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Barać U, Perc M, Gosak M. Determinants of collective failure in excitable networks. CHAOS (WOODBURY, N.Y.) 2023; 33:043120. [PMID: 37097938 DOI: 10.1063/5.0149578] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2023] [Accepted: 03/27/2023] [Indexed: 06/19/2023]
Abstract
We study collective failures in biologically realistic networks that consist of coupled excitable units. The networks have broad-scale degree distribution, high modularity, and small-world properties, while the excitable dynamics is determined by the paradigmatic FitzHugh-Nagumo model. We consider different coupling strengths, bifurcation distances, and various aging scenarios as potential culprits of collective failure. We find that for intermediate coupling strengths, the network remains globally active the longest if the high-degree nodes are first targets for inactivation. This agrees well with previously published results, which showed that oscillatory networks can be highly fragile to the targeted inactivation of low-degree nodes, especially under weak coupling. However, we also show that the most efficient strategy to enact collective failure does not only non-monotonically depend on the coupling strength, but it also depends on the distance from the bifurcation point to the oscillatory behavior of individual excitable units. Altogether, we provide a comprehensive account of determinants of collective failure in excitable networks, and we hope this will prove useful for better understanding breakdowns in systems that are subject to such dynamics.
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Affiliation(s)
- Uroš Barać
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
| | - Matjaž Perc
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
- Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia
- Department of Medical Research, China Medical University Hospital, China Medical University, 40447 Taichung, Taiwan
- Complexity Science Hub Vienna, Josefstädterstrasse 39, 1080 Vienna, Austria
- Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, 02447 Seoul, Republic of Korea
| | - Marko Gosak
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
- Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia
- Institute of Physiology, Faculty of Medicine, University of Maribor, Taborska ulica 8, 2000 Maribor, Slovenia
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2
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Bandyopadhyay B, Banerjee T. Aging transition in coupled quantum oscillators. Phys Rev E 2023; 107:024204. [PMID: 36932509 DOI: 10.1103/physreve.107.024204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2022] [Accepted: 01/27/2023] [Indexed: 06/18/2023]
Abstract
Aging transition is an emergent behavior observed in networks consisting of active (self-oscillatory) and inactive (non-self-oscillatory) nodes, where the network transits from a global oscillatory state to an oscillation collapsed state when the fraction of inactive oscillators surpasses a critical value. However, the aging transition in quantum domain has not been studied yet. In this paper we investigate the quantum manifestation of aging transition in a network of active-inactive quantum oscillators. We show that, unlike classical case, the quantum aging is not characterized by a complete collapse of oscillation but by sufficient reduction in the mean boson number. We identify a critical "knee" value in the fraction of inactive oscillators around which quantum aging occurs in two different ways. Further, in stark contrast to the classical case, quantum aging transition depends upon the nonlinear damping parameter. We also explain the underlying processes leading to quantum aging that have no counterpart in the classical domain.
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Affiliation(s)
- Biswabibek Bandyopadhyay
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan 713 104, West Bengal, India
| | - Tanmoy Banerjee
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan 713 104, West Bengal, India
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3
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Li Y, Shi J, Aihara K. Mean-field analysis of Stuart-Landau oscillator networks with symmetric coupling and dynamical noise. CHAOS (WOODBURY, N.Y.) 2022; 32:063114. [PMID: 35778116 DOI: 10.1063/5.0081295] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2021] [Accepted: 05/09/2022] [Indexed: 06/15/2023]
Abstract
This paper presents analyses of networks composed of homogeneous Stuart-Landau oscillators with symmetric linear coupling and dynamical Gaussian noise. With a simple mean-field approximation, the original system is transformed into a surrogate system that describes uncorrelated oscillation/fluctuation modes of the original system. The steady-state probability distribution for these modes is described using an exponential family, and the dynamics of the system are mainly determined by the eigenvalue spectrum of the coupling matrix and the noise level. The variances of the modes can be expressed as functions of the eigenvalues and noise level, yielding the relation between the covariance matrix and the coupling matrix of the oscillators. With decreasing noise, the leading mode changes from fluctuation to oscillation, generating apparent synchrony of the coupled oscillators, and the condition for such a transition is derived. Finally, the approximate analyses are examined via numerical simulation of the oscillator networks with weak coupling to verify the utility of the approximation in outlining the basic properties of the considered coupled oscillator networks. These results are potentially useful for the modeling and analysis of indirectly measured data of neurodynamics, e.g., via functional magnetic resonance imaging and electroencephalography, as a counterpart of the frequently used Ising model.
