145 research outputs found
Well-posedness and longtime behavior for the modified phase-field crystal equation
We consider a modification of the so-called phase-field crystal (PFC)
equation introduced by K.R. Elder et al. This variant has recently been
proposed by P. Stefanovic et al. to distinguish between elastic relaxation and
diffusion time scales. It consists of adding an inertial term (i.e. a
second-order time derivative) into the PFC equation. The mathematical analysis
of the resulting equation is more challenging with respect to the PFC equation,
even at the well-posedness level. Moreover, its solutions do not regularize in
finite time as in the case of PFC equation. Here we analyze the modified PFC
(MPFC) equation endowed with periodic boundary conditions. We first prove the
existence and uniqueness of a solution with initial data in a bounded energy
space. This solution satisfies some uniform dissipative estimates which allow
us to study the global longtime behavior of the corresponding dynamical system.
In particular, we establish the existence of an exponential attractor. Then we
demonstrate that any trajectory originating from the bounded energy phase space
does converge to a unique equilibrium. This is done by means of a suitable
version of the {\L}ojasiewicz-Simon inequality. A convergence rate estimate is
also given
Coordinated optimization of visual cortical maps (II) Numerical studies
It is an attractive hypothesis that the spatial structure of visual cortical
architecture can be explained by the coordinated optimization of multiple
visual cortical maps representing orientation preference (OP), ocular dominance
(OD), spatial frequency, or direction preference. In part (I) of this study we
defined a class of analytically tractable coordinated optimization models and
solved representative examples in which a spatially complex organization of the
orientation preference map is induced by inter-map interactions. We found that
attractor solutions near symmetry breaking threshold predict a highly ordered
map layout and require a substantial OD bias for OP pinwheel stabilization.
Here we examine in numerical simulations whether such models exhibit
biologically more realistic spatially irregular solutions at a finite distance
from threshold and when transients towards attractor states are considered. We
also examine whether model behavior qualitatively changes when the spatial
periodicities of the two maps are detuned and when considering more than 2
feature dimensions. Our numerical results support the view that neither minimal
energy states nor intermediate transient states of our coordinated optimization
models successfully explain the spatially irregular architecture of the visual
cortex. We discuss several alternative scenarios and additional factors that
may improve the agreement between model solutions and biological observations.Comment: 55 pages, 11 figures. arXiv admin note: substantial text overlap with
arXiv:1102.335
The Spatio-Temporal Structure of Spiral-Defect Chaos
We present a study of the recently discovered spatially-extended chaotic
state known as spiral-defect chaos, which occurs in low-Prandtl-number,
large-aspect-ratio Rayleigh-Benard convection. We employ the modulus squared of
the space-time Fourier transform of time series of two-dimensional shadowgraph
images to construct the structure factor .
This analysis is used to characterize the average spatial and temporal scales
of the chaotic state. We find that the correlation length and time can be
described by power-law dependences on the reduced Rayleigh number .
These power laws have as yet no theoretical explanation.Comment: RevTex 38 pages with 13 figures. Due to their large size, some
figures are stored as separate gif images. The paper with included hi-res eps
figures (981kb compressed, 3.5Mb uncompressed) is available at
ftp://mobydick.physics.utoronto.ca/pub/MBCA96.tar.gz An mpeg movie and
samples of data are also available at
ftp://mobydick.physics.utoronto.ca/pub/. Paper submitted to Physica
Coordinated optimization of visual cortical maps : 2. Numerical studies
In the juvenile brain, the synaptic architecture of the visual cortex remains in a state of flux for months after the natural onset of vision and the initial emergence of feature selectivity in visual cortical neurons. It is an attractive hypothesis that visual cortical architecture is shaped during this extended period of juvenile plasticity by the coordinated optimization of multiple visual cortical maps such as orientation preference (OP), ocular dominance (OD), spatial frequency, or direction preference. In part (I) of this study we introduced a class of analytically tractable coordinated optimization models and solved representative examples, in which a spatially complex organization of the OP map is induced by interactions between the maps. We found that these solutions near symmetry breaking threshold predict a highly ordered map layout. Here we examine the time course of the convergence towards attractor states and optima of these models. In particular, we determine the timescales on which map optimization takes place and how these timescales can be compared to those of visual cortical development and plasticity. We also assess whether our models exhibit biologically more realistic, spatially irregular solutions at a finite distance from threshold, when the spatial periodicities of the two maps are detuned and when considering more than 2 feature dimensions. We show that, although maps typically undergo substantial rearrangement, no other solutions than pinwheel crystals and stripes dominate in the emerging layouts. Pinwheel crystallization takes place on a rather short timescale and can also occur for detuned wavelengths of different maps. Our numerical results thus support the view that neither minimal energy states nor intermediate transient states of our coordinated optimization models successfully explain the architecture of the visual cortex. We discuss several alternative scenarios that may improve the agreement between model solutions and biological observations
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