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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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2
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Shepelev IA, Muni SS, Vadivasova TE. Spatiotemporal patterns in a 2D lattice with linear repulsive and nonlinear attractive coupling. CHAOS (WOODBURY, N.Y.) 2021; 31:043136. [PMID: 34251249 DOI: 10.1063/5.0048324] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2021] [Accepted: 04/07/2021] [Indexed: 06/13/2023]
Abstract
We explore the emergence of a variety of different spatiotemporal patterns in a 2D lattice of self-sustained oscillators, which interact nonlocally through an active nonlinear element. A basic element is a van der Pol oscillator in a regime of relaxation oscillations. The active nonlinear coupling can be implemented by a radiophysical element with negative resistance in its current-voltage curve taking into account nonlinear characteristics (for example, a tunnel diode). We show that such coupling consists of two parts, namely, a repulsive linear term and an attractive nonlinear term. This interaction leads to the emergence of only standing waves with periodic dynamics in time and absence of any propagating wave processes. At the same time, many different spatiotemporal patterns occur when the coupling parameters are varied, namely, regular and complex cluster structures, such as chimera states. This effect is associated with the appearance of new periodic states of individual oscillators by the repulsive part of coupling, while the attractive term attenuates this effect. We also show influence of the coupling nonlinearity on the spatiotemporal dynamics.
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Affiliation(s)
- I A Shepelev
- Institute of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - S S Muni
- School of Fundamental Sciences, Massey University, Palmerston North, New Zealand
| | - T E Vadivasova
- Institute of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
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3
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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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4
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Dixit S, Shrimali MD. Static and dynamic attractive-repulsive interactions in two coupled nonlinear oscillators. CHAOS (WOODBURY, N.Y.) 2020; 30:033114. [PMID: 32237763 DOI: 10.1063/1.5127249] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/2019] [Accepted: 02/16/2020] [Indexed: 06/11/2023]
Abstract
Many systems exhibit both attractive and repulsive types of interactions, which may be dynamic or static. A detailed understanding of the dynamical properties of a system under the influence of dynamically switching attractive or repulsive interactions is of practical significance. However, it can also be effectively modeled with two coexisting competing interactions. In this work, we investigate the effect of time-varying attractive-repulsive interactions as well as the hybrid model of coexisting attractive-repulsive interactions in two coupled nonlinear oscillators. The dynamics of two coupled nonlinear oscillators, specifically limit cycles as well as chaotic oscillators, are studied in detail for various dynamical transitions for both cases. Here, we show that dynamic or static attractive-repulsive interactions can induce an important transition from the oscillatory to steady state in identical nonlinear oscillators due to competitive effects. The analytical condition for the stable steady state in dynamic interactions at the low switching time period and static coexisting interactions are calculated using linear stability analysis, which is found to be in good agreement with the numerical results. In the case of a high switching time period, oscillations are revived for higher interaction strength.
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Affiliation(s)
- Shiva Dixit
- 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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5
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Sathiyadevi K, Chandrasekar VK, Senthilkumar DV, Lakshmanan M. Distinct collective states due to trade-off between attractive and repulsive couplings. Phys Rev E 2018; 97:032207. [PMID: 29776099 DOI: 10.1103/physreve.97.032207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2018] [Indexed: 06/08/2023]
Abstract
We investigate the effect of repulsive coupling together with an attractive coupling in a network of nonlocally coupled oscillators. To understand the complex interaction between these two couplings we introduce a control parameter in the repulsive coupling which plays a crucial role in inducing distinct complex collective patterns. In particular, we show the emergence of various cluster chimera death states through a dynamically distinct transition route, namely the oscillatory cluster state and coherent oscillation death state as a function of the repulsive coupling in the presence of the attractive coupling. In the oscillatory cluster state, the oscillators in the network are grouped into two distinct dynamical states of homogeneous and inhomogeneous oscillatory states. Further, the network of coupled oscillators follow the same transition route in the entire coupling range. Depending upon distinct coupling ranges, the system displays different number of clusters in the death state and oscillatory state. We also observe that the number of coherent domains in the oscillatory cluster state exponentially decreases with increase in coupling range and obeys a power-law decay. Additionally, we show analytical stability for observed solitary state, synchronized state, and incoherent oscillation death state.
