Simulation of motoneuron morphology in three dimensions. I. Building individual dendritic trees

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.

Local Microtubule and F-Actin Distributions Fully Constrain the Spatial Geometry of Drosophila Sensory Dendritic Arbors

Dendritic morphology underlies the source and processing of neuronal signal inputs. Morphology can be broadly described by two types of geometric characteristics. The first is dendrogram topology, defined by the length and frequency of the arbor branches; the second is spatial embedding, mainly determined by branch angles and straightness. We have previously demonstrated that microtubules and actin filaments are associated with arbor elongation and branching, fully constraining dendrogram topology. Here, we relate the local distribution of these two primary cytoskeletal components with dendritic spatial embedding. We first reconstruct and analyze 167 sensory neurons from the Drosophila larva encompassing multiple cell classes and genotypes. We observe that branches with a higher microtubule concentration tend to deviate less from the direction of their parent branch across all neuron types. Higher microtubule branches are also overall straighter. F-actin displays a similar effect on angular deviation and branch straightness, but not as consistently across all neuron types as microtubule. These observations raise the question as to whether the associations between cytoskeletal distributions and arbor geometry are sufficient constraints to reproduce type-specific dendritic architecture. Therefore, we create a computational model of dendritic morphology purely constrained by the cytoskeletal composition measured from real neurons. The model quantitatively captures both spatial embedding and dendrogram topology across all tested neuron groups. These results suggest a common developmental mechanism regulating diverse morphologies, where the local cytoskeletal distribution can fully specify the overall emergent geometry of dendritic arbors.

Local microtubule and F-actin distributions fully determine the spatial geometry of Drosophila sensory dendritic arbors

Dendritic morphology underlies the source and processing of neuronal signal inputs. Morphology can be broadly described by two types of geometric characteristics. The first is dendrogram topology, defined by the length and frequency of the arbor branches; the second is spatial embedding, mainly determined by branch angles and tortuosity. We have previously demonstrated that microtubules and actin filaments are associated with arbor elongation and branching, fully constraining dendrogram topology. Here we relate the local distribution of these two primary cytoskeletal components with dendritic spatial embedding. We first reconstruct and analyze 167 sensory neurons from the Drosophila larva encompassing multiple cell classes and genotypes. We observe that branches with higher microtubule concentration are overall straighter and tend to deviate less from the direction of their parent branch. F-actin displays a similar effect on the angular deviation from the parent branch direction, but its influence on branch tortuosity varies by class and genotype. We then create a computational model of dendritic morphology purely constrained by the cytoskeletal composition imaged from real neurons. The model quantitatively captures both spatial embedding and dendrogram topology across all tested neuron groups. These results suggest a common developmental mechanism regulating diverse morphologies, where the local cytoskeletal distribution can fully specify the overall emergent geometry of dendritic arbors.

AII amacrine cells: quantitative reconstruction and morphometric analysis of electrophysiologically identified cells in live rat retinal slices imaged with multi-photon excitation microscopy

AII amacrine cells have been found in all mammalian retinas examined and play an important role for visual processing under both scotopic and photopic conditions. Whereas ultrastructural investigations have provided a detailed understanding of synaptic connectivity, there is little information available with respect to quantitative properties and variation of cellular morphology. Here, we performed whole-cell recordings from AII amacrine cells in rat retinal slices and filled the cells with fluorescent dyes. Multi-photon excitation microscopy was used to acquire image stacks and after deconvolution, we performed quantitative morphological reconstruction by computer-aided manual tracing. We reconstructed and performed morphometric analysis on 43 AII amacrine cells, with a focus on branching pattern, dendritic lengths and diameters, surface area, and number and distribution of dendritic varicosities. Compared to previous descriptions, the most surprising result was the considerable extent of branching, with the maximum branch order ranging from approximately 10-40. We found that AII amacrine cells conform to a recently described general structural design principle for neural arbors, where arbor density decreases proportionally to increasing territory size. We confirmed and quantified the bi-stratified morphology of AII amacrine cells by analyzing the arborizations as a function of retinal localization or with Sholl spheres. Principal component and cluster analysis revealed no evidence for morphological subtypes of AII amacrines. These results establish a database of morphometric properties important for studies of development, regeneration, degeneration, and disease processes, as well as a workflow compatible with compartmental modeling.

Context-aware modeling of neuronal morphologies

Neuronal morphologies are pivotal for brain functioning: physical overlap between dendrites and axons constrain the circuit topology, and the precise shape and composition of dendrites determine the integration of inputs to produce an output signal. At the same time, morphologies are highly diverse and variant. The variance, presumably, originates from neurons developing in a densely packed brain substrate where they interact (e.g., repulsion or attraction) with other actors in this substrate. However, when studying neurons their context is never part of the analysis and they are treated as if they existed in isolation. Here we argue that to fully understand neuronal morphology and its variance it is important to consider neurons in relation to each other and to other actors in the surrounding brain substrate, i.e., their context. We propose a context-aware computational framework, NeuroMaC, in which large numbers of neurons can be grown simultaneously according to growth rules expressed in terms of interactions between the developing neuron and the surrounding brain substrate. As a proof of principle, we demonstrate that by using NeuroMaC we can generate accurate virtual morphologies of distinct classes both in isolation and as part of neuronal forests. Accuracy is validated against population statistics of experimentally reconstructed morphologies. We show that context-aware generation of neurons can explain characteristics of variation. Indeed, plausible variation is an inherent property of the morphologies generated by context-aware rules. We speculate about the applicability of this framework to investigate morphologies and circuits, to classify healthy and pathological morphologies, and to generate large quantities of morphologies for large-scale modeling.

