1
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Papo D, Buldú JM. Does the brain behave like a (complex) network? I. Dynamics. Phys Life Rev 2024; 48:47-98. [PMID: 38145591 DOI: 10.1016/j.plrev.2023.12.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2023] [Accepted: 12/10/2023] [Indexed: 12/27/2023]
Abstract
Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.
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Affiliation(s)
- D Papo
- Department of Neuroscience and Rehabilitation, Section of Physiology, University of Ferrara, Ferrara, Italy; Center for Translational Neurophysiology, Fondazione Istituto Italiano di Tecnologia, Ferrara, Italy.
| | - J M Buldú
- Complex Systems Group & G.I.S.C., Universidad Rey Juan Carlos, Madrid, Spain
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2
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Zou W. Solvable dynamics of the three-dimensional Kuramoto model with frequency-weighted coupling. Phys Rev E 2024; 109:034215. [PMID: 38632753 DOI: 10.1103/physreve.109.034215] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2024] [Accepted: 03/06/2024] [Indexed: 04/19/2024]
Abstract
The presence of coupling heterogeneity is deemed to be a natural attribute in realistic systems comprised of many interacting agents. In this work, we study dynamics of the 3D Kuramoto model with heterogeneous couplings, where the strength of coupling for each agent is weighted by its intrinsic rotation frequency. The critical coupling strength for the instability of incoherence is rigorously derived to be zero by carrying out a linear stability analysis of an incoherent state. For positive values of the coupling strength, at which the incoherence turns out to be unstable, a self-consistency approach is developed to theoretically predict the degree of global coherence of the model. Our theoretical analyses match well with numerical simulations, which helps us to deepen the understanding of collective behaviors spontaneously emerged in heterogeneously coupled high-dimensional dynamical networks.
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Affiliation(s)
- Wei Zou
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
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3
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Zheng Z, Xu C, Fan J, Liu M, Chen X. Order parameter dynamics in complex systems: From models to data. Chaos 2024; 34:022101. [PMID: 38341762 DOI: 10.1063/5.0180340] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2023] [Accepted: 12/14/2023] [Indexed: 02/13/2024]
Abstract
Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems.
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Affiliation(s)
- Zhigang Zheng
- Institute of Systems Science, Huaqiao University, Xiamen 361021, China and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Can Xu
- Institute of Systems Science, Huaqiao University, Xiamen 361021, China and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Jingfang Fan
- School of Systems Science, Beijing Normal University, Beijing 100875, China and Institute of Nonequilibrium Systems, Beijing Normal University, Beijing 100875, China
| | - Maoxin Liu
- School of Systems Science, Beijing Normal University, Beijing 100875, China and Institute of Nonequilibrium Systems, Beijing Normal University, Beijing 100875, China
| | - Xiaosong Chen
- School of Systems Science, Beijing Normal University, Beijing 100875, China and Institute of Nonequilibrium Systems, Beijing Normal University, Beijing 100875, China
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4
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Qiang C, Li Z, Deng Y. Multifractal analysis of mass function. Soft comput 2023; 27:1-14. [PMID: 37362275 PMCID: PMC10233544 DOI: 10.1007/s00500-023-08502-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 05/02/2023] [Indexed: 06/28/2023]
Abstract
In order to explore the fractal characteristic in Dempster-Shafer evidence theory, a fractal dimension of mass function is proposed recently, to reveal the invariance of scale of belief entropy. When mass function degenerates to probability, the fractal dimension is equivalent to classical Renyi information dimension only with α = 1 , which can measure the change rate of Shannon entropy with the size of framework. For Renyi dimension, different parameters α represent the relationship between different entropies and framework size. However, this compatibility is not shown in existing fractal dimension. Thus, in this paper, we introduce parameter α to generalize the existing dimension. Due to the diversity of the value of α , we name the new dimension: multifractal dimension of mass function. In addition, inspired by multifractal spectrum of Cantor set, we explore the relation between the belief degree of focal element and the number of focal element with same belief degree for some special assignments. Relevant results are also presented by spectrum. We provide a static discounting coefficient generating method to modify mass function to improve the accuracy of classify result. The experiment is conducted in three datasets, and the result shows the effectiveness of our method.
