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We consider graphs Sigma^n in R^m with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau, and Ecker-Huisken, we derive an interior curvature estimate of the form |A|^2<=C/R^2 up to dimension n<=5,…

Matrix visualizations are a useful tool to provide a general overview of a graph's structure. For multivariate graphs, a remaining challenge is to cope with the attributes that are associated with nodes and edges. Addressing this challenge,…

The optimal shapes attained by contractile cells on adhesive substrates are determined by the interplay between intracellular forces and adhesion with the extracellular matrix. We model the cell as a contractile film bounded by an elastic…

Convolutional neural networks (CNNs) have been shown to be state-of-the-art models for visual cortical neurons. Cortical neurons in the primary visual cortex are sensitive to contextual information mediated by extensive horizontal and…

Training convolutional neural networks (CNNs) for segmentation of pulmonary airway, artery, and vein is challenging due to sparse supervisory signals caused by the severe class imbalance between tubular targets and background. We present a…

Feed-forward convolutional neural networks (CNNs) are currently state-of-the-art for object classification tasks such as ImageNet. Further, they are quantitatively accurate models of temporally-averaged responses of neurons in the primate…

This article gives an overview of a normative computational theory of visual receptive fields, by which idealized functional models of early spatial, spatio-chromatic and spatio-temporal receptive fields can be derived in an axiomatic way…

For singular $n$-manifolds in $\mathbb R^{n+k}$ with a corank 1 singular point at $p\in M^n_{\mbox{sing}}$ we define up to $l(n-1)$ different axial curvatures at $p$, where $l=\min\{n,k+1\}$. These curvatures are obtained using the…

For manifolds $\cal M$ of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space $L^2(T^*\cal M)$ and construct an…

Biological cells sense and respond to mechanical forces, but how such a mechanosensing proccess takes place in a nonlinear inhomogeneous fibrous matrix remains unknown. We show that cells in a fibrous matrix induce deformation fields that…

We address the problem of building theoretical models that help elucidate the function of the visual brain at computational/algorithmic and structural/mechanistic levels. We seek to understand how the receptive fields and topographic maps…

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle,…

Geometrical illusions are a subclass of optical illusions in which the geometrical characteristics of patterns such as orientations and angles are distorted and misperceived as the result of low- to high-level retinal/cortical processing.…

Spatial organisation is a hallmark of all living cells, and recreating it in model systems is a necessary step in the creation of synthetic cells. It is therefore of both fundamental and practical interest to better understand the basic…

While analyzing mobile systems we often approximate the actual coverage surface and assume an ideal cell shape. In a multi-cellular network, because of its tessellating nature, a hexagon is more preferred than a circular geometry. Despite…

Neural circuits are able to perform computations under very diverse conditions and requirements. The required computations impose clear constraints on their fine-tuning: a rapid and maximally informative response to stimuli in general…

Foundation models have shown remarkable success in fitting biological visual systems; however, their black-box nature inherently limits their utility for understanding brain function. Here, we peek inside a SOTA foundation model of neural…

Orientation selectivity is a remarkable feature of the neurons located in the primary visual cortex. Provided that the visual neurons acquire orientation selectivity through activity-dependent Hebbian learning, the development process could…

The neural manifold hypothesis postulates that the activity of a neural population forms a low-dimensional manifold whose structure reflects that of the encoded task variables. In this work, we combine topological deep generative models and…

Developments in Carrollian gravity and holography necessitate the use of singular Carroll vector fields, a feature that cannot be accommodated within standard Carrollian geometry. We introduce Carrollian Lie algebroids as a framework to…