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Andrzejak RG, Espinoso A. Chimera states in multiplex networks: Chameleon-like across-layer synchronization. CHAOS (WOODBURY, N.Y.) 2023; 33:2890080. [PMID: 37163994 DOI: 10.1063/5.0146550] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2023] [Accepted: 04/19/2023] [Indexed: 05/12/2023]
Abstract
Different across-layer synchronization types of chimera states in multilayer networks have been discovered recently. We investigate possible relations between them, for example, if the onset of some synchronization type implies the onset of some other type. For this purpose, we use a two-layer network with multiplex inter-layer coupling. Each layer consists of a ring of non-locally coupled phase oscillators. While oscillators in each layer are identical, the layers are made non-identical by introducing mismatches in the oscillators' mean frequencies and phase lag parameters of the intra-layer coupling. We use different metrics to quantify the degree of various across-layer synchronization types. These include phase-locking between individual interacting oscillators, amplitude and phase synchronization between the order parameters of each layer, generalized synchronization between the driver and response layer, and the alignment of the incoherent oscillator groups' position on the two rings. For positive phase lag parameter mismatches, we get a cascaded onset of synchronization upon a gradual increase of the inter-layer coupling strength. For example, the two order parameters show phase synchronization before any of the interacting oscillator pairs does. For negative mismatches, most synchronization types have their onset in a narrow range of the coupling strength. Weaker couplings can destabilize chimera states in the response layer toward an almost fully coherent or fully incoherent motion. Finally, in the absence of a phase lag mismatch, sufficient coupling turns the response dynamics into a replica of the driver dynamics with the phases of all oscillators shifted by a constant lag.
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Affiliation(s)
- Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Anaïs Espinoso
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
- Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute of Science and Technology, Carrer Baldiri Reixac 10-12, 08028 Barcelona, Catalonia, Spain
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2
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Wang G, Fu Y. Spatiotemporal patterns and collective dynamics of bi-layer coupled Izhikevich neural networks with multi-area channels. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:3944-3969. [PMID: 36899611 DOI: 10.3934/mbe.2023184] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
The firing behavior and bifurcation of different types of Izhikevich neurons are analyzed firstly through numerical simulation. Then, a bi-layer neural network driven by random boundary is constructed by means of system simulation, in which each layer is a matrix network composed of 200 × 200 Izhikevich neurons, and the bi-layer neural network is connected by multi-area channels. Finally, the emergence and disappearance of spiral wave in matrix neural network are investigated, and the synchronization property of neural network is discussed. Obtained results show that random boundary can induce spiral waves under appropriate conditions, and it is clear that the emergence and disappearance of spiral wave can be observed only when the matrix neural network is constructed by regular spiking Izhikevich neurons, while it cannot be observed in neural networks constructed by other modes such as fast spiking, chattering and intrinsically bursting. Further research shows that the variation of synchronization factor with coupling strength between adjacent neurons shows an inverse bell-like curve in the form of "inverse stochastic resonance", but the variation of synchronization factor with coupling strength of inter-layer channels is a curve that is approximately monotonically decreasing. More importantly, it is found that lower synchronicity is helpful to develop spatiotemporal patterns. These results enable people to further understand the collective dynamics of neural networks under random conditions.
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Affiliation(s)
- Guowei Wang
- School of Education, Nanchang Institute of Science and Technology, Nanchang 330108, China
| | - Yan Fu
- School of Mathematics and Computer Science, Yuzhang Normal University, Nanchang 330108, China
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A New Memristive Neuron Map Model and Its Network’s Dynamics under Electrochemical Coupling. ELECTRONICS 2022. [DOI: 10.3390/electronics11010153] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
A memristor is a vital circuit element that can mimic biological synapses. This paper proposes the memristive version of a recently proposed map neuron model based on the phase space. The dynamic of the memristive map model is investigated by using bifurcation and Lyapunov exponents’ diagrams. The results prove that the memristive map can present different behaviors such as spiking, periodic bursting, and chaotic bursting. Then, a ring network is constructed by hybrid electrical and chemical synapses, and the memristive neuron models are used to describe the nodes. The collective behavior of the network is studied. It is observed that chemical coupling plays a crucial role in synchronization. Different kinds of synchronization, such as imperfect synchronization, complete synchronization, solitary state, two-cluster synchronization, chimera, and nonstationary chimera, are identified by varying the coupling strengths.
