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Anupong S, Schreiber I, Kheowan OU. Turbulent pattern in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. Phys Chem Chem Phys 2020; 22:28213-28221. [PMID: 33295367 DOI: 10.1039/d0cp04112b] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Chemical turbulence was observed experimentally in the 1,4-cyclohexanedione Belousov-Zhabotinsky (CHD-BZ) reaction in a double layer consisting of a catalyst-loaded gel and uncatalyzed liquid on a Petri dish. The chemical patterns in the CHD-BZ reaction occur spontaneously in various forms as follows: the initial, regular, transient, and turbulent patterns, subsequently. These four patterns are characterized by using the two-dimensional Fourier transform (2D-FT). Mechanism of the onset of the turbulence in the CHD-BZ reaction is proposed. Turbulence in the CHD-BZ reaction is reproducible under a well defined protocol and it exists for a period of time of about 50 minutes, which is sufficiently long to offer a good opportunity to study and control the turbulence in the future. Two models of the BZ reaction were used to simulate the spiral breakup. Both are capable of producing spiral turbulence from initially regular patterns in each layer and reflect certain features of dynamics observed in experiments.
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Affiliation(s)
- Suparinthon Anupong
- Department of Chemistry, Mahidol University, Rama 6 Road, Bangkok 10400, Thailand.
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2
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Gao J, Gu C, Yang H, Weng T. Excited state of spiral waves in oscillatory reaction-diffusion systems caused by a pulse. Phys Rev E 2020; 101:042219. [PMID: 32422728 DOI: 10.1103/physreve.101.042219] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2019] [Accepted: 04/09/2020] [Indexed: 11/07/2022]
Abstract
Previous studies claim that the dynamic behaviors of spiral waves are uniquely determined by the nature of the medium, which can be determined by control parameters. In this article, the authors break from the previous view and present an alternate stable state of spiral waves, named the excited state. The authors find that two states of the spiral wave switch to each other after a one-off pulse is applied to the medium. The dynamic behaviors of the two states are quite different, specifically, the spiral tip trajectory of the original spiral, which is named the ground-state spiral as observed in the previous studies, is a point, while the spiral tip trajectory of the excited-state spiral is a circle. Moreover, the authors study the trajectories of the spiral tip of spiral waves in both states after the pulse is applied and find two types of trajectories, a spiral trajectory and a spiral-inward-petal trajectory. The frequency of the spiral wave in the excited state is less than that in the ground state. The findings enrich the dynamics of pattern formation.
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Affiliation(s)
- Jian Gao
- University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China
| | - Changgui Gu
- University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China
| | - Huijie Yang
- University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China
| | - Tongfeng Weng
- University of Shanghai for Science and Technology, Shanghai 200093, People's Republic of China
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3
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Taylor AF. Mechanism and Phenomenology of an Oscillating Chemical Reaction. PROGRESS IN REACTION KINETICS AND MECHANISM 2019. [DOI: 10.3184/007967402103165414] [Citation(s) in RCA: 97] [Impact Index Per Article: 19.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Chemical reactions, which are far from equilibrium, are capable of displaying oscillations in species concentrations and hence in colour, electrode potential, pH and/or temperature. The oscillations arise from the interplay between positive and negative kinetic feedback. Mechanisms for such reactions are presented, along with the rich phenomenology that these systems exhibit, from complex oscillations and chemical waves, to stationary concentration patterns. This review will focus on the Belousov-Zhabotinksy reaction but reference to other reactions will be made where appropriate.
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4
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Toth R, Taylor AF. The Tris(2,2'-Bipyridyl)Ruthenium-Catalysed Belousov–Zhabotinsky Reaction. PROGRESS IN REACTION KINETICS AND MECHANISM 2019. [DOI: 10.3184/007967406779946928] [Citation(s) in RCA: 36] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
The Belousov – Zhabotinsky (BZ) reaction is the prototypical oscillating chemical reaction. The tris(2,2'-bipyridine)ruthenium-catalysed BZ reaction (often simply referred to as the ruthenium-catalysed BZ reaction) displays photosensitivity and has been widely exploited for examination of the effects of illumination on nonlinear reaction kinetics. In this review, we investigate the behaviour of the ruthenium-catalysed BZ reaction. The mechanism of the reaction is analysed and we examine how light sensitivity is incorporated into kinetic models of the reaction. The temporal dynamics of the photosensitive reaction is presented and, finally, we discuss the extraordinary wealth of behaviour that has been observed in the spatially-distributed system when perturbed by visible light.
