1
|
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.
Collapse
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
| |
Collapse
|
2
|
Skardal PS, Xu C. Tiered synchronization in coupled oscillator populations with interaction delays and higher-order interactions. CHAOS (WOODBURY, N.Y.) 2022; 32:053120. [PMID: 35649978 DOI: 10.1063/5.0086305] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
We study synchronization in large populations of coupled phase oscillators with time delays and higher-order interactions. With each of these effects individually giving rise to bistability between incoherence and synchronization via subcriticality at the onset of synchronization and the development of a saddle node, we find that their combination yields another mechanism behind bistability, where supercriticality at onset may be maintained; instead, the formation of two saddle nodes creates tiered synchronization, i.e., bistability between a weakly synchronized state and a strongly synchronized state. We demonstrate these findings by first deriving the low dimensional dynamics of the system and examining the system bifurcations using a stability and steady-state analysis.
Collapse
Affiliation(s)
| | - Can Xu
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| |
Collapse
|
3
|
Ameli S, Karimian M, Shahbazi F. Time-delayed Kuramoto model in the Watts-Strogatz small-world networks. CHAOS (WOODBURY, N.Y.) 2021; 31:113125. [PMID: 34881592 DOI: 10.1063/5.0064022] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2021] [Accepted: 10/25/2021] [Indexed: 06/13/2023]
Abstract
We study the synchronization of small-world networks of identical coupled phase oscillators through the Kuramoto interaction and uniform time delay. For a given intrinsic frequency and coupling constant, we observe synchronization enhancement in a range of time delays and discontinuous transition from the partially synchronized state with defect patterns to a glassy phase, characterized by a distribution of randomly frozen phase-locked oscillators. By further increasing the time delay, this phase undergoes a discontinuous transition to another partially synchronized state. We found the bimodal frequency distributions and hysteresis loops as indicators of the discontinuous nature of these transitions. Moreover, we found the existence of Chimera states at the onset of transitions.
Collapse
Affiliation(s)
- Sara Ameli
- Max Plank Institute for Physics of Complex Systems, 01187 Dresden, Germany
| | - Maryam Karimian
- Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran
| | - Farhad Shahbazi
- Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran
| |
Collapse
|
4
|
Biga V, Hawley J, Soto X, Johns E, Han D, Bennett H, Adamson AD, Kursawe J, Glendinning P, Manning CS, Papalopulu N. A dynamic, spatially periodic, micro-pattern of HES5 underlies neurogenesis in the mouse spinal cord. Mol Syst Biol 2021; 17:e9902. [PMID: 34031978 PMCID: PMC8144840 DOI: 10.15252/msb.20209902] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2020] [Revised: 03/30/2021] [Accepted: 04/06/2021] [Indexed: 11/12/2022] Open
Abstract
Ultradian oscillations of HES Transcription Factors (TFs) at the single‐cell level enable cell state transitions. However, the tissue‐level organisation of HES5 dynamics in neurogenesis is unknown. Here, we analyse the expression of HES5 ex vivo in the developing mouse ventral spinal cord and identify microclusters of 4–6 cells with positively correlated HES5 level and ultradian dynamics. These microclusters are spatially periodic along the dorsoventral axis and temporally dynamic, alternating between high and low expression with a supra‐ultradian persistence time. We show that Notch signalling is required for temporal dynamics but not the spatial periodicity of HES5. Few Neurogenin 2 cells are observed per cluster, irrespective of high or low state, suggesting that the microcluster organisation of HES5 enables the stable selection of differentiating cells. Computational modelling predicts that different cell coupling strengths underlie the HES5 spatial patterns and rate of differentiation, which is consistent with comparison between the motoneuron and interneuron progenitor domains. Our work shows a previously unrecognised spatiotemporal organisation of neurogenesis, emergent at the tissue level from the synthesis of single‐cell dynamics.
Collapse
Affiliation(s)
- Veronica Biga
- Faculty of Biology Medicine and Health, The University of Manchester, Manchester, UK
| | - Joshua Hawley
- Faculty of Biology Medicine and Health, The University of Manchester, Manchester, UK
| | - Ximena Soto
- Faculty of Biology Medicine and Health, The University of Manchester, Manchester, UK
| | - Emma Johns
- Faculty of Biology Medicine and Health, The University of Manchester, Manchester, UK
| | - Daniel Han
- Department of Mathematics, School of Natural Sciences, Faculty of Science and Engineering, The University of Manchester, Manchester, UK
| | - Hayley Bennett
- Faculty of Biology Medicine and Health, The University of Manchester, Manchester, UK
| | - Antony D Adamson
- Faculty of Biology Medicine and Health, The University of Manchester, Manchester, UK
| | - Jochen Kursawe
- School of Mathematics and Statistics, University of St Andrews, St Andrews, UK
| | - Paul Glendinning
- Department of Mathematics, School of Natural Sciences, Faculty of Science and Engineering, The University of Manchester, Manchester, UK
| | - Cerys S Manning
- Faculty of Biology Medicine and Health, The University of Manchester, Manchester, UK
| | - Nancy Papalopulu
- Faculty of Biology Medicine and Health, The University of Manchester, Manchester, UK
| |
Collapse
|
5
|
Klinshov V, Franović I. Two scenarios for the onset and suppression of collective oscillations in heterogeneous populations of active rotators. Phys Rev E 2020; 100:062211. [PMID: 31962480 DOI: 10.1103/physreve.100.062211] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2019] [Indexed: 11/07/2022]
Abstract
We consider the macroscopic regimes and the scenarios for the onset and the suppression of collective oscillations in a heterogeneous population of active rotators composed of excitable or oscillatory elements. We analyze the system in the continuum limit within the framework of Ott-Antonsen reduction method, determining the states with a constant mean field and their stability boundaries in terms of the characteristics of the rotators' frequency distribution. The system is established to display three macroscopic regimes, namely the homogeneous stationary state, where all the units lie at the resting state, the global oscillatory state, characterized by the partially synchronized local oscillations, and the heterogeneous stationary state, which includes a mixture of resting and asynchronously oscillating units. The transitions between the characteristic domains are found to involve a complex bifurcation structure, organized around three codimension-two bifurcation points: a Bogdanov-Takens point, a cusp point, and a fold-homoclinic point. Apart from the monostable domains, our study also reveals two domains admitting bistable behavior, manifested as coexistence between the two stationary solutions or between a stationary and a periodic solution. It is shown that the collective mode may emerge via two generic scenarios, guided by a saddle-node of infinite period or the Hopf bifurcation, such that the transition from the homogeneous to the heterogeneous stationary state under increasing diversity may follow the classical paradigm, but may also be hysteretic. We demonstrate that the basic bifurcation structure holds qualitatively in the presence of small noise or small coupling delay, with the boundaries of the characteristic domains shifted compared to the noiseless and the delay-free case.