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Zakharova A, Strelkova G, Schöll E, Kurths J. Introduction to focus issue: In memory of Vadim S. Anishchenko: Statistical physics and nonlinear dynamics of complex systems. CHAOS (WOODBURY, N.Y.) 2022; 32:010401. [PMID: 35105142 DOI: 10.1063/5.0082335] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2021] [Accepted: 12/14/2021] [Indexed: 06/14/2023]
Affiliation(s)
- Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Galina Strelkova
- Institute of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Telegraphenberg A 31, 14473 Potsdam, Germany
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Rathore V, Kachhvah AD, Jalan S. Catalytic feed-forward explosive synchronization in multilayer networks. CHAOS (WOODBURY, N.Y.) 2021; 31:123130. [PMID: 34972326 DOI: 10.1063/5.0060803] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2021] [Accepted: 12/01/2021] [Indexed: 06/14/2023]
Abstract
Inhibitory couplings are crucial for the normal functioning of many real-world complex systems. Inhibition in one layer has been shown to induce explosive synchronization in another excitatory (or positive) layer of duplex networks. By extending this framework to multiplex networks, this article shows that inhibition in a single layer can act as a catalyst, leading to explosive synchronization transitions in the rest of the layers feed-forwarded through intermediate layer(s). Considering a multiplex network of coupled Kuramoto oscillators, we demonstrate that the characteristics of the transition emergent in a layer can be entirely controlled by the intra-layer coupling of other layers and the multiplexing strengths. The results presented here are essential to fathom the synchronization behavior of coupled dynamical units in multi-layer systems possessing inhibitory coupling in one of its layers, representing the importance of multiplexing.
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Affiliation(s)
- Vasundhara Rathore
- Department of Biosciences and Biomedical Engineering, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Ajay Deep Kachhvah
- Complex Systems Lab, Department of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Sarika Jalan
- Department of Biosciences and Biomedical Engineering, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
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Sawicki J, Koulen JM, Schöll E. Synchronization scenarios in three-layer networks with a hub. CHAOS (WOODBURY, N.Y.) 2021; 31:073131. [PMID: 34340334 DOI: 10.1063/5.0055835] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2021] [Accepted: 06/22/2021] [Indexed: 06/13/2023]
Abstract
We study various relay synchronization scenarios in a three-layer network, where the middle (relay) layer is a single node, i.e., a hub. The two remote layers consist of non-locally coupled rings of FitzHugh-Nagumo oscillators modeling neuronal dynamics. All nodes of the remote layers are connected to the hub. The role of the hub and its importance for the existence of chimera states are investigated in dependence on the inter-layer coupling strength and inter-layer time delay. Tongue-like regions in the parameter plane exhibiting double chimeras, i.e., chimera states in the remote layers whose coherent cores are synchronized with each other, and salt-and-pepper states are found. At very low intra-layer coupling strength, when chimera states do not exist in single layers, these may be induced by the hub. Also, the influence of the dilution of links between the remote layers and the hub upon the dynamics is investigated. The greatest effect of dilution is observed when links to the coherent domain of the chimeras are removed.
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Affiliation(s)
- Jakub Sawicki
- Potsdam Institute for Climate Impact Research, Telegrafenberg A, 31, 14473 Potsdam, Germany
| | - Julia M Koulen
- Potsdam Institute for Climate Impact Research, Telegrafenberg A, 31, 14473 Potsdam, Germany
| | - Eckehard Schöll
- Potsdam Institute for Climate Impact Research, Telegrafenberg A, 31, 14473 Potsdam, Germany
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Shepelev IA, Muni SS, Schöll E, Strelkova GI. Repulsive inter-layer coupling induces anti-phase synchronization. CHAOS (WOODBURY, N.Y.) 2021; 31:063116. [PMID: 34241296 DOI: 10.1063/5.0054770] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2021] [Accepted: 05/25/2021] [Indexed: 06/13/2023]
Abstract
We present numerical results for the synchronization phenomena in a bilayer network of repulsively coupled 2D lattices of van der Pol oscillators. We consider the cases when the network layers have either different or the same types of intra-layer coupling topology. When the layers are uncoupled, the lattice of van der Pol oscillators with a repulsive interaction typically demonstrates a labyrinth-like pattern, while the lattice with attractively coupled van der Pol oscillators shows a regular spiral wave structure. We reveal for the first time that repulsive inter-layer coupling leads to anti-phase synchronization of spatiotemporal structures for all considered combinations of intra-layer coupling. As a synchronization measure, we use the correlation coefficient between the symmetrical pairs of network nodes, which is always close to -1 in the case of anti-phase synchronization. We also study how the form of synchronous structures depends on the intra-layer coupling strengths when the repulsive inter-layer coupling is varied.
