Arguesian Identities in Invariant Theory

AbstractHaving been motivated by an example of Doubilet, Rota, and Stein [Stud. Appl. Math.56(1976), 185–216], we present a technique for constructing geometric identities in a Grassmann–Cayley algebra. Each identity represents a projective invariant closely related to the Theorem of Desargues in the plane and its generalizations to higher dimensional projective space. The construction employs certain combinatorial properties of matchings in bipartite graphs. We also prove a dimension independence result for Arguesian identities, thereby connecting the identities with lattice theory

Computational neuroanatomy and co-expression of genes in the adult mouse brain, analysis tools for the Allen Brain Atlas

We review quantitative methods and software developed to analyze genome-scale, brain-wide spatially-mapped gene-expression data. We expose new methods based on the underlying high-dimensional geometry of voxel space and gene space, and on simulations of the distribution of co-expression networks of a given size. We apply them to the Allen Atlas of the adult mouse brain, and to the co-expression network of a set of genes related to nicotine addiction retrieved from the NicSNP database. The computational methods are implemented in {\ttfamily{BrainGeneExpressionAnalysis}}, a Matlab toolbox available for download.Comment: 25 pages, 8 figures, accepted in Quantitative Biology (2012) 000

An anatomically comprehensive atlas of the adult human brain transcriptome

Neuroanatomically precise, genome-wide maps of transcript distributions are critical resources to complement genomic sequence data and to correlate functional and genetic brain architecture. Here we describe the generation and analysis of a transcriptional atlas of the adult human brain, comprising extensive histological analysis and comprehensive microarray profiling of ~900 neuroanatomically precise subdivisions in two individuals. Transcriptional regulation varies enormously by anatomical location, with different regions and their constituent cell types displaying robust molecular signatures that are highly conserved between individuals. Analysis of differential gene expression and gene co-expression relationships demonstrates that brain-wide variation strongly reflects the distributions of major cell classes such as neurons, oligodendrocytes, astrocytes and microglia. Local neighbourhood relationships between fine anatomical subdivisions are associated with discrete neuronal subtypes and genes involved with synaptic transmission. The neocortex displays a relatively homogeneous transcriptional pattern, but with distinct features associated selectively with primary sensorimotor cortices and with enriched frontal lobe expression. Notably, the spatial topography of the neocortex is strongly reflected in its molecular topography—the closer two cortical regions, the more similar their transcriptomes. This freely accessible online data resource forms a high-resolution transcriptional baseline for neurogenetic studies of normal and abnormal human brain function

Discovering Neuronal Cell Types and Their Gene Expression Profiles Using a Spatial Point Process Mixture Model

Cataloging the neuronal cell types that comprise circuitry of individual brain regions is a major goal of modern neuroscience and the BRAIN initiative. Single-cell RNA sequencing can now be used to measure the gene expression profiles of individual neurons and to categorize neurons based on their gene expression profiles. While the single-cell techniques are extremely powerful and hold great promise, they are currently still labor intensive, have a high cost per cell, and, most importantly, do not provide information on spatial distribution of cell types in specific regions of the brain. We propose a complementary approach that uses computational methods to infer the cell types and their gene expression profiles through analysis of brain-wide single-cell resolution in situ hybridization (ISH) imagery contained in the Allen Brain Atlas (ABA). We measure the spatial distribution of neurons labeled in the ISH image for each gene and model it as a spatial point process mixture, whose mixture weights are given by the cell types which express that gene. By fitting a point process mixture model jointly to the ISH images, we infer both the spatial point process distribution for each cell type and their gene expression profile. We validate our predictions of cell type-specific gene expression profiles using single cell RNA sequencing data, recently published for the mouse somatosensory cortex. Jointly with the gene expression profiles, cell features such as cell size, orientation, intensity and local density level are inferred per cell type