Collapse
Affiliation(s)
- Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov Street, 603950 Nizhny Novgorod, Russia
| | - Igor Franović
- Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| |
Collapse
|
6
|
Wu H, Kang L, Liu Z, Dhamala M. Exact explosive synchronization transitions in Kuramoto oscillators with time-delayed coupling. Sci Rep 2018; 8:15521. [PMID: 30341395 PMCID: PMC6195523 DOI: 10.1038/s41598-018-33845-6] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2018] [Accepted: 10/04/2018] [Indexed: 11/09/2022] Open
Abstract
Synchronization commonly occurs in many natural and man-made systems, from neurons in the brain to cardiac cells to power grids to Josephson junction arrays. Transitions to or out of synchrony for coupled oscillators depend on several factors, such as individual frequencies, coupling, interaction time delays and network structure-function relation. Here, using a generalized Kuramoto model of time-delay coupled phase oscillators with frequency-weighted coupling, we study the stability of incoherent and coherent states and the transitions to or out of explosive (abrupt, first-order like) phase synchronization. We analytically derive the exact formulas for the critical coupling strengths at different time delays in both directions of increasing (forward) and decreasing (backward) coupling strengths. We find that time-delay does not affect the transition for the backward direction but can shift the transition for the forward direction of increasing coupling strength. These results provide valuable insights into our understanding of dynamical mechanisms for explosive synchronization in presence of often unavoidable time delays present in many physical and biological systems.
Collapse
Affiliation(s)
- Hui Wu
- Department of Mathematical Sciences, Clark Atlanta University, Atlanta, GA 30314, USA
| | - Ling Kang
- Department of Physics, East China Normal University, Shanghai, 200062, China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai, 200062, China
| | - Mukesh Dhamala
- Department of Physics and Astronomy, Neuroscience Institute, Center for Behavioral Neuroscience, Georgia State and Georgia Tech Center for Advanced Brain Imaging, Georgia State University, Atlanta, GA 30303, USA.
| |
Collapse
|
7
|
Petkoski S, Palva JM, Jirsa VK. Phase-lags in large scale brain synchronization: Methodological considerations and in-silico analysis. PLoS Comput Biol 2018; 14:e1006160. [PMID: 29990339 PMCID: PMC6039010 DOI: 10.1371/journal.pcbi.1006160] [Citation(s) in RCA: 44] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2017] [Accepted: 04/29/2018] [Indexed: 01/24/2023] Open
Abstract
Architecture of phase relationships among neural oscillations is central for their functional significance but has remained theoretically poorly understood. We use phenomenological model of delay-coupled oscillators with increasing degree of topological complexity to identify underlying principles by which the spatio-temporal structure of the brain governs the phase lags between oscillatory activity at distant regions. Phase relations and their regions of stability are derived and numerically confirmed for two oscillators and for networks with randomly distributed or clustered bimodal delays, as a first approximation for the brain structural connectivity. Besides in-phase, clustered delays can induce anti-phase synchronization for certain frequencies, while the sign of the lags is determined by the natural frequencies and by the inhomogeneous network interactions. For in-phase synchronization faster oscillators always phase lead, while stronger connected nodes lag behind the weaker during frequency depression, which consistently arises for in-silico results. If nodes are in anti-phase regime, then a distance π is added to the in-phase trends. The statistics of the phases is calculated from the phase locking values (PLV), as in many empirical studies, and we scrutinize the method’s impact. The choice of surrogates do not affects the mean of the observed phase lags, but higher significance levels that are generated by some surrogates, cause decreased variance and might fail to detect the generally weaker coherence of the interhemispheric links. These links are also affected by the non-stationary and intermittent synchronization, which causes multimodal phase lags that can be misleading if averaged. Taken together, the results describe quantitatively the impact of the spatio-temporal connectivity of the brain to the synchronization patterns between brain regions, and to uncover mechanisms through which the spatio-temporal structure of the brain renders phases to be distributed around 0 and π. Trial registration: South African Clinical Trials Register: http://www.sanctr.gov.za/SAClinicalbrnbspTrials/tabid/169/Default.aspx, then link to respiratory tract then link to tuberculosis, pulmonary; and TASK Applied Sciences Clinical Trials, AP-TB-201-16 (ALOPEXX): https://task.org.za/clinical-trials/. Functional connectivity, and in particular, phase coupling between distant brain regions may be fundamental in regulating neuronal processing and communication. However, phase relationships between the nodes of the brain and how they are confined by its spatio-temporal structure, have been mostly overlooked. We use a model of oscillatory dynamics superimposed on the space-time structure defined by the connectome, and we analyze the possible regimes of synchronization. Limitations of data analysis are also considered and we show that the choice of the significance threshold for coherence does not essentially impact the statistics of the observed phase lags, although it is crucial for the right detection of statistically significant coherence. Analytical insights are obtained for networks with heterogeneous time-delays, based on the empirical data from the connectome, and these are confirmed by numerical simulations, which show in- or anti-phase synchronization depending on the frequency and the distribution of time-delays. Phase lags are shown to result from inhomogeneous network interactions, so that stronger connected nodes generally phase lag behind the weaker.
Collapse
Affiliation(s)
- Spase Petkoski
- Aix-Marseille Université, Inserm, INS UMR_S 1106, Marseille, France
- * E-mail: (SP); (VKJ)
| | - J. Matias Palva
- Neuroscience Center, Helsinki Institute of Life Science, University of Helsinki, Helsinki, Finland
| | - Viktor K. Jirsa
- Aix-Marseille Université, Inserm, INS UMR_S 1106, Marseille, France
- * E-mail: (SP); (VKJ)
| |
Collapse
|
8
|
Daffertshofer A, Ton R, Pietras B, Kringelbach ML, Deco G. Scale-freeness or partial synchronization in neural mass phase oscillator networks: Pick one of two? Neuroimage 2018; 180:428-441. [PMID: 29625237 DOI: 10.1016/j.neuroimage.2018.03.070] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2017] [Revised: 03/22/2018] [Accepted: 03/28/2018] [Indexed: 11/18/2022] Open
Abstract
Modeling and interpreting (partial) synchronous neural activity can be a challenge. We illustrate this by deriving the phase dynamics of two seminal neural mass models: the Wilson-Cowan firing rate model and the voltage-based Freeman model. We established that the phase dynamics of these models differed qualitatively due to an attractive coupling in the first and a repulsive coupling in the latter. Using empirical structural connectivity matrices, we determined that the two dynamics cover the functional connectivity observed in resting state activity. We further searched for two pivotal dynamical features that have been reported in many experimental studies: (1) a partial phase synchrony with a possibility of a transition towards either a desynchronized or a (fully) synchronized state; (2) long-term autocorrelations indicative of a scale-free temporal dynamics of phase synchronization. Only the Freeman phase model exhibited scale-free behavior. Its repulsive coupling, however, let the individual phases disperse and did not allow for a transition into a synchronized state. The Wilson-Cowan phase model, by contrast, could switch into a (partially) synchronized state, but it did not generate long-term correlations although being located close to the onset of synchronization, i.e. in its critical regime. That is, the phase-reduced models can display one of the two dynamical features, but not both.
