Statistical morphological analysis of hippocampal principal neurons indicates cell-specific repulsion of dendrites from their own cell

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.

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.

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.

Distinct Relations of Microtubules and Actin Filaments with Dendritic Architecture

Microtubules (MTs) and F-actin (F-act) have long been recognized as key regulators of dendritic morphology. Nevertheless, precisely ascertaining their distinct influences on dendritic trees have been hampered until now by the lack of direct, arbor-wide cytoskeletal quantification. We pair live confocal imaging of fluorescently labeled dendritic arborization (da) neurons in Drosophila larvae with complete multi-signal neural tracing to separately measure MTs and F-act. We demonstrate that dendritic arbor length is highly interrelated with local MT quantity, whereas local F-act enrichment is associated with dendritic branching. Computational simulation of arbor structure solely constrained by experimentally observed subcellular distributions of these cytoskeletal components generated synthetic morphological and molecular patterns statistically equivalent to those of real da neurons, corroborating the efficacy of local MT and F-act in describing dendritic architecture. The analysis and modeling outcomes hold true for the simplest (class I), most complex (class IV), and genetically altered (Formin3 overexpression) da neuron types.

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.

Dendritic branching angles of pyramidal cells across layers of the juvenile rat somatosensory cortex

The characterization of the structural design of cortical microcircuits is essential for understanding how they contribute to function in both health and disease. Since pyramidal neurons represent the most abundant neuronal type and their dendritic spines constitute the major postsynaptic elements of cortical excitatory synapses, our understanding of the synaptic organization of the neocortex largely depends on the available knowledge regarding the structure of pyramidal cells. Previous studies have identified several apparently common rules in dendritic geometry. We study the dendritic branching angles of pyramidal cells across layers to further shed light on the principles that determine the geometric shapes of these cells. We find that the dendritic branching angles of pyramidal cells from layers II-VI of the juvenile rat somatosensory cortex suggest common design principles, despite the particular morphological and functional features that are characteristic of pyramidal cells in each cortical layer. J. Comp. Neurol. 524:2567-2576, 2016. © 2016 Wiley Periodicals, Inc.

Developmental time windows for axon growth influence neuronal network topology

Early brain connectivity development consists of multiple stages: birth of neurons, their migration and the subsequent growth of axons and dendrites. Each stage occurs within a certain period of time depending on types of neurons and cortical layers. Forming synapses between neurons either by growing axons starting at similar times for all neurons (much-overlapped time windows) or at different time points (less-overlapped) may affect the topological and spatial properties of neuronal networks. Here, we explore the extreme cases of axon formation during early development, either starting at the same time for all neurons (parallel, i.e., maximally overlapped time windows) or occurring for each neuron separately one neuron after another (serial, i.e., no overlaps in time windows). For both cases, the number of potential and established synapses remained comparable. Topological and spatial properties, however, differed: Neurons that started axon growth early on in serial growth achieved higher out-degrees, higher local efficiency and longer axon lengths while neurons demonstrated more homogeneous connectivity patterns for parallel growth. Second, connection probability decreased more rapidly with distance between neurons for parallel growth than for serial growth. Third, bidirectional connections were more numerous for parallel growth. Finally, we tested our predictions with C. elegans data. Together, this indicates that time windows for axon growth influence the topological and spatial properties of neuronal networks opening up the possibility to a posteriori estimate developmental mechanisms based on network properties of a developed network.

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.

Branching angles of pyramidal cell dendrites follow common geometrical design principles in different cortical areas

Unraveling pyramidal cell structure is crucial to understanding cortical circuit computations. Although it is well known that pyramidal cell branching structure differs in the various cortical areas, the principles that determine the geometric shapes of these cells are not fully understood. Here we analyzed and modeled with a von Mises distribution the branching angles in 3D reconstructed basal dendritic arbors of hundreds of intracellularly injected cortical pyramidal cells in seven different cortical regions of the frontal, parietal, and occipital cortex of the mouse. We found that, despite the differences in the structure of the pyramidal cells in these distinct functional and cytoarchitectonic cortical areas, there are common design principles that govern the geometry of dendritic branching angles of pyramidal cells in all cortical areas.