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Affiliation(s)
- Rita Toth
- University of the West of England, Bristol, UK
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5
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Jiang H, Hou Z. Formation of spiral waves with substructure in a bursting media. CHAOS (WOODBURY, N.Y.) 2015; 25:123105. [PMID: 26723144 DOI: 10.1063/1.4936794] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Formation of spiral waves in a bursting media is investigated. Due to the multiple timescale oscillation of the local dynamics, an interesting substructure of traveling wave (STW) is observed in the spiral arm. As a result of the special moving media formed by neurons in the spiral arm, STWs propagate from the spiral tip to far field with an increasing wave length and move faster along the front of the spiral arm than along the back, leading to the formation of fragments in STWs. Moreover, we find that a sharp change of stimulus current can lead to backfiring of STWs, which may break the spiral wave front and further result in the formation of a multi-spiral pattern.
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Affiliation(s)
- Huijun Jiang
- Department of Chemical Physics and Hefei National Laboratory for Physical Sciences at Microscales, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zhonghuai Hou
- Department of Chemical Physics and Hefei National Laboratory for Physical Sciences at Microscales, University of Science and Technology of China, Hefei, Anhui 230026, China
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6
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Español MI, Rotstein HG. Complex mixed-mode oscillatory patterns in a periodically forced excitable Belousov-Zhabotinsky reaction model. CHAOS (WOODBURY, N.Y.) 2015; 25:064612. [PMID: 26117137 DOI: 10.1063/1.4922715] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The Oregonator is the simplest chemically plausible model for the Belousov-Zhabotinsky reaction. We investigate the response of the Oregonator to sinusoidal inputs with amplitudes and frequencies within plausible ranges. We focus on a regime where the unforced Oregonator is excitable (with no sustained oscillations). We use numerical simulations and dynamical systems tools to both characterize the response patterns and explain the underlying dynamic mechanisms.
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Affiliation(s)
- Malena I Español
- Department of Mathematics, The University of Akron, Akron, Ohio 44325, USA
| | - Horacio G Rotstein
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
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7
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Majumder R, Pandit R, Panfilov AV. Turbulent electrical activity at sharp-edged inexcitable obstacles in a model for human cardiac tissue. Am J Physiol Heart Circ Physiol 2014; 307:H1024-35. [DOI: 10.1152/ajpheart.00593.2013] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
Abstract
Wave propagation around various geometric expansions, structures, and obstacles in cardiac tissue may result in the formation of unidirectional block of wave propagation and the onset of reentrant arrhythmias in the heart. Therefore, we investigated the conditions under which reentrant spiral waves can be generated by high-frequency stimulation at sharp-edged obstacles in the ten Tusscher-Noble-Noble-Panfilov (TNNP) ionic model for human cardiac tissue. We show that, in a large range of parameters that account for the conductance of major inward and outward ionic currents of the model [fast inward Na+ current ( INa), L—type slow inward Ca2+ current ( ICaL), slow delayed-rectifier current ( IKs), rapid delayed-rectifier current ( IKr), inward rectifier K+ current ( IK1)], the critical period necessary for spiral formation is close to the period of a spiral wave rotating in the same tissue. We also show that there is a minimal size of the obstacle for which formation of spirals is possible; this size is ∼2.5 cm and decreases with a decrease in the excitability of cardiac tissue. We show that other factors, such as the obstacle thickness and direction of wave propagation in relation to the obstacle, are of secondary importance and affect the conditions for spiral wave initiation only slightly. We also perform studies for obstacle shapes derived from experimental measurements of infarction scars and show that the formation of spiral waves there is facilitated by tissue remodeling around it. Overall, we demonstrate that the formation of reentrant sources around inexcitable obstacles is a potential mechanism for the onset of cardiac arrhythmias in the presence of a fast heart rate.
