Automatic sorting of point pattern sets using Minkowski functionals

Fractured columnar small-world functional network organization in volumes of L2/3 of mouse auditory cortex

The sensory cortices of the brain exhibit large-scale functional topographic organization, such as the tonotopic organization of the primary auditory cortex (A1) according to sound frequency. However, at the level of individual neurons, layer 2/3 (L2/3) A1 appears functionally heterogeneous. To identify if there exists a higher-order functional organization of meso-scale neuronal networks within L2/3 that bridges order and disorder, we used in vivo two-photon calcium imaging of pyramidal neurons to identify networks in three-dimensional volumes of L2/3 A1 in awake mice. Using tonal stimuli, we found diverse receptive fields with measurable colocalization of similarly tuned neurons across depth but less so across L2/3 sublayers. These results indicate a fractured microcolumnar organization with a column radius of ∼50 µm, with a more random organization of the receptive field over larger radii. We further characterized the functional networks formed within L2/3 by analyzing the spatial distribution of signal correlations (SCs). Networks show evidence of Rentian scaling in physical space, suggesting effective spatial embedding of subnetworks. Indeed, functional networks have characteristics of small-world topology, implying that there are clusters of functionally similar neurons with sparse connections between differently tuned neurons. These results indicate that underlying the regularity of the tonotopic map on large scales in L2/3 is significant tuning diversity arranged in a hybrid organization with microcolumnar structures and efficient network topologies.

From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns

Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.

Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

A continuous measure of symmetry and the Voronoi entropy of 2D patterns representing Voronoi diagrams emerging from the Penrose tiling were calculated. A given Penrose tiling gives rise to a diversity of the Voronoi diagrams when the centers, vertices, and the centers of the edges of the Penrose rhombs are taken as the seed points (or nuclei). Voronoi diagrams keep the initial symmetry group of the Penrose tiling. We demonstrate that the continuous symmetry measure and the Voronoi entropy of the studied sets of points, generated by the Penrose tiling, do not necessarily correlate. Voronoi diagrams emerging from the centers of the edges of the Penrose rhombs, considered nuclei, deny the hypothesis that the continuous measure of symmetry and the Voronoi entropy are always correlated. The Voronoi entropy of this kind of tiling built of asymmetric convex quadrangles equals zero, whereas the continuous measure of symmetry of this pattern is high. Voronoi diagrams generate new types of Penrose tiling, which are different from the classical Penrose tessellation.

Multiple Particle Correlation Analysis of Many-Particle Systems: Formalism and Application to Active Matter

We introduce a fast spatial point pattern analysis technique that is suitable for systems of many identical particles giving rise to multiparticle correlations up to arbitrary order. The obtained correlation parameters allow us to quantify the quality of mean field assumptions or theories that incorporate correlations of limited order. We study the Vicsek model of self-propelled particles and create a correlation map marking the required correlation order for each point in phase space incorporating up to ten-particle correlations. We find that multiparticle correlations are important even in a large part of the disordered phase. Furthermore, the two-particle correlation parameter serves as an excellent order parameter to locate both phase transitions of the system, whereas two different order parameters were required before.

Characterization of Self-Assembled 2D Patterns with Voronoi Entropy

The Voronoi entropy is a mathematical tool for quantitative characterization of the orderliness of points distributed on a surface. The tool is useful to study various surface self-assembly processes. We provide the historical background, from Kepler and Descartes to our days, and discuss topological properties of the Voronoi tessellation, upon which the entropy concept is based, and its scaling properties, known as the Lewis and Aboav–Weaire laws. The Voronoi entropy has been successfully applied to recently discovered self-assembled structures, such as patterned microporous polymer surfaces obtained by the breath figure method and levitating ordered water microdroplet clusters.

InterCells: A Generic Monte-Carlo Simulation of Intercellular Interfaces Captures Nanoscale Patterning at the Immune Synapse

Molecular interactions across intercellular interfaces serve to convey information between cells and to trigger appropriate cell functions. Examples include cell development and growth in tissues, neuronal and immune synapses (ISs). Here, we introduce an agent-based Monte-Carlo simulation of user-defined cellular interfaces. The simulation allows for membrane molecules, embedded at intercellular contacts, to diffuse and interact, while capturing the topography and energetics of the plasma membranes of the interface. We provide a detailed example related to pattern formation in the early IS. Using simulation predictions and three-color single molecule localization microscopy (SMLM), we detected the intricate mutual patterning of T cell antigen receptors (TCRs), integrins and glycoproteins in early T cell contacts with stimulating coverslips. The simulation further captures the dynamics of the patterning under the experimental conditions and at the IS with antigen presenting cells (APCs). Thus, we provide a generic tool for simulating realistic cell-cell interfaces, which can be used for critical hypothesis testing and experimental design in an iterative manner.

Super-resolution Analysis of TCR-Dependent Signaling: Single-Molecule Localization Microscopy

Single-molecule localization microscopy (SMLM) comprises methods that produce super-resolution images from molecular locations of single molecules. These techniques mathematically determine the center of a diffraction-limited spot produced by a fluorescent molecule, which represents the most likely location of the molecule. Only a small cohort of well-separated molecules is visualized in a single image, and then many images are obtained from a single sample. The localizations from all the images are combined to produce a super-resolution picture of the sample. Here we describe the application of two methods, photoactivation localization microscopy (PALM) and direct stochastic optical reconstruction microscopy (dSTORM), to the study of signaling microclusters in T cells.

Automatic sorting of point pattern sets using Minkowski functionals

Point pattern sets arise in many different areas of physical, biological, and applied research, representing many random realizations of underlying pattern formation mechanisms. These pattern sets can be heterogeneous with respect to underlying spatial processes, which may not be visually distiguishable. This heterogeneity can be elucidated by looking at statistical measures of the patterns sets and using these measures to divide the pattern sets into distinct groups representing like spatial processes. We introduce here a numerical procedure for sorting point pattern sets into spatially homogenous groups using functional principal component analysis (FPCA) applied to the approximated Minkowski functionals of each pattern. We demonstrate that this procedure correctly sorts pattern sets into similar groups both when the patterns are drawn from similar processes and when the second-order characteristics of the pattern are identical. We highlight this routine for distinguishing the molecular patterning of fluorescently labeled cell membrane proteins, a subject of much interest in studies investigating complex spatial signaling patterns involved in the human immune response.