Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics

This paper is devoted to estimates of the exponential decay of eigenfunctions of difference operators on the lattice Z^n which are discrete analogs of the Schr\"{o}dinger, Dirac and square-root Klein-Gordon operators. Our investigation of the essential spectra and the exponential decay of eigenfunctions of the discrete spectra is based on the calculus of so-called pseudodifference operators (i.e., pseudodifferential operators on the group Z^n) with analytic symbols and on the limit operators method. We obtain a description of the location of the essential spectra and estimates of the eigenfunctions of the discrete spectra of the main lattice operators of quantum mechanics, namely: matrix Schr\"{o}dinger operators on Z^n, Dirac operators on Z^3, and square root Klein-Gordon operators on Z^n

Origin of coherent structures in a discrete chaotic medium

Using as an example a large lattice of locally interacting Hindmarsh-Rose chaotic neurons, we disclose the origin of ordered structures in a discrete nonequilibrium medium with fast and slow chaotic oscillations. The origin of the ordering mechanism is related to the appearance of a periodic average dynamics in the group of chaotic neurons whose individual slow activity is significantly synchronized by the group mean field. Introducing the concept of a "coarse grain" as a cluster of neuron elements with periodic averaged behavior allows consideration of the dynamics of a medium composed of these clusters. A study of this medium reveals spatially ordered patterns in the periodic and slow dynamics of the coarse grains that are controlled by the average intensity of the fast chaotic pulsation

Essential spectra of difference operators on \sZ^n-periodic graphs

Let (\cX, \rho) be a discrete metric space. We suppose that the group \sZ^n acts freely on $X$ and that the number of orbits of $X$ with respect to this action is finite. Then we call $X$ a \sZ^n-periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on $l^p(X)$ where $X$ is a \sZ^n-periodic discrete metric space. Our approach is based on the theory of band-dominated operators on \sZ^n and their limit operators. In case $X$ is the set of vertices of a combinatorial graph, the graph structure defines a Schr\"{o}dinger operator on $l^p(X)$ in a natural way. We illustrate our approach by determining the essential spectra of Schr\"{o}dinger operators with slowly oscillating potential both on zig-zag and on hexagonal graphs, the latter being related to nano-structures

Regulation of eosinophilia and allergic airway inflammation by the glycan-binding protein galectin-1

Galectin-1 (Gal-1), a glycan-binding protein with broad antiinflammatory activities, functions as a proresolving mediator in autoimmune and chronic inflammatory disorders. However, its role in allergic airway inflammation has not yet been elucidated. We evaluated the effects of Gal-1 on eosinophil function and its role in a mouse model of allergic asthma. Allergen exposure resulted in airway recruitment of Gal-1-expressing inflammatory cells, including eosinophils, as well as increased Gal-1 in extracellular spaces in the lungs. In vitro, extracellular Gal-1 exerted divergent effects on eosinophils that were N-glycan- And dose-dependent. At concentrations ≤0.25 μM, Gal-1 increased eosinophil adhesion to vascular cell adhesion molecule-1, caused redistribution of integrin CD49d to the periphery and cell clustering, but inhibited ERK(1/2) activation and eotaxin-1-induced migration. Exposure to concentrations ≥1 μM resulted in ERK(1/2)- dependent apoptosis and disruption of the F- Actin cytoskeleton. At lower concentrations, Gal-1 did not alter expression of adhesion molecules (CD49d, CD18, CD11a, CD11b, L-selectin) or of the chemokine receptor CCR3, but decreased CD49d and CCR3 was observed in eosinophils treated with higher concentrations of this lectin. In vivo, allergen-challenged Gal-1-deficient mice exhibited increased recruitment of eosinophils and CD3+ T lymphocytes in the airways as well as elevated peripheral blood and bone marrow eosinophils relative to corresponding WT mice. Further, these mice had an increased propensity to develop airway hyperresponsiveness and displayed significantly elevated levels of TNF-α in lung tissue. This study suggests that Gal-1 can limit eosinophil recruitment to allergic airways and suppresses airway inflammation by inhibiting cell migration and promoting eosinophil apoptosis.Fil: Ge, Xiao Na. University of Minnesota; Estados UnidosFil: Ha, Sung Gil. University of Minnesota; Estados UnidosFil: Greenberg, Yana G.. University of Minnesota; Estados UnidosFil: Rao, Amrita. University of Minnesota; Estados UnidosFil: Bastan, Idil. University of Minnesota; Estados UnidosFil: Blidner, Ada Gabriela. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental. Fundación de Instituto de Biología y Medicina Experimental. Instituto de Biología y Medicina Experimental; ArgentinaFil: Rao, Savita P.. University of Minnesota; Estados UnidosFil: Rabinovich, Gabriel Adrián. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental. Fundación de Instituto de Biología y Medicina Experimental. Instituto de Biología y Medicina Experimental; ArgentinaFil: Sriramarao, P.. University of Minnesota; Estados Unido

