Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset \Z^d$, are independent and their variances satisfy \sigma_{xy}^2:=\E \abs{H_{xy}}^2 = W^{-d} f((x - y)/W) for some probability density $f$. We assume that the law of each matrix element $H_{xy}$ is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on time scales $t\ll W^{d/3}$. We also show that the localization length of the eigenvectors of $H$ is larger than a factor $W^{d/6}$ times the band width $W$. All results are uniform in the size \abs{\Lambda} of the matrix. This extends our recent result \cite{erdosknowles} to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying $\sum_x\sigma_{xy}^2=1$ for all $y$, the largest eigenvalue of $H$ is bounded with high probability by $2 + M^{-2/3 + \epsilon}$ for any $\epsilon > 0$, where M \deq 1 / (\max_{x,y} \sigma_{xy}^2).Comment: Corrected typos and some inaccuracies in appendix

Population Structure and Dispersal of the Coral-Excavating Sponge Cliona delitrix

Some excavating sponges of the genus Cliona compete with live reef corals, often killing and bioeroding entire colonies. Important aspects affecting distribution of these species, such as dispersal capability and population structure, remain largely unknown. Thus, the aim of this study was to determine levels of genetic connectivity and dispersal of Cliona delitrix across the Greater Caribbean (Caribbean Sea, Bahamas and Florida), to understand current patterns and possible future trends in their distribution and effects on coral reefs. Using ten species-specific microsatellite markers, we found high levels of genetic differentiation between six genetically distinct populations: one in the Atlantic (Florida-Bahamas), one specific to Florida and four in the South Caribbean Sea. In Florida, two independent breeding populations are likely separated by depth. Gene flow and ecological dispersal occur among other populations in the Florida reef tract, and between some Florida locations and the Bahamas. Similarly, gene flow occurs between populations in the South Caribbean Sea, but appears restricted between the Caribbean Sea and the Atlantic (Florida-Bahamas). Dispersal of C. delitrix was farther than expected for a marine sponge and favoured in areas where currents are strong enough to transport sponge eggs or larvae over longer distances. Our results support the influence of ocean current patterns on genetic connectivity, and constitute a baseline to monitor future C. delitrix trends under climate change

Low Genetic Differentiation across Three Major Ocean Populations of the Whale Shark, Rhincodon typus

BACKGROUND:Whale sharks are a declining species for which little biological data is available. While these animals are protected in many parts of their range, they are fished legally and illegally in some countries. Baseline biological and ecological data are needed to allow the formulation of an effective conservation plan for whale sharks. It is not known, for example, whether the whale shark is represented by a single worldwide panmictic population or by numerous, reproductively isolated populations. Genetic analysis of population structure is one essential component of the baseline data required for whale shark conservation. METHODOLOGY/PRINCIPAL FINDINGS:We have identified 8 polymorphic microsatellites in the whale shark and used these markers to assess genetic variation and population structure in a panel of whale sharks covering a broad geographic region. This is the first record of microsatellite loci in the whale shark, which displayed an average of 9 alleles per locus and mean H(o) = 0.66 and H(e) = 0.69. All but one of the eight loci meet the expectations of Hardy-Weinberg equilibrium. Analysis of these loci in whale sharks representing three major portions of their range, the Pacific (P), Caribbean (C), and Indian (I) Oceans, determined that there is little population differentiation between animals sampled in different geographic regions, indicating historical gene flow between populations. F(ST) values for inter-ocean comparisons were low (PxC = 0.0387, CxI = 0.0296 and PxI = -0.0022), and only CxI approached statistical significance (p = 0.0495). CONCLUSIONS/SIGNIFICANCE:We have shown only low levels of genetic differentiation between geographically distinct whale shark populations. Existing satellite tracking data have revealed both regional and long-range migration of whale sharks throughout their range, which supports the finding of gene flow between populations. Whale sharks traverse geographic and political boundaries during their life history and interbreed with animals from distant populations; conservation efforts must therefore target international protection for this species

Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the a limiting horizontal mean counting-measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the mean zero-counting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic function with symmetry around the real axis. These results extend a work by Norbert Wiener.Comment: 24 pages, 1 figure. Some corrections were made and presentation was improve

