A Topological Representation of Branching Neuronal Morphologies

Graph Representation Learning for Large-Scale Neuronal Morphological Analysis

The analysis of neuronal morphological data is essential to investigate the neuronal properties and brain mechanisms. The complex morphologies, absence of annotations, and sheer volume of these data pose significant challenges in neuronal morphological analysis, such as identifying neuron types and large-scale neuron retrieval, all of which require accurate measuring and efficient matching algorithms. Recently, many studies have been conducted to describe neuronal morphologies quantitatively using predefined measurements. However, hand-crafted features are usually inadequate for distinguishing fine-grained differences among massive neurons. In this article, we propose a novel morphology-aware contrastive graph neural network (MACGNN) for unsupervised neuronal morphological representation learning. To improve the retrieval efficiency in large-scale neuronal morphological datasets, we further propose Hash-MACGNN by introducing an improved deep hash algorithm to train the network end-to-end to learn binary hash representations of neurons. We conduct extensive experiments on the largest dataset, NeuroMorpho, which contains more than 100 000 neurons. The experimental results demonstrate the effectiveness and superiority of our MACGNN and Hash-MACGNN for large-scale neuronal morphological analysis.

Regulation of Diseases-Associated Microglia in the Optic Nerve by Lipoxin B4 and Ocular Hypertension

Background

The resident astrocyte-retinal ganglion cell (RGC) lipoxin circuit is impaired during retinal stress, which includes ocular hypertension-induced neuropathy. Lipoxin B4 produced by homeostatic astrocytes directly acts on RGCs to increase survival and function in ocular hypertension-induced neuropathy. RGC death in the retina and axonal degeneration in the optic nerve are driven by the complex interactions between microglia and macroglia. Whether LXB4 neuroprotective actions include regulation of other cell types in the retina and/or optic nerve is an important knowledge gap.

Methods

Cellular targets and signaling of LXB4 in the retina were defined by single-cell RNA sequencing. Retinal neurodegeneration was induced by injecting silicone oil into the anterior chamber of the mouse eyes, which induced sustained and stable ocular hypertension. Morphological characterization of microglia populations in the retina and optic nerve was established by MorphOMICs and pseudotime trajectory analyses. The pathways and mechanisms of action of LXB4 in the optic nerve were investigated using bulk RNA sequencing. Transcriptomics data was validated by qPCR and immunohistochemistry. Differences between experimental groups was assessed by Student's t-test and one-way ANOVA.

Results

Single-cell transcriptomics identified microglia as a primary target for LXB4 in the healthy retina. LXB4 downregulated genes that drive microglia environmental sensing and reactivity responses. Analysis of microglial function revealed that ocular hypertension induced distinct, temporally defined, and dynamic phenotypes in the retina and, unexpectedly, in the distal myelinated optic nerve. Microglial expression of CD74, a marker of disease-associated microglia in the brain, was only induced in a unique population of optic nerve microglia, but not in the retina. Genetic deletion of lipoxin formation correlated with the presence of a CD74 optic nerve microglia population in normotensive eyes, while LXB4 treatment during ocular hypertension shifted optic nerve microglia toward a homeostatic morphology and non-reactive state and downregulated the expression of CD74. Furthermore, we identified a correlation between CD74 and phospho-phosphoinositide 3-kinases (p-PI3K) expression levels in the optic nerve, which was reduced by LXB4 treatment.

Conclusion

We identified early and dynamic changes in the microglia functional phenotype, reactivity, and induction of a unique CD74 microglia population in the distal optic nerve as key features of ocular hypertension-induced neurodegeneration. Our findings establish microglia regulation as a novel LXB4 target in the retina and optic nerve. LXB4 maintenance of a homeostatic optic nerve microglia phenotype and inhibition of a disease-associated phenotype are potential neuroprotective mechanisms for the resident LXB4 pathway.

DSM: Deep sequential model for complete neuronal morphology representation and feature extraction

The full morphology of single neurons is indispensable for understanding cell types, the basic building blocks in brains. Projecting trajectories are critical to extracting biologically relevant information from neuron morphologies, as they provide valuable information for both connectivity and cell identity. We developed an artificial intelligence method, deep sequential model (DSM), to extract concise, cell-type-defining features from projections across brain regions. DSM achieves more than 90% accuracy in classifying 12 major neuron projection types without compromising performance when spatial noise is present. Such remarkable robustness enabled us to efficiently manage and analyze several major full-morphology data sources, showcasing how characteristic long projections can define cell identities. We also succeeded in applying our model to both discovering previously unknown neuron subtypes and analyzing exceptional co-expressed genes involved in neuron projection circuits.

Translating single-neuron axonal reconstructions into meso-scale connectivity statistics in the mouse somatosensory thalamus

Characterizing the connectomic and morphological diversity of thalamic neurons is key for better understanding how the thalamus relays sensory inputs to the cortex. The recent public release of complete single-neuron morphological reconstructions enables the analysis of previously inaccessible connectivity patterns from individual neurons. Here we focus on the Ventral Posteromedial (VPM) nucleus and characterize the full diversity of 257 VPM neurons, obtained by combining data from the MouseLight and Braintell projects. Neurons were clustered according to their most dominantly targeted cortical area and further subdivided by their jointly targeted areas. We obtained a 2D embedding of morphological diversity using the dissimilarity between all pairs of axonal trees. The curved shape of the embedding allowed us to characterize neurons by a 1-dimensional coordinate. The coordinate values were aligned both with the progression of soma position along the dorsal-ventral and lateral-medial axes and with that of axonal terminals along the posterior-anterior and medial-lateral axes, as well as with an increase in the number of branching points, distance from soma and branching width. Taken together, we have developed a novel workflow for linking three challenging aspects of connectomics, namely the topography, higher order connectivity patterns and morphological diversity, with VPM as a test-case. The workflow is linked to a unified access portal that contains the morphologies and integrated with 2D cortical flatmap and subcortical visualization tools. The workflow and resulting processed data have been made available in Python, and can thus be used for modeling and experimentally validating new hypotheses on thalamocortical connectivity.

