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Cytrynbaum EN, MacKay V, Nahman-Lévesque O, Dobbs M, Bub G, Shrier A, Glass L. Double-wave reentry in excitable media. CHAOS (WOODBURY, N.Y.) 2019; 29:073103. [PMID: 31370409 DOI: 10.1063/1.5092982] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2019] [Accepted: 06/13/2019] [Indexed: 06/10/2023]
Abstract
A monolayer of chick embryo cardiac cells grown in an annular geometry supports two simultaneous reentrant excitation waves that circulate as a doublet. We propose a mechanism that can lead to such behavior. The velocity restitution gives the instantaneous velocity of a wave as a function of the time since the passage of the previous wave at a given point in space. Nonmonotonic restitution relationships will lead to situations in which various spacings between circulating waves are possible. In cardiology, the situation in which two waves travel in an anatomically defined circuit is referred to as double-wave reentry. Since double-wave reentry may arise as a consequence of pacing during cardiac arrhythmias, understanding the dynamic features of double-wave reentry may be helpful in understanding the physiological properties of cardiac tissue and in the design of therapy.
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Affiliation(s)
- Eric N Cytrynbaum
- Department of Mathematics, University of British Columbia, 1984 Mathematics Rd #121, Vancouver, British Columbia V6T 1Z2, Canada
| | - Vincent MacKay
- Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec H3G 1Y6, Canada
| | - Olivier Nahman-Lévesque
- Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec H3G 1Y6, Canada
| | - Matt Dobbs
- Department of Physics, McGill University, 3600 rue University, Montreal, Quebec H3A 2T8, Canada
| | - Gil Bub
- Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec H3G 1Y6, Canada
| | - Alvin Shrier
- Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec H3G 1Y6, Canada
| | - Leon Glass
- Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec H3G 1Y6, Canada
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Yochelis A. Catalytic Membrane Reactor Model as a Laboratory for Pattern Emergence in Reaction-diffusion-advection Media. Isr J Chem 2018. [DOI: 10.1002/ijch.201700145] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Arik Yochelis
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research (SIDEER), Blaustein Institutes for Desert Research (BIDR); Ben-Gurion University of the Negev, Sede Boqer Campus; Midreshet Ben-Gurion 8499000 Israel
- Department of Physics; Ben-Gurion University of the Negev; Be'er Sheva 8410501 Israel
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Yochelis A, Knobloch E, Köpf MH. Origin of finite pulse trains: Homoclinic snaking in excitable media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032924. [PMID: 25871189 DOI: 10.1103/physreve.91.032924] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2013] [Indexed: 06/04/2023]
Abstract
Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customized initial stimuli, thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.
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Affiliation(s)
- Arik Yochelis
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Jacob Blaustein Institutes for Desert Research (BIDR), Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990, Israel
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Michael H Köpf
- Département de Physique, École Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
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Meijer HGE, Coombes S. Travelling waves in models of neural tissue: from localised structures to periodic waves. ACTA ACUST UNITED AC 2014. [DOI: 10.1140/epjnbp16] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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Travelling waves in a neural field model with refractoriness. J Math Biol 2013; 68:1249-68. [PMID: 23546637 PMCID: PMC3948616 DOI: 10.1007/s00285-013-0670-x] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2012] [Revised: 03/07/2013] [Indexed: 11/24/2022]
Abstract
At one level of abstraction neural tissue can be regarded as a medium for turning local synaptic activity into output signals that propagate over large distances via axons to generate further synaptic activity that can cause reverberant activity in networks that possess a mixture of excitatory and inhibitory connections. This output is often taken to be a firing rate, and the mathematical form for the evolution equation of activity depends upon a spatial convolution of this rate with a fixed anatomical connectivity pattern. Such formulations often neglect the metabolic processes that would ultimately limit synaptic activity. Here we reinstate such a process, in the spirit of an original prescription by Wilson and Cowan (Biophys J 12:1–24, 1972), using a term that multiplies the usual spatial convolution with a moving time average of local activity over some refractory time-scale. This modulation can substantially affect network behaviour, and in particular give rise to periodic travelling waves in a purely excitatory network (with exponentially decaying anatomical connectivity), which in the absence of refractoriness would only support travelling fronts. We construct these solutions numerically as stationary periodic solutions in a co-moving frame (of both an equivalent delay differential model as well as the original delay integro-differential model). Continuation methods are used to obtain the dispersion curve for periodic travelling waves (speed as a function of period), and found to be reminiscent of those for spatially extended models of excitable tissue. A kinematic analysis (based on the dispersion curve) predicts the onset of wave instabilities, which are confirmed numerically.