Collapse
Affiliation(s)
- Andreas Daffertshofer
- Institute for Brain and Behavior Amsterdam & Amsterdam Movement Sciences, Faculty of Behavioural and Movement Sciences, Vrije Universiteit Amsterdam, van der Boechorststraat 9, 1081BT, Amsterdam, The Netherlands.
| | - Robert Ton
- Institute for Brain and Behavior Amsterdam & Amsterdam Movement Sciences, Faculty of Behavioural and Movement Sciences, Vrije Universiteit Amsterdam, van der Boechorststraat 9, 1081BT, Amsterdam, The Netherlands; Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Carrer Tanger 122-140, 08018, Barcelona, Spain
| | - Bastian Pietras
- Institute for Brain and Behavior Amsterdam & Amsterdam Movement Sciences, Faculty of Behavioural and Movement Sciences, Vrije Universiteit Amsterdam, van der Boechorststraat 9, 1081BT, Amsterdam, The Netherlands; Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
| | - Morten L Kringelbach
- University Department of Psychiatry, University of Oxford, Oxford, OX3 7JX, UK; Center for Music in the Brain, Department of Clinical Medicine, Aarhus University, Denmark
| | - Gustavo Deco
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Carrer Tanger 122-140, 08018, Barcelona, Spain; Institució Catalana de la Recerca i Estudis Avanats (ICREA), Universitat Pompeu Fabra, Carrer Tanger 122-140, 08018, Barcelona, Spain
| |
Collapse
|
9
|
Kim M, Noh JD. Time-Delay Induced Dimensional Crossover in the Voter Model. PHYSICAL REVIEW LETTERS 2017; 118:168302. [PMID: 28474930 DOI: 10.1103/physrevlett.118.168302] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2016] [Indexed: 06/07/2023]
Abstract
We investigate the ordering dynamics of the voter model with time-delayed interactions. The dynamical process in the d-dimensional lattice is shown to be equivalent to the first passage problem of a random walker in the (d+1)-dimensional strip of a finite width determined by the delay time. The equivalence reveals that the time delay leads to the dimensional crossover from the (d+1)-dimensional scaling behavior at a short time to the d-dimensional scaling behavior at a long time. The scaling property in both regimes and the crossover time scale are obtained analytically, which are confirmed with the numerical simulation results.
Collapse
Affiliation(s)
- Mina Kim
- Department of Physics, University of Seoul, Seoul 02504, Korea
| | - Jae Dong Noh
- Department of Physics, University of Seoul, Seoul 02504, Korea
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
| |
Collapse
|
10
|
Petkoski S, Spiegler A, Proix T, Aram P, Temprado JJ, Jirsa VK. Heterogeneity of time delays determines synchronization of coupled oscillators. Phys Rev E 2016; 94:012209. [PMID: 27575125 DOI: 10.1103/physreve.94.012209] [Citation(s) in RCA: 32] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2015] [Indexed: 05/01/2023]
Abstract
Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.
Collapse
Affiliation(s)
- Spase Petkoski
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
- Aix-Marseille Université, CNRS, ISM UMR 7287, 13288, Marseille, France
| | - Andreas Spiegler
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
| | - Timothée Proix
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
| | - Parham Aram
- Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield S10 2TN, United Kingdom
| | | | - Viktor K Jirsa
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
| |
Collapse
|
11
|
Pazó D, Montbrió E. From Quasiperiodic Partial Synchronization to Collective Chaos in Populations of Inhibitory Neurons with Delay. PHYSICAL REVIEW LETTERS 2016; 116:238101. [PMID: 27341262 DOI: 10.1103/physrevlett.116.238101] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2015] [Indexed: 06/06/2023]
Abstract
Collective chaos is shown to emerge, via a period-doubling cascade, from quasiperiodic partial synchronization in a population of identical inhibitory neurons with delayed global coupling. This system is thoroughly investigated by means of an exact model of the macroscopic dynamics, valid in the thermodynamic limit. The collective chaotic state is reproduced numerically with a finite population, and persists in the presence of weak heterogeneities. Finally, the relationship of the model's dynamics with fast neuronal oscillations is discussed.
Collapse
Affiliation(s)
- Diego Pazó
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
| | - Ernest Montbrió
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
| |
Collapse
|
12
|
Sadilek M, Thurner S. Physiologically motivated multiplex Kuramoto model describes phase diagram of cortical activity. Sci Rep 2015; 5:10015. [PMID: 25996547 PMCID: PMC4650820 DOI: 10.1038/srep10015] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2014] [Accepted: 03/18/2015] [Indexed: 12/04/2022] Open
Abstract
We derive a two-layer multiplex Kuramoto model from Wilson-Cowan type physiological equations that describe neural activity on a network of interconnected cortical regions. This is mathematically possible due to the existence of a unique, stable limit cycle, weak coupling, and inhibitory synaptic time delays. We study the phase diagram of this model numerically as a function of the inter-regional connection strength that is related to cerebral blood flow, and a phase shift parameter that is associated with synaptic GABA concentrations. We find three macroscopic phases of cortical activity: background activity (unsynchronized oscillations), epileptiform activity (highly synchronized oscillations) and resting-state activity (synchronized clusters/chaotic behaviour). Previous network models could hitherto not explain the existence of all three phases. We further observe a shift of the average oscillation frequency towards lower values together with the appearance of coherent slow oscillations at the transition from resting-state to epileptiform activity. This observation is fully in line with experimental data and could explain the influence of GABAergic drugs both on gamma oscillations and epileptic states. Compared to previous models for gamma oscillations and resting-state activity, the multiplex Kuramoto model not only provides a unifying framework, but also has a direct connection to measurable physiological parameters.
Collapse
Affiliation(s)
- Maximilian Sadilek
- Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria
| | - Stefan Thurner
- Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria
- Santa Fe Institute, 1399 Hyde Park Road, New Mexico 87501, USA
- IIASA, Schlossplatz 1, A-2361 Laxenburg, Austria
| |
Collapse
|
13
|
Barabash ML, Petkoski S, Stefanovska A. Homogeneous delays in the Kuramoto model with time-variable parameters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052903. [PMID: 25493854 DOI: 10.1103/physreve.90.052903] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2014] [Indexed: 06/04/2023]
Abstract
The Kuramoto model with time-varying parameters has been extended to consider the effect of delay in couplings. A collective dynamics arises from the interplay between the time scales of the original system, the external forcing, and the delays. This complex low-dimensional dynamics is described, uncovering an echo effect near the synchronization threshold. Hence, the delayed couplings substantially alter the dynamics of what is an open system and should be taken into consideration, depending on the ensemble's evolution time scale. We also introduce a first-harmonic approximation for the evolution of the mean field under harmonic forcing, valid for any delays and forcing of a coherent population.
