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Inhibitor-Induced Wavetrains and Spiral Waves in an Extended FitzHugh–Nagumo Model of Nerve Cell Dynamics. Bull Math Biol 2022; 84:145. [DOI: 10.1007/s11538-022-01100-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2022] [Accepted: 10/12/2022] [Indexed: 11/10/2022]
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2
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Guckenheimer J, Krauskopf B, Osinga HM, Sandstede B. Invariant manifolds and global bifurcations. CHAOS (WOODBURY, N.Y.) 2015; 25:097604. [PMID: 26428557 DOI: 10.1063/1.4915528] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems.
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Affiliation(s)
- John Guckenheimer
- Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
| | - Bernd Krauskopf
- Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
| | - Hinke M Osinga
- Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
| | - Björn Sandstede
- Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912, USA
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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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Kneer F, Obermayer K, Dahlem MA. Analyzing critical propagation in a reaction-diffusion-advection model using unstable slow waves. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2015; 38:95. [PMID: 25704900 DOI: 10.1140/epje/i2015-15010-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/05/2014] [Revised: 11/18/2014] [Accepted: 01/21/2015] [Indexed: 06/04/2023]
Abstract
The effect of advection on the propagation and in particular on the critical minimal speed of traveling waves in a reaction-diffusion model is studied. Previous theoretical studies estimated this effect on the velocity of stable fast waves and predicted the existence of a critical advection strength below which propagating waves are not supported anymore. In this paper, an analytical expression for the advection-velocity relation of the unstable slow wave is derived. In addition, the critical advection strength is calculated taking into account the unstable slow wave solution. We also analyze a two-variable reaction-diffusion-advection model numerically in a wide parameter range. Due to the new control parameter (advection) we can find stable wave propagation in the otherwise non-excitable parameter regime, if the advection strength exceeds a critical value. Comparing theoretical predictions to numerical results, we find that they are in good agreement. Theory provides an explanation for the observed behaviour.
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Affiliation(s)
- Frederike Kneer
- Department of Software Engineering and Theoretical Computer Science, Technische Universität Berlin, Ernst-Reuter-Platz 7, D-10587, Berlin, Germany,
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Siebert J, Alonso S, Bär M, Schöll E. Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052909. [PMID: 25353863 DOI: 10.1103/physreve.89.052909] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2014] [Indexed: 06/04/2023]
Abstract
A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.
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Affiliation(s)
- Julien Siebert
- Technische Universität, Institut für Theoretische Physik, Hardenberstrasse 36, 10623 Berlin, Germany
| | - Sergio Alonso
- Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin, Germany
| | - Markus Bär
- Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin, Germany
| | - Eckehard Schöll
- Technische Universität, Institut für Theoretische Physik, Hardenberstrasse 36, 10623 Berlin, Germany
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Colet P, Matías MA, Gelens L, Gomila D. Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012914. [PMID: 24580304 DOI: 10.1103/physreve.89.012914] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2013] [Indexed: 05/23/2023]
Abstract
The present work studies the influence of nonlocal spatial coupling on the existence of localized structures in one-dimensional extended systems. We consider systems described by a real field with a nonlocal coupling that has a linear dependence on the field. Leveraging spatial dynamics we provide a general framework to understand the effect of the nonlocality on the shape of the fronts connecting two stable states. In particular we show that nonlocal terms can induce spatial oscillations in the front tails, allowing for the creation of localized structures, that emerge from pinning between two fronts. In parameter space the region where fronts are oscillatory is limited by three transitions: the modulational instability of the homogeneous state, the Belyakov-Devaney transition in which monotonic fronts acquire spatial oscillations with infinite wavelength, and a crossover in which monotonically decaying fronts develop spatial oscillations with a finite wavelength. We show how these transitions are organized by codimension 2 and 3 points and illustrate how by changing the parameters of the nonlocal coupling it is possible to bring the system into the region where localized structures can be formed.
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Affiliation(s)
- Pere Colet
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Manuel A Matías
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Lendert Gelens
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain and Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
| | - Damià Gomila
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
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Gelens L, Matías MA, Gomila D, Dorissen T, Colet P. Formation of localized structures in bistable systems through nonlocal spatial coupling. II. The nonlocal Ginzburg-Landau equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012915. [PMID: 24580305 DOI: 10.1103/physreve.89.012915] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2013] [Indexed: 05/23/2023]
Abstract
We study the influence of a linear nonlocal spatial coupling on the interaction of fronts connecting two equivalent stable states in the prototypical 1-dimensional real Ginzburg-Landau equation. While for local coupling the fronts are always monotonic and therefore the dynamical behavior leads to coarsening and the annihilation of pairs of fronts, nonlocal terms can induce spatial oscillations in the front, allowing for the creation of localized structures, emerging from pinning between two fronts. We show this for three different nonlocal influence kernels. The first two, mod-exponential and Gaussian, are positive definite and decay exponentially or faster, while the third one, a Mexican-hat kernel, is not positive definite.
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Affiliation(s)
- Lendert Gelens
- Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium and IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Manuel A Matías
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Damià Gomila
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Tom Dorissen
- Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
| | - Pere Colet
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
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Stich M, Mikhailov AS, Kuramoto Y. Self-organized pacemakers and bistability of pulses in an excitable medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:026110. [PMID: 19391809 DOI: 10.1103/physreve.79.026110] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2008] [Indexed: 05/27/2023]
Abstract
Pattern formation in an excitable medium described by a three-component reaction-diffusion system is investigated. Our focus is on stable self-organized pacemakers which give rise to spatially extended target patterns. Bistability of pulse solutions in the excitable regime is also reported, and interactions of the different pulses with each other and the pacemaker are studied. Self-organized pacemakers are created by a suitable perturbation from the steady state or through interaction of pulses. Bound states of one-dimensional pacemakers and phase flips are also observed.
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Affiliation(s)
- Michael Stich
- Centro de Astrobiología (CSIC/INTA), Ctra de Ajalvir km. 4, 28850 Torrejón de Ardoz, Madrid, Spain.
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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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