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Affiliation(s)
- Yang Li
- International Research Center for Neurointelligence (WPI-IRCN), The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Jifan Shi
- International Research Center for Neurointelligence (WPI-IRCN), The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Kazuyuki Aihara
- International Research Center for Neurointelligence (WPI-IRCN), The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
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Sharma A, Rakshit B. Enhancement of dynamical robustness in a mean-field coupled network through self-feedback delay. CHAOS (WOODBURY, N.Y.) 2021; 31:013114. [PMID: 33754750 DOI: 10.1063/5.0015821] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2020] [Accepted: 12/16/2020] [Indexed: 06/12/2023]
Abstract
The network of self-sustained oscillators plays an important role in exploring complex phenomena in many areas of science and technology. The aging of an oscillator is referred to as turning non-oscillatory due to some local perturbations that might have adverse effects in macroscopic dynamical activities of a network. In this article, we propose an efficient technique to enhance the dynamical activities for a network of coupled oscillators experiencing aging transition. In particular, we present a control mechanism based on delayed negative self-feedback, which can effectively enhance dynamical robustness in a mean-field coupled network of active and inactive oscillators. Even for a small value of delay, robustness gets enhanced to a significant level. In our proposed scheme, the enhancing effect is more pronounced for strong coupling. To our surprise even if all the oscillators perturbed to equilibrium mode were delayed negative self-feedback is able to restore oscillatory activities in the network for strong coupling strength. We demonstrate that our proposed mechanism is independent of coupling topology. For a globally coupled network, we provide numerical and analytical treatment to verify our claim. To show that our scheme is independent of network topology, we also provide numerical results for the local mean-field coupled complex network. Also, for global coupling to establish the generality of our scheme, we validate our results for both Stuart-Landau limit cycle oscillators and chaotic Rössler oscillators.
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Affiliation(s)
- Amit Sharma
- Department of Physics, Central University of Rajasthan, Ajmer 305 817, India
| | - Biswambhar Rakshit
- Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641 112, India
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Dixit S, Asir M P, Dev Shrimali M. Aging in global networks with competing attractive-Repulsive interaction. CHAOS (WOODBURY, N.Y.) 2020; 30:123112. [PMID: 33380009 DOI: 10.1063/5.0026968] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2020] [Accepted: 11/10/2020] [Indexed: 06/12/2023]
Abstract
We study the dynamical inactivity of the global network of identical oscillators in the presence of mixed attractive and repulsive coupling. We consider that the oscillators are a priori in all to all attractive coupling and then upon increasing the number of oscillators interacting via repulsive interaction, the whole network attains a steady state at a critical fraction of repulsive nodes, pc. The macroscopic inactivity of the network is found to follow a typical aging transition due to competition between attractive-repulsive interactions. The analytical expression connecting the coupling strength and pc is deduced and corroborated with numerical outcomes. We also study the influence of asymmetry in the attractive-repulsive interaction, which leads to symmetry breaking. We detect chimera-like and mixed states for a certain ratio of coupling strengths. We have verified sequential and random modes to choose the repulsive nodes and found that the results are in agreement. The paradigmatic networks with diverse dynamics, viz., limit cycle (Stuart-Landau), chaos (Rössler), and bursting (Hindmarsh-Rose neuron), are analyzed.
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Affiliation(s)
- Shiva Dixit
- Department of Physics, Central University of Rajasthan, NH-8, Bandar Sindri, Ajmer 305 817, India
| | - Paul Asir M
- Department of Physics, Central University of Rajasthan, NH-8, Bandar Sindri, Ajmer 305 817, India
| | - Manish Dev Shrimali
- Department of Physics, Central University of Rajasthan, NH-8, Bandar Sindri, Ajmer 305 817, India
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Rakshit B, Rajendrakumar N, Balaram B. Abnormal route to aging transition in a network of coupled oscillators. CHAOS (WOODBURY, N.Y.) 2020; 30:101101. [PMID: 33138470 DOI: 10.1063/5.0022499] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2020] [Accepted: 09/17/2020] [Indexed: 06/11/2023]
Abstract
In this article, we investigate the dynamical robustness in a network of Van der Pol oscillators. In particular, we consider a network of diffusively coupled Van der Pol oscillators to explore the aging transition phenomena. Our investigation reveals that the route to aging transition in a network of Van der Pol oscillator is different from that of typical sinusoidal oscillators such as Stuart-Landau oscillators. Unlike sinusoidal oscillators, the order parameter does not follow smooth second-order phase transition. Rather, we observe an abnormal phase transition of the order parameter due to the sudden appearance of unbounded trajectories at a critical point. We provide detailed bifurcation analysis of such an abnormal phase transition. We show that the boundary crisis of a limit-cycle oscillator is at the helm of such an unusual discontinuous path of aging transition.
