Branch of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences
| | - Vladimir I. Ponomarenko
- Saratov Branch of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences
| | | | - Boris P. Bezruchko
- Saratov Branch of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences
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29
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Rosenblum M. Controlling collective synchrony in oscillatory ensembles by precisely timed pulses. CHAOS (WOODBURY, N.Y.) 2020; 30:093131. [PMID: 33003901 DOI: 10.1063/5.0019823] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2020] [Accepted: 09/01/2020] [Indexed: 06/11/2023]
Abstract
We present an efficient technique for control of synchrony in a globally coupled ensemble by pulsatile action. We assume that we can observe the collective oscillation and can stimulate all elements of the ensemble simultaneously. We pay special attention to the minimization of intervention into the system. The key idea is to stimulate only at the most sensitive phase. To find this phase, we implement an adaptive feedback control. Estimating the instantaneous phase of the collective mode on the fly, we achieve efficient suppression using a few pulses per oscillatory cycle. We discuss the possible relevance of the results for neuroscience, namely, for the development of advanced algorithms for deep brain stimulation, a medical technique used to treat Parkinson's disease.
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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
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30
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Lukarski D, Ginovska M, Spasevska H, Stankovski T. Time Window Determination for Inference of Time-Varying Dynamics: Application to Cardiorespiratory Interaction. Front Physiol 2020; 11:341. [PMID: 32411009 PMCID: PMC7198895 DOI: 10.3389/fphys.2020.00341] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2019] [Accepted: 03/24/2020] [Indexed: 11/13/2022] Open
Abstract
Interacting dynamical systems abound in nature, with examples ranging from biology and population dynamics, through physics and chemistry, to communications and climate. Often their states, parameters and functions are time-varying, because such systems interact with other systems and the environment, exchanging information and matter. A common problem when analysing time-series data from dynamical systems is how to determine the length of the time window for the analysis. When one needs to follow the time-variability of the dynamics, or the dynamical parameters and functions, the time window needs to be resolved first. We tackled this problem by introducing a method for adaptive determination of the time window for interacting oscillators, as modeled and scaled for the cardiorespiratory interaction. By investigating a system of coupled phase oscillators and utilizing the Dynamical Bayesian Inference method, we propose a procedure to determine the time window and the propagation parameter of the covariance matrix. The optimal values are determined so that the inferred parameters follow the dynamics of the actual ones and at the same time the error of the inference represented by the covariance matrix is minimal. The effectiveness of the methodology is presented on a system of coupled limit-cycle oscillators and on the cardiorespiratory interaction. Three cases of cardiorespiratory interaction were considered-measurement with spontaneous free breathing, one with periodic sine breathing and one with a-periodic time-varying breathing. The results showed that the cardiorespiratory coupling strength and similarity of form of coupling functions have greater values for slower breathing, and this variability follows continuously the change of the breathing frequency. The method can be applied effectively to other time-varying oscillatory interactions and carries important implications for analysis of general dynamical systems.
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Affiliation(s)
- Dushko Lukarski
- Faculty of Medicine, Ss. Cyril and Methodius University, Skopje, Macedonia
- University Clinic for Radiotherapy and Oncology, Skopje, Macedonia
| | - Margarita Ginovska
- Faculty of Electrical Engineering and Information Technologies, Ss. Cyril and Methodius University, Skopje, Macedonia
| | - Hristina Spasevska
- Faculty of Electrical Engineering and Information Technologies, Ss. Cyril and Methodius University, Skopje, Macedonia
| | - Tomislav Stankovski
- Faculty of Medicine, Ss. Cyril and Methodius University, Skopje, Macedonia
- Department of Physics, Lancaster University, Lancaster, United Kingdom
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31
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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: 19] [Impact Index Per Article: 4.8] [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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Lin C, Lin PF, Wang CH, Juan CH, Tran TT, Pham VT, Nien CT, Lin YJ, Wang CY, Yeh CH, Lo MT. Probing age-related changes in cardio-respiratory dynamics by multimodal coupling assessment. CHAOS (WOODBURY, N.Y.) 2020; 30:033118. [PMID: 32237792 DOI: 10.1063/1.5134868] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2019] [Accepted: 02/13/2020] [Indexed: 06/11/2023]