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Affiliation(s)
- Rupamanjari Majumder
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, India
- Laboratory of Experimental Cardiology, Department of Cardiology, Leiden University Medical Center, Leiden, The Netherlands
| | - Rahul Pandit
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, India
- Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India
| | - A. V. Panfilov
- Department of Physics and Astronomy, Gent University, Ghent, Belgium; and
- Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow Region, Russia
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8
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Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study. INTERNATIONAL SCHOLARLY RESEARCH NOTICES 2014; 2014:459675. [PMID: 27379274 PMCID: PMC4897501 DOI: 10.1155/2014/459675] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/11/2014] [Revised: 09/29/2014] [Accepted: 10/03/2014] [Indexed: 11/29/2022]
Abstract
The determination of the mechanisms of spiral breakup in excitable media is still an open problem for researchers. In the context of cardiac electrophysiological activities, spiral breakup exhibits complex spatiotemporal pattern known as ventricular fibrillation. The latter is the major cause of sudden cardiac deaths all over the world. In this paper, we numerically study the instability of periodic planar traveling wave solution in two dimensions. The emergence of stable spiral pattern is observed in the considered model. This pattern occurs when the heart is malfunctioning (i.e., ventricular tachycardia). We show that the spiral wave breakup is a consequence of the transverse instability of the planar traveling wave solutions. The alternans, that is, the oscillation of pulse widths, is observed in our simulation results. Moreover, we calculate the widths of spiral pulses numerically and observe that the stable spiral pattern bifurcates to an oscillatory wave pattern in a one-parameter family of solutions. The spiral breakup occurs far below the bifurcation when the maximum and the minimum excited states become more distinct, and hence the alternans becomes more pronounced.
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9
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Dähmlow P, Vanag VK, Müller SC. Effect of solvents on the pattern formation in a Belousov-Zhabotinsky reaction embedded into a microemulsion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:010902. [PMID: 24580160 DOI: 10.1103/physreve.89.010902] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2013] [Indexed: 06/03/2023]
Abstract
Using the ferroin- and the bathoferroin-catalyzed Belousov-Zhabotinsky (BZ) reaction embedded in the sodium-bis (2-ethylhexyl) sulfosuccinate (AOT) water-in-oil microemulsion, we observed different patterns occurring in two different solvents, hexane and octane. Turing patterns were found in both solvents with ferroin. They differ in their interaction with coexisting bulk oscillations, such that a new excitation front was formed around the evolving Turing patterns in hexane. However, in octane, the bulk oscillation merged with the evolving patterns, forming a new excitation front, which propagated into two directions: towards the existing patterns and away from them. For the bathoferroin-catalyzed BZ reaction, patterns like dash waves, jumping waves, and bubble waves were found in both solvents having different wavelengths. A curvature dependence of the splitting and merging of dashes was found.
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Affiliation(s)
- Patricia Dähmlow
- Institute of Experimental Physics, Otto-von-Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
| | - Vladimir K Vanag
- Chemical-Biological Institute, Immanuel Kant Baltic Federal University, A. Nevskogo 14, 236041 Kaliningrad, Russia
| | - Stefan C Müller
- Institute of Experimental Physics, Otto-von-Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
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10
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Mahmud F, Shiozawa N, Makikawa M, Nomura T. Reentrant excitation in an analog-digital hybrid circuit model of cardiac tissue. CHAOS (WOODBURY, N.Y.) 2011; 21:023121. [PMID: 21721763 DOI: 10.1063/1.3597645] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/14/2023]
Abstract
We propose an analog-digital hybrid circuit model of one-dimensional cardiac tissue with hardware implementation that allows us to perform real-time simulations of spatially conducting cardiac action potentials. Each active nodal compartment of the tissue model is designed using analog circuits and a dsPIC microcontroller, by which the time-dependent and time-independent nonlinear current-voltage relationships of six types of ion channel currents employed in the Luo-Rudy phase I (LR-I) model for a single mammalian cardiac ventricular cell can be reproduced quantitatively. Here, we perform real-time simulations of reentrant excitation conduction in a ring-shaped tissue model that includes eighty nodal compartments. In particular, we show that the hybrid tissue model can exhibit real-time dynamics for initiation of reentries induced by uni-directional block, as well as those for phase resetting that leads to annihilation of the reentry in response to impulsive current stimulations at appropriate nodes and timings. The dynamics of the hybrid model are comparable to those of a spatially distributed tissue model with LR-I compartments. Thus, it is conceivable that the hybrid model might be a useful tool for large scale simulations of cardiac tissue dynamics, as an alternative to numerical simulations, leading toward further understanding of the reentrant mechanisms.
