Understanding visual map formation through vortex dynamics of spin Hamiltonian models

The pattern formation in orientation and ocular dominance columns is one of the most investigated problems in the brain. From a known cortical structure, we build spinlike Hamiltonian models with long-range interactions of the Mexican hat type. These Hamiltonian models allow a coherent interpretation of the diverse phenomena in the visual map formation with the help of relaxation dynamics of spin systems. In particular, we explain various phenomena of self-organization in orientation and ocular dominance map formation including the pinwheel annihilation and its dependency on the columnar wave vector and boundary conditions.     

Different ocular dominance map formation influenced by orientation preference columns in visual cortices

In animal experiments, the observed orientation preference and ocular dominance columns in the visual cortex of the brain show various pattern types. Here, we show that the different visual map formations in various species are due to the crossover behavior in anisotropic systems composed of orientational and scalar components such as easy-plane Heisenberg models. We predict the transition boundary between different pattern types with the anisotropy as a main bifurcation parameter, which is consistent with experimental observations.     

Turing instability for a semi-discrete Gierer-Meinhardt system

In this paper, we investigate the spatial patterns of a Gierer-Meinhardt system where the space is discrete in two dimensions with the periodic boundary condition and time is continuous, in contrast to the continuum models. The conditions of Turing instability are obtained by linear analysis and a series of numerical simulations are performed. In the instability region, we have shown that this system can produce a number of different patterns such as stripes and snowflake pattern, other than ubiquitous fix-spotted patterns. As mentioned, the results are substantiated only by means of snapshots of the apatial grid. However, we also give some analysis by using the time series at three random grids and of the average value of states, that is, the stable state patterns can be observed. On the other hand, the effects of varying parameters on pattern formation are also discussed. Moreover, simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns. The patterns can form as a consequence of cellular interaction. So patterns arising from a semi-discrete model can present simulations on a geometrically accurate representation in biology. As a result, our work is interesting and important in ecology.     

Instability of spatial patterns and its ambiguous impact on species diversity

Self-arrangement of individuals into spatial patterns often accompanies and promotes species diversity in ecological systems. Here, we investigate pattern formation arising from cyclic dominance of three species, operating near a bifurcation point. In its vicinity, an Eckhaus instability occurs, leading to convectively unstable "blurred" patterns. At the bifurcation point, stochastic effects dominate and induce counterintuitive effects on diversity: Large patterns, emerging for medium values of individuals' mobility, lead to rapid species extinction, while small patterns (low mobility) promote diversity, and high mobilities render spatial structures irrelevant. We provide a quantitative analysis of these phenomena, employing a complex Ginzburg-Landau equation.     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.     

Detection of pseudoperiodic patterns using partial acquisition of magnetic resonance images

Improving the resolution of magnetic resonance imaging (MRI), or, alternatively, reducing the acquisition time, can be quite beneficial for many applications. The main motivation of this work is the assumption that any information that is a priori available on the target image could be used to achieve this goal. In order to demonstrate this approach, we present a novel partial acquisition strategy and reconstruction algorithm, suitable for the special case of detection of pseudoperiodic patterns. Pseudoperiodic patterns are frequently encountered in the cerebral cortex due to its columnar functional organization (best exemplified by orientation columns and ocular dominance columns of the visual cortex). We present a new MRI research methodology, in which we seek an activity pattern, and a pattern-specific experiment is devised to detect it. Such specialized experiments extend the limits of conventional MRI experiments by substantially reducing the scan time. Using the fact that pseudoperiodic patterns are localized in the Fourier domain, we present an optimality criterion for partial acquisition of the MR signal and a strategy for obtaining the optimal discrete Fourier transform (DFT) coefficients. A by-product of this strategy is an optimal linear extrapolation estimate. We also present a nonlinear spectral extrapolation algorithm, based on projections onto convex sets (POCSs), used to perform the actual reconstruction. The proposed strategy was tested and analyzed on simulated signals and in MRI phantom experiments.     

