The Tree-Edit-Distance, a Measure for Quantifying Neuronal Morphology

Novel metrics reveal new structure and unappreciated heterogeneity in Caenorhabditis elegans development

High throughput experimental approaches are increasingly allowing for the quantitative description of cellular and organismal phenotypes. Distilling these large volumes of complex data into meaningful measures that can drive biological insight remains a central challenge. In the quantitative study of development, for instance, one can resolve phenotypic measures for single cells onto their lineage history, enabling joint consideration of heritable signals and cell fate decisions. Most attempts to analyze this type of data, however, discard much of the information content contained within lineage trees. In this work we introduce a generalized metric, which we term the branch edit distance, that allows us to compare any two embryos based on phenotypic measurements in individual cells. This approach aligns those phenotypic measurements to the underlying lineage tree, providing a flexible and intuitive framework for quantitative comparisons between, for instance, Wild-Type (WT) and mutant developmental programs. We apply this novel metric to data on cell-cycle timing from over 1300 WT and RNAi-treated Caenorhabditis elegans embryos. Our new metric revealed surprising heterogeneity within this data set, including subtle batch effects in WT embryos and dramatic variability in RNAi-induced developmental phenotypes, all of which had been missed in previous analyses. Further investigation of these results suggests a novel, quantitative link between pathways that govern cell fate decisions and pathways that pattern cell cycle timing in the early embryo. Our work demonstrates that the branch edit distance we propose, and similar metrics like it, have the potential to revolutionize our quantitative understanding of organismal phenotype.

Novel metrics reveal new structure and unappreciated heterogeneity in C. elegans development

High throughput experimental approaches are increasingly allowing for the quantitative description of cellular and organismal phenotypes. Distilling these large volumes of complex data into meaningful measures that can drive biological insight remains a central challenge. In the quantitative study of development, for instance, one can resolve phenotypic measures for single cells onto their lineage history, enabling joint consideration of heritable signals and cell fate decisions. Most attempts to analyze this type of data, however, discard much of the information content contained within lineage trees. In this work we introduce a generalized metric, which we term the branch distance, that allows us to compare any two embryos based on phenotypic measurements in individual cells. This approach aligns those phenotypic measurements to the underlying lineage tree, providing a flexible and intuitive framework for quantitative comparisons between, for instance, Wild-Type (WT) and mutant developmental programs. We apply this novel metric to data on cell-cycle timing from over 1300 WT and RNAi-treated Caenorhabditis elegans embryos. Our new metric revealed surprising heterogeneity within this data set, including subtle batch effects in WT embryos and dramatic variability in RNAi-induced developmental phenotypes, all of which had been missed in previous analyses. Further investigation of these results suggests a novel, quantitative link between pathways that govern cell fate decisions and pathways that pattern cell cycle timing in the early embryo. Our work demonstrates that the branch distance we propose, and similar metrics like it, have the potential to revolutionize our quantitative understanding of organismal phenotype.

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.

Fitting Splines to Axonal Arbors Quantifies Relationship Between Branch Order and Geometry

Neuromorphology is crucial to identifying neuronal subtypes and understanding learning. It is also implicated in neurological disease. However, standard morphological analysis focuses on macroscopic features such as branching frequency and connectivity between regions, and often neglects the internal geometry of neurons. In this work, we treat neuron trace points as a sampling of differentiable curves and fit them with a set of branching B-splines. We designed our representation with the Frenet-Serret formulas from differential geometry in mind. The Frenet-Serret formulas completely characterize smooth curves, and involve two parameters, curvature and torsion. Our representation makes it possible to compute these parameters from neuron traces in closed form. These parameters are defined continuously along the curve, in contrast to other parameters like tortuosity which depend on start and end points. We applied our method to a dataset of cortical projection neurons traced in two mouse brains, and found that the parameters are distributed differently between primary, collateral, and terminal axon branches, thus quantifying geometric differences between different components of an axonal arbor. The results agreed in both brains, further validating our representation. The code used in this work can be readily applied to neuron traces in SWC format and is available in our open-source Python package brainlit: http://brainlit.neurodata.io/.