Dynamical Encoding by Networks of Competing Neuron Groups: Winnerless Competition

Following studies of olfactory processing in insects and fish, we investigate neural networks whose dynamics in phase space is represented by orbits near the heteroclinic connections between saddle regions (fixed points or limit cycles). These networks encode input information as trajectories along the heteroclinic connections. If there are N neurons in the network, the capacity is approximately e(N-1)!, i.e., much larger than that of most traditional network structures. We show that a small winnerless competition network composed of FitzHugh-Nagumo spiking neurons efficiently transforms input information into a spatiotemporal output

Enhancement of synchronization in a hybrid neural circuit by spike timing dependent plasticity

Synchronization of neural activity is fundamental for many functions of the brain. We demonstrate that spike-timing dependent plasticity (STDP) enhances synchronization (entrainment) in a hybrid circuit composed of a spike generator, a dynamic clamp emulating an excitatory plastic synapse, and a chemically isolated neuron from the Aplysia abdominal ganglion. Fixed-phase entrainment of the Aplysia neuron to the spike generator is possible for a much wider range of frequency ratios and is more precise and more robust with the plastic synapse than with a nonplastic synapse of comparable strength. Further analysis in a computational model of HodgkinHuxley-type neurons reveals the mechanism behind this significant enhancement in synchronization. The experimentally observed STDP plasticity curve appears to be designed to adjust synaptic strength to a value suitable for stable entrainment of the postsynaptic neuron. One functional role of STDP might therefore be to facilitate synchronization or entrainment of nonidentical neurons

Winnerless competition between sensory neurons generates chaos: A possible mechanism for molluscan hunting behavior

© 2002 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.In the presence of prey, the marine mollusk Clione limacina exhibits search behavior, i.e., circular motions whose plane and radius change in a chaotic-like manner. We have formulated a dynamical model of the chaotic hunting behavior of Clione based on physiological in vivo and in vitroexperiments. The model includes a description of the action of the cerebral hunting interneuron on the receptor neurons of the gravity sensory organ, the statocyst. A network of six receptor model neurons with Lotka–Volterra-type dynamics and nonsymmetric inhibitory interactions has no simple static attractors that correspond to winner take all phenomena. Instead, the winnerless competition induced by the hunting neuron displays hyperchaos with two positive Lyapunov exponents. The origin of the chaos is related to the interaction of two clusters of receptor neurons that are described with two heteroclinic loops in phase space. We hypothesize that the chaotic activity of the receptor neurons can drive the complex behavior of Clione observed during hunting.Support for this work came from NIH Grant No. 2R01 NS38022- 05A1. P.V. acknowledges support from MCT BFI2000-0157. M.R. acknowledges support from U.S. Department of Energy Grant No. DE-FG03-96ER14592

Pattern formation without heating in an evaporative convection experiment

We present an evaporation experiment in a single fluid layer. When latent heat associated to the evaporation is large enough, the heat flow through the free surface of the layer generates temperature gradients that can destabilize the conductive motionless state giving rise to convective cellular structures without any external heating. The sequence of convective patterns obtained here without heating, is similar to that obtained in B\'enard-Marangoni convection. This work present the sequence of spatial bifurcations as a function of the layer depth. The transition between square to hexagonal pattern, known from non-evaporative experiments, is obtained here with a similar change in wavelength.Comment: Submitted to Europhysics Letter

Coarsening in potential and nonpotential models of oblique stripe patterns

We study the coarsening of two-dimensional oblique stripe patterns by numerically solving potential and nonpotential anisotropic Swift-Hohenberg equations. Close to onset, all models exhibit isotropic coarsening with a single characteristic length scale growing in time as $t^{1/2}$. Further from onset, the characteristic lengths along the preferred directions $\hat{x}$ and $\hat{y}$ grow with different exponents, close to 1/3 and 1/2, respectively. In this regime, one-dimensional dynamical scaling relations hold. We draw an analogy between this problem and Model A in a stationary, modulated external field. For deep quenches, nonpotential effects produce a complicated dislocation dynamics that can lead to either arrested or faster-than-power-law growth, depending on the model considered. In the arrested case, small isolated domains shrink down to a finite size and fail to disappear. A comparison with available experimental results of electroconvection in nematics is presented.Comment: 13 pages, 13 figures. To appear in Phys. Rev.