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Early intervention in psychosis is crucial to improving patient response to treatment and the functional deficits that critically affect their long-term quality of life. Stratification tools are needed to personalize functional deficit prevention strategies at an early stage. In the present study, we applied topological tools to analyze symptoms of early psychosis patients, and detected a clear stratification of the cohort into three groups. One of the groups had a significantly better psychosocial outcome than the others after a 3-year clinical follow-up. This group was characterized by a metabolic profile indicative of an activated antioxidant response, while that of the groups with poorer outcome was indicative of oxidative stress. We replicated in a second cohort the finding that the three distinct clinical profiles at baseline were associated with distinct outcomes at follow-up, thus validating the predictive value of this new stratification. This approach could assist in personalizing treatment strategies.

Environmental cues influence the highly dynamic morphology of microglia. Strategies to characterize these changes usually involve user-selected morphometric features, which preclude the identification of a spectrum of context-dependent morphological phenotypes. Here we develop MorphOMICs, a topological data analysis approach, which enables semiautomatic mapping of microglial morphology into an atlas of cue-dependent phenotypes and overcomes feature-selection biases and biological variability. We extract spatially heterogeneous and sexually dimorphic morphological phenotypes for seven adult mouse brain regions. This sex-specific phenotype declines with maturation but increases over the disease trajectories in two neurodegeneration mouse models, with females showing a faster morphological shift in affected brain regions. Remarkably, microglia morphologies reflect an adaptation upon repeated exposure to ketamine anesthesia and do not recover to control morphologies. Finally, we demonstrate that both long primary processes and short terminal processes provide distinct insights to morphological phenotypes. MorphOMICs opens a new perspective to characterize microglial morphology.

The relationship between the structure and network dynamics within the striatum is currently not well understood. We have applied algebraic topology to investigate the local structural connectivity in the striatum, and then used simulations to predict how structure shapes network dynamics. We used a full-scale digital reconstruction of the mouse striatal microcircuitry: both healthy and at different stages of Parkinson’s Disease (PD). These stages are characterized by successively modified healthy morphologies of the striatal projection neurons (SPN), including changes in dendritic spine count. We compared the distribution of topological motifs, in the form of directed cliques, between these microcircuits. The distribution of directed cliques in the healthy striatal microcircuits showed that striatal interneurons, despite only accounting for 5%, are crucial for the construction of high dimensional directed cliques. In PD networks the presence of directed cliques drastically decreased with the disease progression. We then used simulations to investigate whether these changes in structural connectivity affect functional connectivity. Signal transfer, especially correlation transfer, in the corticostriatal system was affected. We also found that the resulting changes in intrastriatal inhibition altered the correlations between the striatal projection neurons. Directed cliques already provided insight on structural and functional properties of neocortical micrucircuitry. Here we applied this topological approach to investigate striatal networks and highligthed important differences with respect to neocortex. Combining theory with simulations using data-driven in silico reconstructions will allow us to form quantitative predictions on how structure and network dynamics relate in health and disease.

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

In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction., Comment: Found Comput Math (2016)

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.

Background: Early interventions in psychosis are key to improving patients’ functional outcome, engagement, and quality of treatment. In this context, biomarkers that quantify disease state, course, or response to medication should be developed to aid in diagnostics and treatment decision at early stages of the disease. The present study aims at identifying metabolites and pathways that are early makers of psychosis and that distinguish biomarker-based categories (biotypes) of patients.

We study the topological variation in Markov processes and networks due to a coarse-graining procedure able to preserve the Markovian property. Such coarse-graining method simplifies master equation by neglecting the fast states and significantly reduces the network size by decimating weak nodes.

Persistent homology is an emerging tool to identify robust topological features underlying the structure of high-dimensional data and complex dynamical systems (such as brain dynamics, molecular folding, distributed sensing).