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

Dendritic and axonal targeting patterns of a genetically-specified class of retinal ganglion cells that participate in image-forming circuits.

BackgroundThere are numerous functional types of retinal ganglion cells (RGCs), each participating in circuits that encode a specific aspect of the visual scene. This functional specificity is derived from distinct RGC morphologies and selective synapse formation with other retinal cell types; yet, how these properties are established during development remains unclear. Islet2 (Isl2) is a LIM-homeodomain transcription factor expressed in the developing retina, including approximately 40% of all RGCs, and has previously been implicated in the subtype specification of spinal motor neurons. Based on this, we hypothesized that Isl2+ RGCs represent a related subset that share a common function.ResultsWe morphologically and molecularly characterized Isl2+ RGCs using a transgenic mouse line that expresses GFP in the cell bodies, dendrites and axons of Isl2+ cells (Isl2-GFP). Isl2-GFP RGCs have distinct morphologies and dendritic stratification patterns within the inner plexiform layer and project to selective visual nuclei. Targeted filling of individual cells reveals that the majority of Isl2-GFP RGCs have dendrites that are monostratified in layer S3 of the IPL, suggesting they are not ON-OFF direction-selective ganglion cells. Molecular analysis shows that most alpha-RGCs, indicated by expression of SMI-32, are also Isl2-GFP RGCs. Isl2-GFP RGCs project to most retino-recipient nuclei during early development, but specifically innervate the dorsal lateral geniculate nucleus and superior colliculus (SC) at eye opening. Finally, we show that the segregation of Isl2+ and Isl2- RGC axons in the SC leads to the segregation of functional RGC types.ConclusionsTaken together, these data suggest that Isl2+ RGCs comprise a distinct class and support a role for Isl2 as an important component of a transcription factor code specifying functional visual circuits. Furthermore, this study describes a novel genetically-labeled mouse line that will be a valuable resource in future investigations of the molecular mechanisms of visual circuit formation

Loss-of-function analysis of EphA receptors in retinotectal mapping

Peer reviewedPublisher PD

A Phase transition in zero count probability for Stationary Gaussian Processes

We study the probability that a real stationary Gaussian process has at least $\eta T$ zeros in $[0,T]$ (overcrowding), or at most this number (undercrowding). We show that if the spectral measure of the process is supported on $\pm[B,A]$, overcrowding probability transitions from exponential decay to Gaussian decay at $\eta=\tfrac{A}{\pi}$, while undercrowding probability undergoes the reverse transition at $\eta=\tfrac{B}{\pi}$.Comment: 17 page

On the Importance of Countergradients for the Development of Retinotopy: Insights from a Generalised Gierer Model

During the development of the topographic map from vertebrate retina to superior colliculus (SC), EphA receptors are expressed in a gradient along the nasotemporal retinal axis. Their ligands, ephrin-As, are expressed in a gradient along the rostrocaudal axis of the SC. Countergradients of ephrin-As in the retina and EphAs in the SC are also expressed. Disruption of any of these gradients leads to mapping errors. Gierer's (1981) model, which uses well-matched pairs of gradients and countergradients to establish the mapping, can account for the formation of wild type maps, but not the double maps found in EphA knock-in experiments. I show that these maps can be explained by models, such as Gierer's (1983), which have gradients and no countergradients, together with a powerful compensatory mechanism that helps to distribute connections evenly over the target region. However, this type of model cannot explain mapping errors found when the countergradients are knocked out partially. I examine the relative importance of countergradients as against compensatory mechanisms by generalising Gierer's (1983) model so that the strength of compensation is adjustable. Either matching gradients and countergradients alone or poorly matching gradients and countergradients together with a strong compensatory mechanism are sufficient to establish an ordered mapping. With a weaker compensatory mechanism, gradients without countergradients lead to a poorer map, but the addition of countergradients improves the mapping. This model produces the double maps in simulated EphA knock-in experiments and a map consistent with the Math5 knock-out phenotype. Simulations of a set of phenotypes from the literature substantiate the finding that countergradients and compensation can be traded off against each other to give similar maps. I conclude that a successful model of retinotopy should contain countergradients and some form of compensation mechanism, but not in the strong form put forward by Gierer

Long-term site fidelity of endangered smalltooth sawfish (Pristis pectinata) from different mothers

Understanding how endangered species use nursery habitats is vital for recovery planning. Research on the smalltooth sawfish (Pristis pectinata) has shown that areas of estuarine nurseries, called hotspots, are used consistently. The objectives of our study were 1) to determine whether 10 young-of-the-year smalltooth sawfish in an artificial, non-main-stem portion (i.e., a seawall canal system) of a hotspot were descended from one or different mothers and 2) to document long-term habitat use by these individuals. At least 4 mothers contributed to the group, which comprised siblings, half-siblings, and unrelated individuals. Young sawfish exhibited site fidelity to their capture location, spending 61% of their time there. Continuous residency lasted as long as 86 days, but these fish made small-scale diel

Anthropogenic disturbance and evolutionary parameters: a lemon shark population experiencing habitat loss

The level of genetic variation in natural populations influences evolutionary potential, and may therefore influence responses to selection in the face of future environmental changes. By combining long-term monitoring of marked individuals with genetic pedigree reconstruction, we assessed whether habitat loss influenced genetic variation in a lemon shark (Negaprion brevirostris) population at an isolated nursery lagoon (Bimini, Bahamas). We also tracked changes in the strength and direction of natural selection. Contrary to initial expectations, we found that after the habitat loss neutral genetic variation increased, as did additive genetic variance for juvenile morphological traits (body length and mass). We hypothesize that these effects might result from philopatric behavior in females coupled with a possible influx of male genotypes from other nursery sites. We also found changes in the strength of selection on morphological traits, which weakened considerably after the disturbance; habitat loss therefore changed the phenotypes favored by natural selection. Because such human-induced shifts in the adaptive landscape may be common, we suggest that conservation biologists should not simply focus on neutral genetic variation per se, but also on assessing and preserving evolutionary parameters, such as additive genetic variation and selection

A quantitative genetic approach to assess the evolutionary potential of a coastal marine fish to ocean acidification

Assessing the potential of marine organisms to adapt genetically to increasing oceanic CO2 levels requires proxies such as heritability of fitness-related traits under ocean acidification (OA). We applied a quantitative genetic method to derive the first heritability estimate of survival under elevated CO2 conditions in a metazoan. Specifically, we reared offspring, selected from a wild coastal fish population (Atlantic silverside, Menidia menidia), at high CO2 conditions (~2300 μatm) from fertilization to 15 days posthatch, which significantly reduced survival compared to controls. Perished and surviving offspring were quantitatively sampled and genotyped along with their parents, using eight polymorphic microsatellite loci, to reconstruct a parent–offspring pedigree and estimate variance components. Genetically related individuals were phenotypically more similar (i.e., survived similarly long at elevated CO2 conditions) than unrelated individuals, which translated into a significantly nonzero heritability (0.20 ± 0.07). The contribution of maternal effects was surprisingly small (0.05 ± 0.04) and nonsignificant. Survival among replicates was positively correlated with genetic diversity, particularly with observed heterozygosity. We conclude that early life survival of M. menidia under high CO2 levels has a significant additive genetic component that could elicit an evolutionary response to OA, depending on the strength and direction of future selection