EphA3 Expressed in the Chicken Tectum Stimulates Nasal Retinal Ganglion Cell Axon Growth and Is Required for Retinotectal Topographic Map Formation

BACKGROUND: Retinotopic projection onto the tectum/colliculus constitutes the most studied model of topographic mapping and Eph receptors and their ligands, the ephrins, are the best characterized molecular system involved in this process. Ephrin-As, expressed in an increasing rostro-caudal gradient in the tectum/colliculus, repel temporal retinal ganglion cell (RGC) axons from the caudal tectum and inhibit their branching posterior to their termination zones. However, there are conflicting data regarding the nature of the second force that guides nasal axons to invade and branch only in the caudal tectum/colliculus. The predominant model postulates that this second force is produced by a decreasing rostro-caudal gradient of EphA7 which repels nasal optic fibers and prevents their branching in the rostral tectum/colliculus. However, as optic fibers invade the tectum/colliculus growing throughout this gradient, this model cannot explain how the axons grow throughout this repellent molecule. METHODOLOGY/PRINCIPAL FINDINGS: By using chicken retinal cultures we showed that EphA3 ectodomain stimulates nasal RGC axon growth in a concentration dependent way. Moreover, we showed that nasal axons choose growing on EphA3-expressing cells and that EphA3 diminishes the density of interstitial filopodia in nasal RGC axons. Accordingly, in vivo EphA3 ectodomain misexpression directs nasal optic fibers toward the caudal tectum preventing their branching in the rostral tectum. CONCLUSIONS: We demonstrated in vitro and in vivo that EphA3 ectodomain (which is expressed in a decreasing rostro-caudal gradient in the tectum) is necessary for topographic mapping by stimulating the nasal axon growth toward the caudal tectum and inhibiting their branching in the rostral tectum. Furthermore, the ability of EphA3 of stimulating axon growth allows understanding how optic fibers invade the tectum growing throughout this molecular gradient. Therefore, opposing tectal gradients of repellent ephrin-As and of axon growth stimulating EphA3 complement each other to map optic fibers along the rostro-caudal tectal axis

Modeling Activity and Target-Dependent Developmental Cell Death of Mouse Retinal Ganglion Cells Ex Vivo

Programmed cell death is widespread during the development of the central nervous system and serves multiple purposes including the establishment of neural connections. In the mouse retina a substantial reduction of retinal ganglion cells (RGCs) occurs during the first postnatal week, coinciding with the formation of retinotopic maps in the superior colliculus (SC). We previously established a retino-collicular culture preparation which recapitulates the progressive topographic ordering of RGC projections during early post-natal life. Here, we questioned whether this model could also be suitable to examine the mechanisms underlying developmental cell death of RGCs. Brn3a was used as a marker of the RGCs. A developmental decline in the number of Brn3a-immunolabelled neurons was found in the retinal explant with a timing that paralleled that observed in vivo. In contrast, the density of photoreceptors or of starburst amacrine cells increased, mimicking the evolution of these cell populations in vivo. Blockade of neural activity with tetrodotoxin increased the number of surviving Brn3a-labelled neurons in the retinal explant, as did the increase in target availability when one retinal explant was confronted with 2 or 4 collicular slices. Thus, this ex vivo model reproduces the developmental reduction of RGCs and recapitulates its regulation by neural activity and target availability. It therefore offers a simple way to analyze developmental cell death in this classic system. Using this model, we show that ephrin-A signaling does not participate to the regulation of the Brn3a population size in the retina, indicating that eprhin-A-mediated elimination of exuberant projections does not involve developmental cell death