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Affiliation(s)
- Biswambhar Rakshit
- Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641 112, India
| | - Niveditha Rajendrakumar
- Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
| | - Bipin Balaram
- Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641 112, India
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Huang K, Sorrentino F, Hossein-Zadeh M. Experimental observations of synchronization between two bidirectionally coupled physically dissimilar oscillators. Phys Rev E 2020; 102:042215. [PMID: 33212708 DOI: 10.1103/physreve.102.042215] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2020] [Accepted: 09/10/2020] [Indexed: 06/11/2023]
Abstract
We experimentally study the complex dynamics of two mutually coupled physically dissimilar oscillators with two different kinds of coupling mechanisms. Specifically, an optoelectronic oscillator is coupled to a Colpitts oscillator via optical power and the Colpitts oscillator is coupled back to the optoelectronic oscillator via electric voltage. We investigate and characterize phase synchronization and generalized chaos synchronization in this coupled system. Phase synchronization is observed when both oscillators are preset to oscillate periodically prior to coupling while generalized chaos synchronization is observed when both oscillators are preset to oscillate chaotically prior to coupling. In the periodical oscillation regime, we observe a linear relationship between the strengths of the two unidirectional coupling factors at which the system transitions to a synchronized state. In the chaotic regime, we observe a transition from hyperchaos to chaos associated with the onset of generalized synchronization.
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Affiliation(s)
- Ke Huang
- Center for High Technology Materials, The University of New Mexico, 1313 Goddard Street SE, Albuquerque, New Mexico 87106, USA
| | - Francesco Sorrentino
- Department of Mechanical Engineering, The University of New Mexico, Albuquerque, New Mexico 87131, USA
| | - Mani Hossein-Zadeh
- Center for High Technology Materials, The University of New Mexico, 1313 Goddard Street SE, Albuquerque, New Mexico 87106, USA
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Sathiyadevi K, Gowthaman I, Senthilkumar DV, Chandrasekar VK. Aging transition in the absence of inactive oscillators. CHAOS (WOODBURY, N.Y.) 2019; 29:123117. [PMID: 31893654 DOI: 10.1063/1.5121565] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2019] [Accepted: 12/02/2019] [Indexed: 06/10/2023]
Abstract
The role of counter-rotating oscillators in an ensemble of coexisting co- and counter-rotating oscillators is examined by increasing the proportion of the latter. The phenomenon of aging transition was identified at a critical value of the ratio of the counter-rotating oscillators, which was otherwise realized only by increasing the number of inactive oscillators to a large extent. The effect of the mean-field feedback strength in the symmetry preserving coupling is also explored. The parameter space of aging transition was increased abruptly even for a feeble decrease in the feedback strength, and, subsequently, aging transition was observed at a critical value of the feedback strength surprisingly without any counter-rotating oscillators. Further, the study was extended to symmetry breaking coupling using conjugate variables, and it was observed that the symmetry breaking coupling can facilitate the onset of aging transition even in the absence of counter-rotating oscillators and for the unit value of the feedback strength. In general, the parameter space of aging transition was found to increase by increasing the frequency of oscillators and by increasing the proportion of the counter-rotating oscillators in both symmetry preserving and symmetry breaking couplings. Further, the transition from oscillatory to aging occurs via a Hopf bifurcation, while the transition from aging to oscillation death state emerges via the pitchfork bifurcation. Analytical expressions for the critical ratio of the counter-rotating oscillators are deduced to find the stable boundaries of the aging transition.
