scholarly journals Almost tight upper bounds for the single cell and zone problems in three dimensions

1995 ◽  
Vol 14 (4) ◽  
pp. 385-410 ◽  
Author(s):  
D. Halperin ◽  
M. Sharif

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.



2019 ◽  
Author(s):  
FK Janiak ◽  
P Bartel ◽  
MR Bale ◽  
T Yoshimatsu ◽  
E Komulainen ◽  
...  

ABSTACTIn neuroscience, diffraction limited two-photon (2P) microscopy is a cornerstone technique that permits minimally invasive optical monitoring of neuronal activity. However, most conventional 2P microscopes impose significant constraints on the size of the imaging field-of-view and the specific shape of the effective excitation volume, thus limiting the scope of biological questions that can be addressed and the information obtainable. Here, employing ‘divergent beam optics’ (DBO), we present an ultra-low-cost, easily implemented and flexible solution to address these limitations, offering a several-fold expanded three-dimensional field of view that also maintains single-cell resolution. We show that this implementation increases both the space-bandwidth product and effective excitation power, and allows for straight-forward tailoring of the point-spread-function. Moreover, rapid laser-focus control via an electrically tunable lens now allows near-simultaneous imaging of remote regions separated in three dimensions and permits the bending of imaging planes to follow natural curvatures in biological structures. Crucially, our core design is readily implemented (and reversed) within a matter of hours, and fully compatible with a wide range of existing 2P customizations, making it highly suitable as a base platform for further development. We demonstrate the application of our system for imaging neuronal activity in a variety of examples in mice, zebrafish and fruit flies.



2020 ◽  
Author(s):  
Peter Quicke ◽  
Carmel L. Howe ◽  
Pingfan Song ◽  
Herman Verinaz Jadan ◽  
Chenchen Song ◽  
...  

AbstractLight field microscopy (LFM) enables high signal-to-noise ratio (SNR), light efficient volume imaging at fast frame rates, and has been successfully applied to single-cell resolution functional neuronal calcium imaging. Voltage imaging with genetically encoded voltage indicators (GEVIs) stands to particularly benefit from light field microscopy’s volumetric imaging capability due to high required sampling rates, and limited probe brightness and functional sensitivity. Previous LFM studies have imaged GEVIs to track population-level interactions only in invertebrate preparations and without single cell resolution. Here we demonstrate sub-cellular resolution GEVI light field imaging in acute mouse brain slices resolving dendritic voltage signals localized in three dimensions. We characterize the effects of different light field reconstruction techniques on the SNR and signal localization and compare the SNR to fluorescence transients imaged in wide field. Our results demonstrate the potential of light field voltage imaging for studying dendritic integration and action potential propagation and backpropagation in 3 spatial dimensions.



2021 ◽  
Author(s):  
Carla A. Gonçalves ◽  
Michael Larsen ◽  
Sascha Jung ◽  
Johannes Stratmann ◽  
Akiko Nakamura ◽  
...  

Abstract Human organogenesis remains relatively unexplored for ethical and practical reasons. Here we report the establishment of a single cell transcriptome atlas of the human fetal pancreas between 7 and 10 post-conceptional weeks of development. To interrogate cell-cell interactions we developed InterCom, an R-Package for identifying receptors-ligand pairs and their downstream effects. We further report the establishment of a human pancreas culture system starting from fetal tissue or human pluripotent stem cells, enabling the long-term maintenance of pancreas progenitors in a minimal, defined medium in three-dimensions. Benchmarking the cells produced in 2D and those expanded in 3D to fetal tissue reveals that progenitors expanded in 3D are transcriptionally closer to the fetal pancreas. We further demonstrate the potential of this system as a screening platform and identify the importance of the EGF and FGF pathways controlling human pancreas progenitor expansion.



2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Jing Huang ◽  
Huafeng Liu

In this paper, we consider divisor problems related to Hecke eigenvalues in three dimensions. We establish upper bounds and asymptotic formulas for these problems on average.



2021 ◽  
Author(s):  
Tara Chari ◽  
Joeyta Banerjee ◽  
Lior Pachter

Dimensionality reduction is standard practice for filtering noise and identifying relevant dimensions in large-scale data analyses. In biology, single-cell expression studies almost always begin with reduction to two or three dimensions to produce 'all-in-one' visuals of the data that are amenable to the human eye, and these are subsequently used for qualitative and quantitative analysis of cell relationships. However, there is little theoretical support for this practice. We examine the theoretical and practical implications of low-dimensional embedding of single-cell data, and find extensive distortions incurred on the global and local properties of biological patterns relative to the high-dimensional, ambient space. In lieu of this, we propose semi-supervised dimension reduction to higher dimension, and show that such targeted reduction guided by the metadata associated with single-cell experiments provides useful latent space representations for hypothesis-driven biological discovery.



10.37236/8253 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Thao Do

Given $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ ($d\geqslant 3)$, if there are many incidences, we expect to find a big cluster $K_{r,s}$ in their incidence graph. Apfelbaum and Sharir found lower and upper bounds for the largest size of $rs$, which match (up to a constant) only in three dimensions. In this paper we close the gap in four and five dimensions, up to some polylogarithmic factors.



2021 ◽  
Vol 4 (5) ◽  
pp. 1-25
Author(s):  
Kyungkeun Kang ◽  
◽  
Dongkwang Kim

<abstract><p>We construct generalized solutions for the Keller-Segel system with a degradation source coupled to Navier Stokes equations in three dimensions, in case that the power of degradation is smaller than quadratic. Furthermore, if the logistic type source is purely damping with no growing effect, we prove that solutions converge to zero in some norms and provide upper bounds of convergence rates in time.</p></abstract>



2021 ◽  
Author(s):  
Carla A. Gonçalves ◽  
Michael Larsen ◽  
Sascha Jung ◽  
Johannes Stratmann ◽  
Akiko Nakamura ◽  
...  

Abstract Human organogenesis remains relatively unexplored for ethical and practical reasons. Here we report the establishment of a single cell transcriptome atlas of the human fetal pancreas between 7 and 10 post-conceptional weeks of development. To interrogate cell-cell interactions we developed InterCom, an R-Package for identifying receptors-ligand pairs and their downstream effects. We further report the establishment of a human pancreas culture system starting from fetal tissue or human pluripotent stem cells, enabling the long-term maintenance of pancreas progenitors in a minimal, defined medium in three-dimensions. Benchmarking the cells produced in 2D and those expanded in 3D to fetal tissue reveals that progenitors expanded in 3D are transcriptionally closer to the fetal pancreas. We further demonstrate the potential of this system as a screening platform and identify the importance of the EGF and FGF pathways controlling human pancreas progenitor expansion.



Sign in / Sign up

Export Citation Format

Share Document