Collapse
Affiliation(s)
- Miraslau L Barabash
- Department of Theoretical Physics and Astrophysics, Physics Faculty, Belarusian State University, Nezavisimosty Av., 4, 220030, Minsk, Belarus
| | - Spase Petkoski
- Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
| | - Aneta Stefanovska
- Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
| |
Collapse
|
14
|
Mather W, Hasty J, Tsimring LS. Synchronization of degrade-and-fire oscillations via a common activator. PHYSICAL REVIEW LETTERS 2014; 113:128102. [PMID: 25279645 PMCID: PMC4494757 DOI: 10.1103/physrevlett.113.128102] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2014] [Indexed: 06/03/2023]
Abstract
The development of synthetic gene oscillators has not only demonstrated our ability to forward engineer reliable circuits in living cells, but it has also proven to be an excellent testing ground for the statistical behavior of coupled noisy oscillators. Previous experimental studies demonstrated that a shared positive feedback can reliably synchronize such oscillators, though the theoretical mechanism was not studied in detail. In the present work, we examine an experimentally motivated stochastic model for coupled degrade-and-fire gene oscillators, where a core delayed negative feedback establishes oscillations within each cell, and a shared delayed positive feedback couples all cells. We use analytic and numerical techniques to investigate conditions for one cluster and multicluster synchrony. A nonzero delay in the shared positive feedback, as expected for the experimental systems, is found to be important for synchrony to occur.
Collapse
Affiliation(s)
- William Mather
- Department of Physics, Virginia Tech, 850 West Campus Drive, Blacksburg, Virginia 24061-0435, USA and Department of Biological Sciences, Virginia Tech, 1405 Perry Street, Blacksburg, Virginia 24061-0406, USA
| | - Jeff Hasty
- Department of Bioengineering, UCSD, 9500 Gilman Drive, La Jolla, California 92093-0412, USA and Molecular Biology Section, Division of Biology, UCSD, 9500 Gilman Drive, La Jolla, California 92093-0368, USA and BioCircuits Institute, UCSD, 9500 Gilman Drive, La Jolla, California 92093-0328, USA
| | - Lev S Tsimring
- BioCircuits Institute, UCSD, 9500 Gilman Drive, La Jolla, California 92093-0328, USA
| |
Collapse
|
15
|
Ton R, Deco G, Daffertshofer A. Structure-function discrepancy: inhomogeneity and delays in synchronized neural networks. PLoS Comput Biol 2014; 10:e1003736. [PMID: 25078715 PMCID: PMC4117423 DOI: 10.1371/journal.pcbi.1003736] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2013] [Accepted: 06/06/2014] [Indexed: 11/19/2022] Open
Abstract
The discrepancy between structural and functional connectivity in neural systems forms the challenge in understanding general brain functioning. To pinpoint a mapping between structure and function, we investigated the effects of (in)homogeneity in coupling structure and delays on synchronization behavior in networks of oscillatory neural masses by deriving the phase dynamics of these generic networks. For homogeneous delays, the structural coupling matrix is largely preserved in the coupling between phases, resulting in clustered stationary phase distributions. Accordingly, we found only a small number of synchronized groups in the network. Distributed delays, by contrast, introduce inhomogeneity in the phase coupling so that clustered stationary phase distributions no longer exist. The effect of distributed delays mimicked that of structural inhomogeneity. Hence, we argue that phase (de-)synchronization patterns caused by inhomogeneous coupling cannot be distinguished from those caused by distributed delays, at least not by the naked eye. The here-derived analytical expression for the effective coupling between phases as a function of structural coupling constitutes a direct relationship between structural and functional connectivity. Structural connectivity constrains synchronizability that may be modified by the delay distribution. This explains why structural and functional connectivity bear much resemblance albeit not a one-to-one correspondence. We illustrate this in the context of resting-state activity, using the anatomical connectivity structure reported by Hagmann and others.
Collapse
Affiliation(s)
- Robert Ton
- MOVE Research Institute Amsterdam, VU University, Amsterdam, The Netherlands
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
- * E-mail:
| | - Gustavo Deco
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
- Institució Catalana de la Recerca i Estudis Avançats (ICREA), Universitat Pompeu Fabra, Barcelona, Spain
| | | |
Collapse
|
16
|
Breathing synchronization in interconnected networks. Sci Rep 2013; 3:3289. [PMID: 24256765 PMCID: PMC3836035 DOI: 10.1038/srep03289] [Citation(s) in RCA: 45] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2013] [Accepted: 11/06/2013] [Indexed: 12/03/2022] Open
Abstract
Global synchronization in a complex network of oscillators emerges from the interplay between its topology and the dynamics of the pairwise interactions among its numerous components. When oscillators are spatially separated, however, a time delay appears in the interaction which might obstruct synchronization. Here we study the synchronization properties of interconnected networks of oscillators with a time delay between networks and analyze the dynamics as a function of the couplings and communication lag. We discover a new breathing synchronization regime, where two groups appear in each network synchronized at different frequencies. Each group has a counterpart in the opposite network, one group is in phase and the other in anti-phase with their counterpart. For strong couplings, instead, networks are internally synchronized but a phase shift between them might occur. The implications of our findings on several socio-technical and biological systems are discussed.
Collapse
|
17
|
Synchronization of Nonidentical Coupled Phase Oscillators in the Presence of Time Delay and Noise. ACTA ACUST UNITED AC 2013. [DOI: 10.1155/2013/591513] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
We have studied in this paper the dynamics of globally coupled phase oscillators having the Lorentzian
frequency distribution with zero mean in the presence of both time delay and noise. Noise may
be Gaussian or non-Gaussian in characteristics. In the limit of zero noise strength, we find that
the critical coupling strength (CCS) increases linearly as a function of time delay. Thus the role
of time delay in the dynamics for the deterministic system is qualitatively equivalent to the effect
of frequency fluctuations of the phase oscillators by additive white noise in absence of time delay.
But for the stochastic model, the critical coupling strength grows nonlinearly with the increase of
the time delay. The linear dependence of the critical coupling strength on the noise intensity also
changes to become nonlinear due to creation of additional phase difference among the oscillators
by the time delay. We find that the creation of phase difference plays an important role in the
dynamics of the system when the intrinsic correlation induced by the finite correlation time of the
noise is small. We also find that the critical coupling is higher for the non-Gaussian noise compared
to the Gaussian one due to higher effective noise strength.