Shape, connectedness and dynamics in neuronal networks

The morphology of neurons is directly related to several aspects of the nervous system, including its connectedness, health, development, evolution, dynamics and, ultimately, behavior. Such interplays of the neuronal morphology can be understood within the more general shape-function paradigm. The current article reviews, in an introductory way, some key issues regarding the role of neuronal morphology in the nervous system, with emphasis on works developed in the authors' group. The following topics are addressed: (a) characterization of neuronal shape; (b) stochastic synthesis of neurons and neuronal systems; (c) characterization of the connectivity of neuronal networks by using complex networks concepts; and (d) investigations of influences of neuronal shape on network dynamics. The presented concepts and methods are useful also for several other multiple object systems, such as protein-protein interaction, tissues, aggregates and polymers.

Growth dynamics explain the development of spatiotemporal burst activity of young cultured neuronal networks in detail

A typical property of isolated cultured neuronal networks of dissociated rat cortical cells is synchronized spiking, called bursting, starting about one week after plating, when the dissociated cells have sufficiently sent out their neurites and formed enough synaptic connections. This paper is the third in a series of three on simulation models of cultured networks. Our two previous studies [26], [27] have shown that random recurrent network activity models generate intra- and inter-bursting patterns similar to experimental data. The networks were noise or pacemaker-driven and had Izhikevich-neuronal elements with only short-term plastic (STP) synapses (so, no long-term potentiation, LTP, or depression, LTD, was included). However, elevated pre-phases (burst leaders) and after-phases of burst main shapes, that usually arise during the development of the network, were not yet simulated in sufficient detail. This lack of detail may be due to the fact that the random models completely missed network topology .and a growth model. Therefore, the present paper adds, for the first time, a growth model to the activity model, to give the network a time dependent topology and to explain burst shapes in more detail. Again, without LTP or LTD mechanisms. The integrated growth-activity model yielded realistic bursting patterns. The automatic adjustment of various mutually interdependent network parameters is one of the major advantages of our current approach. Spatio-temporal bursting activity was validated against experiment. Depending on network size, wave reverberation mechanisms were seen along the network boundaries, which may explain the generation of phases of elevated firing before and after the main phase of the burst shape.In summary, the results show that adding topology and growth explain burst shapes in great detail and suggest that young networks still lack/do not need LTP or LTD mechanisms.

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.

Geometric theory predicts bifurcations in minimal wiring cost trees in biology are flat

The complex three-dimensional shapes of tree-like structures in biology are constrained by optimization principles, but the actual costs being minimized can be difficult to discern. We show that despite quite variable morphologies and functions, bifurcations in the scleractinian coral Madracis and in many different mammalian neuron types tend to be planar. We prove that in fact bifurcations embedded in a spatial tree that minimizes wiring cost should lie on planes. This biologically motivated generalization of the classical mathematical theory of Euclidean Steiner trees is compatible with many different assumptions about the type of cost function. Since the geometric proof does not require any correlation between consecutive planes, we predict that, in an environment without directional biases, consecutive planes would be oriented independently of each other. We confirm this is true for many branching corals and neuron types. We conclude that planar bifurcations are characteristic of wiring cost optimization in any type of biological spatial tree structure.

Morphology is constrained by function and vice-versa. Often, intricate morphology can be explained by optimization of a cost. However, in biology, the exact form of the cost function is seldom clear. Previously, for many different natural trees authors have reported that most bifurcations are planar and we confirm this here for branching corals and mammalian neurons. In a three-dimensional space, where bifurcations can have many shapes, it is not clear why they are mostly planar. We show, using a geometric proof, that bifurcations that are part of an optimal wiring cost tree should be planar. We demonstrate this by proving that a bifurcation that is not planar cannot be part of an optimal wiring cost tree, using a very general form of wiring cost which applies even when the exact form of the cost function is not known. We conclude that nature selects for developmental mechanisms which produce planar bifurcations.

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.

Quantifying neuronal size: summing up trees and splitting the branch difference

Neurons vary greatly in size, shape, and complexity depending on their underlying function. Overall size of neuronal trees affects connectivity, area of influence, and other biophysical properties. Relative distributions of neuronal extent, such as the difference between subtrees at branch points, are also critically related to function and activity. This review covers neuromorphological research that analyzes shape and size to elucidate their functional role for different neuron types. We also introduce a novel morphometric, "caulescence", capturing the extent to which trees exhibit a main path. Neuronal tree types differ vastly in caulescence, suggesting potential neurocomputational correlates of this property.

