Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

This in silico study predicts the impact that the single-cell neuronal morphological alterations will have on the striatal microcircuit connectivity. We find that the richness in the topological striatal motifs is significantly reduced in Parkinson's disease (PD), highlighting that just measuring the pairwise connectivity between neurons gives an incomplete description of network connectivity. Moreover, we predict how the resulting electrophysiological changes of striatal projection neuron excitability together with their reduced number of dendritic branches affect their response to the glutamatergic drive from the cortex and thalamus. We find that the effective glutamatergic drive is likely significantly increased in PD, in accordance with the hyperglutamatergic hypothesis.

We present the shift-dimension of multipersistence modules and investigate its algebraic properties. This gives rise to a new invariant of multigraded modules over the multivariate polynomial ring arising from the hierarchical stabilization of the zeroth total multigraded Betti number. We give a fast algorithm for the computation of the shift-dimension of interval modules in the bivariate case. We construct multipersistence contours that are parameterized by multivariate functions and hence provide a large class of feature maps for machine learning tasks.

Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is subsampled in various ways. In our framework the dataset serves as a reference object; we then consider different points in the ambient space and endow them with a geometry defined in relation to the reference object, for instance by subsampling the dataset proportionally to the distance between its elements and the point under consideration. We illustrate how this process can be used to extract rich geometrical information, allowing for example to classify points coming from different data distributions.

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.

Motivated by applications in Topological Data Analysis, we consider decompositionsof a simplicial complex induced by a cover of its vertices. We study how the homotopytype of such decompositions approximates the homotopy of the simplicial complexitself. The difference between the simplicial complex and such an approximationis quantitatively measured by means of the so called obstruction complexes. Ourgeneral machinery is then specialized to clique complexes, Vietoris-Rips complexesand Vietoris-Rips complexes of metric gluings.

We set the foundations for a new approach to Topological Data Analysis (TDA) based on homotopical methods at the chain complex level. We present the category of tame parametrised chain complexes as a comprehensive environment that includes several cases that usually TDA handles separately, such as persistence modules, zigzag modules, and commutative ladders. We extract new invariants in this category using a model structure and various minimal cofibrant approximations. Such approximations and their invariants retain some of the topological, and not just homological, aspects of the objects they approximate.

Exciting recent developments in Topological Data Analysis have aimed at combining homology-based invariants with Machine Learning. In this article, we use hierarchical stabilization to bridge between persistence and kernel-based methods by introducing the so-called stable rank kernels. A fundamental property of the stable rank kernels is that they depend on metrics to compare persistence modules. We illustrate their use on artificial and real-world datasets and show that by varying the metric we can improve accuracy in classification tasks.