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Affiliation(s)
- Chenhui Qiang
- Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China, Chengdu, 610054 China
- Yingcai Honors College, University of Electronic Science and Technology of China, Chengdu, 610054 China
| | - Zhen Li
- China Mobile Information Technology Center, Beijing, 100029 China
| | - Yong Deng
- School of Medicine, Vanderbilt University, Nashville, TN 37240 USA
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5
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Meng X, Lin J, Fan Y, Gao F, Fenoaltea EM, Cai Z, Si S. Coupled disease-vaccination behavior dynamic analysis and its application in COVID-19 pandemic. Chaos Solitons Fractals 2023; 169:113294. [PMID: 36891356 PMCID: PMC9977628 DOI: 10.1016/j.chaos.2023.113294] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/07/2022] [Revised: 01/20/2023] [Accepted: 02/19/2023] [Indexed: 06/18/2023]
Abstract
Predicting the evolutionary dynamics of the COVID-19 pandemic is a complex challenge. The complexity increases when the vaccination process dynamic is also considered. In addition, when applying a voluntary vaccination policy, the simultaneous behavioral evolution of individuals who decide whether and when to vaccinate must be included. In this paper, a coupled disease-vaccination behavior dynamic model is introduced to study the coevolution of individual vaccination strategies and infection spreading. We study disease transmission by a mean-field compartment model and introduce a non-linear infection rate that takes into account the simultaneity of interactions. Besides, the evolutionary game theory is used to investigate the contemporary evolution of vaccination strategies. Our findings suggest that sharing information with the entire population about the negative and positive consequences of infection and vaccination is beneficial as it boosts behaviors that can reduce the final epidemic size. Finally, we validate our transmission mechanism on real data from the COVID-19 pandemic in France.
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Affiliation(s)
- Xueyu Meng
- Department of Industrial Engineering, School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
- Ministry of Industry and Information Technology Key Laboratory of Industrial Engineering and Intelligent Manufacturing, Northwestern Polytechnical University, Xi'an 710072, China
- Department of Physics, University of Fribourg, Fribourg 1700, Switzerland
| | - Jianhong Lin
- Department of Physics, University of Fribourg, Fribourg 1700, Switzerland
- Department of Management, Technology and Economics, ETH Zürich, Scheuchzerstrasse 7, CH-8092 Zürich, Switzerland
| | - Yufei Fan
- Department of Industrial Engineering, School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
- Ministry of Industry and Information Technology Key Laboratory of Industrial Engineering and Intelligent Manufacturing, Northwestern Polytechnical University, Xi'an 710072, China
| | - Fujuan Gao
- Department of Physics, University of Fribourg, Fribourg 1700, Switzerland
| | | | - Zhiqiang Cai
- Department of Industrial Engineering, School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
- Ministry of Industry and Information Technology Key Laboratory of Industrial Engineering and Intelligent Manufacturing, Northwestern Polytechnical University, Xi'an 710072, China
| | - Shubin Si
- Department of Industrial Engineering, School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
- Ministry of Industry and Information Technology Key Laboratory of Industrial Engineering and Intelligent Manufacturing, Northwestern Polytechnical University, Xi'an 710072, China
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6
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Zou W, He S, Senthilkumar DV, Kurths J. Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators. Phys Rev Lett 2023; 130:107202. [PMID: 36962012 DOI: 10.1103/physrevlett.130.107202] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2022] [Accepted: 02/10/2023] [Indexed: 06/18/2023]
Abstract
We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the D-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of D=3, the incoherence of the model is rigorously proved to be stable for negative coupling (K<0) but unstable for positive coupling (K>0); the locked states are shown to exist if K>0; in particular, the onset of amplitude death is theoretically predicted. For D≥2, the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed D-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the D-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for D-dimensional limit-cycle oscillators.