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Affiliation(s)
- K Sathiyadevi
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695016, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli-620 024, Tamil Nadu, India
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6
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Sathiyadevi K, Karthiga S, Chandrasekar VK, Senthilkumar DV, Lakshmanan M. Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings. Phys Rev E 2017; 95:042301. [PMID: 28505842 DOI: 10.1103/physreve.95.042301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2016] [Indexed: 06/07/2023]
Abstract
Spontaneous symmetry breaking is an important phenomenon observed in various fields including physics and biology. In this connection, we here show that the trade-off between attractive and repulsive couplings can induce spontaneous symmetry breaking in a homogeneous system of coupled oscillators. With a simple model of a system of two coupled Stuart-Landau oscillators, we demonstrate how the tendency of attractive coupling in inducing in-phase synchronized (IPS) oscillations and the tendency of repulsive coupling in inducing out-of-phase synchronized oscillations compete with each other and give rise to symmetry breaking oscillatory states and interesting multistabilities. Further, we provide explicit expressions for synchronized and antisynchronized oscillatory states as well as the so called oscillation death (OD) state and study their stability. If the Hopf bifurcation parameter (λ) is greater than the natural frequency (ω) of the system, the attractive coupling favors the emergence of an antisymmetric OD state via a Hopf bifurcation whereas the repulsive coupling favors the emergence of a similar state through a saddle-node bifurcation. We show that an increase in the repulsive coupling not only destabilizes the IPS state but also facilitates the reentrance of the IPS state.
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Affiliation(s)
- K Sathiyadevi
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamil Nadu, India
| | - S Karthiga
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamil Nadu, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695 016, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
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7
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Farajzadeh Tehrani N, Razvan M. Bifurcation structure of two coupled FHN neurons with delay. Math Biosci 2015; 270:41-56. [PMID: 26476143 DOI: 10.1016/j.mbs.2015.09.008] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2015] [Revised: 09/12/2015] [Accepted: 09/28/2015] [Indexed: 11/17/2022]
Abstract
This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified.
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Affiliation(s)
| | - MohammadReza Razvan
- Department of Mathematical Sciences, Sharif University of Technology, Tehran 1497945961, Iran
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Yao C, Yi M, Shuai J. Time delay induced different synchronization patterns in repulsively coupled chaotic oscillators. CHAOS (WOODBURY, N.Y.) 2013; 23:033140. [PMID: 24089976 DOI: 10.1063/1.4821942] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Time delayed coupling plays a crucial role in determining the system's dynamics. We here report that the time delay induces transition from the asynchronous state to the complete synchronization (CS) state in the repulsively coupled chaotic oscillators. In particular, by changing the coupling strength or time delay, various types of synchronous patterns, including CS, antiphase CS, antiphase synchronization (ANS), and phase synchronization, can be generated. In the transition regions between different synchronous patterns, bistable synchronous oscillators can be observed. Furthermore, we show that the time-delay-induced phase flip bifurcation is of key importance for the emergence of CS. All these findings may light on our understanding of neuronal synchronization and information processing in the brain.
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Affiliation(s)
- Chenggui Yao
- Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, People's Republic of China
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9
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Ciszak M, Euzzor S, Arecchi FT, Meucci R. Experimental study of firing death in a network of chaotic FitzHugh-Nagumo neurons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022919. [PMID: 23496603 DOI: 10.1103/physreve.87.022919] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2012] [Revised: 11/13/2012] [Indexed: 06/01/2023]
Abstract
The FitzHugh-Nagumo neurons driven by a periodic forcing undergo a period-doubling route to chaos and a transition to mixed-mode oscillations. When coupled, their dynamics tend to be synchronized. We show that the chaotically spiking neurons change their internal dynamics to subthreshold oscillations, the phenomenon referred to as firing death. These dynamical changes are observed below the critical coupling strength at which the transition to full chaotic synchronization occurs. Moreover, we find various dynamical regimes in the subthreshold oscillations, namely, regular, quasiperiodic, and chaotic states. We show numerically that these dynamical states may coexist with large-amplitude spiking regimes and that this coexistence is characterized by riddled basins of attraction. The reported results are obtained for neurons implemented in the electronic circuits as well as for the model equations. Finally, we comment on the possible scenarios where the coupling-induced firing death could play an important role in biological systems.
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Affiliation(s)
- Marzena Ciszak
- CNR-Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Florence, Italy.