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Affiliation(s)
- Farhanahani Mahmud
- Graduate School of Engineering Science, Osaka University, Osaka 5608534, Japan
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11
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Roncaglia DI, Carballido-Landeira J, Muñuzuri AP. Pattern formation in the Belousov–Zhabotinsky-PAMAM dendrimer system. Phys Chem Chem Phys 2011; 13:7426-32. [DOI: 10.1039/c0cp02656e] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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12
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Gao Q, Zhang L, Wang Q, Epstein IR. Arm splitting and backfiring of spiral waves in media displaying local mixed-mode oscillations. CHAOS (WOODBURY, N.Y.) 2009; 19:013135. [PMID: 19334999 DOI: 10.1063/1.3093047] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
The behavior of spiral waves is investigated in a model of reaction-diffusion media supporting local mixed-mode oscillations for a range of values of a control parameter. This local behavior is accompanied by the formation of nodes, at which the arms of the simple spiral waves begin to split. With further parameter changes, this nodal structure loses stability, becoming quite irregular, eventually evolving into turbulence, while the local dynamics increases in complexity. The breakup of the spiral waves arises from a backfiring instability of the nodes induced by the arm splitting. This process of spiral breakup in the presence of mixed-mode oscillations represents an alternative to previously described scenarios of instability of line defects and superspirals in media with period-doubling and quasiperiodic oscillations, respectively.
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Affiliation(s)
- Qingyu Gao
- College of Chemical Engineering, China University of Mining and Technology, Xuzhou, People's Republic of China
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13
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Sang H, He K. Property change of unstable fixed point and phase synchronization in controlling spatiotemporal chaos by a periodic signal. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:036202. [PMID: 17930318 DOI: 10.1103/physreve.76.036202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2007] [Indexed: 05/25/2023]
Abstract
Mechanisms for the suppression of spatiotemporal chaos (STC) in one-dimensional driven drift-wave system to a spatially regular state by a periodic signal are investigated. In the driving wave coordinate, by transforming the system to a set of coupled oscillators (modes) moving in a periodic potential, it is found that the modes can be enslaved one by one through phase synchronization (PS) by the control signal; for some modes frequency-locking occurs while the other modes display multilooping PS without frequency-locking. Further study of the linear behavior of the modes shows that the saddle point embedded in the STC is changed to an unstable focus, which makes it possible for the imperfect PS to change to a perfect functional one, leading to the suppression of the STC.
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Affiliation(s)
- Haibo Sang
- The Key Laboratory of Beam and Materials Modification of Education Ministry, Beijing Normal University, 100875, Beijing, China
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14
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Zhang L, Gao Q, Wang Q, Wang H, Wang J. Spiral turbulence developed through the formation of superimposed target waves in an oscillatory reaction-diffusion medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:046112. [PMID: 17155138 DOI: 10.1103/physreve.74.046112] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2006] [Revised: 07/28/2006] [Indexed: 05/12/2023]
Abstract
An approach leading to the development of spiral turbulence is reported here in an oscillatory reaction-diffusion medium, which is through the spontaneous formation of targetlike waves near the core of a spiral wave. The newly formed target wave emerges with its own characteristic frequency and propagates on top of the original spiral wave, which eventually leads to the breakup of the spiral at a location far from the spiral center. The radius of the surviving spiral segment decreases rapidly with the bifurcation control parameter. Calculation of power spectra suggests that the meandering of the spiral tip is responsible for the onset of the superimposed target and the phase desynchronization of the superimposed target waves.