Switch and template pattern formation in a discrete reaction-diffusion system inspired by the Drosophila eye

We examine a spatially discrete reaction-diffusion model based on the interactions that create a periodic pattern in the Drosophila eye imaginal disc. This model is known to be capable of generating a regular hexagonal pattern of gene expression behind a moving front, as observed in the fly system. In order to better understand the novel “switch and template” mechanism behind this pattern formation, we present here a detailed study of the model's behavior in one dimension, using a combination of analytic methods and numerical searches of parameter space. We find that patterns are created robustly, provided that there is an appropriate separation of timescales and that self-activation is sufficiently strong, and we derive expressions in this limit for the front speed and the pattern wavelength. Moving fronts in pattern-forming systems near an initial linear instability generically select a unique pattern, but our model operates in a strongly nonlinear regime where the final pattern depends on the initial conditions as well as on parameter values. Our work highlights the important role that cellularization and cell-autonomous feedback can play in biological pattern formation.     

Analysis of dynamic pattern formation in nonlinear Fabry-Perot resonators

We present a detailed analysis of transverse effects and pattern formation in bistable optical elements. The system we investigate consists of a Fabry-Perot resonator for the optical feedback element with a nematic liquid-crystal cell used as an optically nonlinear intracavity medium. On illumination with a cw-laser beam, the system causes the beam to break up into several individual spots, passing through several transitions before finally reaching a stationary state. We devise a theoretical model which is used as the basis for numerical simulations of the system. The simulation results are in good agreement with experiment. Finally, we characterize the principal instability of the system using a linear stability analysis of the theoretical model.     

Instabilities of concentration stripe patterns in ferrocolloids

Equations describing the kinetics of the phase separation in ferrocolloids in a Hele-Shaw cell under the action of a rotating magnetic field are proposed. Numerical simulation on the basis of a pseudospectral technique demonstrates that upon the action of a rotating field on a magnetic colloid which undergoes the phase separation a periodical system of stripes parallel to the plane of a rotating magnetic field stripes is created. The period of a structure found numerically satisfactorily corresponds to the one calculated on the basis of the energy minimum. Thus, the undulation instability leading to the formation of chevron structures takes place if the tangential component of a rotating magnetic field is eliminated, whereas the normal component is increased at the same time. If during the development of the undulation deformations of a concentration pattern the magnetic Bond number is large enough the secondary instabilities may occur leading to the fingering of stripes to bring about merging and break-up of stripes. It is shown that an increase in the magnetic Bond number leads to the onset of the instability at the boundaries between the regions with homogeneous orientation of stripes as well as to formation of the characteristic hairpin patterns.     

Modification of pattern formation in doubly resonant second-harmonic generation by competing parametric oscillation

We analyze pattern formation in doubly resonant intracavity second-harmonic generation in the presence of competing nondegenerate parametric downconversion. We show that for positive cavity detuning of the fundamental frequency the threshold for parametric oscillation is lower than that of transverse, pattern forming instabilities. The parametric oscillation strongly modifies the pattern dynamics found previously in a simplified analysis that neglects parametric instability [Phys. Rev. E 56, 4803 (1997)]. Stationary and dynamic patterns in the presence of parametric oscillation are found numerically.     

Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system

The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have influences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will convert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.     

Initiation and evolution of current ripples on a flat sand bed under turbulent water flow

We investigate the formation and dynamics of sand ripples under a turbulent water flow. Our experiments were conducted in an open flume with spherical glass beads between 100 and 500μm in diameter. The flow Reynolds number is of the order of 10 000 and the particle Reynolds number of the order of 1 to 10. We study the development of ripples by measuring their wavelength and amplitude in course of time and investigate the influence of the grain size and the flow properties. In particular, we demonstrate two different regimes according to the grain size. For fine grains, a slow coarsening process (i.e., a logarithmic increase of the wavelength and amplitude) takes place, while for coarser grains, this process occurs at a much faster rate (i.e., with a linear growth) and stops after a finite time. In the later case, a stable pattern is eventually observed. Besides, we carefully analyze the wavelength of ripples in the first stages of the instability as a function of the grain size and the shear velocity of the flow, and compare our results with other available experimental data and with theoretical predictions based on linear stability analyses.     

Pattern formation in mixtures of ultracold atoms in optical lattices

Regular pattern formation is ubiquitous in nature; it occurs in biological, physical, and materials science systems. Here we propose a set of experiments with ultracold atoms that show how to examine different types of pattern formation. In particular, we show how one can see the analog of labyrinthine patterns (so-called quantum emulsions) in mixtures of light and heavy atoms (that tend to phase separate) by tuning the trap potential and we show how complex geometrically ordered patterns emerge (when the mixtures do not phase separate), which could be employed for low-temperature thermometry. The complex physical mechanisms for the pattern formation at zero temperature are understood within a theoretical analysis called the local density approximation.     

