A scaling law derived from optimal dendritic wiring

Single-neuron analysis of axon arbors reveals distinct presynaptic organizations between feedforward and feedback projections

The morphology and spatial distribution of axon arbors and boutons are crucial for neuron presynaptic functions. However, the principles governing their whole-brain organization at the single-neuron level remain unclear. We developed a machine-learning method to separate axon arbors from passing axons in single-neuron reconstruction from fluorescence micro-optical sectioning tomography imaging data and obtained 62,374 axon arbors that displayed distinct morphology, spatial patterns, and scaling laws dependent on neuron types and targeted brain areas. Focusing on the axon arbors in the thalamus and cortex, we revealed the segregated spatial distributions and distinct morphology but shared topographic gradients between feedforward and feedback projections. Furthermore, we uncovered an association between arbor complexity and microglia density. Finally, we found that the boutons on terminal arbors show branch-specific clustering with a log-normal distribution that again differed between feedforward and feedback terminal arbors. Together, our study revealed distinct presynaptic structural organizations underlying diverse functional innervation of single projection neurons.

Fractal Resonance: Can Fractal Geometry Be Used to Optimize the Connectivity of Neurons to Artificial Implants?

In parallel to medical applications, exploring how neurons interact with the artificial interface of implants in the human body can be used to learn about their fundamental behavior. For both fundamental and applied research, it is important to determine the conditions that encourage neurons to maintain their natural behavior during these interactions. Whereas previous biocompatibility studies have focused on the material properties of the neuron-implant interface, here we discuss the concept of fractal resonance - the possibility that favorable connectivity properties might emerge by matching the fractal geometry of the implant surface to that of the neurons.To investigate fractal resonance, we first determine the degree to which neurons are fractal and the impact of this fractality on their functionality. By analyzing three-dimensional images of rat hippocampal neurons, we find that the way their dendrites fork and weave through space is important for generating their fractal-like behavior. By modeling variations in neuron connectivity along with the associated energetic and material costs, we highlight how the neurons' fractal dimension optimizes these constraints. To simulate neuron interactions with implant interfaces, we distort the neuron models away from their natural form by modifying the dendrites' fork and weaving patterns. We find that small deviations can induce large changes in fractal dimension, causing the balance between connectivity and cost to deteriorate rapidly. We propose that implant surfaces should be patterned to match the fractal dimension of the neurons, allowing them to maintain their natural functionality as they interact with the implant.

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.

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.

Single-neuron analysis of dendrites and axons reveals the network organization in mouse prefrontal cortex

The structures of dendrites and axons form the basis for the connectivity of neural network, but their precise relationship at single-neuron level remains unclear. Here we report the complete dendrite and axon morphology of nearly 2,000 neurons in mouse prefrontal cortex (PFC). We identified morphological variations of somata, dendrites and axons across laminar layers and PFC subregions and the general rules of somatodendritic scaling with cytoarchitecture. We uncovered 24 morphologically distinguishable dendrite subtypes in 1,515 pyramidal projection neurons and 405 atypical pyramidal projection neurons and spiny stellate neurons with unique axon projection patterns. Furthermore, correspondence analysis among dendrites, local axons and long-range axons revealed coherent morphological changes associated with electrophysiological phenotypes. Finally, integrative dendrite-axon analysis uncovered the organization of potential intra-column, inter-hemispheric and inter-column connectivity among projection neuron types in PFC. Together, our study provides a comprehensive structural repertoire for the reconstruction and analysis of PFC neural network.

Brain-wide dendrites in a near-optimal performance of dynamic range and information transmission

Dendrites receive and process signals from other neurons. The range of signal intensities that can be robustly distinguished by dendrites is quantified by the dynamic range. We investigate the dynamic range and information transmission efficiency of dendrites in relation to dendritic morphology. We model dendrites in a neuron as multiple excitable binary trees connected to the soma where each node in a tree can be excited by external stimulus or by receiving signals transmitted from adjacent excited nodes. It has been known that larger dendritic trees have a higher dynamic range. We show that for dendritic tress of the same number of nodes, the dynamic range increases with the number of somatic branches and decreases with the asymmetry of dendrites, and the information transmission is more efficient for dendrites with more somatic branches. Moreover, our simulated data suggest that there is an exponential association (decay resp.) of overall relative energy consumption (dynamic range resp.) in relation to the number of somatic branches. This indicates that further increasing the number of somatic branches (e.g. beyond 10 somatic branches) has limited ability to improve the transmission efficiency. With brain-wide neuron digital reconstructions of the pyramidal cells, 90% of neurons have no more than 10 dendrites. These suggest that actual brain-wide dendritic morphology is near optimal in terms of both dynamic range and information transmission.