Abstract
Quantifying respiratory sinus arrhythmia (RSA) can provide an index of parasympathetic function. Fourier spectral analysis, the most widely used approach, estimates the power of the heart rate variability in the frequency band of breathing. However, it neglects the time-varying characteristics of the transitions as well as the nonlinear properties of the cardio-respiratory coupling. Here, we propose a novel approach based on Hilbert-Huang transform, called the multimodal coupling analysis (MMCA) method, to assess cardio-respiratory dynamics by examining the instantaneous nonlinear phase interactions between two interconnected signals (i.e., heart rate and respiration) and compare with the counterparts derived from the wavelet-based method. We used an online database. The corresponding RSA components of the 90-min ECG and respiratory signals of 20 young and 20 elderly healthy subjects were extracted and quantified. A cycle-based analysis and a synchro-squeezed wavelet transform were also introduced to assess the amplitude or phase changes of each respiratory cycle. Our results demonstrated that the diminished mean and standard deviation of the derived dynamical RSA activities can better discriminate between elderly and young subjects. Moreover, the degree of nonlinearity of the cycle-by-cycle RSA waveform derived from the differences between the instantaneous frequency and the mean frequency of each respiratory cycle was significantly decreased in the elderly subjects by the MMCA method. The MMCA method in combination with the cycle-based analysis can potentially be a useful tool to depict the aging changes of the parasympathetic function as well as the waveform nonlinearity of RSA compared to the Fourier-based high-frequency power and the wavelet-based method.
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Affiliation(s)
- Chen Lin
- Department of Biomedical Sciences and Engineering, National Central University, Taoyuan 320, Taiwan
| | - Pei-Feng Lin
- Department of Geriatrics, Tainan Hospital, Ministry of Health and Welfare, Tainan 700, Taiwan
| | - Chen-Hsu Wang
- Department of Biomedical Sciences and Engineering, National Central University, Taoyuan 320, Taiwan
| | - Chung-Hau Juan
- Department of Biomedical Sciences and Engineering, National Central University, Taoyuan 320, Taiwan
| | - Thi-Thao Tran
- Department of Biomedical Sciences and Engineering, National Central University, Taoyuan 320, Taiwan
| | - Van-Truong Pham
- Department of Biomedical Sciences and Engineering, National Central University, Taoyuan 320, Taiwan
| | - Chun-Tung Nien
- Department of Biomedical Sciences and Engineering, National Central University, Taoyuan 320, Taiwan
| | - Yenn-Jiang Lin
- Heart Rhythm Center, Division of Cardiology, Department of Medicine, Taipei Veterans General Hospital, Taipei 112, Taiwan
| | - Cheng-Yen Wang
- Department of Biomedical Sciences and Engineering, National Central University, Taoyuan 320, Taiwan
| | - Chien-Hung Yeh
- School of information and Electronics Engineering, Beijing Institute of Technology, Beijing 100081, China
| | - Men-Tzung Lo
- Department of Biomedical Sciences and Engineering, National Central University, Taoyuan 320, Taiwan
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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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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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Stankovski T, Pereira T, McClintock PVE, Stefanovska A. Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190039. [PMID: 31656134 PMCID: PMC6834002 DOI: 10.1098/rsta.2019.0039] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 08/13/2019] [Indexed: 06/10/2023]
Abstract
Dynamical systems are widespread, with examples in physics, chemistry, biology, population dynamics, communications, climatology and social science. They are rarely isolated but generally interact with each other. These interactions can be characterized by coupling functions-which contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how each interaction occurs. Coupling functions can be used, not only to understand, but also to control and predict the outcome of the interactions. This theme issue assembles ground-breaking work on coupling functions by leading scientists. After overviewing the field and describing recent advances in the theory, it discusses novel methods for the detection and reconstruction of coupling functions from measured data. It then presents applications in chemistry, neuroscience, cardio-respiratory physiology, climate, electrical engineering and social science. Taken together, the collection summarizes earlier work on coupling functions, reviews recent developments, presents the state of the art, and looks forward to guide the future evolution of the field. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.