Pattern formation induced by additive noise: a moment-based analysis

We analyze the condition for instability and pattern formation induced by additive noise in spatially extended systems. The approach is based on consideration of higher moments which extract out nonlinearities of appropriate order. Our analysis reveals that cubic nonlinearity plays a crucial role for the additive noise to a leading order that determines the instability threshold which is corroborated by numerical simulation in two specific reaction-diffusion systems.     

Pattern formation in a predator–prey model with spatial effect

In this paper, spatial dynamics in the Beddington–DeAngelis predator–prey model with self-diffusion and cross-diffusion is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. Numerical simulations show that this system exhibits complex dynamical behaviors. In the Turing space, we find three types of typical patterns. One is the coexistence of hexagon patterns and stripe patterns. The other two are hexagon patterns of different types. The obtained results well enrich the finding in predator–prey models with Beddington–DeAngelis functional response.     

Spatial pattern formation of a ratio-dependent predator-prey model

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator-prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.     

The evolution of chemical patterns in reactive liquids,driven by hydrodynamic instabilities

We summarize our activity in unveiling a very wide phenomenon: When a chemical reaction takes place at a liquid interface, spectacular patterns of product form (see Plate 1). The pattern formation phenomenon is general, and is observed in reactions between liquids separated by a membrane, in liquids subjected to gaseous reactants, and in photoreactive liquids. We have demonstrated the phenomenon on over 100 different reactions of all types, thus discovering what we believe to be one of the widest macroscopic pattern formation processes known to chemistry. As can be seen in the accompanying pictures, the richness, beauty, and variations in types of patterns can be breathtaking. Two important aspects of these patterns are noted: First, the patterns are true far-from-equilibrium structures, which are maintained only as long as reactants are available, or only as long as light energy is supplied to the system; and second, the chemical products that form the patterns are not precipitates, but are entirely soluble in the liquid in which they form. Thus, if the containers in which the patterns form are shaken or stirred, a homogeneous solution results. Our research of this phenomenon concentrated on three main aspects. The first one was phenomenological. Here we explored the scope and generality of the phenomenon, motivated both by the aesthetic appeal of the phenomenon, and by the puzzle of how is it that such a wide-scope, experimentally simple phenomenon, has by and large, escaped the attention of the scientific community.The second aspect was devoted to the understanding of the underlying general mechanism. Of the many mechanisms we analyzed and tested, some very complex, others quite trivial, the one that fits the majority of the physical and chemical observations is the following: By performing a reaction through a liquid interface, a concentration gradient of the product forms near the interface. We have shown that in many cases, these gradients lead to hydrodynamic instabilities, which then break nonlinearly into a pattern which onsets slow convections. In other words, we found that these patterns mark the route along which a chemical instability relaxes. The third aspect of our research was theoretical. Here we concentrated in depth on one of the reactions (the Fe(+2)/Fe(+3) photoredox reaction), determined all its important physical parameters, and modeled its behavior theoretically. Our model, which was based on the instability buildup described above, was solved numerically, and its results compared with computerized image analysis of the evolving patterns; very good agreement between theory and experiment, was obtained. (c) 1995 American Institute of Physics.     

Stratified layer formation in particle suspensions

Initially homogeneous suspensions of colloidal particles often develop patterns during sedimentation. Commonly, the concentration profile of the particles evolves into a “staircase”: layers of nearly constant concentration, separated by sharp boundaries between successive layers, with the concentration of each successive layer increasing with depth. Siano [1] has demonstrated experimentally that uphill diffusion, diffusion against the concentration gradient, occurs during this pattern formation. Thus, these patterns appear to be the result of spinodal decomposition. We find that these staircase patterns cannot be explained by the classical spinodal decomposition theory of Cahn and Hilliard, but that they can be explained if the linear gradient-energy term of Tiller, Pound, and Hirth is added to the free energy. Such a term plays a central role in the faceting of crystals. In the present application we believe that the physical origin of this extra term may be the Rayleigh—Taylor instability.