The fractal brain: scale-invariance in structure and dynamics

The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

The branching code: A model of actin-driven dendrite arborization

The cytoskeleton is crucial for defining neuronal-type-specific dendrite morphologies. To explore how the complex interplay of actin-modulatory proteins (AMPs) can define neuronal types in vivo, we focused on the class III dendritic arborization (c3da) neuron of Drosophila larvae. Using computational modeling, we reveal that the main branches (MBs) of c3da neurons follow general models based on optimal wiring principles, while the actin-enriched short terminal branches (STBs) require an additional growth program. To clarify the cellular mechanisms that define this second step, we thus concentrated on STBs for an in-depth quantitative description of dendrite morphology and dynamics. Applying these methods systematically to mutants of six known and novel AMPs, we revealed the complementary roles of these individual AMPs in defining STB properties. Our data suggest that diverse dendrite arbors result from a combination of optimal-wiring-related growth and individualized growth programs that are neuron-type specific.

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.

Single-neuron projectome of mouse prefrontal cortex

Prefrontal cortex (PFC) is the cognitive center that integrates and regulates global brain activity. However, the whole-brain organization of PFC axon projections remains poorly understood. Using single-neuron reconstruction of 6,357 mouse PFC projection neurons, we identified 64 projectome-defined subtypes. Each of four previously known major cortico-cortical subnetworks was targeted by a distinct group of PFC subtypes defined by their first-order axon collaterals. Further analysis unraveled topographic rules of soma distribution within PFC, first-order collateral branch point-dependent target selection and terminal arbor distribution-dependent target subdivision. Furthermore, we obtained a high-precision hierarchical map within PFC and three distinct functionally related PFC modules, each enriched with internal recurrent connectivity. Finally, we showed that each transcriptome subtype corresponds to multiple projectome subtypes found in different PFC subregions. Thus, whole-brain single-neuron projectome analysis reveals organization principles of axon projections within and outside PFC and provides the essential basis for elucidating neuronal connectivity underlying diverse PFC functions.

A general principle of dendritic constancy: A neuron's size- and shape-invariant excitability

Reducing neuronal size results in less membrane and therefore lower input conductance. Smaller neurons are thus more excitable, as seen in their responses to somatic current injections. However, the impact of a neuron's size and shape on its voltage responses to dendritic synaptic activation is much less understood. Here we use analytical cable theory to predict voltage responses to distributed synaptic inputs in unbranched cables, showing that these are entirely independent of dendritic length. For a given synaptic density, neuronal responses depend only on the average dendritic diameter and intrinsic conductivity. This remains valid for a wide range of morphologies irrespective of their arborization complexity. Spiking models indicate that morphology-invariant numbers of spikes approximate the percentage of active synapses. In contrast to spike rate, spike times do depend on dendrite morphology. In summary, neuronal excitability in response to distributed synaptic inputs is largely unaffected by dendrite length or complexity.

Drosophila Dendritic Arborisation Neurons: Fantastic Actin Dynamics and Where to Find Them

Neuronal dendrites receive, integrate, and process numerous inputs and therefore serve as the neuron's "antennae". Dendrites display extreme morphological diversity across different neuronal classes to match the neuron's specific functional requirements. Understanding how this structural diversity is specified is therefore important for shedding light on information processing in the healthy and diseased nervous system. Popular models for in vivo studies of dendrite differentiation are the four classes of dendritic arborization (c1da-c4da) neurons of Drosophila larvae with their class-specific dendritic morphologies. Using da neurons, a combination of live-cell imaging and computational approaches have delivered information on the distinct phases and the time course of dendrite development from embryonic stages to the fully developed dendritic tree. With these data, we can start approaching the basic logic behind differential dendrite development. A major role in the definition of neuron-type specific morphologies is played by dynamic actin-rich processes and the regulation of their properties. This review presents the differences in the growth programs leading to morphologically different dendritic trees, with a focus on the key role of actin modulatory proteins. In addition, we summarize requirements and technological progress towards the visualization and manipulation of such actin regulators in vivo.