Collapse
|
18
|
Nishikawa I, Tanaka G, Aihara K. Nonstandard scaling law of fluctuations in finite-size systems of globally coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:024102. [PMID: 24032967 DOI: 10.1103/physreve.88.024102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2013] [Indexed: 06/02/2023]
Abstract
Universal scaling laws form one of the central issues in physics. A nonstandard scaling law or a breakdown of a standard scaling law, on the other hand, can often lead to the finding of a new universality class in physical systems. Recently, we found that a statistical quantity related to fluctuations follows a nonstandard scaling law with respect to the system size in a synchronized state of globally coupled nonidentical phase oscillators [I. Nishikawa et al., Chaos 22, 013133 (2012)]. However, it is still unclear how widely this nonstandard scaling law is observed. In the present paper, we discuss the conditions required for the unusual scaling law in globally coupled oscillator systems and validate the conditions by numerical simulations of several different models.
Collapse
Affiliation(s)
- Isao Nishikawa
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
| | | | | |
Collapse
|
19
|
Ton R, Deco G, Daffertshofer A. The effects of time delays on synchronization properties in a network of neural mass models. BMC Neurosci 2013. [PMCID: PMC3704775 DOI: 10.1186/1471-2202-14-s1-p78] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022] Open
|
20
|
Zhdanov VP, Kasemo B. Synchronization of metabolic oscillations:two cells and ensembles of adsorbed cells. J Biol Phys 2013; 27:295-311. [PMID: 23345750 DOI: 10.1023/a:1014209410688] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We treat synchronization of metabolic oscillations in two cellsand in ensembles of cells adsorbed at the liquid-solid interface.(i) Synchronization of oscillations in two cells is assumedto occur via perturbation of the metabolite concentration nearone cell due to the metabolite diffusion flux from another cell.This direct channel of synchronization may be important ifthe distance between two cells is comparable with the cell diameter.The corresponding coupling coefficient is found to be proportionalto the metabolite diffusion coefficient and inversely proportionalto the cell radius and the distance between the cells.(ii) In the case of ensembles of adsorbed cells, synchronizationof oscillations is considered to be indirect, i.e., to occur viathe metabolite concentration formed outside the cells nearthe interface due to metabolite diffusion from the cells. We havederived a general integral equation relating the metaboliteconcentration near the interface with concentrations inside the cells.PACS: 82.37.-j, 82.40.Bj.
Collapse
|
21
|
Mier-Y-Teran-Romero L, Lindley B, Schwartz IB. Statistical multimoment bifurcations in random-delay coupled swarms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:056202. [PMID: 23214852 PMCID: PMC3845360 DOI: 10.1103/physreve.86.056202] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2012] [Indexed: 06/01/2023]
Abstract
We study the effects of discrete, randomly distributed time delays on the dynamics of a coupled system of self-propelling particles. Bifurcation analysis on a mean field approximation of the system reveals that the system possesses patterns with certain universal characteristics that depend on distinguished moments of the time delay distribution. Specifically, we show both theoretically and numerically that although bifurcations of simple patterns, such as translations, change stability only as a function of the first moment of the time delay distribution, more complex patterns arising from Hopf bifurcations depend on all of the moments.
Collapse
Affiliation(s)
- Luis Mier-Y-Teran-Romero
- U.S. Naval Research Laboratory, Code 6792, Nonlinear System Dynamics Section, Plasma Physics Division, Washington, DC 20375, USA
| | | | | |
Collapse
|
22
|
Peron TKD, Rodrigues FA. Explosive synchronization enhanced by time-delayed coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:016102. [PMID: 23005486 DOI: 10.1103/physreve.86.016102] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/07/2011] [Indexed: 06/01/2023]
Abstract
This paper deals with the emergence of explosive synchronization in scale-free networks by considering the Kuramoto model of coupled phase oscillators. The natural frequencies of oscillators are assumed to be correlated with their degrees, and a time delay is included in the system. This assumption allows enhancing the explosive transition to reach a synchronous state. We provide an analytical treatment developed in a star graph, which reproduces results obtained in scale-free networks. Our findings have important implications in understanding the synchronization of complex networks since the time delay is present in most real-world complex systems due to the finite speed of the signal transmission over a distance.
Collapse
Affiliation(s)
- Thomas Kauê Dal'Maso Peron
- Instituto de Física de São Carlos, Universidade de São Paulo, Avenida Trabalhador São Carlense 400, Caixa Postal 369, CEP 13560-970, São Carlos, São Paulo, Brazil
| | | |
Collapse
|
23
|
Lee WS, Restrepo JG, Ott E, Antonsen TM. Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times. CHAOS (WOODBURY, N.Y.) 2011; 21:023122. [PMID: 21721764 DOI: 10.1063/1.3596697] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We consider systems of many spatially distributed phase oscillators that interact with their neighbors. Each oscillator is allowed to have a different natural frequency, as well as a different response time to the signals it receives from other oscillators in its neighborhood. Using the ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)] and adopting a strategy similar to that employed in the recent work of Laing [Physica D 238, 1569 (2009)], we reduce the microscopic dynamics of these systems to a macroscopic partial-differential-equation description. Using this macroscopic formulation, we numerically find that finite oscillator response time leads to interesting spatiotemporal dynamical behaviors including propagating fronts, spots, target patterns, chimerae, spiral waves, etc., and we study interactions and evolutionary behaviors of these spatiotemporal patterns.
Collapse
Affiliation(s)
- Wai Shing Lee
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
| | | | | | | |
Collapse
|
24
|
Popovych OV, Krachkovskyi V, Tass PA. Phase-locking swallows in coupled oscillators with delayed feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:046203. [PMID: 21230361 DOI: 10.1103/physreve.82.046203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2010] [Indexed: 05/30/2023]
Abstract
We show that a nonlinear coupling with delayed feedback between two limit-cycle oscillators can lead to phase-locked, periodically modulated, and chaotic phase synchronization as well as to desynchronization. Parameter regions with stable phase-locked states attain the well-known form of the swallows or shrimps found and studied for nonlinear maps. We demonstrate that the swallow regions can be accompanied by a different bifurcation scenario where the periodic orbits of the phase-locked states undergo a torus bifurcation instead of a previously reported period-doubling bifurcation. This property has an impact on the spatial organization of the swallows in the parameter space. The swallow regions contribute to the synchronization domain of the considered system, and we analytically approximate the parameter synchronization threshold.
Collapse
Affiliation(s)
- Oleksandr V Popovych
- Institute of Neuroscience and Medicine-Neuromodulation (INM-7), Research Center Jülich, 52425 Jülich, Germany
| | | | | |
Collapse
|
25
|
Perlikowski P, Yanchuk S, Popovych OV, Tass PA. Periodic patterns in a ring of delay-coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:036208. [PMID: 21230162 DOI: 10.1103/physreve.82.036208] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2010] [Revised: 08/24/2010] [Indexed: 05/30/2023]
Abstract
We describe the appearance and stability of spatiotemporal periodic patterns (rotating waves) in unidirectional rings of coupled oscillators with delayed couplings. We show how delays in the coupling lead to the splitting of each rotating wave into several new ones. The appearance of rotating waves is mediated by the Hopf bifurcations of the symmetric equilibrium. We also conclude that the coupling delays can be effectively replaced by increasing the number of oscillators in the chain. The phenomena are shown for the Stuart-Landau oscillators as well as for the coupled FitzHugh-Nagumo systems modeling an ensemble of spiking neurons interacting via excitatory chemical synapses.