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Affiliation(s)
- Igor A Shepelev
- Institute of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - Sishu S Muni
- School of Fundamental Sciences, Massey University, Palmerston North 4410, New Zealand
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Galina I Strelkova
- Institute of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
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Andrzejak RG. Chimeras confined by fractal boundaries in the complex plane. CHAOS (WOODBURY, N.Y.) 2021; 31:053104. [PMID: 34240923 DOI: 10.1063/5.0049631] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2021] [Accepted: 04/16/2021] [Indexed: 06/13/2023]
Abstract
Complex-valued quadratic maps either converge to fixed points, enter into periodic cycles, show aperiodic behavior, or diverge to infinity. Which of these scenarios takes place depends on the map's complex-valued parameter c and the initial conditions. The Mandelbrot set is defined by the set of c values for which the map remains bounded when initiated at the origin of the complex plane. In this study, we analyze the dynamics of a coupled network of two pairs of two quadratic maps in dependence on the parameter c. Across the four maps, c is kept the same whereby the maps are identical. In analogy to the behavior of individual maps, the network iterates either diverge to infinity or remain bounded. The bounded solutions settle into different stable states, including full synchronization and desynchronization of all maps. Furthermore, symmetric partially synchronized states of within-pair synchronization and across-pair synchronization as well as a symmetry broken chimera state are found. The boundaries between bounded and divergent solutions in the domain of c are fractals showing a rich variety of intriguingly esthetic patterns. Moreover, the set of bounded solutions is divided into countless subsets throughout all length scales in the complex plane. Each individual subset contains only one state of synchronization and is enclosed within fractal boundaries by c values leading to divergence.
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Affiliation(s)
- Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Catalonia, Spain
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Bashkirtseva I, Ryashko L. Chaotic transients, riddled basins, and stochastic transitions in coupled periodic logistic maps. CHAOS (WOODBURY, N.Y.) 2021; 31:053101. [PMID: 34240946 DOI: 10.1063/5.0050613] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2021] [Accepted: 04/12/2021] [Indexed: 06/13/2023]
Abstract
A system of two coupled map-based oscillators is studied. As units, we use identical logistic maps in two-periodic modes. In this system, increasing coupling strength significantly changes deterministic regimes of collective dynamics with coexisting periodic, quasiperiodic, and chaotic attractors. We study how random noise deforms these dynamical regimes in parameter zones of mono- and bistability, causes "order-chaos" transformations, and destroys regimes of in-phase and anti-phase synchronization. In the analytical study of these noise-induced phenomena, a stochastic sensitivity technique and a method of confidence domains for periodic and multi-band chaotic attractors are used. In this analysis, a key role of chaotic transients and geometry of "riddled" basins is revealed.
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Affiliation(s)
- Irina Bashkirtseva
- Department of Theoretical and Mathematical Physics, Ural Federal University, Lenina, 51, 620000 Ekaterinburg, Russia
| | - Lev Ryashko
- Department of Theoretical and Mathematical Physics, Ural Federal University, Lenina, 51, 620000 Ekaterinburg, Russia
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Shepelev IA, Muni SS, Vadivasova TE. Synchronization of wave structures in a heterogeneous multiplex network of 2D lattices with attractive and repulsive intra-layer coupling. CHAOS (WOODBURY, N.Y.) 2021; 31:021104. [PMID: 33653058 DOI: 10.1063/5.0044327] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2021] [Accepted: 02/04/2021] [Indexed: 06/12/2023]
Abstract
We explore numerically the synchronization effects in a heterogeneous two-layer network of two-dimensional (2D) lattices of van der Pol oscillators. The inter-layer coupling of the multiplex network has an attractive character. One layer of 2D lattices is characterized by attractive coupling of oscillators and demonstrates a spiral wave regime for both local and nonlocal interactions. The oscillators in the second layer are coupled through active elements and the interaction between them has repulsive character. We show that the lattice with the repulsive type of coupling demonstrates complex spatiotemporal cluster structures, which can be called labyrinth-like structures. We show for the first time that this multiplex network with fundamentally various types of intra-layer coupling demonstrates mutual synchronization and a competition between two types of structures. Our numerical study indicates that the synchronization threshold and the type of spatiotemporal patterns in both layers strongly depend on the ratio of the intra-layer coupling strength of the two lattices. We also analyze the impact of intra-layer coupling ranges on the synchronization effects.