How neurons exploit fractal geometry to optimize their network connectivity

We investigate the degree to which neurons are fractal, the origin of this fractality, and its impact on functionality. By analyzing three-dimensional images of rat neurons, we show the way their dendrites fork and weave through space is unexpectedly important for generating fractal-like behavior well-described by an ‘effective’ fractal dimension D. This discovery motivated us to create distorted neuron models by modifying the dendritic patterns, so generating neurons across wide ranges of D extending beyond their natural values. By charting the D-dependent variations in inter-neuron connectivity along with the associated costs, we propose that their D values reflect a network cooperation that optimizes these constraints. We discuss the implications for healthy and pathological neurons, and for connecting neurons to medical implants. Our automated approach also facilitates insights relating form and function, applicable to individual neurons and their networks, providing a crucial tool for addressing massive data collection projects (e.g. connectomes).

Achieving functional neuronal dendrite structure through sequential stochastic growth and retraction

Class I ventral posterior dendritic arborisation (c1vpda) proprioceptive sensory neurons respond to contractions in the Drosophila larval body wall during crawling. Their dendritic branches run along the direction of contraction, possibly a functional requirement to maximise membrane curvature during crawling contractions. Although the molecular machinery of dendritic patterning in c1vpda has been extensively studied, the process leading to the precise elaboration of their comb-like shapes remains elusive. Here, to link dendrite shape with its proprioceptive role, we performed long-term, non-invasive, in vivo time-lapse imaging of c1vpda embryonic and larval morphogenesis to reveal a sequence of differentiation stages. We combined computer models and dendritic branch dynamics tracking to propose that distinct sequential phases of stochastic growth and retraction achieve efficient dendritic trees both in terms of wire and function. Our study shows how dendrite growth balances structure-function requirements, shedding new light on general principles of self-organisation in functionally specialised dendrites.

Excess Neuronal Branching Allows for Local Innervation of Specific Dendritic Compartments in Mature Cortex

The connectivity of cortical microcircuits is a major determinant of brain function; defining how activity propagates between different cell types is key to scaling our understanding of individual neuronal behavior to encompass functional networks. Furthermore, the integration of synaptic currents within a dendrite depends on the spatial organization of inputs, both excitatory and inhibitory. We identify a simple equation to estimate the number of potential anatomical contacts between neurons; finding a linear increase in potential connectivity with cable length and maximum spine length, and a decrease with overlapping volume. This enables us to predict the mean number of candidate synapses for reconstructed cells, including those realistically arranged. We identify an excess of potential local connections in mature cortical data, with densities of neurite higher than is necessary to reliably ensure the possible implementation of any given axo-dendritic connection. We show that the number of local potential contacts allows specific innervation of distinct dendritic compartments.

Flexible neural connectivity under constraints on total connection strength

Neural computation is determined by neurons’ dynamics and circuit connectivity. Uncertain and dynamic environments may require neural hardware to adapt to different computational tasks, each requiring different connectivity configurations. At the same time, connectivity is subject to a variety of constraints, placing limits on the possible computations a given neural circuit can perform. Here we examine the hypothesis that the organization of neural circuitry favors computational flexibility: that it makes many computational solutions available, given physiological constraints. From this hypothesis, we develop models of connectivity degree distributions based on constraints on a neuron’s total synaptic weight. To test these models, we examine reconstructions of the mushroom bodies from the first instar larva and adult Drosophila melanogaster. We perform a Bayesian model comparison for two constraint models and a random wiring null model. Overall, we find that flexibility under a homeostatically fixed total synaptic weight describes Kenyon cell connectivity better than other models, suggesting a principle shaping the apparently random structure of Kenyon cell wiring. Furthermore, we find evidence that larval Kenyon cells are more flexible earlier in development, suggesting a mechanism whereby neural circuits begin as flexible systems that develop into specialized computational circuits.