Topology recapitulates morphogenesis of neuronal dendrites

Branching allows neurons to make synaptic contacts with large numbers of other neurons, facilitating the high connectivity of nervous systems. Neuronal arbors have geometric properties such as branch lengths and diameters that are optimal in that they maximize signaling speeds while minimizing construction costs. In this work, we asked whether neuronal arbors have topological properties that may also optimize their growth or function. We discovered that for a wide range of invertebrate and vertebrate neurons the distributions of their subtree sizes follow power laws, implying that they are scale invariant. The power-law exponent distinguishes different neuronal cell types. Postsynaptic spines and branchlets perturb scale invariance. Through simulations, we show that the subtree-size distribution depends on the symmetry of the branching rules governing arbor growth and that optimal morphologies are scale invariant. Thus, the subtree-size distribution is a topological property that recapitulates the functional morphology of dendrites.

Predicting Affinity Through Homology (PATH): Interpretable Binding Affinity Prediction with Persistent Homology

Accurate binding affinity prediction is crucial to structure-based drug design. Recent work used computational topology to obtain an effective representation of protein-ligand interactions. Although persistent homology encodes geometric features, previous works on binding affinity prediction using persistent homology employed uninterpretable machine learning models and failed to explain the underlying geometric and topological features that drive accurate binding affinity prediction. In this work, we propose a novel, interpretable algorithm for protein-ligand binding affinity prediction. Our algorithm achieves interpretability through an effective embedding of distances across bipartite matchings of the protein and ligand atoms into real-valued functions by summing Gaussians centered at features constructed by persistent homology. We name these functions internuclear persistent contours (IPCs) . Next, we introduce persistence fingerprints , a vector with 10 components that sketches the distances of different bipartite matching between protein and ligand atoms, refined from IPCs. Let the number of protein atoms in the protein-ligand complex be n , number of ligand atoms be m , and ω ≈ 2.4 be the matrix multiplication exponent. We show that for any 0 < ε < 1, after an 𝒪 ( mn log( mn )) preprocessing procedure, we can compute an ε -accurate approximation to the persistence fingerprint in 𝒪 ( m log 6 ω ( m/" )) time, independent of protein size. This is an improvement in time complexity by a factor of 𝒪 (( m + n ) 3 ) over any previous binding affinity prediction that uses persistent homology. We show that the representational power of persistence fingerprint generalizes to protein-ligand binding datasets beyond the training dataset. Then, we introduce PATH , Predicting Affinity Through Homology, an interpretable, small ensemble of shallow regression trees for binding affinity prediction from persistence fingerprints. We show that despite using 1,400-fold fewer features, PATH has comparable performance to a previous state-of-the-art binding affinity prediction algorithm that uses persistent homology features. Moreover, PATH has the advantage of being interpretable. Finally, we visualize the features captured by persistence fingerprint for variant HIV-1 protease complexes and show that persistence fingerprint captures binding-relevant structural mutations. The source code for PATH is released open-source as part of the osprey protein design software package.

Controlling morpho-electrophysiological variability of neurons with detailed biophysical models

Variability, which is known to be a universal feature among biological units such as neuronal cells, holds significant importance, as, for example, it enables a robust encoding of a high volume of information in neuronal circuits and prevents hypersynchronizations. While most computational studies on electrophysiological variability in neuronal circuits were done with single-compartment neuron models, we instead focus on the variability of detailed biophysical models of neuron multi-compartmental morphologies. We leverage a Markov chain Monte Carlo method to generate populations of electrical models reproducing the variability of experimental recordings while being compatible with a set of morphologies to faithfully represent specifi morpho-electrical type. We demonstrate our approach on layer 5 pyramidal cells and study the morpho-electrical variability and in particular, find that morphological variability alone is insufficient to reproduce electrical variability. Overall, this approach provides a strong statistical basis to create detailed models of neurons with controlled variability.

A robust approach to 3D neuron shape representation for quantification and classification

We consider the problem of finding an accurate representation of neuron shapes, extracting sub-cellular features, and classifying neurons based on neuron shapes. In neuroscience research, the skeleton representation is often used as a compact and abstract representation of neuron shapes. However, existing methods are limited to getting and analyzing "curve" skeletons which can only be applied for tubular shapes. This paper presents a 3D neuron morphology analysis method for more general and complex neuron shapes. First, we introduce the concept of skeleton mesh to represent general neuron shapes and propose a novel method for computing mesh representations from 3D surface point clouds. A skeleton graph is then obtained from skeleton mesh and is used to extract sub-cellular features. Finally, an unsupervised learning method is used to embed the skeleton graph for neuron classification. Extensive experiment results are provided and demonstrate the robustness of our method to analyze neuron morphology.

The shape of cancer relapse: Topological data analysis predicts recurrence in paediatric acute lymphoblastic leukaemia

Although children and adolescents with acute lymphoblastic leukaemia (ALL) have high survival rates, approximately 15-20% of patients relapse. Risk of relapse is routinely estimated at diagnosis by biological factors, including flow cytometry data. This high-dimensional data is typically manually assessed by projecting it onto a subset of biomarkers. Cell density and "empty spaces" in 2D projections of the data, i.e. regions devoid of cells, are then used for qualitative assessment. Here, we use topological data analysis (TDA), which quantifies shapes, including empty spaces, in data, to analyse pre-treatment ALL datasets with known patient outcomes. We combine these fully unsupervised analyses with Machine Learning (ML) to identify significant shape characteristics and demonstrate that they accurately predict risk of relapse, particularly for patients previously classified as 'low risk'. We independently confirm the predictive power of CD10, CD20, CD38, and CD45 as biomarkers for ALL diagnosis. Based on our analyses, we propose three increasingly detailed prognostic pipelines for analysing flow cytometry data from ALL patients depending on technical and technological availability: 1. Visual inspection of specific biological features in biparametric projections of the data; 2. Computation of quantitative topological descriptors of such projections; 3. A combined analysis, using TDA and ML, in the four-parameter space defined by CD10, CD20, CD38 and CD45. Our analyses readily extend to other haematological malignancies.