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Affiliation(s)
- Lu Zhang
- College of Chemical Engineering, China University of Mining and Technology, Xuzhou 221008, People's Republic of China
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Cassidy I, Müller SC. Desensitization effects in the ruthenium-catalyzed Belousov-Zhabotinsky reaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:026206. [PMID: 17025524 DOI: 10.1103/physreve.74.026206] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2005] [Revised: 05/11/2006] [Indexed: 05/12/2023]
Abstract
The influence of visible light on the velocities of spiral waves in the [Ru(bpy)](3)(2+) -catalyzed Belousov-Zhabotinsky (BZ) reaction is well documented. However, there are only few reports showing the effect of the way a change in the applied intensity is made, or on "desensitization" or "memory" type phenomena. In this paper, we present observations showing significant changes in spiral tip dynamics without varying the light intensity during the course of the experiments. We produce further evidence showing that changes in wave velocity and inhibitory effects are depending on whether the increase in intensity is applied in one large step or in a number of smaller steps. Also, the tip trajectories before and after the spiral waves have been subjected to an increase and subsequent decrease in intensity levels are different, suggesting a change in excitation of the system. The experimental results are separated into two groups depending on the light sensitivity of the system and the behavior of the spiral tip. Simulation results demonstrate that the different tip trajectories observed in the experiments can be modeled by only varying the excitation threshold. Our observations indicate that there must be at least two different competing pathways for the reaction mechanism not only in the oscillatory BZ system but also in excitable media and that intermediates may also play an important part in determining the excitation of the system and not just the initial concentrations of the reactants.
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Affiliation(s)
- Iris Cassidy
- Institut für Experimentelle Physik, Otto-von-Guericke-Universität Magdeburg, D-39106 Magdeburg, Germany
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16
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Yang L, Berenstein I, Epstein IR. Segmented waves from a spatiotemporal transverse wave instability. PHYSICAL REVIEW LETTERS 2005; 95:038303. [PMID: 16090777 DOI: 10.1103/physrevlett.95.038303] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2005] [Indexed: 05/03/2023]
Abstract
We observe traveling waves emitted from Turing spots in the chlorine dioxide-iodine-malonic acid reaction. The newborn waves are continuous, but they break into segments as they propagate, and the propagation of these segments ultimately gives rise to spatiotemporal chaos. We model the wave-breaking process and the motion of the chaotic segments. We find stable segmented spirals as well. We attribute the segmentation to an interaction between front rippling via a transverse instability and front symmetry breaking by a fast-diffusing inhibitor far from the codimension-2 Hopf-Turing bifurcation, and the chaos to a secondary instability of the periodic segmentation.
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Affiliation(s)
- Lingfa Yang
- Department of Chemistry and Volen Center for Complex Systems, MS 015, Brandeis University, Waltham, MA 02454-9110, USA
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17
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Marts B, Martinez K, Lin AL. Front dynamics in an oscillatory bistable Belousov-Zhabotinsky chemical reaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:056223. [PMID: 15600744 DOI: 10.1103/physreve.70.056223] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2004] [Indexed: 05/24/2023]
Abstract
We observe breathing front dynamics which select three distinct types of bistable patterns in the 2:1 resonance regime of the periodically forced oscillatory Belousov-Zhabotinsky reaction. We measure the curvature-driven shrinking of a circular domain R approximately t(1/2) at forcing frequencies below a specific value, and show that the fast time scale front oscillations (breathing) drive this slow time scale shrinking. Above a specific frequency, we observe fronts of higher curvature grow instead of shrink and labyrinth patterns form. Just below the transition frequency is a relatively narrow range of frequencies where the curvature-driven coarsening is balanced by a competing front interaction, which leads to a pattern of localized structures. The length scale of the localized structure and labyrinth patterns is set by the front interactions.