High-throughput electron microscopic anatomical experiments have begun to yield detailed maps of neural circuit connectivity. Uncovering the principles that govern these circuit structures is a major challenge for systems neuroscience. Healthy neural circuits must be able to perform computational tasks while satisfying physiological constraints. Those constraints can restrict a neuron’s possible connectivity, and thus potentially restrict its computation. Here we examine simple models of constraints on total synaptic weights, and calculate the number of circuit configurations they allow: a simple measure of their computational flexibility. We propose probabilistic models of connectivity that weight the number of synaptic partners according to computational flexibility under a constraint and test them using recent wiring diagrams from a learning center, the mushroom body, in the fly brain. We compare constraints that fix or bound a neuron’s total connection strength to a simple random wiring null model. Of these models, the fixed total connection strength matched the overall connectivity best in mushroom bodies from both larval and adult flies. We also provide evidence suggesting that neural circuits are more flexible in early stages of development and lose this flexibility as they grow towards specialized function.

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.

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.

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 regularity index for dendrites - local statistics of a neuron's input space

Neurons collect their inputs from other neurons by sending out arborized dendritic structures. However, the relationship between the shape of dendrites and the precise organization of synaptic inputs in the neural tissue remains unclear. Inputs could be distributed in tight clusters, entirely randomly or else in a regular grid-like manner. Here, we analyze dendritic branching structures using a regularity index R, based on average nearest neighbor distances between branch and termination points, characterizing their spatial distribution. We find that the distributions of these points depend strongly on cell types, indicating possible fundamental differences in synaptic input organization. Moreover, R is independent of cell size and we find that it is only weakly correlated with other branching statistics, suggesting that it might reflect features of dendritic morphology that are not captured by commonly studied branching statistics. We then use morphological models based on optimal wiring principles to study the relation between input distributions and dendritic branching structures. Using our models, we find that branch point distributions correlate more closely with the input distributions while termination points in dendrites are generally spread out more randomly with a close to uniform distribution. We validate these model predictions with connectome data. Finally, we find that in spatial input distributions with increasing regularity, characteristic scaling relationships between branching features are altered significantly. In summary, we conclude that local statistics of input distributions and dendrite morphology depend on each other leading to potentially cell type specific branching features.

Dendritic tree structures of nerve cells are built to optimally collect inputs from other cells in the circuit. By looking at how regularly the branch and termination points of dendrites are distributed, we find characteristic differences between cell types that correlate little with other traditional branching statistics and affect their scaling properties. Using computational models based on optimal wiring principles, we then show that termination points of dendrites generally spread more randomly than the inputs that they receive while branch points follow more closely the underlying input organization. Existing connectome data validate these predictions indicating the importance of our findings for large scale neural circuit analysis.

Modelling brain-wide neuronal morphology via rooted Cayley trees

Neuronal morphology is an essential element for brain activity and function. We take advantage of current availability of brain-wide neuron digital reconstructions of the Pyramidal cells from a mouse brain, and analyze several emergent features of brain-wide neuronal morphology. We observe that axonal trees are self-affine while dendritic trees are self-similar. We also show that tree size appear to be random, independent of the number of dendrites within single neurons. Moreover, we consider inhomogeneous branching model which stochastically generates rooted 3-Cayley trees for the brain-wide neuron topology. Based on estimated order-dependent branching probability from actual axonal and dendritic trees, our inhomogeneous model quantitatively captures a number of topological features including size and shape of both axons and dendrites. This sheds lights on a universal mechanism behind the topological formation of brain-wide axonal and dendritic trees.

T2N as a new tool for robust electrophysiological modeling demonstrated for mature and adult-born dentate granule cells

Compartmental models are the theoretical tool of choice for understanding single neuron computations. However, many models are incomplete, built ad hoc and require tuning for each novel condition rendering them of limited usability. Here, we present T2N, a powerful interface to control NEURON with Matlab and TREES toolbox, which supports generating models stable over a broad range of reconstructed and synthetic morphologies. We illustrate this for a novel, highly detailed active model of dentate granule cells (GCs) replicating a wide palette of experiments from various labs. By implementing known differences in ion channel composition and morphology, our model reproduces data from mouse or rat, mature or adult-born GCs as well as pharmacological interventions and epileptic conditions. This work sets a new benchmark for detailed compartmental modeling. T2N is suitable for creating robust models useful for large-scale networks that could lead to novel predictions. We discuss possible T2N application in degeneracy studies.