Models of Purkinje cell dendritic tree selection during early cerebellar development

We investigate the relationship between primary dendrite selection of Purkinje cells and migration of their presynaptic partner granule cells during early cerebellar development. During postnatal development, each Purkinje cell grows more than three dendritic trees, from which a primary tree is selected for development, whereas the others completely retract. Experimental studies suggest that this selection process is coordinated by physical and synaptic interactions with granule cells, which undergo a massive migration at the same time. However, technical limitations hinder continuous experimental observation of multiple cell populations. To explore possible mechanisms underlying this selection process, we constructed a computational model using a new computational framework, NeuroDevSim. The study presents the first computational model that simultaneously simulates Purkinje cell growth and the dynamics of granule cell migrations during the first two postnatal weeks, allowing exploration of the role of physical and synaptic interactions upon dendritic selection. The model suggests that interaction with parallel fibers is important to establish the distinct planar morphology of Purkinje cell dendrites. Specific rules to select which dendritic trees to keep or retract result in larger winner trees with more synaptic contacts than using random selection. A rule based on afferent synaptic activity was less effective than rules based on dendritic size or numbers of synapses.

CAJAL enables analysis and integration of single-cell morphological data using metric geometry

High-resolution imaging has revolutionized the study of single cells in their spatial context. However, summarizing the great diversity of complex cell shapes found in tissues and inferring associations with other single-cell data remains a challenge. Here, we present CAJAL, a general computational framework for the analysis and integration of single-cell morphological data. By building upon metric geometry, CAJAL infers cell morphology latent spaces where distances between points indicate the amount of physical deformation required to change the morphology of one cell into that of another. We show that cell morphology spaces facilitate the integration of single-cell morphological data across technologies and the inference of relations with other data, such as single-cell transcriptomic data. We demonstrate the utility of CAJAL with several morphological datasets of neurons and glia and identify genes associated with neuronal plasticity in C. elegans. Our approach provides an effective strategy for integrating cell morphology data into single-cell omics analyses.

BigNeuron: a resource to benchmark and predict performance of algorithms for automated tracing of neurons in light microscopy datasets

BigNeuron is an open community bench-testing platform with the goal of setting open standards for accurate and fast automatic neuron tracing. We gathered a diverse set of image volumes across several species that is representative of the data obtained in many neuroscience laboratories interested in neuron tracing. Here, we report generated gold standard manual annotations for a subset of the available imaging datasets and quantified tracing quality for 35 automatic tracing algorithms. The goal of generating such a hand-curated diverse dataset is to advance the development of tracing algorithms and enable generalizable benchmarking. Together with image quality features, we pooled the data in an interactive web application that enables users and developers to perform principal component analysis, t-distributed stochastic neighbor embedding, correlation and clustering, visualization of imaging and tracing data, and benchmarking of automatic tracing algorithms in user-defined data subsets. The image quality metrics explain most of the variance in the data, followed by neuromorphological features related to neuron size. We observed that diverse algorithms can provide complementary information to obtain accurate results and developed a method to iteratively combine methods and generate consensus reconstructions. The consensus trees obtained provide estimates of the neuron structure ground truth that typically outperform single algorithms in noisy datasets. However, specific algorithms may outperform the consensus tree strategy in specific imaging conditions. Finally, to aid users in predicting the most accurate automatic tracing results without manual annotations for comparison, we used support vector machine regression to predict reconstruction quality given an image volume and a set of automatic tracings.

Strong and reliable synaptic communication between pyramidal neurons in adult human cerebral cortex

Synaptic transmission constitutes the primary mode of communication between neurons. It is extensively studied in rodent but not human neocortex. We characterized synaptic transmission between pyramidal neurons in layers 2 and 3 using neurosurgically resected human middle temporal gyrus (MTG, Brodmann area 21), which is part of the distributed language circuitry. We find that local connectivity is comparable with mouse layer 2/3 connections in the anatomical homologue (temporal association area), but synaptic connections in human are 3-fold stronger and more reliable (0% vs 25% failure rates, respectively). We developed a theoretical approach to quantify properties of spinous synapses showing that synaptic conductance and voltage change in human dendritic spines are 3-4-folds larger compared with mouse, leading to significant NMDA receptor activation in human unitary connections. This model prediction was validated experimentally by showing that NMDA receptor activation increases the amplitude and prolongs decay of unitary excitatory postsynaptic potentials in human but not in mouse connections. Since NMDA-dependent recurrent excitation facilitates persistent activity (supporting working memory), our data uncovers cortical microcircuit properties in human that may contribute to language processing in MTG.

Cell morphologies in the nervous system: Glia steal the limelight

Neurons and glia have distinct yet interactive functions but are both characterized by branching morphology. Dendritic trees have been digitally traced for over 40 years in many animal species, anatomical regions, and neuron types. Recently, long-range axons also are being reconstructed throughout the brain of many organisms from invertebrates to primates. In contrast, less attention has been paid until lately to glial morphology. Thus, although glia and neurons are similarly abundant in the nervous systems of humans and most animal models, glia have traditionally been much less represented than neurons in morphological reconstruction repositories such as NeuroMorpho.Org. This is rapidly changing with the advent of high-throughput glia tracing. NeuroMorpho.Org introduced glial cells in 2017 and today they constitute nearly a third of the database content. It took NeuroMorpho.Org 10 years to collect the first 40,000 neurons and now that amount of data can be produced in a single publication. This not only demonstrates the spectacular technological progress in data production, but also demands a corresponding advancement in informatics processing. At the same time, these publicly available data also open new opportunities for quantitative analysis and computational modeling to identify universal or cell-type-specific design principles in the cellular architecture of nervous systems. As a first application, we demonstrated that supervised machine learning of tree geometry classifies neurons and glia with practically perfect accuracy. Furthermore, we discovered a new morphometric biomarker capable of robustly separating these cell classes across multiple species, brain regions, and experimental preparations, with only sparse sampling of branch measurements.