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Affiliation(s)
- Bradley Marts
- Center for Nonlinear and Complex Systems and Department of Physics, Duke University, Durham, North Carolina 27708, USA
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18
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Vanag VK, Epstein IR. From the Cover: Segmented spiral waves in a reaction-diffusion system. Proc Natl Acad Sci U S A 2003; 100:14635-8. [PMID: 14645709 PMCID: PMC299750 DOI: 10.1073/pnas.2534816100] [Citation(s) in RCA: 106] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2003] [Indexed: 11/18/2022] Open
Abstract
Pattern formation in reaction-diffusion systems is often invoked as a mechanism for biological morphogenesis. Patterns in chemical systems typically occur either as propagating waves or as stationary, spatially periodic, Turing structures. The spiral and concentric (target) waves found to date in spatially extended chemical or physical systems are smooth and continuous; only living systems, such as seashells, lichens, pine cones, or flowers, have been shown to demonstrate segmentation of these patterns. Here, we report observations of segmented spiral and target waves in the Belousov-Zhabotinsky reaction dispersed in water nanodroplets of a water-in-oil microemulsion. These highly ordered chemical patterns, consisting of short wave segments regularly separated by gaps, form a link between Turing and trigger wave patterns and narrow the disparity between chemistry and biology. They exhibit aspects of such fundamental biological behavior as self-replication of structural elements and preservation of morphology during evolutionary development from a simpler precursor to a more complex structure.
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Affiliation(s)
- Vladimir K Vanag
- Department of Chemistry and Volen Center for Complex Systems, Brandeis University, Waltham, MA 02454, USA.
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19
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Vanag VK, Epstein IR. Dash waves in a reaction-diffusion system. PHYSICAL REVIEW LETTERS 2003; 90:098301. [PMID: 12689258 DOI: 10.1103/physrevlett.90.098301] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2002] [Indexed: 05/24/2023]
Abstract
Patterns in reaction-diffusion systems generally consist of smooth traveling waves or of stationary, discontinuous Turing structures. Hybrid patterns that blend the properties of waves and Turing structures have not previously been observed. We report observation of dash waves, which consist of wave segments regularly separated by gaps, moving coherently in the Belousov-Zhabotinsky system dispersed in water-in-oil microemulsion. Dash waves emerge from the interaction between excitable and pseudo-Turing-unstable steady states. We are able to generate dash waves in simulations with simple models.
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Affiliation(s)
- Vladimir K Vanag
- Department of Chemistry and Volen Center for Complex Systems, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110, USA
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20
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Neufeld Z, López C, Hernández-García E, Piro O. Excitable media in open and closed chaotic flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:066208. [PMID: 12513385 DOI: 10.1103/physreve.66.066208] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2002] [Indexed: 05/24/2023]
Abstract
We investigate the response of an excitable medium to a localized perturbation in the presence of a two-dimensional smooth chaotic flow. Two distinct types of flows are numerically considered: open and closed. For both of them three distinct regimes are found, depending on the relative strengths of the stirring and the rate of the excitable reaction. In order to clarify and understand the role of the many competing mechanisms present, simplified models of the process are introduced. They are one-dimensional baker-map models for the flow and a one-dimensional approximation for the transverse profile of the filaments observed in the concentration patterns.
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Affiliation(s)
- Zoltán Neufeld
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, United Kingdom
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21
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Woltering M, Markus M. Oscillations and turbulence induced by an activating agent in an active medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:045601. [PMID: 11690082 DOI: 10.1103/physreve.64.045601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2001] [Indexed: 05/23/2023]
Abstract
An excitable Belousov-Zhabotinsky reagent becomes oscillatory above a threshold of methanol concentration [Me]. The oscillation period decreases with increasing [Me]. A model describes these observations quantitatively. In a spatiotemporal setup, a [Me] gradient causes waves with spatially varying properties; this leads to wave breaks that end up in turbulence, both in experiments and in simulations with partial differential equations.
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Affiliation(s)
- M Woltering
- Max-Planck-Institut für Molekulare Physiologie, Postfach 500247, D-44202 Dortmund, Germany.