Non-parametric algorithmic generation of neuronal morphologies

Generation algorithms allow for the generation of Virtual Neurons (VNs) from a small set of morphological properties. The set describes the morphological properties of real neurons in terms of statistical descriptors such as the number of branches and segment lengths (among others). The majority of reconstruction algorithms use the observed properties to estimate the parameters of a priori fixed probability distributions in order to construct statistical descriptors that fit well with the observed data. In this article, we present a non-parametric generation algorithm based on kernel density estimators (KDEs). The new algorithm is called KDE-NEURON: and has three advantages over parametric reconstruction algorithms: (1) no a priori specifications about the distributions underlying the real data, (2) peculiarities in the biological data will be reflected in the VNs, and (3) ability to reconstruct different cell types. We experimentally generated motor neurons and granule cells, and statistically validated the obtained results. Moreover, we assessed the quality of the prototype data set and observed that our generated neurons are as good as the prototype data in terms of the used statistical descriptors. The opportunities and limitations of data-driven algorithmic reconstruction of neurons are discussed.

A comparative computer simulation of dendritic morphology

Computational modeling of neuronal morphology is a powerful tool for understanding developmental processes and structure-function relationships. We present a multifaceted approach based on stochastic sampling of morphological measures from digital reconstructions of real cells. We examined how dendritic elongation, branching, and taper are controlled by three morphometric determinants: Branch Order, Radius, and Path Distance from the soma. Virtual dendrites were simulated starting from 3,715 neuronal trees reconstructed in 16 different laboratories, including morphological classes as diverse as spinal motoneurons and dentate granule cells. Several emergent morphometrics were used to compare real and virtual trees. Relating model parameters to Branch Order best constrained the number of terminations for most morphological classes, except pyramidal cell apical trees, which were better described by a dependence on Path Distance. In contrast, bifurcation asymmetry was best constrained by Radius for apical, but Path Distance for basal trees. All determinants showed similar performance in capturing total surface area, while surface area asymmetry was best determined by Path Distance. Grouping by other characteristics, such as size, asymmetry, arborizations, or animal species, showed smaller differences than observed between apical and basal, pointing to the biological importance of this separation. Hybrid models using combinations of the determinants confirmed these trends and allowed a detailed characterization of morphological relations. The differential findings between morphological groups suggest different underlying developmental mechanisms. By comparing the effects of several morphometric determinants on the simulation of different neuronal classes, this approach sheds light on possible growth mechanism variations responsible for the observed neuronal diversity.

Neurons in the brain have a variety of complex arbor shapes that help determine both their interconnectivity and functional roles. Molecular biology is beginning to uncover important details on the development of these tree-like structures, but how and why vastly different shapes arise is still largely unknown. We developed a novel set of computer models of branching in which measurements of real nerve cell structures digitally traced from microscopic imaging are resampled to create virtual trees. The different rules that the models use to create the most similar virtual trees to the real data support specific hypotheses regarding development. Surprisingly, the arborizations that differed most in the optimal rules were found on opposite sides of the same type of neuron, namely apical and basal trees of pyramidal cells. The details of the rules suggest that pyramidal cell trees may respond in unique and complex ways to their external environment. By better understanding how these trees are formed in the brain, we can learn more about their normal function and why they are often malformed in neurological diseases.

A cost-benefit analysis of neuronal morphology

Over hundreds of millions of years, evolution has optimized brain design to maximize its functionality while minimizing costs associated with building and maintenance. This observation suggests that one can use optimization theory to rationalize various features of brain design. Here, we attempt to explain the dimensions and branching structure of dendritic arbors by minimizing dendritic cost for given potential synaptic connectivity. Assuming only that dendritic cost increases with total dendritic length and path length from synapses to soma, we find that branching, planar, and compact dendritic arbors, such as those belonging to Purkinje cells in the cerebellum, are optimal. The theory predicts that adjacent Purkinje dendritic arbors should spatially segregate. In addition, we propose two explicit cost function expressions, falsifiable by measuring dendritic caliber near bifurcations.

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.

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

We have developed a computational method that accurately reproduces the three-dimensional (3D) morphology of individual dendritic trees of six cat alpha motoneurons. The first step was simulation of trees with straight branches based on the branch lengths and topology of actual trees. A second step introduced the meandering, or wandering, trajectories observed in natural dendritic branches into the straight-branch tree simulations. These two steps each required only two parameters, one extracted from the data on actual motoneuron dendrites and the other adjusted by comparing simulated and observed trees, using measurements that were independent of the model specifications (i.e., emergent properties). The results suggest that: 1) there is a somatofugal "tropism" (a bias introduced by the environment that affects the trajectory of dendritic branches) that tends to constrain the lateral expansion of alpha motoneuron dendrites; and 2) that most of the meandering of natural dendritic branches can be described by assuming that they are fractal objects with an average fractal dimension D of about 1.05. When analyzed in the same way, the dendrites of gamma motoneurons showed no evidence of a similar tropism, although they had the same fractal dimension of branch meandering.

