The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence toward peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities.

Many biological systems consist of branching structures that exhibit a wide variety of shapes. Our understanding of their systematic roles is hampered from the start by the lack of a fundamental means of standardizing the description of complex branching patterns, such as those of neuronal trees. To solve this problem, we have invented the Topological Morphology Descriptor (TMD), a method for encoding the spatial structure of any tree as a “barcode”, a unique topological signature. As opposed to traditional morphometrics, the TMD couples the topology of the branches with their spatial extents by tracking their topological evolution in 3-dimensional space. We prove that neuronal trees, as well as stochastically generated trees, can be accurately categorized based on their TMD profiles. The TMD retains sufficient global and local information to create an unbiased benchmark test for their categorization and is able to quantify and characterize the structural differences between distinct morphological groups. The use of this mathematically rigorous method will advance our understanding of the anatomy and diversity of branching morphologies.Electronic supplementary material The online version of this article (10.1007/s12021-017-9341-1) contains supplementary material, which is available to authorized users.

In most applications of nanoporous materials the pore structure is as important as the chemical composition as a determinant of performance. For example, one can alter performance in applications like carbon capture or methane storage by orders of magnitude by only modifying the pore structure. For these applications it is therefore important to identify the optimal pore geometry and use this information to find similar materials. However, the mathematical language and tools to identify materials with similar pore structures, but different composition, has been lacking. We develop a pore recognition approach to quantify similarity of pore structures and classify them using topological data analysis. This allows us to identify materials with similar pore geometries, and to screen for materials that are similar to given top-performing structures. Using methane storage as a case study, we also show that materials can be divided into topologically distinct classes requiring different optimization strategies.

Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning. We give efficient algorithms for calculating persistence landscapes, their averages, and distances between such averages. We discuss an implementation of these algorithms and some related procedures. These are intended to facilitate the combination of statistics and machine learning with topological data analysis. We present an experiment showing that the low-dimensional persistence landscapes of points sampled from spheres (and boxes) of varying dimensions differ.Comment: 24 page

The materials genome initiative has led to the creation of a large (over a million) database of different classes of nanoporous materials. As the number of hypothetical materials that can, in principle, be experimentally synthesized is infinite, a bottleneck in the use of these databases for the discovery of novel materials is the lack of efficient computational tools to analyze them. Current approaches use brute-force molecular simulations to generate thermodynamic data needed to predict the performance of these materials in different applications, but this approach is limited to the analysis of tens of thousands of structures due to computational intractability. As such, it is conceivable and even likely that the best nanoporous materials for any given application have yet to be discovered both experimentally and theoretically. In this article, we seek a computational approach to tackle this issue by transitioning away from brute-force characterization to high-throughput screening methods based on big-data analysis, using the zeolite database as an example. For identifying and comparing zeolites, we used a topological data analysis-based descriptor (TD) recognizing pore shapes. For methane storage and carbon capture applications, our analyses seeking pairs of highly similar zeolites discovered good correlations between performance properties of a seed zeolite and the corresponding pair, which demonstrates the capability of TD to predict performance properties. It was also shown that when some top zeolites are known, TD can be used to detect other high-performing materials as their neighbors with high probability. Finally, we performed high-throughput screening of zeolites based on TD. For methane storage (or carbon capture) applications, the promising sets from our screenings contained high-percentages of top-performing zeolites: 45% (or 23%) of the top 1% zeolites in the entire set. This result shows that our screening approach using TD is highly efficient in finding high-performing materials. We expect that this approach could easily be extended to other applications by simply adjusting one parameter, the size of the target gas molecule.

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We describe an algorithm for computing a finite, and typically small, presentation of the fundamental group of a finite regular CW-space. The algorithm is based on the construction of a discrete vector field on the 3-skeleton of the space. A variant yields the homomorphism of fundamental groups induced by a cellular map of spaces. We illustrate how the algorithm can be used to infer information about the fundamental group π 1 (K ) of a metric space K using only a finite point cloud X sampled from the space. In the special case where K is a d-dimensional compact manifold K ⊂ R d , we consider the closure of the complement of K in the d-sphere M K = S d \ K . For a base-point x in the boundary ∂ M K of the manifold M K one can attempt to determine, from the point cloud X , the induced homomorphism of fundamental groups φ : π 1 (∂ M K , x) → π 1 (M K , x) in the category of finitely presented groups. We illustrate a computer implementation for K a small closed tubular neigh-P.D. 123 28 P. Brendel et al.bourhood of a tame knot in R 3 . In this case the homomorphism φ is known to be a complete ambient isotopy invariant of the knot. We observe that low-index subgroups of finitely presented groups provide useful invariants of φ. In particular, the first integral homology of subgroups G < π 1 (M K ) of index at most six suffices to distinguish between all prime knots with 11 or fewer crossings (ignoring chirality). We plan to provide formal time estimates for our algorithm and characteristics of a high performance C++ implementation in a subsequent paper. The prototype computer implementation of the present paper has been written in the interpreted gap programming language for computational algebra.

Recent work on algebraic-topological methods for verifying coverage in planar sensor networks relied exclusively on centralized computation: a limiting constraint for large networks. This paper presents a distributed algorithm for homology computation over a sensor network, for purposes of verifying coverage. The techniques involve reduction and coreduction of simplicial complexes, and are of independent interest. Verification of the ensuing algorithms is proved, and simulations detail the improved network efficiency and performance.