Collapse
Affiliation(s)
- P Perlikowski
- Institute of Mathematics, Humboldt University of Berlin, 10099 Berlin, Germany
| | | | | | | |
Collapse
|
26
|
Lee WS, Ott E, Antonsen TM. Large coupled oscillator systems with heterogeneous interaction delays. PHYSICAL REVIEW LETTERS 2009; 103:044101. [PMID: 19659358 DOI: 10.1103/physrevlett.103.044101] [Citation(s) in RCA: 58] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2009] [Indexed: 05/08/2023]
Abstract
In order to discover generic effects of heterogeneous communication delays on the dynamics of large systems of coupled oscillators, this Letter studies a modification of the Kuramoto model incorporating a distribution of interaction delays. Corresponding to the case of a large number N of oscillators, we consider the continuum limit (i.e., N --> infinity). By focusing attention on the reduced dynamics on an invariant manifold of the original system, we derive governing equations for the system which we use to study the stability of the incoherent states and the dynamical transitional behavior from stable incoherent states to stable coherent states. We find that spread in the distribution function of delays can greatly alter the system dynamics.
Collapse
Affiliation(s)
- Wai Shing Lee
- University of Maryland, College Park, Maryland 20742, USA
| | | | | |
Collapse
|
27
|
Ott E, Antonsen TM. Long time evolution of phase oscillator systems. CHAOS (WOODBURY, N.Y.) 2009; 19:023117. [PMID: 19566252 DOI: 10.1063/1.3136851] [Citation(s) in RCA: 237] [Impact Index Per Article: 15.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
It is shown, under weak conditions, that the dynamical evolution of large systems of globally coupled phase oscillators with Lorentzian distributed oscillation frequencies is, in an appropriate physical sense, time-asymptotically attracted toward a reduced manifold of the system states. This manifold was previously known and used to facilitate the discovery of attractors and bifurcations of such systems. The result of this paper establishes that attractors for the order parameter dynamics obtained by restriction to this reduced manifold are, in fact, the only such attractors of the full system. Thus all long time dynamical behaviors of the order parameters of these systems can be obtained by restriction to the reduced manifold.
Collapse
Affiliation(s)
- Edward Ott
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
| | | |
Collapse
|
28
|
Ambika G, Amritkar RE. Anticipatory synchronization with variable time delay and reset. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:056206. [PMID: 19518535 DOI: 10.1103/physreve.79.056206] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2008] [Revised: 02/17/2009] [Indexed: 05/27/2023]
Abstract
A method to synchronize two chaotic systems with anticipation or lag, coupled in the drive response mode, is proposed. The coupling involves variable delay with three time scales. The method has the advantage that synchronization is realized with intermittent information about the driving system at intervals fixed by a reset time. The stability of the synchronization manifold is analyzed with the resulting discrete error dynamics. The numerical calculations in standard systems such as the Rössler and Lorenz systems are used to demonstrate the method and the results of the analysis.
Collapse
Affiliation(s)
- G Ambika
- Indian Institute of Science Education and Research, Pune 411 021, India
| | | |
Collapse
|
29
|
Ermentrout B, Ko TW. Delays and weakly coupled neuronal oscillators. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2009; 367:1097-115. [PMID: 19218153 DOI: 10.1098/rsta.2008.0259] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
We use weakly coupled oscillator theory to study the effects of delays on coupled systems of neuronal oscillators. We explore, first, simple pairs with constant delays and then examine the role of distributed delays as would occur in systems with dendritic branches or in networks where there is a distance-dependent conductance delay. In the latter, we use mean field theory to show the emergence of travelling waves and the loss of synchronization. Next, we consider phase models with stronger coupling and delays in the state variables. We show that they have a richer dynamics but one that is still similar to the weakly coupled case.
Collapse
Affiliation(s)
- Bard Ermentrout
- Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
| | | |
Collapse
|
30
|
Ott E, Antonsen TM. Low dimensional behavior of large systems of globally coupled oscillators. CHAOS (WOODBURY, N.Y.) 2008; 18:037113. [PMID: 19045487 DOI: 10.1063/1.2930766] [Citation(s) in RCA: 471] [Impact Index Per Article: 29.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.
Collapse
Affiliation(s)
- Edward Ott
- University of Maryland, College Park, Maryland 20742, USA
| | | |
Collapse
|
31
|
Ko TW, Ermentrout GB. Effects of axonal time delay on synchronization and wave formation in sparsely coupled neuronal oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:056206. [PMID: 18233738 DOI: 10.1103/physreve.76.056206] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2007] [Revised: 06/29/2007] [Indexed: 05/25/2023]
Abstract
We investigate the effects of axonal time delay when the neuronal oscillators are coupled by sparse and random connections. Using phase-reduced models with general coupling functions, we show that a small fraction of connections with time delay can destabilize synchronous states and induce near-regular wave states. An order parameter is introduced to characterize those states. We analyze the systems using mean-field-type approximation.
Collapse
Affiliation(s)
- Tae-Wook Ko
- Department of Mathematics, University of Pittsburgh, Pennsylvania 15260, USA.
| | | |
Collapse
|
32
|
Wood K, Van den Broeck C, Kawai R, Lindenberg K. Continuous and discontinuous phase transitions and partial synchronization in stochastic three-state oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041132. [PMID: 17994961 DOI: 10.1103/physreve.76.041132] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2007] [Indexed: 05/25/2023]
Abstract
We investigate both continuous (second-order) and discontinuous (first-order) transitions to macroscopic synchronization within a single class of discrete, stochastic (globally) phase-coupled oscillators. We provide analytical and numerical evidence that the continuity of the transition depends on the coupling coefficients and, in some nonuniform populations, on the degree of quenched disorder. Hence, in a relatively simple setting this class of models exhibits the qualitative behaviors characteristic of a variety of considerably more complicated models. In addition, we study the microscopic basis of synchronization above threshold and detail the counterintuitive subtleties relating measurements of time-averaged frequencies and mean-field oscillations. Most notably, we observe a state of suprathreshold partial synchronization in which time-averaged frequency measurements from individual oscillators do not correspond to the frequency of macroscopic oscillations observed in the population.