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Affiliation(s)
- Tomislav Stankovski
- Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
- Faculty of Medicine, Ss Cyril and Methodius University, Skopje 1000, Macedonia
| | - Tiago Pereira
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
- Institute of Mathematical and Computer Sciences, University of Sao Paulo, Sao Carlos 13566-590, Brazil
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Hagos Z, Stankovski T, Newman J, Pereira T, McClintock PVE, Stefanovska A. Synchronization transitions caused by time-varying coupling functions. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190275. [PMID: 31656137 PMCID: PMC6834000 DOI: 10.1098/rsta.2019.0275] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 09/09/2019] [Indexed: 06/10/2023]
Abstract
Interacting dynamical systems are widespread in nature. The influence that one such system exerts on another is described by a coupling function; and the coupling functions extracted from the time-series of interacting dynamical systems are often found to be time-varying. Although much effort has been devoted to the analysis of coupling functions, the influence of time-variability on the associated dynamics remains largely unexplored. Motivated especially by coupling functions in biology, including the cardiorespiratory and neural delta-alpha coupling functions, this paper offers a contribution to the understanding of effects due to time-varying interactions. Through both numerics and mathematically rigorous theoretical consideration, we show that for time-variable coupling functions with time-independent net coupling strength, transitions into and out of phase- synchronization can occur, even though the frozen coupling functions determine phase-synchronization solely by virtue of their net coupling strength. Thus the information about interactions provided by the shape of coupling functions plays a greater role in determining behaviour when these coupling functions are time-variable. 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)
- Zeray Hagos
- Institute of Mathematical and Computer Sciences, University of São Paulo, São Carlos 13566-590, Brazil
- Department of Mathematics, Mekelle University, Mekelle, Ethiopia
| | - Tomislav Stankovski
- Faculty of Medicine, Ss Cyril and Methodius University, 50 Divizija 6, Skopje, North Macedonia
- Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
| | - Julian Newman
- Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
| | - Tiago Pereira
- Institute of Mathematical and Computer Sciences, University of São Paulo, São Carlos 13566-590, Brazil
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
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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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Chakrabarti B, Saintillan D. Hydrodynamic Synchronization of Spontaneously Beating Filaments. PHYSICAL REVIEW LETTERS 2019; 123:208101. [PMID: 31809101 DOI: 10.1103/physrevlett.123.208101] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2019] [Revised: 09/12/2019] [Indexed: 06/10/2023]
Abstract
Using a geometric feedback model of the flagellar axoneme accounting for dynein motor kinetics, we study elastohydrodynamic phase synchronization in a pair of spontaneously beating filaments with waveforms ranging from sperm to cilia and Chlamydomonas. Our computations reveal that both in-phase and antiphase synchrony can emerge for asymmetric beats while symmetric waveforms go in phase, and elucidate the mechanism for phase slips due to biochemical noise. Model predictions agree with recent experiments and illuminate the crucial roles of hydrodynamics and mechanochemical feedback in synchronization.
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Affiliation(s)
- Brato Chakrabarti
- Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
| | - David Saintillan
- Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
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Peter F, Gong CC, Pikovsky A. Microscopic correlations in the finite-size Kuramoto model of coupled oscillators. Phys Rev E 2019; 100:032210. [PMID: 31639966 DOI: 10.1103/physreve.100.032210] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/09/2019] [Indexed: 11/07/2022]
Abstract
Supercritical Kuramoto oscillators with distributed frequencies can be separated into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators-at least so in the thermodynamic limit. In finite ensembles, in contrast, such clear separation fails: The mean field fluctuates due to finite-size effects and thereby induces order in the disordered group. This publication demonstrates this effect, similar to noise-induced synchronization, in a purely deterministic system. We start by modeling the situation as a stationary mean field with additional white noise acting on a pair of unlocked Kuramoto oscillators. An analytical expression shows that the cross-correlation between the two increases with decreasing ratio of natural frequency difference and noise intensity. In a deterministic finite Kuramoto model, the strength of the mean-field fluctuations is inextricably linked to the typical natural frequency difference. Therefore, we let a fluctuating mean field, generated by a finite ensemble of active oscillators, act on pairs of passive oscillators with a microscopic natural frequency difference between which we then measure the cross-correlation, at both super- and subcritical coupling.