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.

Maximization of the connectivity repertoire as a statistical principle governing the shapes of dendritic arbors

The shapes of dendritic arbors are fascinating and important, yet the principles underlying these complex and diverse structures remain unclear. Here, we analyzed basal dendritic arbors of 2,171 pyramidal neurons sampled from mammalian brains and discovered 3 statistical properties: the dendritic arbor size scales with the total dendritic length, the spatial correlation of dendritic branches within an arbor has a universal functional form, and small parts of an arbor are self-similar. We proposed that these properties result from maximizing the repertoire of possible connectivity patterns between dendrites and surrounding axons while keeping the cost of dendrites low. We solved this optimization problem by drawing an analogy with maximization of the entropy for a given energy in statistical physics. The solution is consistent with the above observations and predicts scaling relations that can be tested experimentally. In addition, our theory explains why dendritic branches of pyramidal cells are distributed more sparsely than those of Purkinje cells. Our results represent a step toward a unifying view of the relationship between neuronal morphology and function.

L-Measure: a web-accessible tool for the analysis, comparison and search of digital reconstructions of neuronal morphologies

L-Measure (LM) is a freely available software tool for the quantitative characterization of neuronal morphology. LM computes a large number of neuroanatomical parameters from 3D digital reconstruction files starting from and combining a set of core metrics. After more than six years of development and use in the neuroscience community, LM enables the execution of commonly adopted analyses as well as of more advanced functions. This report illustrates several LM protocols: (i) extraction of basic morphological parameters, (ii) computation of frequency distributions, (iii) measurements from user-specified subregions of the neuronal arbors, (iv) statistical comparison between two groups of cells and (v) filtered selections and searches from collections of neurons based on any Boolean combination of the available morphometric measures. These functionalities are easily accessed and deployed through a user-friendly graphical interface and typically execute within few minutes on a set of approximately 20 neurons. The tool is available at http://krasnow.gmu.edu/cn3 for either online use on any Java-enabled browser and platform or download for local execution under Windows and Linux.

Successes and rewards in sharing digital reconstructions of neuronal morphology

The computer-assisted three-dimensional reconstruction of neuronal morphology is becoming an increasingly popular technique to quantify the arborization patterns of dendrites and axons. The resulting digital files are suitable for comprehensive morphometric analyses as well as for building anatomically realistic compartmental models of membrane biophysics and neuronal electrophysiology. The digital tracings acquired in a lab for a specific purpose can be often re-used by a different research group to address a completely unrelated scientific question, if the original investigators are willing to share the data. Since reconstructing neuronal morphology is a labor-intensive process, data sharing and re-analysis is particularly advantageous for the neuroscience and biomedical communities. Here we present numerous cases of "success stories" in which digital reconstructions of neuronal morphology were shared and re-used, leading to additional, independent discoveries and publications, and thus amplifying the impact of the "source" study for which the data set was first collected. In particular, we overview four main applications of this kind of data: comparative morphometric analyses, statistical estimation of potential synaptic connectivity, morphologically accurate electrophysiological simulations, and computational models of neuronal shape and development.

Simulation of motoneuron morphology in three dimensions. II. Building complete neurons

By using dendrogram data from six adult cat alpha motoneurons, we have constructed computer simulations of these cells in three dimensions (3D) by "growing" their dendritic trees from stem branches that were oriented as in the original cells. Individual trees were simulated by using the algorithms and parameters discussed in the companion paper (Marks and Burke [2007] J. Comp. Neurol. 503:685-700). It was not possible to distinguish real from simulated motoneurons by visual inspection of 3D drawings. Simulated cells were compared quantitatively with their actual exemplars by using features that were measured in spherical shells at various radii centered on the soma. These included nearest neighbor distances (NNDs) between branches, the sizes and overlaps between the territories of individual dendrites measured as convex hulls (polygons that enclose all branches passing through a shell), and the sizes of circular zones that contained no branches. We also compared the 3D fractal dimensions and lacunarity (a measure of the 3D dispersion of branches) in actual cells and their simulations. The statistical properties of these quantitative measures were not significantly different, suggesting that the simulation algorithm was quite successful. However, there were three exceptions: 1) there were more NNDs at distances < 50 microm in simulated than in actual motoneurons; 2) average overlaps between the territories of different dendrites were almost twice as large in simulated compared with actual motoneurons; and 3) estimates of lacunarity were also larger in simulated cells. These exceptions suggest that dendritic branches in actual motoneurons tend to avoid one another. We discuss possible interpretations of these results.