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Neufeld Z. Excitable media in a chaotic flow. PHYSICAL REVIEW LETTERS 2001; 87:108301. [PMID: 11531505 DOI: 10.1103/physrevlett.87.108301] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2000] [Revised: 03/26/2001] [Indexed: 05/23/2023]
Abstract
The response to a localized perturbation of an excitable medium under stirring by chaotic advection is investigated. It is found that below a critical stirring rate a localized perturbation produces a coherent global excitation of the system. For very slow stirring, however, the coherence of the global excitation is gradually lost. We propose a simple model to describe the effect of the flow on the excitable dynamics, and explain the observed behavior as a consequence of a steady excited filament state found in the reduced problem.
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Affiliation(s)
- Z Neufeld
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom
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Perez-Villar V, Munuzuri AP, Perez-Munuzuri V. Convective structures in a two-layer gel-liquid excitable medium. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:3771-3776. [PMID: 11088156 DOI: 10.1103/physreve.61.3771] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/1999] [Indexed: 05/23/2023]
Abstract
The onset of convection due to wave propagation is investigated in the framework of the Belousov-Zhabotinsky reaction. Numerical calculations are based on a three variable Oregonator model coupled with the Navier-Stokes hydrodynamic equations under the Boussineq approximation for a system consisting of two layers, a liquid and a gel, both in close contact through an interface where chemical concentration exchange is allowed. The influence on the formation of convective rolls associated to wave front propagation is studied in terms of the exchange rate through the interface, the liquid layer width, and the coupling strength between the fluid flow and chemical dynamics. Waves are initiated on the surface of the gel and this perturbation is allowed to propagate into the liquid initiating either two counterrotating convective cells (at both sides of the front) or a disordered pattern.
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Affiliation(s)
- V Perez-Villar
- Group of Nonlinear Physics, Faculty of Physics, University of Santiago de Compostela, 15706 Santiago de Compostela, Spain
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Woltering M, Markus M. Riddled-like basins of transient chaos. PHYSICAL REVIEW LETTERS 2000; 84:630-633. [PMID: 11017333 DOI: 10.1103/physrevlett.84.630] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/23/1999] [Indexed: 05/23/2023]
Abstract
The set of initial conditions leading to transient chaos in the neighborhood of a crisis is shown to display riddled-like behavior for finite, arbitrarily small accuracy. Calculations are performed with the logistic equation, as well as with an experimentally verifiable, quantitative description of a chemical reaction. Limitations in computational or experimental accuracy make the apparent riddling of initial conditions a phenomenon that is indistinguishable from riddled basins.
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Affiliation(s)
- M Woltering
- Max Planck Institut fur Molekulare Physiologie, Postfach 500247, 44202 Dortmund, Germany
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26
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Qu Z, Weiss JN, Garfinkel A. From local to global spatiotemporal chaos in a cardiac tissue model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:727-732. [PMID: 11046316 DOI: 10.1103/physreve.61.727] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/1999] [Indexed: 05/23/2023]
Abstract
Two kinds of chaos can occur in cardiac tissue, chaotic meander of a single intact spiral wave and chaotic spiral wave breakup. We studied these behaviors in a model of two-dimensional cardiac tissue based on the Luo-Rudy I action potential model. In the chaotic meander regime, chaos is spatially localized to the core of the spiral wave. When persistent spiral wave breakup occurs, there is a transition from local to global spatiotemporal chaos.
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Affiliation(s)
- Z Qu
- Department of Medicine (Cardiology), University of California at Los Angeles, Los Angeles, California 90095, USA
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Biktashev VN, Biktasheva IV, Holden AV, Tsyganov MA, Brindley J, Hill NA. Spatiotemporal irregularity in an excitable medium with shear flow. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:1897-900. [PMID: 11969978 DOI: 10.1103/physreve.60.1897] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/1998] [Revised: 03/02/1999] [Indexed: 04/18/2023]
Abstract
We consider an excitable medium moving with relative shear, subjected to a localized disturbance that in a stationary medium would produce a pair of spiral waves. The spiral waves so created are distorted and then broken by the motion of the medium. Such breaks generate new spiral waves, and so a "chain reaction" of spiral wave births and deaths is observed. This leads to a complicated spatiotemporal pattern, the "frazzle gas" [term suggested by Markus et al., Nature (London) 371, 402 (1994)], which eventually fills the whole medium. In this paper, we display and interpret the main features of the pattern.
