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Schmidt H, Avitabile D. Bumps and oscillons in networks of spiking neurons. CHAOS (WOODBURY, N.Y.) 2020; 30:033133. [PMID: 32237760 DOI: 10.1063/1.5135579] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2019] [Accepted: 03/03/2020] [Indexed: 06/11/2023]
Abstract
We study localized patterns in an exact mean-field description of a spatially extended network of quadratic integrate-and-fire neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range, where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases, we observe period-doubling cascades leading to chaotic oscillations.
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Affiliation(s)
- Helmut Schmidt
- Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstrasse 1a, 04103 Leipzig, Germany
| | - Daniele Avitabile
- Department of Mathematics, Faculteit der Exacte Wetenschappen, Vrije Universiteit (VU University Amsterdam), De Boelelaan 1081a, 1081 HV Amsterdam, Netherlands and Mathneuro Team, Inria Sophia Antipolis, 2004 Rue des Lucioles, Sophia Antipolis, 06902 Cedex, France
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Knobloch E, Uecker H, Wetzel D. Defectlike structures and localized patterns in the cubic-quintic-septic Swift-Hohenberg equation. Phys Rev E 2019; 100:012204. [PMID: 31499926 DOI: 10.1103/physreve.100.012204] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2019] [Indexed: 11/07/2022]
Abstract
We study numerically the cubic-quintic-septic Swift-Hohenberg (SH357) equation on bounded one-dimensional domains. Under appropriate conditions stripes with wave number k≈1 bifurcate supercritically from the zero state and form S-shaped branches resulting in bistability between small and large amplitude stripes. Within this bistability range we find stationary heteroclinic connections or fronts between small and large amplitude stripes, and demonstrate that the associated spatially localized defectlike structures either snake or fall on isolas. In other parameter regimes we also find heteroclinic connections to spatially homogeneous states and a multitude of dynamically stable steady states consisting of patches of small and large amplitude stripes with different wave numbers or of spatially homogeneous patches. The SH357 equation is thus extremely rich in the types of patterns it exhibits. Some of the features of the bifurcation diagrams obtained by numerical continuation can be understood using a conserved quantity, the spatial Hamiltonian of the system.
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Affiliation(s)
- Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Hannes Uecker
- Institut für Mathematik, Universität Oldenburg, D26111 Oldenburg, Germany
| | - Daniel Wetzel
- Institut für Mathematik, Universität Oldenburg, D26111 Oldenburg, Germany
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Lecoanet D, Kerswell RR. Connection between nonlinear energy optimization and instantons. Phys Rev E 2018; 97:012212. [PMID: 29448452 DOI: 10.1103/physreve.97.012212] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2017] [Indexed: 11/07/2022]
Abstract
How systems transit between different stable states under external perturbation is an important practical issue. We discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems. In the context of the one-dimensional Swift-Hohenberg equation, which has multiple stable equilibria, we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances-minimal seeds-for transition to specific attractors from the ground state. Then, after generalizing the technique to consider multiple, equally spaced-in-time perturbations, it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations. Importantly, we find that the key features of the instanton can be captured by a low number of discrete perturbations (typically one perturbation per basin of attraction crossed). This suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton.
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Affiliation(s)
- Daniel Lecoanet
- Physics Department, University of California, Berkeley, California 94720, USA; Astronomy Department and Theoretical Astrophysics Center, University of California, Berkeley, California 94720, USA; Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA; Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA; and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
| | - Rich R Kerswell
- Centre for Mathematical Sciences, Cambridge University, Cambridge CB3 0WA, United Kingdom; School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom; and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
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Pattern formation – A missing link in the study of ecosystem response to environmental changes. Math Biosci 2016; 271:1-18. [DOI: 10.1016/j.mbs.2015.10.015] [Citation(s) in RCA: 56] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2015] [Revised: 10/17/2015] [Accepted: 10/23/2015] [Indexed: 11/18/2022]
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Haim L, Hagberg A, Meron E. Non-monotonic resonance in a spatially forced Lengyel-Epstein model. CHAOS (WOODBURY, N.Y.) 2015; 25:064307. [PMID: 26117118 DOI: 10.1063/1.4921768] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
We study resonant spatially periodic solutions of the Lengyel-Epstein model modified to describe the chlorine dioxide-iodine-malonic acid reaction under spatially periodic illumination. Using multiple-scale analysis and numerical simulations, we obtain the stability ranges of 2:1 resonant solutions, i.e., solutions with wavenumbers that are exactly half of the forcing wavenumber. We show that the width of resonant wavenumber response is a non-monotonic function of the forcing strength, and diminishes to zero at sufficiently strong forcing. We further show that strong forcing may result in a π/2 phase shift of the resonant solutions, and argue that the nonequilibrium Ising-Bloch front bifurcation can be reversed. We attribute these behaviors to an inherent property of forcing by periodic illumination, namely, the increase of the mean spatial illumination as the forcing amplitude is increased.
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Affiliation(s)
- Lev Haim
- Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
| | - Aric Hagberg
- Center for Nonlinear Studies, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Ehud Meron
- Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
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Haim L, Mau Y, Meron E. Spatial forcing of pattern-forming systems that lack inversion symmetry. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022904. [PMID: 25215796 DOI: 10.1103/physreve.90.022904] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2014] [Indexed: 06/03/2023]
Abstract
The entrainment of periodic patterns to spatially periodic parametric forcing is studied. Using a weak nonlinear analysis of a simple pattern formation model we study the resonant responses of one-dimensional systems that lack inversion symmetry. Focusing on the first three n:1 resonances, in which the system adjusts its wavenumber to one nth of the forcing wavenumber, we delineate commonalities and differences among the resonances. Surprisingly, we find that all resonances show multiplicity of stable phase states, including the 1:1 resonance. The phase states in the 2:1 and 3:1 resonances, however, differ from those in the 1:1 resonance in remaining symmetric even when the inversion symmetry is broken. This is because of the existence of a discrete translation symmetry in the forced system. As a consequence, the 2:1 and 3:1 resonances show stationary phase fronts and patterns, whereas phase fronts within the 1:1 resonance are propagating and phase patterns are transients. In addition, we find substantial differences between the 2:1 resonance and the other two resonances. While the pattern forming instability in the 2:1 resonance is supercritical, in the 1:1 and 3:1 resonances it is subcritical, and while the inversion asymmetry extends the ranges of resonant solutions in the 1:1 and 3:1 resonances, it has no effect on the 2:1 resonance range. We conclude by discussing a few open questions.
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Affiliation(s)
- Lev Haim
- Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel and Department of Oncology, Soroka University Medical Center, Beer Sheva, 84101, Israel
| | - Yair Mau
- Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel and Department of Civil and Environmental Engineering, Duke University, Durham, North Carolina 27708, USA
| | - Ehud Meron
- Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel and Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University of the Negev, Sede Boqer Campus, 84990, Israel
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