Mapping function onto neuronal morphology

Neurons have a wide range of dendritic morphologies the functions of which are largely unknown. We used an optimization procedure to find neuronal morphological structures for two computational tasks: first, neuronal morphologies were selected for linearly summing excitatory synaptic potentials (EPSPs); second, structures were selected that distinguished the temporal order of EPSPs. The solutions resembled the morphology of real neurons. In particular the neurons optimized for linear summation electrotonically separated their synapses, as found in avian nucleus laminaris neurons, and neurons optimized for spike-order detection had primary dendrites of significantly different diameter, as found in the basal and apical dendrites of cortical pyramidal neurons. This similarity makes an experimentally testable prediction of our theoretical approach, which is that pyramidal neurons can act as spike-order detectors for basal and apical inputs. The automated mapping between neuronal function and structure introduced here could allow a large catalog of computational functions to be built indexed by morphological structure.

Mathematical modelling and numerical simulation of the morphological development of neurons

Background

The morphological development of neurons is a very complex process involving both genetic and environmental components. Mathematical modelling and numerical simulation are valuable tools in helping us unravel particular aspects of how individual neurons grow their characteristic morphologies and eventually form appropriate networks with each other.

Methods

A variety of mathematical models that consider (1) neurite initiation (2) neurite elongation (3) axon pathfinding, and (4) neurite branching and dendritic shape formation are reviewed. The different mathematical techniques employed are also described.

Results

Some comparison of modelling results with experimental data is made. A critique of different modelling techniques is given, leading to a proposal for a unified modelling environment for models of neuronal development.

Conclusion

A unified mathematical and numerical simulation framework should lead to an expansion of work on models of neuronal development, as has occurred with compartmental models of neuronal electrical activity.

Spatial embedding of neuronal trees modeled by diffusive growth

The relative importance of the intrinsic and extrinsic factors determining the variety of geometric shapes exhibited by dendritic trees remains unclear. This question was addressed by developing a model of the growth of dendritic trees based on diffusion-limited aggregation process. The model reproduces diverse neuronal shapes (i.e., granule cells, Purkinje cells, the basal and apical dendrites of pyramidal cells, and the axonal trees of interneurons) by changing only the size of the growth area, the time span of pruning, and the spatial concentration of 'neurotrophic particles'. Moreover, the presented model shows how competition between neurons can affect the shape of the dendritic trees. The model reveals that the creation of complex (but reproducible) dendrite-like trees does not require precise guidance or an intrinsic plan of the dendrite geometry. Instead, basic environmental factors and the simple rules of diffusive growth adequately account for the spatial embedding of different types of dendrites observed in the cortex. An example demonstrating the broad applicability of the algorithm to model diverse types of tree structures is also presented.

Morphological homeostasis in cortical dendrites

Neurons have significant potential for the homeostatic regulation of a broad range of functional features, from gene expression to synaptic excitability. In this article, we show that dendritic morphology may also be under intrinsic homeostatic control. We present the results from a statistical analysis of a large collection of digitally reconstructed neurons, demonstrating that fluctuations in dendritic size in one given portion of a neuron are systematically counterbalanced by the remaining dendrites in the same cell. As a result, the total dendritic measure (e.g., number of branches, length, and surface area) of each neuron in a given morphological class is, on average, significantly less random than would be expected if trees (and their parts) were regulated independently during development. This observation is general across scales that range from gross basal/apical subdivisions to individual branches and bifurcations, and its statistical significance is robust among various brain regions, cell types, and experimental conditions. Given the pivotal dendritic role in signal integration, synaptic plasticity, and network connectivity, these findings add a dimension to the functional characterization of neuronal homeostasis.

Local diameter fully constrains dendritic size in basal but not apical trees of CA1 pyramidal neurons

Computational modeling of dendritic morphology is a powerful tool for quantitatively describing complex geometrical relationships, uncovering principles of dendritic development, and synthesizing virtual neurons to systematically investigate cellular biophysics and network dynamics. A feature common to many morphological models is a dependence of the branching probability on local diameter. Previous models of this type have been able to recreate a wide variety of dendritic morphologies. However, these diameter-dependent models have so far failed to properly constrain branching when applied to hippocampal CA1 pyramidal cells, leading to explosive growth. Here we present a simple modification of this basic approach, in which all parameter sampling, not just bifurcation probability, depends on branch diameter. This added constraint prevents explosive growth in both apical and basal trees of simulated CA1 neurons, yielding arborizations with average numbers and patterns of bifurcations extremely close to those observed in real cells. However, simulated apical trees are much more varied in size than the corresponding real dendrites. We show that, in this model, the excessive variability of simulated trees is a direct consequence of the natural variability of diameter changes at and between bifurcations observed in apical, but not basal, dendrites. Conversely, some aspects of branch distribution were better matched by virtual apical trees than by virtual basal trees. Dendritic morphometrics related to spatial position, such as path distance from the soma or branch order, may be necessary to fully constrain CA1 apical tree size and basal branching pattern.