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Affiliation(s)
- K Sathiyadevi
- Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur, Tamil Nadu 613 401, India
| | - I Gowthaman
- Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur, Tamil Nadu 613 401, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram, Kerala 695551, India
| | - V K Chandrasekar
- Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur, Tamil Nadu 613 401, India
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Sun Z, Liu Y, Liu K, Yang X, Xu W. Aging transition in mixed active and inactive fractional-order oscillators. CHAOS (WOODBURY, N.Y.) 2019; 29:103150. [PMID: 31675845 DOI: 10.1063/1.5114998] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2019] [Accepted: 10/09/2019] [Indexed: 06/10/2023]
Abstract
Recently, a large number of studies have concentrated on aging transition, but they have so far been restricted to coupled integer-order oscillators. Here, we report the first study of aging transition in mixed active and inactive fractional-order oscillators. It has been demonstrated that while the heterogeneity is caused by the distance parameter, both the coupling strength and the fractional-order derivative can modulate the critical ratio at which aging transition occurs. In addition, a small fractional-order derivative may ruin the ability of oscillation and, thus, reduce the critical ratio in globally coupled fractional-order Stuart-Landau oscillators. Remarkably, the larger the natural frequency is the more easily the aging transition occurs in coupled fractional-order oscillators. Further studies have shown that, being diverse from an integer-order Stuart-Landau oscillator, the natural frequency may induce a Hopf bifurcation in a fractional-order Stuart-Landau oscillator, accordingly, introducing a new heterogeneity in the coupled fractional-order Stuart-Landau oscillators. Therein, a counterintuitive phenomenon has been found that the critical ratio depends unmonotonously on the coupling strength, which implies that the coupled fractional-order Stuart-Landau oscillators possess the weakest robustness of oscillation at a certain level of coupling strength.
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Affiliation(s)
- Zhongkui Sun
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
| | - Yuanyuan Liu
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
| | - Ke Liu
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
| | - Xiaoli Yang
- College of Mathematics and Information Science, Shaan'xi Normal University, Xi'an 710062, People's Republic of China
| | - Wei Xu
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
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Kundu S, Majhi S, Sasmal SK, Ghosh D, Rakshit B. Survivability of a metapopulation under local extinctions. Phys Rev E 2017; 96:062212. [PMID: 29347456 DOI: 10.1103/physreve.96.062212] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2017] [Indexed: 06/07/2023]
Abstract
A metapopulation structure in landscape ecology comprises a group of interacting spatially separated subpopulations or patches of the same species that may experience several local extinctions. This makes the investigation of survivability (in the form of global oscillation) of a metapopulation on top of diverse dispersal topologies extremely crucial. However, among various dispersal topologies in ecological networks, which one can provide higher metapopulation survivability under local extinction is still not well explored. In this article, we scrutinize the robustness of an ecological network consisting of prey-predator patches having Holling type I functional response, against progressively extinct population patches. We present a comprehensive study on this while considering global, small-world, and scale-free dispersal of the subpopulations. Furthermore, we extend our work in enhancing survivability in the form of sustained global oscillation by introducing asymmetries in the dispersal rates of the considered species. Our findings affirm that the asynchrony among the patches plays an important role in the survivability of a metapopulation. In order to demonstrate the model independence of the observed phenomenon, we perform a similar analysis for patches exhibiting Holling type II functional response. On the grounds of the obtained results, our work is expected to provide a better perception of the influence of dispersal arrangements on the global survivability of ecological networks.
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Affiliation(s)
- Srilena Kundu
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Soumen Majhi
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Sourav Kumar Sasmal
- Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Biswambhar Rakshit
- Department of Mathematics, Amrita School of Engineering-Coimbatore Amrita Vishwa Vidyapeetham, India
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Yuan T, Tanaka G. Robustness of coupled oscillator networks with heterogeneous natural frequencies. CHAOS (WOODBURY, N.Y.) 2017; 27:123105. [PMID: 29289041 DOI: 10.1063/1.4991742] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2017] [Accepted: 11/23/2017] [Indexed: 06/07/2023]
Abstract
Robustness of coupled oscillator networks against local degradation of oscillators has been intensively studied in this decade. The oscillation behavior on the whole network is typically reduced with an increase in the fraction of degraded (inactive) oscillators. The critical fraction of inactive oscillators, at which a transition from an oscillatory to a quiescent state occurs, has been used as a measure for the network robustness. The larger (smaller) this measure is, the more robust (fragile) the oscillatory behavior on the network is. Most previous studies have used oscillators with identical natural frequencies, for which the oscillators are necessarily synchronized and thereby the analysis is simple. In contrast, we focus on the effect of heterogeneity in the natural frequencies on the network robustness. First, we analytically derive the robustness measure for the coupled oscillator models with heterogeneous natural frequencies under some conditions. Then, we show that increasing the heterogeneity in natural frequencies makes the network fragile. Moreover, we discuss the optimal parameter condition to maximize the network robustness.
