Symmetric Jacobians

This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over C such that JH satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d >= 3.Comment: 18 pages, minor corrections, grayscale eepic boxes have been replaced by colorful tikz boxe

A Conceptual Cortical Surface Atlas

Volumetric, slice-based, 3-D atlases are invaluable tools for understanding complex cortical convolutions. We present a simple scheme to convert a slice-based atlas to a conceptual surface atlas that is easier to visualize and understand. The key idea is to unfold each slice into a one-dimensional vector, and concatenate a succession of these vectors – while maintaining as much spatial contiguity as possible – into a 2-D matrix. We illustrate our methodology using a coronal slice-based atlas of the Rhesus Monkey cortex. The conceptual surface-based atlases provide a useful complement to slice-based atlases for the purposes of indexing and browsing

Long Range Magnetic Order and the Darwin Lagrangian

We simulate a finite system of $N$ confined electrons with inclusion of the Darwin magnetic interaction in two- and three-dimensions. The lowest energy states are located using the steepest descent quenching adapted for velocity dependent potentials. Below a critical density the ground state is a static Wigner lattice. For supercritical density the ground state has a non-zero kinetic energy. The critical density decreases with $N$ for exponential confinement but not for harmonic confinement. The lowest energy state also depends on the confinement and dimension: an antiferromagnetic cluster forms for harmonic confinement in two dimensions.Comment: 5 figure

Optic Flow Stimuli in and Near the Visual Field Centre: A Group fMRI Study of Motion Sensitive Regions

Motion stimuli in one visual hemifield activate human primary visual areas of the contralateral side, but suppress activity of the corresponding ipsilateral regions. While hemifield motion is rare in everyday life, motion in both hemifields occurs regularly whenever we move. Consequently, during motion primary visual regions should simultaneously receive excitatory and inhibitory inputs. A comparison of primary and higher visual cortex activations induced by bilateral and unilateral motion stimuli is missing up to now. Many motion studies focused on the MT+ complex in the parieto-occipito-temporal cortex. In single human subjects MT+ has been subdivided in area MT, which was activated by motion stimuli in the contralateral visual field, and area MST, which responded to motion in both the contra- and ipsilateral field. In this study we investigated the cortical activation when excitatory and inhibitory inputs interfere with each other in primary visual regions and we present for the first time group results of the MT+ subregions, allowing for comparisons with the group results of other motion processing studies. Using functional magnetic resonance imaging (fMRI), we investigated whole brain activations in a large group of healthy humans by applying optic flow stimuli in and near the visual field centre and performed a second level analysis. Primary visual areas were activated exclusively by motion in the contralateral field but to our surprise not by central flow fields. Inhibitory inputs to primary visual regions appear to cancel simultaneously occurring excitatory inputs during central flow field stimulation. Within MT+ we identified two subregions. Putative area MST (pMST) was activated by ipsi- and contralateral stimulation and located in the anterior part of MT+. The second subregion was located in the more posterior part of MT+ (putative area MT, pMT)

Potency of transgenic effectors for neurogenetic manipulation in Drosophila larvae

Genetic manipulations of neuronal activity are a cornerstone of studies aimed to identify the functional impact of defined neurons for animal behavior. With its small nervous system, rapid life cycle, and genetic amenability, the fruit fly Drosophila melanogaster provides an attractive model system to study neuronal circuit function. In the past two decades, a large repertoire of elegant genetic tools has been developed to manipulate and study neural circuits in the fruit fly. Current techniques allow genetic ablation, constitutive silencing, or hyperactivation of neuronal activity and also include conditional thermogenetic or optogenetic activation or inhibition. As for all genetic techniques, the choice of the proper transgenic tool is essential for behavioral studies. Potency and impact of effectors may vary in distinct neuron types or distinct types of behavior. We here systematically test genetic effectors for their potency to alter the behavior of Drosophila larvae, using two distinct behavioral paradigms: general locomotor activity and directed, visually guided navigation. Our results show largely similar but not equal effects with different effector lines in both assays. Interestingly, differences in the magnitude of induced behavioral alterations between different effector lines remain largely consistent between the two behavioral assays. The observed potencies of the effector lines in aminergic and cholinergic neurons assessed here may help researchers to choose the best-suited genetic tools to dissect neuronal networks underlying the behavior of larval fruit flies

Modelling the Distribution of 3D Brain MRI using a 2D Slice VAE

Probabilistic modelling has been an essential tool in medical image analysis, especially for analyzing brain Magnetic Resonance Images (MRI). Recent deep learning techniques for estimating high-dimensional distributions, in particular Variational Autoencoders (VAEs), opened up new avenues for probabilistic modeling. Modelling of volumetric data has remained a challenge, however, because constraints on available computation and training data make it difficult effectively leverage VAEs, which are well-developed for 2D images. We propose a method to model 3D MR brain volumes distribution by combining a 2D slice VAE with a Gaussian model that captures the relationships between slices. We do so by estimating the sample mean and covariance in the latent space of the 2D model over the slice direction. This combined model lets us sample new coherent stacks of latent variables to decode into slices of a volume. We also introduce a novel evaluation method for generated volumes that quantifies how well their segmentations match those of true brain anatomy. We demonstrate that our proposed model is competitive in generating high quality volumes at high resolutions according to both traditional metrics and our proposed evaluation.Comment: accepted for publication at MICCAI 2020. Code available https://github.com/voanna/slices-to-3d-brain-vae