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10
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Horikawa Y, Kitajima H. Transient chaotic rotating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol oscillators near a codimension-two bifurcation point. CHAOS (WOODBURY, N.Y.) 2012; 22:033115. [PMID: 23020454 DOI: 10.1063/1.4737430] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Propagating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol (BVP) oscillators were studied. The parameter values of the BVP oscillators were near a codimension-two bifurcation point around which oscillatory, monostable, and bistable states coexist. Bifurcations of periodic, quasiperiodic, and chaotic rotating waves were found in a ring of three oscillators. In rings of large numbers of oscillators with small coupling strength, transient chaotic waves were found and their duration increased exponentially with the number of oscillators. These exponential chaotic transients could be described by a coupled map model derived from the Poincaré map of a ring of three oscillators. The quasiperiodic rotating waves due to the mode interaction near the codimension-two bifurcation point were evidently responsible for the emergence of the transient chaotic rotating waves.
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Affiliation(s)
- Yo Horikawa
- Faculty of Engineering, Kagawa University, Takamatsu 761-0396, Japan.
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11
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Wang Q, Chen G, Perc M. Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling. PLoS One 2011; 6:e15851. [PMID: 21253015 PMCID: PMC3017050 DOI: 10.1371/journal.pone.0015851] [Citation(s) in RCA: 249] [Impact Index Per Article: 19.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2010] [Accepted: 11/25/2010] [Indexed: 11/18/2022] Open
Abstract
This paper investigates the dependence of synchronization transitions of bursting oscillations on the information transmission delay over scale-free neuronal networks with attractive and repulsive coupling. It is shown that for both types of coupling, the delay always plays a subtle role in either promoting or impairing synchronization. In particular, depending on the inherent oscillation period of individual neurons, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions are manifested as well-expressed minima in the measure for spatiotemporal synchrony. For attractive coupling, the minima appear at every integer multiple of the average oscillation period, while for the repulsive coupling, they appear at every odd multiple of the half of the average oscillation period. The obtained results are robust to the variations of the dynamics of individual neurons, the system size, and the neuronal firing type. Hence, they can be used to characterize attractively or repulsively coupled scale-free neuronal networks with delays.
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Affiliation(s)
- Qingyun Wang
- Department of Dynamics and Control, Beihang University, Beijing, China
| | - Guanrong Chen
- Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
| | - Matjaž Perc
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia
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12
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Chen Y, Xiao J, Liu W, Li L, Yang Y. Dynamics of chaotic systems with attractive and repulsive couplings. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:046206. [PMID: 19905414 DOI: 10.1103/physreve.80.046206] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2009] [Revised: 08/01/2009] [Indexed: 05/28/2023]
Abstract
Together with attractive couplings, repulsive couplings play crucial roles in determining important evolutions in natural systems, such as in learning and oscillatory processes of neural networks. The complex interactions between them have great influence on the systems. A detailed understanding of the dynamical properties under this type of couplings is of practical significance. In this paper, we propose a model to investigate the dynamics of attractive and repulsive couplings, which give rise to rich phenomena, especially for amplitude death (AD). The relationship among various dynamics and possible transitions to AD are illustrated. When the system is in the maximally stable AD, we observe the transient behavior of in-phase (high frequency) and out-of-phase (low frequency) motions. The mechanism behind the phenomenon is given.
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Affiliation(s)
- Yuehua Chen
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
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13
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Ciszak M, Montina A, Arecchi FT. Control of transient synchronization with external stimuli. CHAOS (WOODBURY, N.Y.) 2009; 19:015104. [PMID: 19335008 DOI: 10.1063/1.3080195] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
A network of coupled chaotic oscillators can switch spontaneously to a state of collective synchronization at some critical coupling strength. We show that for a locally coupled network of units with coexisting quiescence and chaotic spiking states, set slightly below the critical coupling value, the collective excitable or bistable states of synchronization arise in response to a stimulus applied to a single node. We provide an explanation of this behavior and show that it is due to a combination of the dynamical properties of a single node and the coupling topology. By the use of entropy as a collective indicator, we present a new method for controlling the transient synchronization.