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Affiliation(s)
- Franziska Peter
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24-25, 14476 Potsdam, Germany
| | - Chen Chris Gong
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24-25, 14476 Potsdam, Germany
| | - Arkady Pikovsky
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24-25, 14476 Potsdam, Germany.,Department of Control Theory, Nizhny Novgorod State University, Gagarin Av. 23, 606950, Nizhny Novgorod, Russia
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40
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Dhingra RR, Furuya WI, Galán RF, Dutschmann M. Excitation-inhibition balance regulates the patterning of spinal and cranial inspiratory motor outputs in rats in situ. Respir Physiol Neurobiol 2019; 266:95-102. [DOI: 10.1016/j.resp.2019.05.001] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2019] [Revised: 04/11/2019] [Accepted: 05/02/2019] [Indexed: 11/25/2022]
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Dhingra RR, Furuya WI, Bautista TG, Dick TE, Galán RF, Dutschmann M. Increasing Local Excitability of Brainstem Respiratory Nuclei Reveals a Distributed Network Underlying Respiratory Motor Pattern Formation. Front Physiol 2019; 10:887. [PMID: 31396094 PMCID: PMC6664290 DOI: 10.3389/fphys.2019.00887] [Citation(s) in RCA: 35] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2019] [Accepted: 06/26/2019] [Indexed: 11/18/2022] Open
Abstract
The core circuit of the respiratory central pattern generator (rCPG) is located in the ventrolateral medulla, especially in the pre-Bötzinger complex (pre-BötC) and the neighboring Bötzinger complex (BötC). To test the hypothesis that this core circuit is embedded within an anatomically distributed pattern-generating network, we investigated whether local disinhibition of the nucleus tractus solitarius (NTS), the Kölliker-Fuse nuclei (KFn), or the midbrain periaqueductal gray area (PAG) can similarly affect the respiratory pattern compared to disinhibition of the pre-BötC/BötC core. In arterially-perfused brainstem preparations of rats, we recorded the three-phase respiratory pattern (inspiration, post-inspiration and late-expiration) from phrenic and vagal nerves before and after bilateral microinjections of the GABA(A)R antagonist bicuculline (50 nl, 10 mM). Local disinhibition of either NTS, pre-BötC/BötC, or KFn, but not PAG, triggered qualitatively similar disruptions of the respiratory pattern resulting in a highly significant increase in the variability of the respiratory cycle length, including inspiratory and expiratory phase durations. To quantitatively analyze these motor pattern perturbations, we measured the strength of phase synchronization between phrenic and vagal motor outputs. This analysis showed that local disinhibition of all brainstem target nuclei, but not the midbrain PAG, significantly decreased the strength of phase synchronization. The convergent perturbations of the respiratory pattern suggest that the rCPG expands rostrally and dorsally from the designated core but does not include higher mid-brain structures. Our data also suggest that excitation-inhibition balance of respiratory network synaptic interactions critically determines the network dynamics that underlie vital respiratory rhythm and pattern formation.
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Affiliation(s)
- Rishi R Dhingra
- Division of Systems Neurophysiology, The Florey Institute of Neuroscience and Mental Health, Parkville, VIC, Australia
| | - Werner I Furuya
- Division of Systems Neurophysiology, The Florey Institute of Neuroscience and Mental Health, Parkville, VIC, Australia
| | - Tara G Bautista
- Division of Systems Neurophysiology, The Florey Institute of Neuroscience and Mental Health, Parkville, VIC, Australia
| | - Thomas E Dick
- Division of Pulmonary, Critical Care and Sleep, Department of Medicine, Case Western Reserve University, Cleveland, OH, United States
| | - Roberto F Galán
- Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH, United States
| | - Mathias Dutschmann
- Division of Systems Neurophysiology, The Florey Institute of Neuroscience and Mental Health, Parkville, VIC, Australia
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Negrello M, Warnaar P, Romano V, Owens CB, Lindeman S, Iavarone E, Spanke JK, Bosman LWJ, De Zeeuw CI. Quasiperiodic rhythms of the inferior olive. PLoS Comput Biol 2019; 15:e1006475. [PMID: 31059498 PMCID: PMC6538185 DOI: 10.1371/journal.pcbi.1006475] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2018] [Revised: 05/28/2019] [Accepted: 04/16/2019] [Indexed: 12/13/2022] Open
Abstract
Inferior olivary activity causes both short-term and long-term changes in cerebellar output underlying motor performance and motor learning. Many of its neurons engage in coherent subthreshold oscillations and are extensively coupled via gap junctions. Studies in reduced preparations suggest that these properties promote rhythmic, synchronized output. However, the interaction of these properties with torrential synaptic inputs in awake behaving animals is not well understood. Here we combine electrophysiological recordings in awake mice with a realistic tissue-scale computational model of the inferior olive to study the relative impact of intrinsic and extrinsic mechanisms governing its activity. Our data and model suggest that if subthreshold oscillations are present in the awake state, the period of these oscillations