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.

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.

Metrics for comparing neuronal tree shapes based on persistent homology

As more and more neuroanatomical data are made available through efforts such as NeuroMorpho.Org and FlyCircuit.org, the need to develop computational tools to facilitate automatic knowledge discovery from such large datasets becomes more urgent. One fundamental question is how best to compare neuron structures, for instance to organize and classify large collection of neurons. We aim to develop a flexible yet powerful framework to support comparison and classification of large collection of neuron structures efficiently. Specifically we propose to use a topological persistence-based feature vectorization framework. Existing methods to vectorize a neuron (i.e, convert a neuron to a feature vector so as to support efficient comparison and/or searching) typically rely on statistics or summaries of morphometric information, such as the average or maximum local torque angle or partition asymmetry. These simple summaries have limited power in encoding global tree structures. Based on the concept of topological persistence recently developed in the field of computational topology, we vectorize each neuron structure into a simple yet informative summary. In particular, each type of information of interest can be represented as a descriptor function defined on the neuron tree, which is then mapped to a simple persistence-signature. Our framework can encode both local and global tree structure, as well as other information of interest (electrophysiological or dynamical measures), by considering multiple descriptor functions on the neuron. The resulting persistence-based signature is potentially more informative than simple statistical summaries (such as average/mean/max) of morphometric quantities-Indeed, we show that using a certain descriptor function will give a persistence-based signature containing strictly more information than the classical Sholl analysis. At the same time, our framework retains the efficiency associated with treating neurons as points in a simple Euclidean feature space, which would be important for constructing efficient searching or indexing structures over them. We present preliminary experimental results to demonstrate the effectiveness of our persistence-based neuronal feature vectorization framework.

BlastNeuron for Automated Comparison, Retrieval and Clustering of 3D Neuron Morphologies

Characterizing the identity and types of neurons in the brain, as well as their associated function, requires a means of quantifying and comparing 3D neuron morphology. Presently, neuron comparison methods are based on statistics from neuronal morphology such as size and number of branches, which are not fully suitable for detecting local similarities and differences in the detailed structure. We developed BlastNeuron to compare neurons in terms of their global appearance, detailed arborization patterns, and topological similarity. BlastNeuron first compares and clusters 3D neuron reconstructions based on global morphology features and moment invariants, independent of their orientations, sizes, level of reconstruction and other variations. Subsequently, BlastNeuron performs local alignment between any pair of retrieved neurons via a tree-topology driven dynamic programming method. A 3D correspondence map can thus be generated at the resolution of single reconstruction nodes. We applied BlastNeuron to three datasets: (1) 10,000+ neuron reconstructions from a public morphology database, (2) 681 newly and manually reconstructed neurons, and (3) neurons reconstructions produced using several independent reconstruction methods. Our approach was able to accurately and efficiently retrieve morphologically and functionally similar neuron structures from large morphology database, identify the local common structures, and find clusters of neurons that share similarities in both morphology and molecular profiles.

Topological characterization of neuronal arbor morphology via sequence representation: II--global alignment

BACKGROUND

The increasing abundance of neuromorphological data provides both the opportunity and the challenge to compare massive numbers of neurons from a wide diversity of sources efficiently and effectively. We implemented a modified global alignment algorithm representing axonal and dendritic bifurcations as strings of characters. Sequence alignment quantifies neuronal similarity by identifying branch-level correspondences between trees.

RESULTS

The space generated from pairwise similarities is capable of classifying neuronal arbor types as well as, or better than, traditional topological metrics. Unsupervised cluster analysis produces groups that significantly correspond with known cell classes for axons, dendrites, and pyramidal apical dendrites. Furthermore, the distinguishing consensus topology generated by multiple sequence alignment of a group of neurons reveals their shared branching blueprint. Interestingly, the axons of dendritic-targeting interneurons in the rodent cortex associates with pyramidal axons but apart from the (more topologically symmetric) axons of perisomatic-targeting interneurons.