Semi-Automated Quantitative Evaluation of Neuron Developmental Morphology In Vitro Using the Change-Point Test

Neuron morphology gives rise to distinct axons and dendrites and plays an essential role in neuronal functionality and circuit dynamics. In rat hippocampal neurons, morphological development occurs over roughly one week in vitro. This development has been qualitatively described as occurring in 5 stages. Still, there is a need to quantify cell growth to monitor cell culture health, understand cell responses to sensory cues, and compare experimental results and computational growth model predictions. To address this need, embryonic rat hippocampal neurons were observed in vitro over six days, and their processes were quantified using both standard morphometrics (degree, number of neurites, total length, and tortuosity) and new metrics (distance between change points, relative turning angle, and the number of change points) based on the Change-Point Test to track changes in path trajectories. Of the standard morphometrics, the total length of neurites per cell and the number of endpoints were significantly different between 0.5, 1.5, and 4 days in vitro, which are typically associated with Stages 2-4. Using the Change-Point Test, the number of change points and the average distance between change points per cell were also significantly different between those key time points. This work highlights key quantitative characteristics, both among common and novel morphometrics, that can describe neuron development in vitro and provides a foundation for analyzing directional changes in neurite growth for future studies.

Neuron tracing from light microscopy images: automation, deep learning and bench testing

MOTIVATION

Large-scale neuronal morphologies are essential to neuronal typing, connectivity characterization and brain modeling. It is widely accepted that automation is critical to the production of neuronal morphology. Despite previous survey papers about neuron tracing from light microscopy data in the last decade, thanks to the rapid development of the field, there is a need to update recent progress in a review focusing on new methods and remarkable applications.

RESULTS

This review outlines neuron tracing in various scenarios with the goal to help the community understand and navigate tools and resources. We describe the status, examples and accessibility of automatic neuron tracing. We survey recent advances of the increasingly popular deep-learning enhanced methods. We highlight the semi-automatic methods for single neuron tracing of mammalian whole brains as well as the resulting datasets, each containing thousands of full neuron morphologies. Finally, we exemplify the commonly used datasets and metrics for neuron tracing bench testing.

Model simulations unveil the structure-function-dynamics relationship of the cerebellar cortical microcircuit

The cerebellar network is renowned for its regular architecture that has inspired foundational computational theories. However, the relationship between circuit structure, function and dynamics remains elusive. To tackle the issue, we developed an advanced computational modeling framework that allows us to reconstruct and simulate the structure and function of the mouse cerebellar cortex using morphologically realistic multi-compartmental neuron models. The cerebellar connectome is generated through appropriate connection rules, unifying a collection of scattered experimental data into a coherent construct and providing a new model-based ground-truth about circuit organization. Naturalistic background and sensory-burst stimulation are used for functional validation against recordings in vivo, monitoring the impact of cellular mechanisms on signal propagation, inhibitory control, and long-term synaptic plasticity. Our simulations show how mossy fibers entrain the local neuronal microcircuit, boosting the formation of columns of activity travelling from the granular to the molecular layer providing a new resource for the investigation of local microcircuit computation and of the neural correlates of behavior.

A new measure for the attitude to mobility of Italian students and graduates: a topological data analysis approach

Students' and graduates' mobility is an interesting topic of discussion especially for the Italian education system and universities. The main reasons for migration and for the so called brain drain, can be found in the socio-economic context and in the famous North-South divide. Measuring mobility and understanding its dynamic over time and space are not trivial tasks. Most of the studies in the related literature focus on the determinants of such phenomenon, in this paper, instead, combining tools coming from graph theory and Topological Data Analysis we propose a new measure for the attitude to mobility. Each mobility trajectory is represented by a graph and the importance of the features constituting the graph are evaluated over time using persistence diagrams. The attitude to mobility of the students is then ranked computing the distance between the individual persistence diagram and the theoretical persistence diagram of the stayer student. The new approach is used for evaluating the mobility of the students that in 2008 enrolled in an Italian university. The relation between attitude to mobility and the main socio-demographic variables is investigated.

A tool for mapping microglial morphology, morphOMICs, reveals brain-region and sex-dependent phenotypes

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.

Dynamical systems defined on simplicial complexes: Symmetries, conjugacies, and invariant subspaces

We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.

Large scale similarity search across digital reconstructions of neural morphology

Most functions of the nervous system depend on neuronal and glial morphology. Continuous advances in microscopic imaging and tracing software have provided an increasingly abundant availability of 3D reconstructions of arborizing dendrites, axons, and processes, allowing their detailed study. However, efficient, large-scale methods to rank neural morphologies by similarity to an archetype are still lacking. Using the NeuroMorpho.Org database, we present a similarity search software enabling fast morphological comparison of hundreds of thousands of neural reconstructions from any species, brain regions, cell types, and preparation protocols. We compared the performance of different morphological measurements: 1) summary morphometrics calculated by L-Measure, 2) persistence vectors, a vectorized descriptor of branching structure, 3) the combination of the two. In all cases, we also investigated the impact of applying dimensionality reduction using principal component analysis (PCA). We assessed qualitative performance by gauging the ability to rank neurons in order of visual similarity. Moreover, we quantified information content by examining explained variance and benchmarked the ability to identify occasional duplicate reconstructions of the same specimen. We also compared two different methods for selecting the number of principal components using this benchmark. The results indicate that combining summary morphometrics and persistence vectors with applied PCA using maximum likelihood based automatic dimensionality selection provides an information rich characterization that enables efficient and precise comparison of neural morphology. We have deployed the similarity search as open-source online software both through a user-friendly graphical interface and as an API for programmatic access.