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Affiliation(s)
- Wei Zou
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
| | - Sujuan He
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, Kerala, India
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Telegraphenberg, Potsdam D-14415, Germany
- Institute of Physics, Humboldt University Berlin, Berlin D-12489, Germany
- Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China
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7
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Calmon L, Krishnagopal S, Bianconi G. Local Dirac Synchronization on networks. Chaos 2023; 33:033117. [PMID: 37003807 DOI: 10.1063/5.0132468] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2022] [Accepted: 02/13/2023] [Indexed: 06/19/2023]
Abstract
We propose Local Dirac Synchronization that uses the Dirac operator to capture the dynamics of coupled nodes and link signals on an arbitrary network. In Local Dirac Synchronization, the harmonic modes of the dynamics oscillate freely while the other modes are interacting non-linearly, leading to a collectively synchronized state when the coupling constant of the model is increased. Local Dirac Synchronization is characterized by discontinuous transitions and the emergence of a rhythmic coherent phase. In this rhythmic phase, one of the two complex order parameters oscillates in the complex plane at a slow frequency (called emergent frequency) in the frame in which the intrinsic frequencies have zero average. Our theoretical results obtained within the annealed approximation are validated by extensive numerical results on fully connected networks and sparse Poisson and scale-free networks. Local Dirac Synchronization on both random and real networks, such as the connectome of Caenorhabditis Elegans, reveals the interplay between topology (Betti numbers and harmonic modes) and non-linear dynamics. This unveils how topology might play a role in the onset of brain rhythms.
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Affiliation(s)
- Lucille Calmon
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
| | - Sanjukta Krishnagopal
- Department of Electrical Engineering and Computer Science, University of California Berkeley, California 94720, USA
| | - Ginestra Bianconi
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
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8
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Liu W, Xie J, Liu H, Xiao J. Heterogeneity induced splay state of amplitude envelope in globally coupled oscillators. Chaos 2022; 32:123117. [PMID: 36587328 DOI: 10.1063/5.0130753] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2022] [Accepted: 11/11/2022] [Indexed: 06/17/2023]
Abstract
Splay states of the amplitude envelope are stably observed as a heterogenous node is introduced into the globally coupled identical oscillators with repulsive coupling. With the increment of the frequency mismatches between the heterogenous nodes and the rest identical globally coupled oscillators, the formal stable splay state based on the time series becomes unstable, while a splay state based on the new-born amplitude envelopes of time series is stably observed among the rest identical oscillators. The characteristics of the splay state based on the amplitude envelope are numerically and theoretically presented for different parameters of the coupling strength ϵ and the frequency mismatches Δω for small coupling strength and large frequency mismatches. We expect that all these results could reveal the generality of splay states in coupled nonidentical oscillators and help to understand the rich dynamics of amplitude envelopes in multidisciplinary fields.
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Affiliation(s)
- Weiqing Liu
- School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
| | - Jiangnan Xie
- School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
| | - Hanchang Liu
- School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
| | - Jinghua Xiao
- School of Science, Beijing University of Posts and Communications, Beijing 100876, China
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9
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Zhou H, Liu Z, Chu D, Li W. Sampled-data synchronization of complex network based on periodic self-triggered intermittent control and its application to image encryption. Neural Netw 2022; 152:419-433. [DOI: 10.1016/j.neunet.2022.05.004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2021] [Revised: 04/10/2022] [Accepted: 05/08/2022] [Indexed: 10/18/2022]
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10
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Shen H, Huang Z, Wu Z, Cao J, Park JH. Nonfragile H ∞ Synchronization of BAM Inertial Neural Networks Subject to Persistent Dwell-Time Switching Regularity. IEEE Trans Cybern 2022; 52:6591-6602. [PMID: 34705662 DOI: 10.1109/tcyb.2021.3119199] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
This article concentrates on the synchronization of discrete-time persistent dwell-time (PDT) switched bidirectional associative memory inertial neural networks with time-varying delays. Through the use of the switched system theory related to the PDT, the convex optimization technique together with some straightforward decoupling methods, an appropriate mode-dependent controller with nonfragility is developed to acclimatize itself to some practical circumstances. Simultaneously, sufficient conditions of ensuring the H∞ performance and exponential stability for the resulting switched synchronization error system are derived. Finally, a numerical example is utilized to show the validity of the model constructed and the influence of the PDT on the H∞ performance. In addition, an image encryption example is employed to show the potential application prospect of the investigated system.