CONCLUSIONS

Global pairwise and multiple sequence alignment of neurite topologies enables detailed comparison of neurites and identification of conserved topological features in alignment-defined clusters. The methods presented also provide a framework for incorporation of additional branch-level morphological features. Moreover, comparison of multiple alignment with motif analysis shows that the two techniques provide complementary information respectively revealing global and local features.

Models and simulation of 3D neuronal dendritic trees using Bayesian networks

Neuron morphology is crucial for neuronal connectivity and brain information processing. Computational models are important tools for studying dendritic morphology and its role in brain function. We applied a class of probabilistic graphical models called Bayesian networks to generate virtual dendrites from layer III pyramidal neurons from three different regions of the neocortex of the mouse. A set of 41 morphological variables were measured from the 3D reconstructions of real dendrites and their probability distributions used in a machine learning algorithm to induce the model from the data. A simulation algorithm is also proposed to obtain new dendrites by sampling values from Bayesian networks. The main advantage of this approach is that it takes into account and automatically locates the relationships between variables in the data instead of using predefined dependencies. Therefore, the methodology can be applied to any neuronal class while at the same time exploiting class-specific properties. Also, a Bayesian network was defined for each part of the dendrite, allowing the relationships to change in the different sections and to model heterogeneous developmental factors or spatial influences. Several univariate statistical tests and a novel multivariate test based on Kullback-Leibler divergence estimation confirmed that virtual dendrites were similar to real ones. The analyses of the models showed relationships that conform to current neuroanatomical knowledge and support model correctness. At the same time, studying the relationships in the models can help to identify new interactions between variables related to dendritic morphology.

NetMets: software for quantifying and visualizing errors in biological network segmentation

One of the major goals in biomedical image processing is accurate segmentation of networks embedded in volumetric data sets. Biological networks are composed of a meshwork of thin filaments that span large volumes of tissue. Examples of these structures include neurons and microvasculature, which can take the form of both hierarchical trees and fully connected networks, depending on the imaging modality and resolution. Network function depends on both the geometric structure and connectivity. Therefore, there is considerable demand for algorithms that segment biological networks embedded in three-dimensional data. While a large number of tracking and segmentation algorithms have been published, most of these do not generalize well across data sets. One of the major reasons for the lack of general-purpose algorithms is the limited availability of metrics that can be used to quantitatively compare their effectiveness against a pre-constructed ground-truth. In this paper, we propose a robust metric for measuring and visualizing the differences between network models. Our algorithm takes into account both geometry and connectivity to measure network similarity. These metrics are then mapped back onto an explicit model for visualization.

The DIADEM metric: comparing multiple reconstructions of the same neuron

Digital reconstructions of neuronal morphology are used to study neuron function, development, and responses to various conditions. Although many measures exist to analyze differences between neurons, none is particularly suitable to compare the same arborizing structure over time (morphological change) or reconstructed by different people and/or software (morphological error). The metric introduced for the DIADEM (DIgital reconstruction of Axonal and DEndritic Morphology) Challenge quantifies the similarity between two reconstructions of the same neuron by matching the locations of bifurcations and terminations as well as their topology between the two reconstructed arbors. The DIADEM metric was specifically designed to capture the most critical aspects in automating neuronal reconstructions, and can function in feedback loops during algorithm development. During the Challenge, the metric scored the automated reconstructions of best-performing algorithms against manually traced gold standards over a representative data set collection. The metric was compared with direct quality assessments by neuronal reconstruction experts and with clocked human tracing time saved by automation. The results indicate that relevant morphological features were properly quantified in spite of subjectivity in the underlying image data and varying research goals. The DIADEM metric is freely released open source ( http://diademchallenge.org ) as a flexible instrument to measure morphological error or change in high-throughput reconstruction projects.