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Affiliation(s)
- Tianyu Yuan
- Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
| | - Gouhei Tanaka
- Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
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Kharche SR, Vigmond E, Efimov IR, Dobrzynski H. Computational assessment of the functional role of sinoatrial node exit pathways in the human heart. PLoS One 2017; 12:e0183727. [PMID: 28873427 PMCID: PMC5584965 DOI: 10.1371/journal.pone.0183727] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2017] [Accepted: 08/09/2017] [Indexed: 11/19/2022] Open
Abstract
AIM The human right atrium and sinoatrial node (SAN) anatomy is complex. Optical mapping experiments suggest that the SAN is functionally insulated from atrial tissue except at discrete SAN-atrial electrical junctions called SAN exit pathways, SEPs. Additionally, histological imaging suggests the presence of a secondary pacemaker close to the SAN. We hypothesise that a) an insulating border-SEP anatomical configuration is related to SAN arrhythmia; and b) a secondary pacemaker, the paranodal area, is an alternate pacemaker but accentuates tachycardia. A 3D electro-anatomical computational model was used to test these hypotheses. METHODS A detailed 3D human SAN electro-anatomical mathematical model was developed based on our previous anatomical reconstruction. Electrical activity was simulated using tissue specific variants of the Fenton-Karma action potential equations. Simulation experiments were designed to deploy this complex electro-anatomical system to assess the roles of border-SEPs and paranodal area by mimicking experimentally observed SAN arrhythmia. Robust and accurate numerical algorithms were implemented for solving the mono domain reaction-diffusion equation implicitly, calculating 3D filament traces, and computing dominant frequency among other quantitative measurements. RESULTS A centre to periphery gradient of increasing diffusion was sufficient to permit initiation of pacemaking at the centre of the 3D SAN. Re-entry within the SAN, micro re-entry, was possible by imposing significant SAN fibrosis in the presence of the insulating border. SEPs promoted the micro re-entry to generate more complex SAN-atrial tachycardia. Simulation of macro re-entry, i.e. re-entry around the SAN, was possible by inclusion of atrial fibrosis in the presence of the insulating border. The border shielded the SAN from atrial tachycardia. However, SAN micro-structure intercellular gap junctional coupling and the paranodal area contributed to prolonged atrial fibrillation. Finally, the micro-structure was found to be sufficient to explain shifts of leading pacemaker site location. CONCLUSIONS The simulations establish a relationship between anatomy and SAN electrical function. Microstructure, in the form of intercellular gap junction coupling, was found to regulate SAN function and arrhythmia.
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Affiliation(s)
- Sanjay R. Kharche
- Institute of Cardiovascular Sciences, School of Medical Sciences, University of Manchester, Manchester, United Kingdom
| | - Edward Vigmond
- University of Bordeaux, IMB, UMR 5251, Talence, France
- IHU Liryc, Electrophysiology and Heart Modeling Institute, Fondation Bordeaux Université, Pessac- Bordeaux, France
| | - Igor R. Efimov
- Department of Biomedical Engineering, The George Washington University, Washington, DC, United States of America
| | - Halina Dobrzynski
- Institute of Cardiovascular Sciences, School of Medical Sciences, University of Manchester, Manchester, United Kingdom
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13
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Rahman B, Blyuss KB, Kyrychko YN. Aging transition in systems of oscillators with global distributed-delay coupling. Phys Rev E 2017; 96:032203. [PMID: 29347035 DOI: 10.1103/physreve.96.032203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2016] [Indexed: 06/07/2023]
Abstract
We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.