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Affiliation(s)
- I A Shepelev
- Department of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - S S Muni
- School of Fundamental Sciences, Massey University, Palmerston North 4442, New Zealand
| | - T E Vadivasova
- Department of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
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Vadivasova TE, Slepnev AV, Zakharova A. Control of inter-layer synchronization by multiplexing noise. CHAOS (WOODBURY, N.Y.) 2020; 30:091101. [PMID: 33003909 DOI: 10.1063/5.0023071] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2020] [Accepted: 08/25/2020] [Indexed: 06/11/2023]
Abstract
We study the synchronization of spatio-temporal patterns in a two-layer network of coupled chaotic maps, where each layer is represented by a nonlocally coupled ring. In particular, we focus on noisy inter-layer communication that we call multiplexing noise. We show that noisy modulation of inter-layer coupling strength has a significant impact on the dynamics of the network and specifically on the degree of synchronization of spatio-temporal patterns of interacting layers initially (in the absence of interaction) exhibiting chimera states. Our goal is to develop control strategies based on multiplexing noise for both identical and non-identical layers. We find that for the appropriate choice of intensity and frequency characteristics of parametric noise, complete or partial synchronization of the layers can be observed. Interestingly, for achieving inter-layer synchronization through multiplexing noise, it is crucial to have colored noise with intermediate spectral width. In the limit of white noise, the synchronization is destroyed. These results are the first step toward understanding the role of noisy inter-layer communication for the dynamics of multilayer networks.
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Affiliation(s)
- T E Vadivasova
- Department of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - A V Slepnev
- Department of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - A Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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Kuznetsov M, Polezhaev A. Widening the criteria for emergence of Turing patterns. CHAOS (WOODBURY, N.Y.) 2020; 30:033106. [PMID: 32237770 DOI: 10.1063/1.5140520] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/28/2019] [Accepted: 02/17/2020] [Indexed: 06/11/2023]
Abstract
The classical concept for emergence of Turing patterns in reaction-diffusion systems requires that a system should be composed of complementary subsystems, one of which is unstable and diffuses sufficiently slowly while the other one is stable and diffuses sufficiently rapidly. In this work, the phenomena of emergence of Turing patterns are studied and do not fit into this concept, yielding the following results. (1) The criteria are derived, under which a reaction-diffusion system with immobile species should spontaneously produce Turing patterns under any diffusion coefficients of its mobile species. It is shown for such systems that under certain sets of types of interactions between their species, Turing patterns should be produced under any parameter values, at least provided that the corresponding spatially non-distributed system is stable. (2) It is demonstrated that in a reaction-diffusion system, which contains more than two species and is stable in absence of diffusion, the presence of a sufficiently slowly diffusing unstable subsystem is already sufficient for diffusion instability (i.e., Turing or wave instability), while its complementary subsystem can also be unstable. (3) It is shown that the presence of an immobile unstable subsystem, which leads to destabilization of waves within an infinite range of wavenumbers, in a spatially discrete case can result in the generation of large-scale stationary or oscillatory patterns. (4) It is demonstrated that under the presence of subcritical Turing and supercritical wave bifurcations, the interaction of two diffusion instabilities can result in the spontaneous formation of Turing structures outside the region of Turing instability.
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Affiliation(s)
- Maxim Kuznetsov
- P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 53 Leninskiy Prospekt, Moscow119991, Russian Federation
| | - Andrey Polezhaev
- P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 53 Leninskiy Prospekt, Moscow119991, Russian Federation
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