Dynamic instability of dendrite tips generates the highly branched morphologies of sensory neurons

The highly ramified arbors of neuronal dendrites provide the substrate for the high connectivity and computational power of the brain. Altered dendritic morphology is associated with neuronal diseases. Many molecules have been shown to play crucial roles in shaping and maintaining dendrite morphology. However, the underlying principles by which molecular interactions generate branched morphologies are not understood. To elucidate these principles, we visualized the growth of dendrites throughout larval development of Drosophila sensory neurons and found that the tips of dendrites undergo dynamic instability, transitioning rapidly and stochastically between growing, shrinking, and paused states. By incorporating these measured dynamics into an agent-based computational model, we showed that the complex and highly variable dendritic morphologies of these cells are a consequence of the stochastic dynamics of their dendrite tips. These principles may generalize to branching of other neuronal cell types, as well as to branching at the subcellular and tissue levels.

Topological Sholl descriptors for neuronal clustering and classification

Neuronal morphology is a fundamental factor influencing information processing within neurons and networks. Dendritic morphology in particular can widely vary among cell classes, brain regions, and animal species. Thus, accurate quantitative descriptions allowing classification of large sets of neurons is essential for their structural and functional characterization. Current robust and unbiased computational methods that characterize groups of neurons are scarce. In this work, we introduce a novel technique to study dendritic morphology, complementing and advancing many of the existing techniques. Our approach is to conceptualize the notion of a Sholl descriptor and associate, for each morphological feature, and to each neuron, a function of the radial distance from the soma, taking values in a metric space. Functional distances give rise to pseudo-metrics on sets of neurons which are then used to perform the two distinct tasks of clustering and classification. To illustrate the use of Sholl descriptors, four datasets were retrieved from the large public repository https://neuromorpho.org/ comprising neuronal reconstructions from different species and brain regions. Sholl descriptors were subsequently computed, and standard clustering methods enhanced with detection and metric learning algorithms were then used to objectively cluster and classify each dataset. Importantly, our descriptors outperformed conventional morphometric techniques (L-Measure metrics) in several of the tested datasets. Therefore, we offer a novel and effective approach to the analysis of diverse neuronal cell types, and provide a toolkit for researchers to cluster and classify neurons.

Multiscale topology characterizes dynamic tumor vascular networks

Advances in imaging techniques enable high-resolution three-dimensional (3D) visualization of vascular networks over time and reveal abnormal structural features such as twists and loops, and their quantification is an active area of research. Here, we showcase how topological data analysis, the mathematical field that studies the "shape" of data, can characterize the geometric, spatial, and temporal organization of vascular networks. We propose two topological lenses to study vasculature, which capture inherent multiscale features and vessel connectivity, and surpass the single-scale analysis of existing methods. We analyze images collected using intravital and ultramicroscopy modalities and quantify spatiotemporal variation of twists, loops, and avascular regions (voids) in 3D vascular networks. This topological approach validates and quantifies known qualitative trends such as dynamic changes in tortuosity and loops in response to antibodies that modulate vessel sprouting; furthermore, it quantifies the effect of radiotherapy on vessel architecture.

Multiscale topology characterizes dynamic tumor vascular networks

Advances in imaging techniques enable high-resolution three-dimensional (3D) visualization of vascular networks over time and reveal abnormal structural features such as twists and loops, and their quantification is an active area of research. Here, we showcase how topological data analysis, the mathematical field that studies the "shape" of data, can characterize the geometric, spatial, and temporal organization of vascular networks. We propose two topological lenses to study vasculature, which capture inherent multiscale features and vessel connectivity, and surpass the single-scale analysis of existing methods. We analyze images collected using intravital and ultramicroscopy modalities and quantify spatiotemporal variation of twists, loops, and avascular regions (voids) in 3D vascular networks. This topological approach validates and quantifies known qualitative trends such as dynamic changes in tortuosity and loops in response to antibodies that modulate vessel sprouting; furthermore, it quantifies the effect of radiotherapy on vessel architecture.

Computational synthesis of cortical dendritic morphologies

Neuronal morphologies provide the foundation for the electrical behavior of neurons, the connectomes they form, and the dynamical properties of the brain. Comprehensive neuron models are essential for defining cell types, discerning their functional roles, and investigating brain-disease-related dendritic alterations. However, a lack of understanding of the principles underlying neuron morphologies has hindered attempts to computationally synthesize morphologies for decades. We introduce a synthesis algorithm based on a topological descriptor of neurons, which enables the rapid digital reconstruction of entire brain regions from few reference cells. This topology-guided synthesis generates dendrites that are statistically similar to biological reconstructions in terms of morpho-electrical and connectivity properties and offers a significant opportunity to investigate the links between neuronal morphology and brain function across different spatiotemporal scales. Synthesized cortical networks based on structurally altered dendrites associated with diverse brain pathologies revealed principles linking branching properties to the structure of large-scale networks.

Efficient metadata mining of web-accessible neural morphologies

Advancements in neuroscience research have led to steadily accelerating data production and sharing. The online community repository of neural reconstructions NeuroMorpho.Org grew from fewer than 1000 digitally traced neurons in 2006 to more than 140,000 cells today, including glia that now constitute 10.1% of the content. Every reconstruction consists of a detailed 3D representation of branch geometry and connectivity in a standardized format, from which a collection of morphometric features is extracted and stored. Moreover, each entry in the database is accompanied by rich metadata annotation describing the animal subject, anatomy, and experimental details. The rapid expansion of this resource in the past decade was accompanied by a parallel rise in the complexity of the available information, creating both opportunities and challenges for knowledge mining. Here, we introduce a new summary reporting functionality, allowing NeuroMorpho.Org users to efficiently download digests of metadata and morphometrics from multiple groups of similar cells for further analysis. We demonstrate the capabilities of the tool for both glia and neurons and present an illustrative statistical analysis of the resulting data.

Quantitative neuronal morphometry by supervised and unsupervised learning

We present a protocol to characterize the morphological properties of individual neurons reconstructed from microscopic imaging. We first describe a simple procedure to extract relevant morphological features from digital tracings of neural arbors. Then, we provide detailed steps on classification, clustering, and statistical analysis of the traced cells based on morphological features. We illustrate the pipeline design using specific examples from zebrafish anatomy. Our approach can be readily applied and generalized to the characterization of axonal, dendritic, or glial geometry. For complete context and scientific motivation for the studies and datasets used here, refer to Valera et al. (2021).