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Affiliation(s)
- B Rahman
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
| | - K B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
| | - Y N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
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14
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Daido H. Boundary in the dynamic phase of globally coupled oscillatory and excitable units. Phys Rev E 2017; 96:012210. [PMID: 29347184 DOI: 10.1103/physreve.96.012210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2017] [Indexed: 06/07/2023]
Abstract
There is a crucial boundary between dynamic phase 1 and dynamic phase 2 of globally coupled oscillatory and excitable units, where the mean field is constant and oscillates in the former and the latter, respectively. This boundary is theoretically derived here for a large population of dynamical units, each having only a phase variable, where it is assumed that both the coupling strength and the distribution width of bifurcation parameters are equally small. This theory, which is applicable only if all or most of the units are intrinsically oscillatory, is confirmed to agree with simulation results for two different distribution densities.
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Affiliation(s)
- Hiroaki Daido
- Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan
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15
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Krishnagopal S, Lehnert J, Poel W, Zakharova A, Schöll E. Synchronization patterns: from network motifs to hierarchical networks. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2017; 375:20160216. [PMID: 28115613 PMCID: PMC5311436 DOI: 10.1098/rsta.2016.0216] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 11/04/2016] [Indexed: 05/12/2023]
Abstract
We investigate complex synchronization patterns such as cluster synchronization and partial amplitude death in networks of coupled Stuart-Landau oscillators with fractal connectivities. The study of fractal or self-similar topology is motivated by the network of neurons in the brain. This fractal property is well represented in hierarchical networks, for which we present three different models. In addition, we introduce an analytical eigensolution method and provide a comprehensive picture of the interplay of network topology and the corresponding network dynamics, thus allowing us to predict the dynamics of arbitrarily large hierarchical networks simply by analysing small network motifs. We also show that oscillation death can be induced in these networks, even if the coupling is symmetric, contrary to previous understanding of oscillation death. Our results show that there is a direct correlation between topology and dynamics: hierarchical networks exhibit the corresponding hierarchical dynamics. This helps bridge the gap between mesoscale motifs and macroscopic networks.This article is part of the themed issue 'Horizons of cybernetical physics'.
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Affiliation(s)
- Sanjukta Krishnagopal
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
- Department of Physics, Birla Institute for Technology and Science Pilani, Pilani, Goa 403726, India
| | - Judith Lehnert
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Winnie Poel
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
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16
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Sun Z, Ma N, Xu W. Aging transition by random errors. Sci Rep 2017; 7:42715. [PMID: 28198430 PMCID: PMC5309892 DOI: 10.1038/srep42715] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2016] [Accepted: 01/13/2017] [Indexed: 11/12/2022] Open
Abstract
In this paper, the effects of random errors on the oscillating behaviors have been studied theoretically and numerically in a prototypical coupled nonlinear oscillator. Two kinds of noises have been employed respectively to represent the measurement errors accompanied with the parameter specifying the distance from a Hopf bifurcation in the Stuart-Landau model. It has been demonstrated that when the random errors are uniform random noise, the change of the noise intensity can effectively increase the robustness of the system. While the random errors are normal random noise, the increasing of variance can also enhance the robustness of the system under certain conditions that the probability of aging transition occurs reaches a certain threshold. The opposite conclusion is obtained when the probability is less than the threshold. These findings provide an alternative candidate to control the critical value of aging transition in coupled oscillator system, which is composed of the active oscillators and inactive oscillators in practice.
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Affiliation(s)
- Zhongkui Sun
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, China
| | - Ning Ma
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, China
| | - Wei Xu
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, China
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17
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Yuan T, Aihara K, Tanaka G. Robustness and fragility in coupled oscillator networks under targeted attacks. Phys Rev E 2017; 95:012315. [PMID: 28208319 DOI: 10.1103/physreve.95.012315] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2016] [Indexed: 06/06/2023]
Abstract
The dynamical tolerance of coupled oscillator networks against local failures is studied. As the fraction of failed oscillator nodes gradually increases, the mean oscillation amplitude in the entire network decreases and then suddenly vanishes at a critical fraction as a phase transition. This critical fraction, widely used as a measure of the network robustness, was analytically derived for random failures but not for targeted attacks so far. Here we derive the general formula for the critical fraction, which can be applied to both random failures and targeted attacks. We consider the effects of targeting oscillator nodes based on their degrees. First we deal with coupled identical oscillators with homogeneous edge weights. Then our theory is applied to networks with heterogeneous edge weights and to those with nonidentical oscillators. The analytical results are validated by numerical experiments. Our results reveal the key factors governing the robustness and fragility of oscillator networks.