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11
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Sarkar M, Gupta S. Synchronization in the Kuramoto model in presence of stochastic resetting. Chaos 2022; 32:073109. [PMID: 35907730 DOI: 10.1063/5.0090861] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2022] [Accepted: 06/10/2022] [Indexed: 06/15/2023]
Abstract
What happens when the paradigmatic Kuramoto model involving interacting oscillators of distributed natural frequencies and showing spontaneous collective synchronization in the stationary state is subject to random and repeated interruptions of its dynamics with a reset to the initial condition? While resetting to a synchronized state, it may happen between two successive resets that the system desynchronizes, which depends on the duration of the random time interval between the two resets. Here, we unveil how such a protocol of stochastic resetting dramatically modifies the phase diagram of the bare model, allowing, in particular, for the emergence of a synchronized phase even in parameter regimes for which the bare model does not support such a phase. Our results are based on an exact analysis invoking the celebrated Ott-Antonsen ansatz for the case of the Lorentzian distribution of natural frequencies and numerical results for Gaussian frequency distribution. Our work provides a simple protocol to induce global synchrony in the system through stochastic resetting.
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Affiliation(s)
- Mrinal Sarkar
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Shamik Gupta
- Department of Physics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah 711202, India
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12
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Liu Y, Shen B, Zhang P. Synchronization and state estimation for discrete-time coupled delayed complex-valued neural networks with random system parameters. Neural Netw 2022; 150:181-93. [DOI: 10.1016/j.neunet.2022.02.028] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2021] [Revised: 01/07/2022] [Accepted: 02/28/2022] [Indexed: 11/21/2022]
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13
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Palabas T, Longtin A, Ghosh D, Uzuntarla M. Controlling the spontaneous firing behavior of a neuron with astrocyte. Chaos 2022; 32:051101. [PMID: 35649970 DOI: 10.1063/5.0093234] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2022] [Accepted: 04/18/2022] [Indexed: 06/15/2023]
Abstract
Mounting evidence in recent years suggests that astrocytes, a sub-type of glial cells, not only serve metabolic and structural support for neurons and synapses but also play critical roles in the regulation of proper functioning of the nervous system. In this work, we investigate the effect of astrocytes on the spontaneous firing activity of a neuron through a combined model that includes a neuron-astrocyte pair. First, we show that an astrocyte may provide a kind of multistability in neuron dynamics by inducing different firing modes such as random and bursty spiking. Then, we identify the underlying mechanism of this behavior and search for the astrocytic factors that may have regulatory roles in different firing regimes. More specifically, we explore how an astrocyte can participate in the occurrence and control of spontaneous irregular spiking activity of a neuron in random spiking mode. Additionally, we systematically investigate the bursty firing regime dynamics of the neuron under the variation of biophysical facts related to the intracellular environment of the astrocyte. It is found that an astrocyte coupled to a neuron can provide a control mechanism for both spontaneous firing irregularity and burst firing statistics, i.e., burst regularity and size.
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Affiliation(s)
- Tugba Palabas
- Department of Biomedical Engineering, Zonguldak Bulent Ecevit University, 67100 Zonguldak, Turkey
| | - Andre Longtin
- Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Muhammet Uzuntarla
- Department of Bioengineering, Gebze Technical University, 41400 Kocaeli, Turkey
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14
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Zou W, Chen Y, Senthilkumar DV, Kurths J. Oscillation quenching in diffusively coupled dynamical networks with inertial effects. Chaos 2022; 32:041102. [PMID: 35489855 DOI: 10.1063/5.0087839] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2022] [Accepted: 03/22/2022] [Indexed: 06/14/2023]
Abstract
Self-sustained oscillations are ubiquitous and of fundamental importance for a variety of physical and biological systems including neural networks, cardiac dynamics, and circadian rhythms. In this work, oscillation quenching in diffusively coupled dynamical networks including "inertial" effects is analyzed. By adding inertia to diffusively coupled first-order oscillatory systems, we uncover that even small inertia is capable of eradicating the onset of oscillation quenching. We consolidate the generality of inertia in eradicating oscillation quenching by extensively examining diverse quenching scenarios, where macroscopic oscillations are extremely deteriorated and even completely lost in the corresponding models without inertia. The presence of inertia serves as an additional scheme to eradicate the onset of oscillation quenching, which does not need to tailor the coupling functions. Our findings imply that inertia of a system is an enabler against oscillation quenching in coupled dynamical networks, which, in turn, is helpful for understanding the emergence of rhythmic behaviors in complex coupled systems with amplitude degree of freedom.