Quantifying neuronal structural changes over time using dynamic morphometrics

Brain circuit development involves tremendous structural formation and rearrangement of dendrites, axons, and the synaptic connections between them. Direct studies of neuronal morphogenesis are now possible through recent developments in multiple technologies, including single-neuron labeling, time-lapse imaging in intact tissues, and 4D rendering software capable of tracking neural growth over periods spanning minutes to days. These methods allow detailed quantification of structural changes of neurons over time, called dynamic morphometrics, providing new insights into fundamental growth patterns, underlying molecular mechanisms, and the intertwined influences of external factors, including neural activity, and intrinsic genetic programs. Here, we review the methods of dynamic morphometrics sampling and analyses, focusing on their applications to studies of activity-driven dendritogenesis in vertebrate systems.

Digital Reconstruction of the Neuro-Glia-Vascular Architecture

Astrocytes connect the vasculature to neurons mediating the supply of nutrients and biochemicals. They are involved in a growing number of physiological and pathophysiological processes that result from biophysical, physiological, and molecular interactions in this neuro-glia-vascular ensemble (NGV). The lack of a detailed cytoarchitecture severely restricts the understanding of how they support brain function. To address this problem, we used data from multiple sources to create a data-driven digital reconstruction of the NGV at micrometer anatomical resolution. We reconstructed 0.2 mm3 of the rat somatosensory cortex with 16 000 morphologically detailed neurons, 2500 protoplasmic astrocytes, and its microvasculature. The consistency of the reconstruction with a wide array of experimental measurements allows novel predictions of the NGV organization, allowing the anatomical reconstruction of overlapping astrocytic microdomains and the quantification of endfeet connecting each astrocyte to the vasculature, as well as the extent to which they cover the latter. Structural analysis showed that astrocytes optimize their positions to provide uniform vascular coverage for trophic support and signaling. However, this optimal organization rapidly declines as their density increases. The NGV digital reconstruction is a resource that will enable a better understanding of the anatomical principles and geometric constraints, which govern how astrocytes support brain function.

Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors

Quantifying and comparing complex spatial biological datasets is crucial for medical applications and remains an active area of research. As datasets become more heterogeneous and complicated, so must the methods that are used to understand them. Multiparameter topology is built upon the assumption that the shape of data depends on multiple parameters, such as scale, outliers, or other parameters (e.g., cell density and oxygen levels in the case of tumors). A key difficulty encountered in multiparameter persistent homology (MPH) is interpreting and comparing data. The present work uses statistical MPH landscapes to overcome this difficulty and quantifies differences in synthetic data of immune cell infiltration as well as clinical tumor histology data of T cells, macrophages, and hypoxia.

Highly resolved spatial data of complex systems encode rich and nonlinear information. Quantification of heterogeneous and noisy data—often with outliers, artifacts, and mislabeled points—such as those from tissues, remains a challenge. The mathematical field that extracts information from the shape of data, topological data analysis (TDA), has expanded its capability for analyzing real-world datasets in recent years by extending theory, statistics, and computation. An extension to the standard theory to handle heterogeneous data is multiparameter persistent homology (MPH). Here we provide an application of MPH landscapes, a statistical tool with theoretical underpinnings. MPH landscapes, computed for (noisy) data from agent-based model simulations of immune cells infiltrating into a spheroid, are shown to surpass existing spatial statistics and one-parameter persistent homology. We then apply MPH landscapes to study immune cell location in digital histology images from head and neck cancer. We quantify intratumoral immune cells and find that infiltrating regulatory T cells have more prominent voids in their spatial patterns than macrophages. Finally, we consider how TDA can integrate and interrogate data of different types and scales, e.g., immune cell locations and regions with differing levels of oxygenation. This work highlights the power of MPH landscapes for quantifying, characterizing, and comparing features within the tumor microenvironment in synthetic and real datasets.

Analysis of Brain Functional Connectivity Neural Circuits in Children With Autism Based on Persistent Homology

Autism spectrum disorder (ASD) is a complex neuropsychiatric disorder with a complex and unknown etiology. Statistics demonstrate that the number of people diagnosed with ASD is increasing in countries around the world. Currently, although many neuroimaging studies indicate that ASD is characterized by abnormal functional connectivity (FC) patterns within brain networks rather than local functional or structural abnormalities, the FC characteristics of ASD are still poorly understood. In this study, a Vietoris-Rips (VR) complex filtration model of the brain functional network was established by using resting-state functional magnetic resonance imaging (fMRI) data of children aged 6–13 years old [including 54 ASD patients and 52 typical development (TD) controls] from the Autism Brain Imaging Data Exchange (ABIDE) public database. VR complex filtration barcodes are calculated by using persistent homology to describe the changes in the FC neural circuits of brain networks. The number of FC neural circuits with different length ranges at different threshold values is calculated by using the barcodes, the different brain regions participating in FC neural circuits are discussed, and the connectivity characteristics of brain FC neural circuits in the two groups are compared and analyzed. Our results show that the number of FC neural circuits with lengths of 8–12 is significantly decreased in the ASD group compared with the TD control group at threshold values of 0.7, 0.8 and 0.9, and there is no significant difference in the number of FC neural circuits with lengths of 4–7 and 13–16 and lengths 16. When the thresholds are 0.7, 0.8, and 0.9, the number of FC neural circuits in some brain regions, such as the right orbital part of the superior frontal gyrus, the left supplementary motor area, the left hippocampus, and the right caudate nucleus, involved in the study is significantly decreased in the ASD group compared with the TD control group. The results of this study indicate that there are significant differences in the FC neural circuits of brain networks in the ASD group compared with the TD control group.