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Affiliation(s)
- Tianyu Yuan
- Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
| | - Kazuyuki Aihara
- Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
| | - Gouhei Tanaka
- Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
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18
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Ratas I, Pyragas K. Macroscopic self-oscillations and aging transition in a network of synaptically coupled quadratic integrate-and-fire neurons. Phys Rev E 2016; 94:032215. [PMID: 27739712 DOI: 10.1103/physreve.94.032215] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2016] [Indexed: 06/06/2023]
Abstract
We analyze the dynamics of a large network of coupled quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rate and the mean membrane potential, which are exact in the infinite-size limit. The bifurcation analysis of the reduced equations reveals a rich scenario of asymptotic behavior, the most interesting of which is the macroscopic limit-cycle oscillations. It is shown that the finite width of synaptic pulses is a necessary condition for the existence of such oscillations. The robustness of the oscillations against aging damage, which transforms spiking neurons into nonspiking neurons, is analyzed. The validity of the reduced equations is confirmed by comparing their solutions with the solutions of microscopic equations for the finite-size networks.
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Affiliation(s)
- Irmantas Ratas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Kestutis Pyragas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
- Department of Theoretical Physics, Faculty of Physics, Vilnius University, LT-10222 Vilnius, Lithuania
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19
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Daido H, Nishio K. Bifurcation and scaling at the aging transition boundary in globally coupled excitable and oscillatory units. Phys Rev E 2016; 93:052226. [PMID: 27300899 DOI: 10.1103/physreve.93.052226] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2016] [Indexed: 06/06/2023]
Abstract
Following a previous paper [Phys. Rev. E 88, 052907 (2013)PLEEE81539-375510.1103/PhysRevE.88.052907], we study in detail the mechanism of aging transition in globally and diffusively coupled excitable and oscillatory units. Here two of the three models taken up in the earlier work are used, each composed of a large number of units with their bifurcation parameters forming a uniform distribution. The control parameters are the coupling strength and the average of the bifurcation parameters. The present work is mostly devoted to a region of the phase diagram near the aging transition boundary with the coupling strength greater than its critical value for the onset of bistability and hysteresis. The bifurcation structure of each system at the aging transition boundary is investigated theoretically as well as numerically. Moreover, we show that critical scaling laws of order parameters S and M used in the previous paper are different depending on which region of the coupling strength to be chosen.
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Affiliation(s)
- Hiroaki Daido
- Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan
| | - Kazuho Nishio
- Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan
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20
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Zou W, Senthilkumar DV, Nagao R, Kiss IZ, Tang Y, Koseska A, Duan J, Kurths J. Restoration of rhythmicity in diffusively coupled dynamical networks. Nat Commun 2015; 6:7709. [PMID: 26173555 PMCID: PMC4518287 DOI: 10.1038/ncomms8709] [Citation(s) in RCA: 120] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2015] [Accepted: 06/01/2015] [Indexed: 12/18/2022] Open
Abstract
Oscillatory behaviour is essential for proper functioning of various physical and biological processes. However, diffusive coupling is capable of suppressing intrinsic oscillations due to the manifestation of the phenomena of amplitude and oscillation deaths. Here we present a scheme to revoke these quenching states in diffusively coupled dynamical networks, and demonstrate the approach in experiments with an oscillatory chemical reaction. By introducing a simple feedback factor in the diffusive coupling, we show that the stable (in)homogeneous steady states can be effectively destabilized to restore dynamic behaviours of coupled systems. Even a feeble deviation from the normal diffusive coupling drastically shrinks the death regions in the parameter space. The generality of our method is corroborated in diverse non-linear systems of diffusively coupled paradigmatic models with various death scenarios. Our study provides a general framework to strengthen the robustness of dynamic activity in diffusively coupled dynamical networks.