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Affiliation(s)
- Wei Zou
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
| | - Yuxuan Chen
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, Kerala, India
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Telegraphenberg, Potsdam D-14415, Germany
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15
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Li R, Wu H, Cao J. Exponential synchronization for variable-order fractional discontinuous complex dynamical networks with short memory via impulsive control. Neural Netw 2022; 148:13-22. [PMID: 35051866 DOI: 10.1016/j.neunet.2021.12.021] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2021] [Revised: 12/17/2021] [Accepted: 12/30/2021] [Indexed: 11/30/2022]
Abstract
This paper considers the exponential synchronization issue for variable-order fractional complex dynamical networks (FCDNs) with short memory and derivative couplings via the impulsive control scheme, where dynamical nodes are modeled to be discontinuous. Firstly, the mathematics model with respect to variable-order fractional systems with short memory is established under the impulsive controller, in which the impulse strength is not only determined by the impulse control gain, but also the order of the control systems. Secondly, the exponential stability criterion for variable-order fractional systems with short memory is developed. Thirdly, the hybrid controller, which consists of the impulsive coupling controller and the discontinuous feedback controller, is designed to realize the synchronization objective. In addition, by constructing Lyapunov functional and applying inequality analysis techniques, the synchronization conditions are achieved in terms of linear matrix inequalities (LMIs). Finally, two simulation examples are performed to verify the effectiveness of the developed synchronization scheme and the theoretical outcomes.
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Affiliation(s)
- Ruihong Li
- School of Science, Yanshan University, Qinhuangdao 066001, China.
| | - Huaiqin Wu
- School of Science, Yanshan University, Qinhuangdao 066001, China.
| | - Jinde Cao
- School of Mathematics, Southeast University, Nanjing 210096, China; Yonsei Frontier Lab, Yonsei University, Seoul 03722, Republic of Korea.
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16
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Ling X, Ju WB, Guo N, Zhu KJ, Wu CY, Hao QY. Effects of topological characteristics on rhythmic states of the D-dimensional Kuramoto model in complex networks. Chaos 2022; 32:013118. [PMID: 35105134 DOI: 10.1063/5.0058747] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2021] [Accepted: 11/24/2021] [Indexed: 06/14/2023]
Abstract
Synchronization is a ubiquitous phenomenon in engineering and natural ecosystems. While the dynamics of synchronization modeled by the Kuramoto model are commonly studied in two dimensions and the state of dynamic units is characterized by a scalar angle variable, we studied the Kuramoto model generalized to D dimensions in the framework of a complex network and utilized the local synchronous order parameter between the agent and its neighbors as the controllable variable to adjust the coupling strength. Here, we reported that average connectivity of networks affects the time-dependent, rhythmic, cyclic state. Importantly, we found that the level of heterogeneity of networks governs the rhythmic state in the transition process. The analytical treatment for observed scenarios in a D-dimensional Kuramoto model at D=3 was provided. These results offered a platform for a better understanding of time-dependent swarming and flocking dynamics in nature.