Noise robustness of persistent homology on greyscale images, across filtrations and signatures

Topological data analysis is a recent and fast growing field that approaches the analysis of datasets using techniques from (algebraic) topology. Its main tool, persistent homology (PH), has seen a notable increase in applications in the last decade. Often cited as the most favourable property of PH and the main reason for practical success are the stability theorems that give theoretical results about noise robustness, since real data is typically contaminated with noise or measurement errors. However, little attention has been paid to what these stability theorems mean in practice. To gain some insight into this question, we evaluate the noise robustness of PH on the MNIST dataset of greyscale images. More precisely, we investigate to what extent PH changes under typical forms of image noise, and quantify the loss of performance in classifying the MNIST handwritten digits when noise is added to the data. The results show that the sensitivity to noise of PH is influenced by the choice of filtrations and persistence signatures (respectively the input and output of PH), and in particular, that PH features are often not robust to noise in a classification task.

Consensus on simplicial complexes: Results on stability and synchronization

We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogs of structures seen in related network models.

A Systematic Evaluation of Interneuron Morphology Representations for Cell Type Discrimination

Quantitative analysis of neuronal morphologies usually begins with choosing a particular feature representation in order to make individual morphologies amenable to standard statistics tools and machine learning algorithms. Many different feature representations have been suggested in the literature, ranging from density maps to intersection profiles, but they have never been compared side by side. Here we performed a systematic comparison of various representations, measuring how well they were able to capture the difference between known morphological cell types. For our benchmarking effort, we used several curated data sets consisting of mouse retinal bipolar cells and cortical inhibitory neurons. We found that the best performing feature representations were two-dimensional density maps, two-dimensional persistence images and morphometric statistics, which continued to perform well even when neurons were only partially traced. Combining these feature representations together led to further performance increases suggesting that they captured non-redundant information. The same representations performed well in an unsupervised setting, implying that they can be suitable for dimensionality reduction or clustering.

From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives

Methods of topological data analysis have been successfully applied in a wide range of fields to provide useful summaries of the structure of complex data sets in terms of topological descriptors, such as persistence diagrams. While there are many powerful techniques for computing topological descriptors, the inverse problem, i.e., recovering the input data from topological descriptors, has proved to be challenging. In this article, we study in detail the Topological Morphology Descriptor (TMD), which assigns a persistence diagram to any tree embedded in Euclidean space, and a sort of stochastic inverse to the TMD, the Topological Neuron Synthesis (TNS) algorithm, gaining both theoretical and computational insights into the relation between the two. We propose a new approach to classify barcodes using symmetric groups, which provides a concrete language to formulate our results. We investigate to what extent the TNS recovers a geometric tree from its TMD and describe the effect of different types of noise on the process of tree generation from persistence diagrams. We prove moreover that the TNS algorithm is stable with respect to specific types of noise.

Event history and topological data analysis

Persistent homology is used to track the appearance and disappearance of features as we move through a nested sequence of topological spaces. Equating the nested sequence to a filtration and the appearance and disappearance of features to events, we show that simple event history methods can be used for the analysis of topological data. We propose a version of the well-known Nelson–Aalen cumulative hazard estimator for the comparison of topological features of random fields and for testing parametric assumptions. We suggest a Cox proportional hazards approach for the analysis of embedded metric trees. The Nelson–Aalen method is illustrated on globally distributed climate data and on neutral hydrogen distribution in the Milky Way. The Cox method is used to compare vascular patterns in fundus images of the eyes of healthy and diabetic retinopathy patients.

Quantifying neurovascular canal branching patterns reveals a shared crocodylian arrangement

Highly branched dendritic structures are common in nature and often difficult to quantify and therefore compare. Cranial neurovascular canals, examples of such structures, are osteological correlates for somatosensory systems and have been explored only qualitatively. Adaptations of traditional stream-ordering methods are applied to representative structures derived from computed tomography-scan data. Applying these methods to crocodylian taxa, this clade demonstrates a shared branching pattern and exemplifies the comparative utility of these methods. Additionally, this pattern corresponds with current understanding of crocodylian sensory abilities and behaviors. The method is applicable to many taxa and anatomical structures and provides evidence for morphology-based hypotheses of sensory and physiological evolution.

Dissecting Sholl Analysis into Its Functional Components

Sholl analysis has been an important technique in dendritic anatomy for more than 60 years. The Sholl intersection profile is obtained by counting the number of dendritic branches at a given distance from the soma and is a key measure of dendritic complexity; it has applications from evaluating the changes in structure induced by pathologies to estimating the expected number of anatomical synaptic contacts. We find that the Sholl intersection profiles of most neurons can be reproduced from three basic, functional measures: the domain spanned by the dendritic arbor, the total length of the dendrite, and the angular distribution of how far dendritic segments deviate from a direct path to the soma (i.e., the root angle distribution). The first two measures are determined by axon location and hence microcircuit structure; the third arises from optimal wiring and represents a branching statistic estimating the need for conduction speed in a neuron.

A topological data analysis based classification method for multiple measurements

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.

New morphological features based on the Sholl analysis for automatic classification of traced neurons

BACKGROUND

This article addresses the automatic classification of reconstructed neurons through their morphological features. The purpose was to extend the capabilities of the L-Measure software.

METHODS

New morphological features were developed, based on modifications of the conventional Sholl analysis. The lengths of the compartments, as well as their volumes, were added to the features used in the classical analysis in order to improve the results during automatic neuron classification. FSM were used to obtain subsets of lower cardinality from the full feature sets and the usefulness of these subsets was tested through their use in supervised classification tasks. The study was based on two types of neurons belonging to mice: pyramidal and GABAergic interneurons. Furthermore, a set of pyramidal neurons belonging to Later 4 and Layer 5 was analyzed.

RESULTS

RF classifier shown the best performance combined with a Wrapper method.U-WNAD set allowed to obtain higher values than WN, A and D in all cases and better results than LM for the filters and wrappers FSM. U-LM-WNAD set, led to the highest AUC values for all the FSM studied. Similar results for different regions of cortex were obtained. Comparison with Existing Methods The new features exhibited high discriminatory power with which the values of AUC and Acc obtained in the experiments exceeded those obtained using only the features provided by L-Measure.