Discovery and integrative neuroscience

Hypothesis driven research has been shown to be an excellent model for pursuing investigations in neuroscience. The Human Genome Project demonstrated the added value of discovery research, especially in areas where large amounts of data are produced. Neuroscience has become a data rich field, and one that would be enhanced by incorporating the discovery approach. Databases, as well as analytical, modeling and simulation tools, will have to be developed, and they will need to be interoperable and federated. This paper presents an overview of the development of the field of neuroscience databases and associate tools: Neuroinformatics. The primary focus is on the impact of NIH funding of this process. The important issues of data sharing, as viewed from the perspective of the scientist and private and public funding organizations, are discussed. Neuroinformatics will provide more than just a sophisticated array of information technologies to help scientists understand and integrate nervous system data. It will make available powerful models of neural functions and facilitate discovery, hypothesis formulation and electronic collaboration.

Statistical determinants of dendritic morphology in hippocampal pyramidal neurons: A hidden Markov model

Dendritic structure is traditionally characterized by distributions and interrelations of morphometric parameters, such as Sholl-like plots of the number of branches versus dendritic path distance. However, how much of a given morphology is effectively captured by any statistical description is generally unknown. In this work, we assemble a small number of standard geometrical parameters measured from experimental data in a simple stochastic algorithm to describe the dendrograms of hippocampal pyramidal cells. The model, consistent with the hidden Markov framework, is feedforward, local, and causal. It relies on two "hidden" local variables: the expected number of terminal tips in a given subtree, and the current path distance from the soma. The algorithm generates dendrograms that statistically reproduce all morphological essentials of dendrites observed in real neurons, including the distributions of branching and termination points, branch lengths, membrane area, topological asymmetry, and (assuming passive membrane parameters within physiological range) electrotonic characteristics. Thus, this algorithm and the small number of its morphometric parameters constitute a remarkably complete description of the dendrograms of hippocampal pyramidal cells. Specifically, it is found that CA3 and CA1 basal dendrites and CA3 apical dendrites can each be described as homogeneous morphological classes. In contrast, the accurate generation of CA1 apical dendrites necessitates the separate sampling of two types of branches, main and oblique, suggesting their derivations from different developmental mechanisms (terminal and interstitial growth, respectively). We further offer a plausible biophysical interpretation of the model hidden variables, relating them to microtubules and other intracellular resources.

Quantitative morphometry of hippocampal pyramidal cells: Differences between anatomical classes and reconstructing laboratories

The dendritic trees of hippocampal pyramidal cells play important roles in the establishment and regulation of network connectivity, synaptic plasticity, and firing dynamics. Several laboratories routinely reconstruct CA3 and CA1 dendrites to correlate their three-dimensional structure with biophysical, electrophysiological, and anatomical observables. To integrate and assess the consistency of the quantitative data available to the scientific community, we exhaustively analyzed 143 completely reconstructed neurons intracellularly filled and digitized in five different laboratories from 10 experimental conditions. Thirty morphometric parameters, including the most common neuroanatomical measurements, were extracted from all neurons. A consistent fraction of parameters (11 of 30) was significantly different between CA3 and CA1 cells. A considerably large number of parameters was also found that discriminated among neurons within the same morphological class, but reconstructed in different laboratories. These interlaboratory differences (8 of 30 parameters) far outweighed the differences between experimental conditions within a single lab, such as aging or preparation method (at most two significant parameters). The set of morphometrics separating anatomical regions and that separating reconstructing laboratories were almost entirely nonoverlapping. CA3 and CA1 neurons could be distinguished by global quantities such as branch order and Sholl distance. Differences among laboratories were largely due to local variables such as branch diameter and local bifurcation angles. Only one parameter (a ratio of branch diameters) separated both morphological classes and reconstructing laboratories. Compartmental simulations of electrophysiological activity showed that both differences between anatomical classes and reconstructing laboratories could dramatically affect the firing rate of these neurons under different experimental conditions.