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Affiliation(s)
- Wei Zou
- School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
- Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China
- Potsdam Institute for Climate Impact Research, Telegraphenberg, D-14415 Potsdam, Germany
| | - D. V. Senthilkumar
- Potsdam Institute for Climate Impact Research, Telegraphenberg, D-14415 Potsdam, Germany
- Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA University, Thanjavur 613401, India
| | - Raphael Nagao
- 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
| | - Yang Tang
- Potsdam Institute for Climate Impact Research, Telegraphenberg, D-14415 Potsdam, Germany
- The Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China
| | - Aneta Koseska
- Department of Systemic Cell Biology, Max Planck Institute of Molecular Physiology, Dortmund D-44227, Germany
- Research Centre for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Skopje, Macedonia
| | - Jinqiao Duan
- School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
- Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Telegraphenberg, D-14415 Potsdam, Germany
- Institute of Physics, Humboldt University Berlin, D-12489 Berlin, Germany
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, UK
- Department of Control Theory, Nizhny Novgorod State University, Gagarin Avenue 23, 606950 Nizhny Novgorod, Russia
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21
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Sasai T, Morino K, Tanaka G, Almendral JA, Aihara K. Robustness of oscillatory behavior in correlated networks. PLoS One 2015; 10:e0123722. [PMID: 25894574 PMCID: PMC4403822 DOI: 10.1371/journal.pone.0123722] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2014] [Accepted: 03/06/2015] [Indexed: 11/23/2022] Open
Abstract
Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.
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Affiliation(s)
- Takeyuki Sasai
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Kai Morino
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Gouhei Tanaka
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
- Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
- * E-mail:
| | - Juan A. Almendral
- Complex Systems Group, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Univ. Politecnica de Madrid, 28223 Pozuelo de Alarcon, Madrid, Spain
| | - Kazuyuki Aihara
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
- Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
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22
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Thakur B, Sharma D, Sen A. Time-delay effects on the aging transition in a population of coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042904. [PMID: 25375564 DOI: 10.1103/physreve.90.042904] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/13/2014] [Indexed: 06/04/2023]
Abstract
We investigate the influence of time-delayed coupling on the nature of the aging transition in a system of coupled oscillators that have a mix of active and inactive oscillators, where the aging transition is defined as the gradual loss of collective synchrony as the proportion of inactive oscillators is increased. We start from a simple model of two time-delay coupled Stuart-Landau oscillators that have identical frequencies but are located at different distances from the Hopf bifurcation point. A systematic numerical and analytic study delineates the dependence of the critical coupling strength (at which the system experiences total loss of synchrony) on time delay and the average distance of the system from the Hopf bifurcation point. We find that time delay can act to facilitate the aging transition by lowering the threshold coupling strength for amplitude death in the system. We then extend our study to larger systems of globally coupled active and inactive oscillators including an infinite system in the thermodynamic limit. Our model system and results can provide a useful paradigm for understanding the functional robustness of diverse physical and biological systems that are prone to aging transitions.
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Affiliation(s)
- Bhumika Thakur
- Institute for Plasma Research, Bhat, Gandhinagar 382428, India
| | - Devendra Sharma
- Institute for Plasma Research, Bhat, Gandhinagar 382428, India
| | - Abhijit Sen
- Institute for Plasma Research, Bhat, Gandhinagar 382428, India
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23
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Huang W, Zhang X, Hu X, Zou Y, Liu Z, Guan S. Variation of critical point of aging transition in a networked oscillators system. CHAOS (WOODBURY, N.Y.) 2014; 24:023122. [PMID: 24985436 DOI: 10.1063/1.4881215] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
In this work, we study the variation of critical point in aging transition in a networked system consisting of both active and inactive oscillators. By theoretical analysis and numerical simulations, we show that the critical point of aging transition actually is determined by the (normalized) cross links between active and inactive subpopulations of oscillators. This reveals how specific configuration of active and inactive oscillators in the network can lead to the variation of transition point. In particular, we investigate how different strategies of targeted inactivation influence the transition point based on the theory. Our results theoretically explain why the low-degree nodes are crucial regarding dynamical robustness in such systems.
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Affiliation(s)
- Wenwen Huang
- Department of Physics, East China Normal University, Shanghai 200241, People's Republic of China
| | - Xiyun Zhang
- Department of Physics, East China Normal University, Shanghai 200241, People's Republic of China
| | - Xin Hu
- Department of Physics, East China Normal University, Shanghai 200241, People's Republic of China
| | - Yong Zou
- Department of Physics, East China Normal University, Shanghai 200241, People's Republic of China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai 200241, People's Republic of China
| | - Shuguang Guan
- Department of Physics, East China Normal University, Shanghai 200241, People's Republic of China
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