The molecular phylogeny of eph receptors and ephrin ligands

<p>Abstract</p> <p>Background</p> <p>The tissue distributions and functions of Eph receptors and their ephrin ligands have been well studied, however less is known about their evolutionary history. We have undertaken a phylogenetic analysis of Eph receptors and ephrins from a number of invertebrate and vertebrate species.</p> <p>Results</p> <p>Our findings indicate that Eph receptors form three major clades: one comprised of non-chordate and cephalochordate Eph receptors, a second comprised of urochordate Eph receptors, and a third comprised of vertebrate Eph receptors. Ephrins, on the other hand, fall into either a clade made up of the non-chordate and cephalochordate ephrins plus the urochordate and vertebrate ephrin-Bs or a clade made up of the urochordate and vertebrate ephrin-As.</p> <p>Conclusion</p> <p>We have concluded that Eph receptors and ephrins diverged into A and B-types at different points in their evolutionary history, such that primitive chordates likely possessed an ancestral ephrin-A and an ancestral ephrin-B, but only a single Eph receptor. Furthermore, ephrin-As appear to have arisen in the common ancestor of urochordates and vertebrates, whereas ephrin-Bs have a more ancient bilaterian origin. Ancestral ephrin-B-like ligands had transmembrane domains; as GPI anchors appear to have arisen or been lost at least 3 times.</p

A Multi-Component Model of the Developing Retinocollicular Pathway Incorporating Axonal and Synaptic Growth

During development, neurons extend axons to different brain areas and produce stereotypical patterns of connections. The mechanisms underlying this process have been intensively studied in the visual system, where retinal neurons form retinotopic maps in the thalamus and superior colliculus. The mechanisms active in map formation include molecular guidance cues, trophic factor release, spontaneous neural activity, spike-timing dependent plasticity (STDP), synapse creation and retraction, and axon growth, branching and retraction. To investigate how these mechanisms interact, a multi-component model of the developing retinocollicular pathway was produced based on phenomenological approximations of each of these mechanisms. Core assumptions of the model were that the probabilities of axonal branching and synaptic growth are highest where the combined influences of chemoaffinity and trophic factor cues are highest, and that activity-dependent release of trophic factors acts to stabilize synapses. Based on these behaviors, model axons produced morphologically realistic growth patterns and projected to retinotopically correct locations in the colliculus. Findings of the model include that STDP, gradient detection by axonal growth cones and lateral connectivity among collicular neurons were not necessary for refinement, and that the instructive cues for axonal growth appear to be mediated first by molecular guidance and then by neural activity. Although complex, the model appears to be insensitive to variations in how the component developmental mechanisms are implemented. Activity, molecular guidance and the growth and retraction of axons and synapses are common features of neural development, and the findings of this study may have relevance beyond organization in the retinocollicular pathway

Ephrin-A5 and EphA5 Interaction Induces Synaptogenesis during Early Hippocampal Development

Synaptogenesis is a fundamental step in neuronal development. For spiny glutamatergic synapses in hippocampus and cortex, synaptogenesis involves adhesion of pre and postsynaptic membranes, delivery and anchorage of pre and postsynaptic structures including scaffolds such as PSD-95 and NMDA and AMPA receptors, which are glutamate-gated ion channels, as well as the morphological maturation of spines. Although electrical activity-dependent mechanisms are established regulators of these processes, the mechanisms that function during early development, prior to the onset of electrical activity, are unclear. The Eph receptors and ephrins provide cell contact-dependent pathways that regulate axonal and dendritic development. Members of the ephrin-A family are glycosyl-phosphatidylinositol-anchored to the cell surface and activate EphA receptors, which are receptor tyrosine kinases.Here we show that ephrin-A5 interaction with the EphA5 receptor following neuron-neuron contact during early development of hippocampus induces a complex program of synaptogenic events, including expression of functional synaptic NMDA receptor-PSD-95 complexes plus morphological spine maturation and the emergence of electrical activity. The program depends upon voltage-sensitive calcium channel Ca2+ fluxes that activate PKA, CaMKII and PI3 kinase, leading to CREB phosphorylation and a synaptogenic program of gene expression. AMPA receptor subunits, their scaffolds and electrical activity are not induced. Strikingly, in contrast to wild type, stimulation of hippocampal slices from P6 EphA5 receptor functional knockout mice yielded no NMDA receptor currents.These studies suggest that ephrin-A5 and EphA5 signals play a necessary, activity-independent role in the initiation of the early phases of synaptogenesis. The coordinated expression of the NMDAR and PSD-95 induced by eprhin-A5 interaction with EphA5 receptors may be the developmental switch that induces expression of AMPAR and their interacting proteins and the transition to activity-dependent synaptic regulation