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Affiliation(s)
- Xiang Ling
- School of Automotive and Transportation Engineering, Hefei University of Technology, 230009 Hefei, People's Republic of China
| | - Wen-Bin Ju
- School of Automotive and Transportation Engineering, Hefei University of Technology, 230009 Hefei, People's Republic of China
| | - Ning Guo
- School of Automotive and Transportation Engineering, Hefei University of Technology, 230009 Hefei, People's Republic of China
| | - Kong-Jin Zhu
- School of Automotive and Transportation Engineering, Hefei University of Technology, 230009 Hefei, People's Republic of China
| | - Chao-Yun Wu
- School of Mathematics and Physics, Anqing Normal University, Anqing 246133, People's Republic of China
| | - Qing-Yi Hao
- School of Mathematics and Physics, Anqing Normal University, Anqing 246133, People's Republic of China
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17
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Xu C, Wang X, Zheng Z, Cai Z. Stability and bifurcation of collective dynamics in phase oscillator populations with general coupling. Phys Rev E 2021; 103:032307. [PMID: 33862749 DOI: 10.1103/physreve.103.032307] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2020] [Accepted: 02/22/2021] [Indexed: 02/03/2023]
Abstract
The Kuramoto model serves as an illustrative paradigm for studying the synchronization transitions and collective behaviors in large ensembles of coupled dynamical units. In this paper, we present a general framework for analytically capturing the stability and bifurcation of the collective dynamics in oscillator populations by extending the global coupling to depend on an arbitrary function of the Kuramoto order parameter. In this generalized Kuramoto model with rotation and reflection symmetry, we show that all steady states characterizing the long-term macroscopic dynamics can be expressed in a universal profile given by the frequency-dependent version of the Ott-Antonsen reduction, and the introduced empirical stability criterion for each steady state degenerates to a remarkably simple expression described by the self-consistent equation [Iatsenko et al., Phys. Rev. Lett. 110, 064101 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064101]. Here, we provide a detailed description of the spectrum structure in the complex plane by performing a rigorous stability analysis of various steady states in the reduced system. More importantly, we uncover that the empirical stability criterion for each steady state involved in the system is completely equivalent to its linear stability condition that is determined by the nontrivial eigenvalues (discrete spectrum) of the linearization. Our study provides a new and widely applicable approach for exploring the stability properties of collective synchronization, which we believe improves the understanding of the underlying mechanisms of phase transitions and bifurcations in coupled dynamical networks.
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Affiliation(s)
- Can Xu
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Xuan Wang
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Zhigang Zheng
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Zongkai Cai
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
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Phogat R, Ray A, Parmananda P, Ghosh D. Phase coalescence in a population of heterogeneous Kuramoto oscillators. Chaos 2021; 31:041104. [PMID: 34251244 DOI: 10.1063/5.0050451] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2021] [Accepted: 04/01/2021] [Indexed: 06/13/2023]
Abstract
Phase coalescence (PC) is an emerging phenomenon in an ensemble of oscillators that manifests itself as a spontaneous rise in the order parameter. This increment in the order parameter is due to the overlaying of oscillator phases to a pre-existing system state. In the current work, we present a comprehensive analysis of the phenomenon of phase coalescence observed in a population of Kuramoto phase oscillators. The given population is divided into responsive and non-responsive oscillators depending on the position of the phases of the oscillators. The responsive set of oscillators is then reset by a pulse perturbation. This resetting leads to a temporary rise in a macroscopic observable, namely, order parameter. The provoked rise thus induced in the order parameter is followed by unprovoked increments separated by a constant time τPC. These unprovoked increments in the order parameter are caused due to a temporary gathering of the oscillator phases in a configuration similar to the initial system state, i.e., the state of the network immediately following the perturbation. A theoretical framework corroborating this phenomenon as well as the corresponding simulation results are presented. Dependence of τPC and the magnitude of spontaneous order parameter augmentation on various network parameters such as coupling strength, network size, degree of the network, and frequency distribution are then explored. The size of the phase resetting region would also affect the magnitude of the order parameter at τPC since it directly affects the number of oscillators reset by the perturbation. Therefore, the dependence of order parameter on the size of the phase resetting region is also analyzed.
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Affiliation(s)
- Richa Phogat
- Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
| | - Arnob Ray
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - P Parmananda
- Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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