Collapse
Affiliation(s)
- Kevin Wood
- Department of Chemistry and Biochemistry and Institute for Nonlinear Science, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093-0340, USA
| | | | | | | |
Collapse
|
33
|
Chen Z, Lin W, Zhou J. Complete and generalized synchronization in a class of noise perturbed chaotic systems. CHAOS (WOODBURY, N.Y.) 2007; 17:023106. [PMID: 17614660 DOI: 10.1063/1.2718491] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
In the paper, in light of the LaSalle-type invariance principle for stochastic differential equations, chaos synchronization is investigated for a class of chaotic systems dissatisfying a globally Lipschitz condition with noise perturbation. Sufficient criteria for both complete synchronization and generalized synchronization are rigorously established and thus successfully applied to realize chaos synchronization in the coupled unified chaotic systems. Furthermore, concrete examples as well as their numerical simulations are provided to illustrate the possible application of the established criteria.
Collapse
Affiliation(s)
- Zhang Chen
- Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Research Center for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
| | | | | |
Collapse
|
34
|
Montbrió E, Pazó D, Schmidt J. Time delay in the Kuramoto model with bimodal frequency distribution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:056201. [PMID: 17279978 DOI: 10.1103/physreve.74.056201] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2006] [Revised: 08/09/2006] [Indexed: 05/08/2023]
Abstract
We investigate the effects of a time-delayed all-to-all coupling scheme in a large population of oscillators with natural frequencies following a bimodal distribution. The regions of parameter space corresponding to synchronized and incoherent solutions are obtained both numerically and analytically for particular frequency distributions. In particular, we find that bimodality introduces a new time scale that results in a quasiperiodic disposition of the regions of incoherence.
Collapse
Affiliation(s)
- Ernest Montbrió
- Computational Neuroscience, Technology Department, Universitat Pompeu Fabra, 08003 Barcelona, Spain
| | | | | |
Collapse
|
35
|
Lin W, Chen G. Using white noise to enhance synchronization of coupled chaotic systems. CHAOS (WOODBURY, N.Y.) 2006; 16:013134. [PMID: 16599765 DOI: 10.1063/1.2183734] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
In the paper, complete synchronization of two chaotic oscillators via unidirectional coupling determined by white noise distribution is investigated. It is analytically proved that chaos synchronization could be achieved with probability one merely via white-noise-based coupling. The established theoretical result supports the observation of an interesting phenomenon that a certain kind of white noise could enhance chaos synchronization between two chaotic oscillators. Furthermore, numerical examples are provided to illustrate some possible applications of the theoretical result.
Collapse
Affiliation(s)
- Wei Lin
- Research Center and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Shanghai 200433, People's Republic of China.
| | | |
Collapse
|
36
|
Lin W, He Y. Complete synchronization of the noise-perturbed Chua's circuits. CHAOS (WOODBURY, N.Y.) 2005; 15:23705. [PMID: 16035895 DOI: 10.1063/1.1938627] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
In this paper, complete synchronization between unidirectionally coupled Chua's circuits within stochastic perturbation is investigated. Sufficient conditions of complete synchronization between these noise-perturbed circuits are established by means of the so-called LaSalle-type invariance principle for stochastic differential equations. Specific examples and their numerical simulations are also provided to demonstrate the feasibility of these conditions. Furthermore, the results obtained for the coupled Chua's circuits are further generalized to the wide class of coupled systems within stochastic perturbation.
Collapse
Affiliation(s)
- Wei Lin
- Research Center and Laboratory of Mathematics for Nonlinear Science, School of Mathematical Sciences, Fudan University, Shanghai 200433, People's Republic of China.
| | | |
Collapse
|
37
|
Ko TW, Jeong SO, Moon HT. Wave formation by time delays in randomly coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:056106. [PMID: 15244882 DOI: 10.1103/physreve.69.056106] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2003] [Indexed: 05/24/2023]
Abstract
We study the dynamics of randomly coupled oscillators when interactions between oscillators are time delayed due to the finite and constant speed of coupling signals. Numerical simulations show that the time delays, proportional to the Euclidean distances between interacting oscillators, can induce near regular waves in addition to near in-phase synchronous oscillations even though oscillators are randomly coupled. We discuss the stability conditions for the wave states and the in-phase synchronous states.
Collapse
Affiliation(s)
- Tae-Wook Ko
- Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea.
| | | | | |
Collapse
|
38
|
Chik DTW, Wang ZD. Postinhibitory rebound delay and weak synchronization in Hodgkin-Huxley neuronal networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:031907. [PMID: 14524803 DOI: 10.1103/physreve.68.031907] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2003] [Indexed: 05/24/2023]
Abstract
Noise-induced weak synchronized oscillatory activities in a globally inhibitory coupled Hodgkin-Huxley neuronal network are studied numerically. A kind of intrinsic delay induced by the postinhibitory rebound is observed and is found to be important in determining the overall frequency of the network. Synchronization occurs in an optimal range of noise intensity with a bell-shaped curve when the inhibitory coupling strength is sufficiently strong. Comparisons with the results for the excitatory coupling are also addressed.
Collapse
Affiliation(s)
- David T W Chik
- Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
| | | |
Collapse
|
39
|
Earl MG, Strogatz SH. Synchronization in oscillator networks with delayed coupling: a stability criterion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:036204. [PMID: 12689150 DOI: 10.1103/physreve.67.036204] [Citation(s) in RCA: 74] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2002] [Revised: 12/04/2002] [Indexed: 05/24/2023]
Abstract
We derive a stability criterion for the synchronous state in networks of identical phase oscillators with delayed coupling. The criterion applies to any network (whether regular or random, low dimensional or high dimensional, directed or undirected) in which each oscillator receives delayed signals from k others, where k is uniform for all oscillators.
Collapse
Affiliation(s)
- Matthew G Earl
- Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, New York 14853-1503, USA.
| | | |
Collapse
|
40
|
Pradhan GR, Chatterjee N, Gupte N. Mode locking of spatiotemporally periodic orbits in coupled sine circle map lattices. PHYSICAL REVIEW E 2002; 65:046227. [PMID: 12005996 DOI: 10.1103/physreve.65.046227] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2001] [Indexed: 11/07/2022]
Abstract
We study the organization of mode-locked intervals corresponding to the stable spatiotemporally periodic solutions in a lattice of diffusively coupled sine circle maps with periodic boundary conditions. Spatially periodic initial conditions settle down to spatiotemporally periodic solutions over large regions of the parameter space. In the case of synchronized solutions resulting from synchronized initial conditions, the mode-locked intervals have been seen to follow strict Farey ordering in the temporal periods. However, the nature of the organization of the mode-locked intervals corresponding to higher spatiotemporal periods is highly dependent on initial conditions and on system parameters. Farey ordering in the temporal periods is seen at low coupling for mode-locked intervals of all spatial periods. On the other hand, stable spatial period two solutions show an interesting reversal of Farey ordering at high values of coupling. Other spatially periodic solutions show a complete departure from Farey ordering at high coupling. We also examine the issue of completeness of the mode-locked intervals via a calculation of the fractal dimension of the complement of the mode-locked intervals as a function of the coupling epsilon and the nonlinearity parameter K. Our results are consistent with completeness over a range of values for these parameters. Spatiotemporally periodic solutions of the traveling wave type have their own organization in the parameter space. Novel bifurcations to other types of solutions are seen in the mode-locked intervals. We discuss various features of these bifurcations. We also define a set of new variables using which an analytic treatment of the bifurcations along the Omega=0 line is carried out.