Branching morphology determines signal propagation dynamics in neurons

Computational modeling of signal propagation in neurons is critical to our understanding of basic principles underlying brain organization and activity. Exploring these models is used to address basic neuroscience questions as well as to gain insights for clinical applications. The seminal Hodgkin Huxley model is a common theoretical framework to study brain activity. It was mainly used to investigate the electrochemical and physical properties of neurons. The influence of neuronal structure on activity patterns was explored, however, the rich dynamics observed in neurons with different morphologies is not yet fully understood. Here, we study signal propagation in fundamental building blocks of neuronal branching trees, unbranched and branched axons. We show how these simple axonal elements can code information on spike trains, and how asymmetric responses can emerge in axonal branching points. This asymmetric phenomenon has been observed experimentally but until now lacked theoretical characterization. Together, our results suggest that axonal morphological parameters are instrumental in activity modulation and information coding. The insights gained from this work lay the ground for better understanding the interplay between function and form in real-world complex systems. It may also supply theoretical basis for the development of novel therapeutic approaches to damaged nervous systems.

Universal features of dendrites through centripetal branch ordering

Dendrites form predominantly binary trees that are exquisitely embedded in the networks of the brain. While neuronal computation is known to depend on the morphology of dendrites, their underlying topological blueprint remains unknown. Here, we used a centripetal branch ordering scheme originally developed to describe river networks—the Horton-Strahler order (SO)–to examine hierarchical relationships of branching statistics in reconstructed and model dendritic trees. We report on a number of universal topological relationships with SO that are true for all binary trees and distinguish those from SO-sorted metric measures that appear to be cell type-specific. The latter are therefore potential new candidates for categorising dendritic tree structures. Interestingly, we find a faithful correlation of branch diameters with centripetal branch orders, indicating a possible functional importance of SO for dendritic morphology and growth. Also, simulated local voltage responses to synaptic inputs are strongly correlated with SO. In summary, our study identifies important SO-dependent measures in dendritic morphology that are relevant for neural function while at the same time it describes other relationships that are universal for all dendrites.

Similarly to river beds, dendritic trees of nerve cells form elaborate networks that branch out to cover extensive areas. In the 1940s, ecologist Robert E. Horton developed an ordering system for branches in river networks that was refined in the 1950s by geoscientist Arthur N. Strahler, the Horton-Strahler order (SO). Branches at the tips start with order 1 and increase their order in a systematic way when encountering new branches on the way to the root. SO relationships have recently become popular for quantifying dendritic morphologies. Various branching statistics can be studied as a function of SO. Here we describe that topological measures such as the number of branches, the branch bifurcation ratio and the size of subtrees exhibit stereotypical relations with SO in dendritic trees independently of cell type, mirroring universal features of binary trees. Other functionally more relevant features such as mean branch lengths, local diameters and simulated voltage responses to synaptic inputs directly correlate with SO in a cell type-specific manner, indicating the importance of SO for understanding dendrite growth as well as neural computation.

Sloppy morphological tuning in identified neurons of the crustacean stomatogastric ganglion

Neuronal physiology depends on a neuron’s ion channel composition and unique morphology. Variable ion channel compositions can produce similar neuronal physiologies across animals. Less is known regarding the morphological precision required to produce reliable neuronal physiology. Theoretical studies suggest that moraphology is tightly tuned to minimize wiring and conduction delay of synaptic events. We utilize high-resolution confocal microscopy and custom computational tools to characterize the morphologies of four neuron types in the stomatogastric ganglion (STG) of the crab Cancer borealis. Macroscopic branching patterns and fine cable properties are variable within and across neuron types. We compare these neuronal structures to synthetic minimal spanning neurite trees constrained by a wiring cost equation and find that STG neurons do not adhere to prevailing hypotheses regarding wiring optimization principles. In this highly modulated and oscillating circuit, neuronal structures appear to be governed by a space-filling mechanism that outweighs the cost of inefficient wiring.