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Zemskov EP, Loskutov AY. Traveling waves in a piecewise-linear reaction-diffusion model of excitable medium. Biophysics (Nagoya-shi) 2010. [DOI: 10.1134/s0006350909050145] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022] Open
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Yochelis A, Sheintuch M. Drifting solitary waves in a reaction-diffusion medium with differential advection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:025203. [PMID: 20365619 DOI: 10.1103/physreve.81.025203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2009] [Revised: 09/21/2009] [Indexed: 05/29/2023]
Abstract
A distinct propagation of solitary waves in the presence of autocatalysis, diffusion, and symmetry-breaking (differential) advection, is being studied. These pulses emerge at lower reaction rates of the autocatalytic activator, i.e., when the advective flow overcomes the fast excitation and induces a fluid type "drifting" behavior, making the phenomenon unique to reaction-diffusion-advection class systems. Using the spatial dynamics analysis of a canonical model, we present the properties and the organization of such drifting pulses. The insights underly a general understanding of localized transport in simple reaction-diffusion-advection models and thus provide a background to potential chemical and biological applications.
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Affiliation(s)
- Arik Yochelis
- Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel.
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Golovin AA, Kanevsky Y, Nepomnyashchy AA. Feedback control of subcritical Turing instability with zero mode. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:046218. [PMID: 19518323 DOI: 10.1103/physreve.79.046218] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2008] [Revised: 03/11/2009] [Indexed: 05/27/2023]
Abstract
A global feedback control of a system that exhibits a subcritical monotonic instability at a nonzero wave number (short-wave or Turing instability) in the presence of a zero mode is investigated using a Ginzburg-Landau equation coupled to an equation for the zero mode. This system is studied analytically and numerically. It is shown that feedback control, based on measuring the maximum of the pattern amplitude over the domain, can stabilize the system and lead to the formation of localized unipulse stationary states or traveling solitary waves. It is found that the unipulse traveling structures result from an instability of the stationary unipulse structures when one of the parameters characterizing the coupling between the periodic pattern and the zero mode exceeds a critical value that is determined by the zero mode damping coefficient.
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Affiliation(s)
- A A Golovin
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
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Yochelis A, Knobloch E, Xie Y, Qu Z, Garfinkel A. Generation of finite wave trains in excitable media. EUROPHYSICS LETTERS 2008; 83:64005p1-64005p6. [PMID: 21572974 PMCID: PMC3092297 DOI: 10.1209/0295-5075/83/64005] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/15/2023]
Abstract
Spatiotemporal control of excitable media is of paramount importance in the development of new applications, ranging from biology to physics. To this end, we identify and describe a qualitative property of excitable media that enables us to generate a sequence of traveling pulses of any desired length, using a one-time initial stimulus. The wave trains are produced by a transient pacemaker generated by a one-time suitably tailored spatially localized finite amplitude stimulus, and belong to a family of fast pulse trains. A second family, of slow pulse trains, is also present. The latter are created through a clumping instability of a traveling wave state (in an excitable regime) and are inaccessible to single localized stimuli of the type we use. The results indicate that the presence of a large multiplicity of stable, accessible, multi-pulse states is a general property of simple models of excitable media.
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Affiliation(s)
- A Yochelis
- Department of Medicine (Cardiology), University of California - Los Angeles, CA 90095, USA
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Bordyugov G, Engel H. Anomalous pulse interaction in dissipative media. CHAOS (WOODBURY, N.Y.) 2008; 18:026104. [PMID: 18601506 DOI: 10.1063/1.2943307] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
We review a number of phenomena occurring in one-dimensional excitable media due to modified decay behind propagating pulses. Those phenomena can be grouped in two categories depending on whether the wake of a solitary pulse is oscillatory or not. Oscillatory decay leads to nonannihilative head-on collision of pulses and oscillatory dispersion relation of periodic pulse trains. Stronger wake oscillations can even result in a bistable dispersion relation. Those effects are illustrated with the help of the Oregonator and FitzHugh-Nagumo models for excitable media. For a monotonic wake, we show that it is possible to induce bound states of solitary pulses and anomalous dispersion of periodic pulse trains by introducing nonlocal spatial coupling to the excitable medium.
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Affiliation(s)
- Grigory Bordyugov
- Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Strasse 24/25, D-14476 Potsdam, Germany.
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Abstract
Scroll waves are three-dimensional excitation patterns that rotate around one-dimensional space curves. Typically these filaments are closed loops or end at the system boundary. However, in excitable media with anomalous dispersion, filaments can be pinned to the wake of traveling wave pulses. This pinning is studied in experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction and a three-variable reaction-diffusion model. We show that wave-pinned filaments are related to the coexistence of rotating and translating wave defects in two dimensions. Filament pinning causes a continuous expansion of the total filament length. It can be ended by annihilating the pinning pulse in a frontal wave collision. Following such an annihilation, the filament connects itself to the system boundary. Its postannihilation shape that is initially the exposed rim of the scroll wave unwinds continuously over numerous rotation periods.
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Affiliation(s)
- Tamás Bánsági
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, USA
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