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Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components.

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.

A multifiltration is a functor indexed by $\mathbb{N}^r$ that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals., Comment: 21 pages, 3 figures

Background Predicting functional outcome is a major challenge in service care of early psychosis patients (EPP). We investigated clinical and metabolic features that enable objective assessment of the risk of poor functional outcomes in early psychosis patients (EPP), using state-of-the-art topological data analysis (TDA). The guiding philosophy of TDA is that the shape of large data sets, encoded by a mathematical signature, should reveal important relations among the data points. Our primary TDA tool is Mapper, which provides unsupervised multivariate pattern analysis of high-dimensional data, producing a compressed visual representation of the data giving a strong indication of where to look for meaningful clustering. Application of Mapper has already led to a number of remarkable results, including the identification of a new subtype of breast cancer and of two phenotypically distinct types of fragile X syndrome. Methods Recruitment: 101 EPP with informed consent were recruited from the Lausanne “Treatment and early Intervention in Psychosis Program”. They had reached the psychosis threshold on the CAARMS scale. Patients were assessed at entrance of the program by PANSS and MADRS, functioning by SOFAS and GAF scales. After 3 years of clinical follow-up, EPP were evaluated for symptoms levels, social and occupational functioning. Metabolomics: Plasma extracts at the program begin were analyzed by liquid-chromatography-mass spectrometry. Features data (m/z, retention time, integrated ion intensity) were extracted using xMSanalyser v1.3.5 with apLCMS v5.9.4. Mapper: Using the Mapper algorithm, we built a topological map representing the EP cohort through the lens of the PANSS scores at baseline. The mapper algorithm captures the shape of high dimensional data through a graph. Nodes represent patients or groups of patients. Edges between nodes represent shared patients. Results Three distinct clusters of patients could be identified: Group A, characterized by low levels of negative (N2) and depression symptoms (G6); group B, characterized by high levels of positive symptoms (P1, 6) having the worst outcome; and group C with high levels of negative (N2) and depression symptoms. Comparison of the groups after 3-years of clinical follow-up showed that group A displayed significantly better social and occupational functioning. The same grouping and predictions were confirmed in a second independent cohort of EPP. Inclusion of the metabolomics data revealed that group A had a distinct profile, which may serve as predictor of functional outcome in EEP. Conclusions These novel findings suggest that topological analysis of a combination of symptoms and blood metabolic profile might contribute to the prediction of functional outcome at early stages of psychosis.