CONCLUSIONS

The highest values of AUC and Acc were obtained from the sets U-WNAD and U-LM-WNAD, evidencing the discriminatory power of the new proposed features.

Density Visualization Pipeline: A Tool for Cellular and Network Density Visualization and Analysis

Neuron classification is an important component in analyzing network structure and quantifying the effect of neuron topology on signal processing. Current quantification and classification approaches rely on morphology projection onto lower-dimensional spaces. In this paper a 3D visualization and quantification tool is presented. The Density Visualization Pipeline (DVP) computes, visualizes and quantifies the density distribution, i.e., the "mass" of interneurons. We use the DVP to characterize and classify a set of GABAergic interneurons. Classification of GABAergic interneurons is of crucial importance to understand on the one hand their various functions and on the other hand their ubiquitous appearance in the neocortex. 3D density map visualization and projection to the one-dimensional x, y, z subspaces show a clear distinction between the studied cells, based on these metrics. The DVP can be coupled to computational studies of the behavior of neurons and networks, in which network topology information is derived from DVP information. The DVP reads common neuromorphological file formats, e.g., Neurolucida XML files, NeuroMorpho.org SWC files and plain ASCII files. Full 3D visualization and projections of the density to 1D and 2D manifolds are supported by the DVP. All routines are embedded within the visual programming IDE VRL-Studio for Java which allows the definition and rapid modification of analysis workflows.

Objective Morphological Classification of Neocortical Pyramidal Cells

A consensus on the number of morphologically different types of pyramidal cells (PCs) in the neocortex has not yet been reached, despite over a century of anatomical studies, due to the lack of agreement on the subjective classifications of neuron types, which is based on expert analyses of neuronal morphologies. Even for neurons that are visually distinguishable, there is no common ground to consistently define morphological types. The objective classification of PCs can be achieved with methods from algebraic topology, and the dendritic arborization is sufficient for the reliable identification of distinct types of cortical PCs. Therefore, we objectively identify 17 types of PCs in the rat somatosensory cortex. In addition, we provide a solution to the challenging problem of whether 2 similar neurons belong to different types or to a continuum of the same type. Our topological classification does not require expert input, is stable, and helps settle the long-standing debate on whether cell-types are discrete or continuous morphological variations of each other.

Generation of Granule Cell Dendritic Morphologies by Estimating the Spatial Heterogeneity of Dendritic Branching

Biological realism of dendritic morphologies is important for simulating electrical stimulation of brain tissue. By adding point process modeling and conditional sampling to existing generation strategies, we provide a novel means of reproducing the nuanced branching behavior that occurs in different layers of granule cell dendritic morphologies. In this study, a heterogeneous Poisson point process was used to simulate branching events. Conditional distributions were then used to select branch angles depending on the orthogonal distance to the somatic plane. The proposed method was compared to an existing generation tool and a control version of the proposed method that used a homogeneous Poisson point process. Morphologies were generated with each method and then compared to a set of digitally reconstructed neurons. The introduction of a conditionally dependent branching rate resulted in the generation of morphologies that more accurately reproduced the emergent properties of dendritic material per layer, Sholl intersections, and proximal passive current flow. Conditional dependence was critically important for the generation of realistic granule cell dendritic morphologies.

The shape of things to come: Topological data analysis and biology, from molecules to organisms

Shape is data and data is shape. Biologists are accustomed to thinking about how the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. Less often do we consider that data itself has shape and structure, or that it is possible to measure the shape of data and analyze it. Here, we review applications of topological data analysis (TDA) to biology in a way accessible to biologists and applied mathematicians alike. TDA uses principles from algebraic topology to comprehensively measure shape in data sets. Using a function that relates the similarity of data points to each other, we can monitor the evolution of topological features-connected components, loops, and voids. This evolution, a topological signature, concisely summarizes large, complex data sets. We first provide a TDA primer for biologists before exploring the use of TDA across biological sub-disciplines, spanning structural biology, molecular biology, evolution, and development. We end by comparing and contrasting different TDA approaches and the potential for their use in biology. The vision of TDA, that data are shape and shape is data, will be relevant as biology transitions into a data-driven era where the meaningful interpretation of large data sets is a limiting factor.

NeuroPath2Path: Classification and elastic morphing between neuronal arbors using path-wise similarity

Neuron shape and connectivity affect function. Modern imaging methods have proven successful at extracting morphological information. One potential path to achieve analysis of this morphology is through graph theory. Encoding by graphs enables the use of high throughput informatic methods to extract and infer brain function. However, the application of graph-theoretic methods to neuronal morphology comes with certain challenges in term of complex subgraph matching and the difficulty in computing intermediate shapes in between two imaged temporal samples. Here we report a novel, efficacious graph-theoretic method that rises to the challenges. The morphology of a neuron, which consists of its overall size, global shape, local branch patterns, and cell-specific biophysical properties, can vary significantly with the cell's identity, location, as well as developmental and physiological state. Various algorithms have been developed to customize shape based statistical and graph related features for quantitative analysis of neuromorphology, followed by the classification of neuron cell types using the features. Unlike the classical feature extraction based methods from imaged or 3D reconstructed neurons, we propose a model based on the rooted path decomposition from the soma to the dendrites of a neuron and extract morphological features from each constituent path. We hypothesize that measuring the distance between two neurons can be realized by minimizing the cost of continuously morphing the set of all rooted paths of one neuron to another. To validate this claim, we first establish the correspondence of paths between two neurons using a modified Munkres algorithm. Next, an elastic deformation framework that employs the square root velocity function is established to perform the continuous morphing, which, as an added benefit, provides an effective visualization tool. We experimentally show the efficacy of NeuroPath2Path, NeuroP2P, over the state of the art.

A Structural Average of Labeled Merge Trees for Uncertainty Visualization

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.