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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. Colombo et al. build a morphological spectrum of over 40,000 microglia across development and disease with a topological data analysis approach that allows mapping of new conditions along these sex-region-specific and brain-region-specific atlases.

Background Machine learning models for repeated measurements are limited. Using topological data analysis (TDA), we present a classifier for repeated measurements which samples from the data space and builds a network graph based on the data topology. A machine learning model with cross-validation is then applied for classification. When test this on three case studies, accuracy exceeds an alternative support vector machine (SVM) voting model in most situations tested, with additional benefits such as reporting data subsets with high purity along with feature values. Results For 100 examples of 3 different tree species, the model reached 80% classification accuracy after 30 datapoints, which was improved to 90% after increased sampling to 400 datapoints. The alternative SVM classifier achieved a maximum accuracy of 68.7%. Using data from 100 examples from each class of 6 different random point processes, the classifier achieved 96.8% accuracy, vastly outperforming the SVM. Using two outcomes in neuron spiking data, the TDA classifier was similarly accurate to the SVM in one case (both converged to 97.8% accuracy), but was outperformed in the other (relative accuracies 79.8% and 92.2%, respectively). Conclusions This algorithm and software can be beneficial for repeated measurement data common in biological sciences, as both an accurate classifier and a feature selection tool.

In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model approximation. A model approximation of a category $\\mathcal{C}$ with a given class of weak equivalences is a model category $\\mathcal{M}$ together with a pair of adjoint functors $\\mathcal{M} \\rightleftarrows \\mathcal{C}$ which satisfy certain properties. Our key result says that if $\\mathcal{C}$ admits a model approximation then so does the functor category $Fun(I, \\mathcal{C})$. From the homotopy theoretical point of view categories with model approximations have similar properties to those of model categories.They admit homotopy categories (localizations with respect to weak equivalences). They also can be used to construct derived functors by taking the analogs of fibrant and cofibrant replacements. A category with weak equivalences can have several useful model approximations. We take advantage of this possibility and in each situation choose one that suits our needs. In this way we prove all the fundamental properties of the homotopy colimit and limit: Fubini Theorem (the homotopy colimit - respectively limit- commutes with itself), Thomason's theorem about diagrams indexed by Grothendieck constructions, and cofinality statements. Since the model approximations we present here consist of certain functors 'indexed by spaces', the key role in all our arguments is played by the geometric nature of the indexing categories.

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R -modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class of “injective objects” I . We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A;I) , we need more: the split error term must vanish. This is the case when I is the class of all injective R -modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4 ^ - n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.

Striatum, the input stage of the basal ganglia, is important for sensory-motor integration, initiation and selection of behavior, as well as reward learning. Striatum receives glutamatergic inputs from mainly cortex and thalamus. In rodents, the striatal projection neurons (SPNs), giving rise to the direct and the indirect pathway (dSPNs and iSPNs, respectively), account for 95% of the neurons, and the remaining 5% are GABAergic and cholinergic interneurons. Interneuron axon terminals as well as local dSPN and iSPN axon collaterals form an intricate striatal network. Following chronic dopamine depletion as in Parkinson’s disease (PD), both morphological and electrophysiological striatal neuronal features have been shown to be altered in rodent models. Our goal with this in silico study is twofold: (a) to predict and quantify how the intrastriatal network connectivity structure becomes altered as a consequence of the morphological changes reported at the single-neuron level and (b) to investigate how the effective glutamatergic drive to the SPNs would need to be altered to account for the activity level seen in SPNs during PD. In summary, we predict that the richness of the connectivity motifs in the striatal network is significantly decreased during PD while, at the same time, a substantial enhancement of the effective glutamatergic drive to striatum is present. 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 identify the homotopy type of the moduli of maps with a given homotopy type of the base and the homotopy fiber. A new model for the space of weak equivalences and its classifying space is given.

The aim of this paper is twofold. On the one hand, we show that the kernel C(A)¯\\overline {C(A)} of the Bousfield periodization functor PAP_A is cellularly generated by a space BB, i.e., we construct a space BB such that the smallest closed class C(B)C(B) containing BB is exactly C(A)¯\\overline {C(A)}. On the other hand, we show that the partial order (Spaces,≫)(Spaces,\\gg ) is a complete lattice, where B≫AB\\gg A if B∈C(A)B\\in C(A). Finally, as a corollary we obtain Bousfield’s theorem, which states that (Spaces,>)(Spaces,>) is a complete lattice, where B>AB>A if B∈C(A)¯B\\in \\overline {C(A)}.

Let ℳ be a simplicial model category and J : ℳ → ℳ a simplicial coaugmented functor. Given an object X, the assignment n↦Jn+1X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define JsX = tots([n] ↦ Jn+1X). An object X is called J-injective if it is a retract of JX in Ho(ℳ) via the natural map. We show that certain homotopy limits of J-injective objects are Js-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) → X which admits a left contraction gives rise to a pro-weak equivalence of towers X(-1)s≥0→totSXs ≥ 0.

In this paper we explain how to convert discrete invariants into stable ones via what we call hierarchical stabilization. We illustrate this process by constructing stable invariants for multi-parameter persistence modules with respect to the interleaving distance and so called simple noise systems. For one parameter, we recover the standard barcode information. For more than one parameter we prove that the constructed invariants are in general NP-hard to calculate. A consequence is that computing the feature counting function, proposed by Scolamiero et. al. (2016), is NP-hard.