352,956 research outputs found
The dynamics of neural fields on bounded domains: an interface approach for Dirichlet boundary conditions
Continuum neural field equations model the large scale spatio-temporal dynamics of interacting neurons on a cortical surface. They have been extensively studied, both analytically and numerically, on bounded as well as unbounded domains. Neural field models do not require the specification of boundary conditions. Relatively little attention has been paid to the imposition of neural activity on the boundary, or to its role in inducing patterned states. Here we redress this imbalance by studying neural field models of Amari type (posed on one- and two-dimensional bounded domains) with Dirichlet boundary conditions. The Amari model has a Heaviside nonlinearity that allows for a description of localised solutions of the neural field with an interface dynamics. We show how to generalise this reduced but exact description by deriving a normal velocity rule for an interface that encapsulates boundary effects. The linear stability analysis of localised states in the interface dynamics is used to understand how spatially extended patterns may develop in the absence and presence of boundary conditions. Theoretical results for pattern formation are shown to be in excellent agreement with simulations of the full neural field model. Furthermore, a numerical scheme for the interface dynamics is introduced and used to probe the way in which a Dirichlet boundary condition can limit the growth of labyrinthine structures
Quasi-Solitons in Dissipative Systems and Exactly Solvable Lattice Models
A system of first-order differential-difference equations with time lag
describes the formation of density waves, called as quasi-solitons for
dissipative systems in this paper. For co-moving density waves, the system
reduces to some exactly solvable lattice models. We construct a shock-wave
solution as well as one-quasi-soliton solution, and argue that there are
pseudo-conserved quantities which characterize the formation of the co-moving
waves. The simplest non-trivial one is given to discuss the presence of a
cascade phenomena in relaxation process toward the pattern formation.Comment: REVTeX, 4 pages, 1 figur
Toda Lattice Solutions of Differential-Difference Equations for Dissipative Systems
In a certain class of differential-difference equations for dissipative
systems, we show that hyperbolic tangent model is the only the nonlinear system
of equations which can admit some particular solutions of the Toda lattice. We
give one parameter family of exact solutions, which include as special cases
the Toda lattice solutions as well as the Whitham's solutions in the Newell's
model. Our solutions can be used to describe temporal-spatial density patterns
observed in the optimal velocity model for traffic flow.Comment: Latex, 13 pages, 1 figur
Localized solutions of Lugiato-Lefever equations with focused pump
Lugiato-Lefever (LL) equations in one and two dimensions (1D and 2D)
accurately describe the dynamics of optical fields in pumped lossy cavities
with the intrinsic Kerr nonlinearity. The external pump is usually assumed to
be uniform, but it can be made tightly focused too -- in particular, for
building small pixels. We obtain solutions of the LL equations, with both the
focusing and defocusing intrinsic nonlinearity, for 1D and 2D confined modes
supported by the localized pump. In the 1D setting, we first develop a simple
perturbation theory, based in the sech ansatz, in the case of weak pump and
loss. Then, a family of exact analytical solutions for spatially confined modes
is produced for the pump focused in the form of a delta-function, with a
nonlinear loss (two-photon absorption) added to the LL model. Numerical
findings demonstrate that these exact solutions are stable, both dynamically
and structurally (the latter means that stable numerical solutions close to the
exact ones are found when a specific condition, necessary for the existence of
the analytical solution, does not hold). In 2D, vast families of stable
confined modes are produced by means of a variational approximation and full
numerical simulations.Comment: 26 pages, 9 figures, accepted for publication in Scientific Report
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Survival benefits in mimicry: a quantitative framework
Mimicry is a resemblance between species that benefits at least one of the
species. It is a ubiquitous evolutionary phenomenon particularly common among
prey species, in which case the advantage involves better protection from
predation. We formulate a mathematical description of mimicry among prey
species, to investigate benefits and disadvantages of mimicry. The basic setup
involves differential equations for quantities representing predator behavior,
namely, the probabilities for attacking prey at the next encounter. Using this
framework, we present new quantitative results, and also provide a unified
description of a significant fraction of the quantitative mimicry literature.
The new results include `temporary' mutualism between prey species, and an
optimal density at which the survival benefit is greatest for the mimic. The
formalism leads naturally to extensions in several directions, such as the
evolution of mimicry, the interplay of mimicry with population dynamics, etc.
We demonstrate this extensibility by presenting some explorations on
spatiotemporal pattern dynamics.Comment: 9 pages, 7 figure
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