Principal component and cluster analysis of layer V pyramidal cells in visual and non-visual cortical areas projecting to the primary visual cortex of the mouse

The long-distance corticocortical connections between visual and nonvisual sensory areas that arise from pyramidal neurons located within layer V can be considered as a subpopulation of feedback connections. The purpose of the present study is to determine if layer V pyramidal neurons from visual and nonvisual sensory cortical areas that project onto the visual cortex (V1) constitute a homogeneous population of cells. Additionally, we ask whether dendritic arborization relates to the target, the sensory modality, the hierarchical level, or laterality of the source cortical area. Complete 3D reconstructions of dendritic arbors of retrogradely labeled layer V pyramidal neurons were performed for neurons of the primary auditory (A1) and somatosensory (S1) cortices and from the lateral (V2L) and medial (V2M) parts of the secondary visual cortices of both hemispheres. The morphological parameters extracted from these reconstructions were subjected to principal component analysis (PCA) and cluster analysis. The PCA showed that neurons are distributed within a continuous range of morphologies and do not form discrete groups. Nevertheless, the cluster analysis defines neuronal groups that share similar features. Each cortical area includes neurons belonging to several clusters. We suggest that layer V feedback connections within a single cortical area comprise several cell types.

Automated reconstruction of neuronal morphology: an overview

Digital reconstruction of neuronal morphology is a powerful technique for investigating the nervous system. This process consists of tracing the axonal and dendritic arbors of neurons imaged by optical microscopy into a geometrical format suitable for quantitative analysis and computational modeling. Algorithmic automation of neuronal tracing promises to increase the speed, accuracy, and reproducibility of morphological reconstructions. Together with recent breakthroughs in cellular imaging and accelerating progress in optical microscopy, automated reconstruction of neuronal morphology will play a central role in the development of high throughput screening and the acquisition of connectomic data. Yet, despite continuous advances in image processing algorithms, to date manual tracing remains the overwhelming choice for digitizing neuronal morphology. We summarize the issues involved in automated reconstruction, overview the available techniques, and provide a realistic assessment of future perspectives.

Neuron tracing in perspective

The study of the structure and function of neuronal cells and networks is of crucial importance in the endeavor to understand how the brain works. A key component in this process is the extraction of neuronal morphology from microscopic imaging data. In the past four decades, many computational methods and tools have been developed for digital reconstruction of neurons from images, with limited success. As witnessed by the growing body of literature on the subject, as well as the organization of challenging competitions in the field, the quest for a robust and fully automated system of more general applicability still continues. The aim of this work, is to contribute by surveying recent developments in the field for anyone interested in taking up the challenge. Relevant aspects discussed in the article include proposed image segmentation methods, quantitative measures of neuronal morphology, currently available software tools for various related purposes, and morphology databases. (c) 2010 International Society for Advancement of Cytometry.

Identifying neuronal lineages of Drosophila by sequence analysis of axon tracts

The Drosophila brain is formed by an invariant set of lineages, each of which is derived from a unique neural stem cell (neuroblast) and forms a genetic and structural unit of the brain. The task of reconstructing brain circuitry at the level of individual neurons can be made significantly easier by assigning neurons to their respective lineages. In this article we address the automation of neuron and lineage identification. We focused on the Drosophila brain lineages at the larval stage when they form easily recognizable secondary axon tracts (SATs) that were previously partially characterized. We now generated an annotated digital database containing all lineage tracts reconstructed from five registered wild-type brains, at higher resolution and including some that were previously not characterized. We developed a method for SAT structural comparisons based on a dynamic programming approach akin to nucleotide sequence alignment and a machine learning classifier trained on the annotated database of reference SATs. We quantified the stereotypy of SATs by measuring the residual variability of aligned wild-type SATs. Next, we used our method for the identification of SATs within wild-type larval brains, and found it highly accurate (93-99%). The method proved highly robust for the identification of lineages in mutant brains and in brains that differed in developmental time or labeling. We describe for the first time an algorithm that quantifies neuronal projection stereotypy in the Drosophila brain and use the algorithm for automatic neuron and lineage recognition.