Collapse
Affiliation(s)
- Gauri R Pradhan
- Department of Physics, University of Pune, Pune 411007, India.
| | | | | |
Collapse
|
41
|
Hong H, Jeon GS, Choi MY. Spontaneous phase oscillation induced by inertia and time delay. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:026208. [PMID: 11863633 DOI: 10.1103/physreve.65.026208] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2001] [Indexed: 05/23/2023]
Abstract
We consider a system of coupled oscillators with finite inertia and time-delayed interaction, and investigate the interplay between inertia and delay both analytically and numerically. The phase velocity of the system is examined; revealed in numerical simulations is the emergence of spontaneous phase oscillation without external driving, which turns out to be in good agreement with analytical results derived in the strong-coupling limit. Such self-oscillation is found to suppress synchronization, and its frequency is observed to decrease with inertia and delay. We obtain the phase diagram, which displays oscillatory and stationary phases in the appropriate regions of the parameters.
Collapse
Affiliation(s)
- H Hong
- School of Physics and Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea
| | | | | |
Collapse
|
42
|
Influence of time delayed global feedback on pattern formation in oscillatory CO oxidation on Pt(1 1 0). Chem Phys Lett 2001. [DOI: 10.1016/s0009-2614(01)00936-8] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
|
43
|
Kozyreff G, Vladimirov AG, Mandel P. Dynamics of a semiconductor laser array with delayed global coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:016613. [PMID: 11461434 DOI: 10.1103/physreve.64.016613] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2001] [Indexed: 05/23/2023]
Abstract
We study the dynamics of an array of single mode semiconductor lasers globally but weakly coupled by a common external feedback mirror and by nearest neighbor interactions. We seek to determine the conditions under which all lasers of the array are in phase, whether in a steady, periodic, quasiperiodic, or chaotic regime, in order to maximize the output far field intensity. We show that the delay may be a useful control parameter to achieve in-phase synchronization. For the in-phase steady state, there is a competition between a delay-induced Hopf bifurcation leading to an in-phase periodic regime and a delay-independent Hopf bifurcation leading to an antiphased periodic regime. Both regimes are described analytically and secondary Hopf bifurcations to quasiperiodic solutions are found. Close to the stable steady state, the array is described by a set of Kuramoto equations for the phases of the fields. Above the first Hopf bifurcation, these equations are generalized by the addition of second and third order time derivatives of the phases.
Collapse
Affiliation(s)
- G Kozyreff
- Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine CP 231, B-1050 Bruxelles, Belgium
| | | | | |
Collapse
|
44
|
Zhdanov VP, Kasemo B. Coupled catalytic oscillators: Beyond the mass-action law. CHAOS (WOODBURY, N.Y.) 2001; 11:335-343. [PMID: 12779467 DOI: 10.1063/1.1368129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We present Monte Carlo simulations of the reaction kinetics corresponding to two coupled catalytic oscillators in the case when oscillations result from the interplay between the reaction steps and adsorbate-induced surface restructuring. The model used is aimed to mimic oscillations on a single nm catalyst particle with two kinds of facets or on two catalyst particles on a support. Specifically, we treat the NO reduction by H(2) on a composite catalyst containing two catalytically active Pt(100) parts connected by an inactive link. The catalyst is represented by a rectangular fragment of a square lattice. The left- and right-hand parts of the lattice mimic Pt(100). With an appropriate choice of the model parameters, these sublattices play a role of catalytic oscillators. The central catalytically inactive sublattice is considered to be able only to adsorb NO reversibly and can be viewed as a Pt(111) facet or a support. The interplay of the reactions running on the catalytically active areas occurs via NO diffusion over the boundaries between the sublattices. Using this model, we show that the coupling of the catalytically active sublattices may synchronize nearly harmonic oscillations observed on these sublattices and also may result in the appearance of aperiodic partly synchronized oscillations. The spatio-temporal patterns corresponding to these regimes are nontrivial. In particular, the model predicts that, due to phase separation, the reaction may be accompanied by the formation of narrow NO-covered zones on the left and right sublattices near the boundaries between these sublattices and the central sublattice. Such patterns cannot be obtained by using the conventional mean-field reaction-diffusion equations based on the mass-action law. The experimental opportunities to observe the predicted phenomena are briefly discussed. (c) 2001 American Institute of Physics.
Collapse
Affiliation(s)
- V. P. Zhdanov
- Department of Applied Physics, Chalmers University of Technology, S-412 96 Goteborg, SwedenBoreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia
| | | |
Collapse
|
45
|
Chatterjee N, Gupte N. Analysis of spatiotemporally periodic behavior in lattices of coupled piecewise monotonic maps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:017202. [PMID: 11304393 DOI: 10.1103/physreve.63.017202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2000] [Indexed: 05/23/2023]
Abstract
We study the stability of spatiotemporally periodic orbits in 1-d lattices of piecewise monotonic maps coupled via translationally invariant coupling and periodic boundary conditions. States of such systems have independent spatial and temporal periodicities and their stability can be studied through the analysis of a single, uniquely identified reduced matrix of size kxk when the system size is MxM, for M=kN, a multiple of k. This result applies for arbitrary temporal periods and is valid for all coupled map lattice systems coupled in a translationally invariant manner with stability matrices which are irreducible and non-negative, as in the present case. Our analysis could be useful in the analysis of stability regions and bifurcation behavior in a variety of spatially extended systems.
Collapse
Affiliation(s)
- N Chatterjee
- Department of Physics and Astronomy, Condensed Matter and Surface Sciences Program, Ohio University, Athens, OH 45701, USA.
| | | |
Collapse
|
46
|
Zanette DH. Propagating structures in globally coupled systems with time delays. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:3167-3172. [PMID: 11088811 DOI: 10.1103/physreve.62.3167] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2000] [Revised: 03/31/2000] [Indexed: 05/23/2023]
Abstract
We consider an ensemble of globally coupled phase oscillators whose interaction is transmitted at finite speed. This introduces time delays, which make the spatial coordinates relevant in spite of the infinite range of the interaction. In the limit of short delays, we show that the ensemble approaches a state of frequency synchronization, where all the oscillators have the same frequency, and can develop a nontrivial distribution of phases over space. Numerical calculations on one-dimensional arrays with periodic boundary conditions reveal that, in such geometry, the phase distribution is a propagating structure.
Collapse
Affiliation(s)
- DH Zanette
- Consejo Nacional de Investigaciones Cientificas y Tecnicas, Centro Atomico Bariloche and Instituto Balseiro, 8400 Bariloche, Argentina
| |
Collapse
|