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Affiliation(s)
- Marzena Ciszak
- CNR-Istituto Nazionale di Ottica Applicata, Firenze, Italy
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14
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Yanagita T, Suetani H, Aihara K. Bifurcation analysis of solitary and synchronized pulses and formation of reentrant waves in laterally coupled excitable fibers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:056208. [PMID: 19113201 DOI: 10.1103/physreve.78.056208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2008] [Indexed: 05/27/2023]
Abstract
We study the dynamics of a reaction-diffusion system comprising two mutually coupled excitable fibers. We consider a case in which the dynamical properties of the two fibers are nonidentical due to the parameter mismatch between them. By using the spatially one-dimensional FitzHugh-Nagumo equations as a model of a single excitable fiber, synchronized pulses are found to be stable in some parameter regime. Furthermore, there exists a critical coupling strength beyond which the synchronized pulses are stable for any amount of parameter mismatch. We show the bifurcation structures of the synchronized and solitary pulses and identify a codimension-2 cusp singularity as the source of the destabilization of synchronized pulses. When stable solitary pulses in both fibers disappear via a saddle-node bifurcation on increasing the coupling strength, a reentrant wave is formed. The parameter region, where a stable reentrant wave is observed in direct numerical simulation, is consistent with that obtained by bifurcation analysis.
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Affiliation(s)
- Tatsuo Yanagita
- Research Institute for Electronic Science, Hokkaido University, Sapporo 001-0020, Japan.
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15
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Ciszak M, Montina A, Arecchi FT. Spike synchronization of chaotic oscillators as a phase transition. Cogn Process 2008; 10 Suppl 1:S33-9. [PMID: 18936995 DOI: 10.1007/s10339-008-0235-x] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2008] [Revised: 09/26/2008] [Accepted: 09/30/2008] [Indexed: 10/21/2022]
Abstract
We study how a locally coupled array of spiking chaotic systems synchronizes to an external driving in a short time. Synchronization means spike separation at adjacent sites much shorter than the average inter-spike interval; a local lack of synchronization is called a defect. The system displays sudden spontaneous defect disappearance at a critical coupling strength suggesting an existence of a phase transition. Below critical coupling, the system reaches order at a definite amplitude of an external input; this order persists for a fixed time slot. Thus, the array behaves as an excitable-like system, even though the single element lacks such a property.
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Affiliation(s)
- M Ciszak
- C.N.R.-Istituto Nazionale di Ottica Applicata, L.go E. Fermi 6, 50125, Florence, Italy.
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16
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Nekhamkina O, Sheintuch M. Boundary-induced patterns in excitable systems: the structure of oscillatory domain. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:056210. [PMID: 17677151 DOI: 10.1103/physreve.75.056210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2006] [Revised: 02/12/2007] [Indexed: 05/16/2023]
Abstract
The present work extends our recently published study [Phys. Rev. E 73, 066224 (2006)] on a mechanism of pattern formation in excitable media due to inhomogeneous boundary conditions (BC). To that end, we analyze a pair of coupled excitable and oscillatory cells, a distributed FitzHugh Nagumo model, and a distributed five-variable model that describes CO catalytic oxidation. For the three systems we determine the structure of the oscillatory domains, composed of bands of complex firing solutions with period-adding bifurcations, and show the commonality of the structures. The obtained results account for the recently reported experimental observations of mixed-mode oscillations showing a period-adding bifurcation during CO oxidation on a disk-shaped catalytic cloth with imposed cold temperature BC.
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Affiliation(s)
- Olga Nekhamkina
- Department of Chemical Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel
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17
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Sato YD, Shiino M. Generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:011909. [PMID: 17358186 DOI: 10.1103/physreve.75.011909] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2006] [Revised: 07/23/2006] [Indexed: 05/14/2023]
Abstract
The integrate-and-fire (IF) neuron model and the piecewise-linear version of FitzHugh-Nagumo (PL) neuron model with a time scale parameter mu are frequently being used in the study of synchronization phenomena. Although the two models are regarded to be different type, we show a certain equivalence between them by deriving the coupled IF model of an improved version with a firing duration from the recovery variable coupled system of the PL model, under taking the limit of mu-->0 without a loss of any coupling properties. In the coupled IF model with the duration time, the synchronization behavior of a pair of neurons with excitatory or inhibitory synaptic coupling can be systematically explored in terms of three parameters in the model (synaptic strength, decaying relaxation rate of the synaptic coupling, and the parameter exhibiting the firing duration). We find some irregularly synchronous behavior with or without constant firing order alternations. We show that the duration of an impulse plays an important role in synchronization phenomena.
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Affiliation(s)
- Yasuomi Daishin Sato
- Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany.
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