will be transient and variable. Accordingly, by using different temporal patterns of sensory stimulation, we found that complex spike rhythmicity was readily evoked but limited to short intervals of no more than a few hundred milliseconds and that the periodicity of this rhythmic activity was not fixed but dynamically related to the synaptic input to the inferior olive as well as to motor output. In contrast, in the long-term, the average olivary spiking activity was not affected by the strength and duration of the sensory stimulation, while the level of gap junctional coupling determined the stiffness of the rhythmic activity in the olivary network during its dynamic response to sensory modulation. Thus, interactions between intrinsic properties and extrinsic inputs can explain the variations of spiking activity of olivary neurons, providing a temporal framework for the creation of both the short-term and long-term changes in cerebellar output. Activity of the inferior olive, transmitted via climbing fibers to the cerebellum, regulates initiation and amplitude of movements, signals unexpected sensory feedback, and directs cerebellar learning. It is characterized by widespread subthreshold oscillations and synchronization promoted by strong electrotonic coupling. In brain slices, subthreshold oscillations gate which inputs can be transmitted by inferior olivary neurons and which will not—dependent on the phase of the oscillation. We tested whether the subthreshold oscillations had a measurable impact on temporal patterning of climbing fiber activity in intact, awake mice. We did so by recording neural activity of the postsynaptic Purkinje cells, in which complex spike firing faithfully represents climbing fiber activity. For short intervals (<300 ms) many Purkinje cells showed spontaneously rhythmic complex spike activity. However, our experiments designed to evoke conditional responses indicated that complex spikes are not predominantly predicated on stimulus history. Our realistic network model of the inferior olive explains the experimental observations via continuous phase modulations of the subthreshold oscillations under the influence of synaptic fluctuations. We conclude that complex spike activity emerges from a quasiperiodic rhythm that is stabilized by electrotonic coupling between its dendrites, yet dynamically influenced by the status of their synaptic inputs.
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Affiliation(s)
- Mario Negrello
- Department of Neuroscience, Erasmus MC, Rotterdam, the Netherlands
- * E-mail: (MN); (LWJB); (CIDZ)
| | - Pascal Warnaar
- Department of Neuroscience, Erasmus MC, Rotterdam, the Netherlands
| | - Vincenzo Romano
- Department of Neuroscience, Erasmus MC, Rotterdam, the Netherlands
| | - Cullen B. Owens
- Department of Neuroscience, Erasmus MC, Rotterdam, the Netherlands
| | - Sander Lindeman
- Department of Neuroscience, Erasmus MC, Rotterdam, the Netherlands
| | | | - Jochen K. Spanke
- Department of Neuroscience, Erasmus MC, Rotterdam, the Netherlands
| | - Laurens W. J. Bosman
- Department of Neuroscience, Erasmus MC, Rotterdam, the Netherlands
- * E-mail: (MN); (LWJB); (CIDZ)
| | - Chris I. De Zeeuw
- Department of Neuroscience, Erasmus MC, Rotterdam, the Netherlands
- Netherlands Institute for Neuroscience, Royal Academy of Arts and Sciences, Amsterdam, the Netherlands
- * E-mail: (MN); (LWJB); (CIDZ)
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Schaworonkow N, Nikulin VV. Spatial neuronal synchronization and the waveform of oscillations: Implications for EEG and MEG. PLoS Comput Biol 2019; 15:e1007055. [PMID: 31086368 PMCID: PMC6534335 DOI: 10.1371/journal.pcbi.1007055] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2018] [Revised: 05/24/2019] [Accepted: 04/26/2019] [Indexed: 11/24/2022] Open
Abstract
Neuronal oscillations are ubiquitous in the human brain and are implicated in virtually all brain functions. Although they can be described by a prominent peak in the power spectrum, their waveform is not necessarily sinusoidal and shows rather complex morphology. Both frequency and temporal descriptions of such non-sinusoidal neuronal oscillations can be utilized. However, in non-invasive EEG/MEG recordings the waveform of oscillations often takes a sinusoidal shape which in turn leads to a rather oversimplified view on oscillatory processes. In this study, we show in simulations how spatial synchronization can mask non-sinusoidal features of the underlying rhythmic neuronal processes. Consequently, the degree of non-sinusoidality can serve as a measure of spatial synchronization. To confirm this empirically, we show that a mixture of EEG components is indeed associated with more sinusoidal oscillations compared to the waveform of oscillations in each constituent component. Using simulations, we also show that the spatial mixing of the non-sinusoidal neuronal signals strongly affects the amplitude ratio of the spectral harmonics constituting the waveform. Finally, our simulations show how spatial mixing can affect the strength and even the direction of the amplitude coupling between constituent neuronal harmonics at different frequencies. Validating these simulations, we also demonstrate these effects in real EEG recordings. Our findings have far reaching implications for the neurophysiological interpretation of spectral profiles, cross-frequency interactions, as well as for the unequivocal determination of oscillatory phase.