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Affiliation(s)
- V N Biktashev
- School of Biomedical Sciences, University of Leeds, Leeds LS2 9JT, United Kingdom
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28
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Woltering M, Girnus R, Markus M. Quantification of Turbulence in the Belousov−Zhabotinsky Reaction by Monitoring Wave Tips. J Phys Chem A 1999. [DOI: 10.1021/jp990447y] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Matthias Woltering
- Max-Planck-Institut für Molekulare Physiologie, Postfach 500247, 44202 Dortmund, Germany
| | - Ralf Girnus
- Max-Planck-Institut für Molekulare Physiologie, Postfach 500247, 44202 Dortmund, Germany
| | - Mario Markus
- Max-Planck-Institut für Molekulare Physiologie, Postfach 500247, 44202 Dortmund, Germany
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29
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Rustici M, Branca M, Brunetti A, Caravati C, Marchettini N. Inverse Ruelle–Takens–Newhouse scenario in a closed unstirred cerium-catalysed Belousov–Zhabotinsky system. Chem Phys Lett 1998. [DOI: 10.1016/s0009-2614(98)00781-7] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Markus M, Czajka A. Oxygen-induced wavefront instabilities and disorder in the Belousov–Zhabotinsky reaction. Chem Phys Lett 1998. [DOI: 10.1016/s0009-2614(98)00392-3] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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31
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Cabo C, Pertsov AM, Davidenko JM, Jalife J. Electrical turbulence as a result of the critical curvature for propagation in cardiac tissue. CHAOS (WOODBURY, N.Y.) 1998; 8:116-126. [PMID: 12779715 DOI: 10.1063/1.166292] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
In cardiac tissue, the propagation of electrical excitation waves is dependent on the active properties of the cell membrane (ionic channels) and the passive electrical properties of cardiac tissue (passive membrane properties, distribution of gap junctions, and cell shapes). Initiation of cardiac arrhythmias is usually associated with heterogeneities in the active and/or passive properties of cardiac tissue. However, as a result of the effect of wave front geometry (curvature) on propagation of cardiac waves, inexcitable anatomical obstacles, like veins and arteries, may cause the formation of self-sustained vortices and uncontrolled high-frequency excitation in normal homogeneous myocardium. (c) 1998 American Institute of Physics.
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Affiliation(s)
- Candido Cabo
- Department of Pharmacology, College of Physicians and Surgeons of Columbia University, New York, New York 10032
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32
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Rustici M, Branca M, Caravati C, Marchettini N. Evidence of a chaotic transient in a closed unstirred cerium catalyzed Belousov-Zhabotinsky system. Chem Phys Lett 1996. [DOI: 10.1016/s0009-2614(96)01249-3] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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33
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Agladze K, Voignier V, Hamm E, Plaza F, Krinsky V. Fast Selective Elimination of Spiral Waves. ACTA ACUST UNITED AC 1996. [DOI: 10.1021/jp961826j] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- K. Agladze
- Institut Non-Lineaire de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
| | - V. Voignier
- Institut Non-Lineaire de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
| | - E. Hamm
- Institut Non-Lineaire de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
| | - F. Plaza
- Institut Non-Lineaire de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
| | - V. Krinsky
- Institut Non-Lineaire de Nice, 1361 Route des Lucioles, 06560 Valbonne, France
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Ševčíková H, Kosek J, Marek M. Splitting of 2D Waves of Excitation in a Direct Current Electric Field. ACTA ACUST UNITED AC 1996. [DOI: 10.1021/jp952004+] [Citation(s) in RCA: 24] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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35
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36
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Ouyang Q, Flesselles JM. Transition from spirals to defect turbulence driven by a convective instability. Nature 1996. [DOI: 10.1038/379143a0] [Citation(s) in RCA: 223] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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37
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Hildebrand M, Bär M, Eiswirth M. Statistics of Topological Defects and Spatiotemporal Chaos in a Reaction-Diffusion System. PHYSICAL REVIEW LETTERS 1995; 75:1503-1506. [PMID: 10060314 DOI: 10.1103/physrevlett.75.1503] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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