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Affiliation(s)
- Natalie Schaworonkow
- Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University, Frankfurt am Main, Germany
- Department of Neurology & Stroke, and Hertie Institute for Clinical Brain Research, University of Tübingen, Tübingen, Germany
| | - Vadim V. Nikulin
- Department of Neurology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
- Centre for Cognition and Decision Making, National Research University Higher School of Economics, Moscow, Russian Federation
- Neurophysics Group, Department of Neurology, Charité-University Medicine Berlin – Campus Benjamin Franklin, Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany
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Thoracic sympathetic chain stimulation modulates and entrains the respiratory pattern. Auton Neurosci 2019; 218:16-24. [DOI: 10.1016/j.autneu.2019.01.008] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2018] [Revised: 01/29/2019] [Accepted: 01/30/2019] [Indexed: 11/21/2022]
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Yeldesbay A, Fink GR, Daun S. Reconstruction of effective connectivity in the case of asymmetric phase distributions. J Neurosci Methods 2019; 317:94-107. [PMID: 30786248 DOI: 10.1016/j.jneumeth.2019.02.009] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2018] [Revised: 02/13/2019] [Accepted: 02/15/2019] [Indexed: 11/16/2022]
Abstract
BACKGROUND The interaction of different brain regions is supported by transient synchronization between neural oscillations at different frequencies. Different measures based on synchronization theory are used to assess the strength of the interactions from experimental data. One method of estimating the effective connectivity between brain regions, within the framework of the theory of weakly coupled phase oscillators, was implemented in Dynamic Causal Modelling (DCM) for phase coupling (Penny et al., 2009). However, the results of such an approach strongly depend on the observables used to reconstruct the equations (Kralemann et al., 2008). In particular, an asymmetric distribution of the observables could result in a false estimation of the effective connectivity between the network nodes. NEW METHOD In this work we built a new modelling part into DCM for phase coupling, and extended it with a distortion function that accommodates departures from purely sinusoidal oscillations. RESULTS By analysing numerically generated data sets with an asymmetric phase distribution, we demonstrated that the extended DCM for phase coupling with the additional modelling component, correctly estimates the coupling functions. COMPARISON WITH EXISTING METHODS The new method allows for different intrinsic frequencies among coupled neuronal populations and provides results that do not depend on the distribution of the observables. CONCLUSIONS The proposed method can be used to analyse effective connectivity between brain regions within and between different frequency bands, to characterize m:n phase coupling, and to unravel underlying mechanisms of the transient synchronization.
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Affiliation(s)
- Azamat Yeldesbay
- University of Cologne, Institute of Zoology, Heisenberg Research Group of Computational Neuroscience - Modeling Neural Network Function, Zülpicher Str. 47b, 50674 Cologne, Germany; Research Centre Jülich, Institute of Neuroscience and Medicine (INM-3), Cognitive Neuroscience, 52425 Jülich, Germany.
| | - Gereon R Fink
- Research Centre Jülich, Institute of Neuroscience and Medicine (INM-3), Cognitive Neuroscience, 52425 Jülich, Germany; University of Cologne, Department of Neurology, Medical Faculty and University Hospital Cologne, Kerpener Str. 62, 50937 Cologne, Germany
| | - Silvia Daun
- University of Cologne, Institute of Zoology, Heisenberg Research Group of Computational Neuroscience - Modeling Neural Network Function, Zülpicher Str. 47b, 50674 Cologne, Germany; Research Centre Jülich, Institute of Neuroscience and Medicine (INM-3), Cognitive Neuroscience, 52425 Jülich, Germany.
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46
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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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47
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Rosenblum M, Pikovsky A. Efficient determination of synchronization domains from observations of asynchronous dynamics. CHAOS (WOODBURY, N.Y.) 2018; 28:106301. [PMID: 30384634 DOI: 10.1063/1.5037012] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2018] [Accepted: 06/28/2018] [Indexed: 05/28/2023]
Abstract
We develop an approach for a fast experimental inference of synchronization properties of an oscillator. While the standard technique for determination of synchronization domains implies that the oscillator under study is forced with many different frequencies and amplitudes, our approach requires only several observations of a driven system. Reconstructing the phase dynamics from data, we successfully determine synchronization domains of noisy and chaotic oscillators. Our technique is especially important for experiments with living systems where an external action can be harmful and shall be minimized.
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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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48
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Inferring the phase response curve from observation of a continuously perturbed oscillator. Sci Rep 2018; 8:13606. [PMID: 30206301 PMCID: PMC6134126 DOI: 10.1038/s41598-018-32069-y] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2018] [Accepted: 08/23/2018] [Indexed: 11/24/2022] Open
Abstract
Phase response curves are important for analysis and modeling of oscillatory dynamics in various applications, particularly in neuroscience. Standard experimental technique for determining them requires isolation of the system and application of a specifically designed input. However, isolation is not always feasible and we are compelled to observe the system in its natural environment under free-running conditions. To that end we propose an approach relying only on passive observations of the system and its input. We illustrate it with simulation results of an oscillator driven by a stochastic force.
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49
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Hannay KM, Forger DB, Booth V. Macroscopic models for networks of coupled biological oscillators. SCIENCE ADVANCES 2018; 4:e1701047. [PMID: 30083596 PMCID: PMC6070363 DOI: 10.1126/sciadv.1701047] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2017] [Accepted: 06/20/2018] [Indexed: 05/20/2023]
Abstract
The study of synchronization of coupled biological oscillators is fundamental to many areas of biology including neuroscience, cardiac dynamics, and circadian rhythms. Mathematical models of these systems may involve hundreds of variables in thousands of individual cells resulting in an extremely high-dimensional description of the system. This often contrasts with the low-dimensional dynamics exhibited on the collective or macroscopic scale for these systems. We introduce a macroscopic reduction for networks of coupled oscillators motivated by an elegant structure we find in experimental measurements of circadian protein expression and several mathematical models for coupled biological oscillators. The observed structure in the collective amplitude of the oscillator population differs from the well-known Ott-Antonsen ansatz, but its emergence can be characterized through a simple argument depending only on general phase-locking behavior in coupled oscillator systems. We further demonstrate its emergence in networks of noisy heterogeneous oscillators with complex network connectivity. Applying this structure, we derive low-dimensional macroscopic models for oscillator population activity. This approach allows for the incorporation of cellular-level experimental data into the macroscopic model whose parameters and variables can then be directly associated with tissue- or organism-level properties, thereby elucidating the core properties driving the collective behavior of the system.
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Affiliation(s)
- Kevin M. Hannay
- Department of Mathematics, Schreiner University, Kerrville, TX 78028, USA
- Corresponding author.
| | - Daniel B. Forger
- Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
- Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI 48109, USA
| | - Victoria Booth
- Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
- Department of Anesthesiology, University of Michigan, Ann Arbor, MI 48109, USA
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50
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Omel'chenko OE, Sebek M, Kiss IZ. Universal relations of local order parameters for partially synchronized oscillators. Phys Rev E 2018; 97:062207. [PMID: 30011585 DOI: 10.1103/physreve.97.062207] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2018] [Indexed: 11/07/2022]
Abstract
Interactions among discrete oscillatory units (e.g., cells) can result in partially synchronized states when some of the units exhibit phase locking and others phase slipping. Such states are typically characterized by a global order parameter that expresses the extent of synchrony in the system. Here we show that such states carry data-rich information of the system behavior, and a local order parameter analysis reveals universal relations through a semicircle representation. The universal relations are derived from thermodynamic limit analysis of a globally coupled Kuramoto-type phase oscillator model. The relations are confirmed with the partially synchronized states in numerical simulations with a model of circadian cells and in laboratory experiments with chemical oscillators. The application of the theory allows direct approximation of coupling strength, the natural frequency of oscillations, and the phase lag parameter without extensive nonlinear fits as well as a self-consistency check for presence of network interactions and higher harmonic components in the phase model.
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Affiliation(s)
| | - Michael Sebek
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
| | - István Z Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
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