DOI:http://dx.doi.org/10.7554/eLife.22352.001

Micro-connectomics: probing the organization of neuronal networks at the cellular scale

Defining the organizational principles of neuronal networks at the cellular scale, or micro-connectomics, is a key challenge of modern neuroscience. In this Review, we focus on graph theoretical parameters of micro-connectome topology, often informed by economical principles that conceptually originated with Ramón y Cajal's conservation laws. First, we summarize results from studies in intact small organisms and in samples from larger nervous systems. We then evaluate the evidence for an economical trade-off between biological cost and functional value in the organization of neuronal networks. Various results suggest that many aspects of neuronal network organization are indeed the outcome of competition between these two fundamental selection pressures.

Dendritic and Axonal Wiring Optimization of Cortical GABAergic Interneurons

The way in which a neuronal tree expands plays an important role in its functional and computational characteristics. We aimed to study the existence of an optimal neuronal design for different types of cortical GABAergic neurons. To do this, we hypothesized that both the axonal and dendritic trees of individual neurons optimize brain connectivity in terms of wiring length. We took the branching points of real three-dimensional neuronal reconstructions of the axonal and dendritic trees of different types of cortical interneurons and searched for the minimal wiring arborization structure that respects the branching points. We compared the minimal wiring arborization with real axonal and dendritic trees. We tested this optimization problem using a new approach based on graph theory and evolutionary computation techniques. We concluded that neuronal wiring is near-optimal in most of the tested neurons, although the wiring length of dendritic trees is generally nearer to the optimum. Therefore, wiring economy is related to the way in which neuronal arborizations grow irrespective of the marked differences in the morphology of the examined interneurons.

A general homeostatic principle following lesion induced dendritic remodeling

INTRODUCTION

Neuronal death and subsequent denervation of target areas are hallmarks of many neurological disorders. Denervated neurons lose part of their dendritic tree, and are considered "atrophic", i.e. pathologically altered and damaged. The functional consequences of this phenomenon are poorly understood.

RESULTS

Using computational modelling of 3D-reconstructed granule cells we show that denervation-induced dendritic atrophy also subserves homeostatic functions: By shortening their dendritic tree, granule cells compensate for the loss of inputs by a precise adjustment of excitability. As a consequence, surviving afferents are able to activate the cells, thereby allowing information to flow again through the denervated area. In addition, action potentials backpropagating from the soma to the synapses are enhanced specifically in reorganized portions of the dendritic arbor, resulting in their increased synaptic plasticity. These two observations generalize to any given dendritic tree undergoing structural changes.

CONCLUSIONS

Structural homeostatic plasticity, i.e. homeostatic dendritic remodeling, is operating in long-term denervated neurons to achieve functional homeostasis.

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.

Cortical Composition Hierarchy Driven by Spine Proportion Economical Maximization or Wire Volume Minimization

The structure and quantitative composition of the cerebral cortex are interrelated with its computational capacity. Empirical data analyzed here indicate a certain hierarchy in local cortical composition. Specifically, neural wire, i.e., axons and dendrites take each about 1/3 of cortical space, spines and glia/astrocytes occupy each about (1/3)², and capillaries around (1/3)⁴. Moreover, data analysis across species reveals that these fractions are roughly brain size independent, which suggests that they could be in some sense optimal and thus important for brain function. Is there any principle that sets them in this invariant way? This study first builds a model of local circuit in which neural wire, spines, astrocytes, and capillaries are mutually coupled elements and are treated within a single mathematical framework. Next, various forms of wire minimization rule (wire length, surface area, volume, or conduction delays) are analyzed, of which, only minimization of wire volume provides realistic results that are very close to the empirical cortical fractions. As an alternative, a new principle called “spine economy maximization” is proposed and investigated, which is associated with maximization of spine proportion in the cortex per spine size that yields equally good but more robust results. Additionally, a combination of wire cost and spine economy notions is considered as a meta-principle, and it is found that this proposition gives only marginally better results than either pure wire volume minimization or pure spine economy maximization, but only if spine economy component dominates. However, such a combined meta-principle yields much better results than the constraints related solely to minimization of wire length, wire surface area, and conduction delays. Interestingly, the type of spine size distribution also plays a role, and better agreement with the data is achieved for distributions with long tails. In sum, these results suggest that for the efficiency of local circuits wire volume may be more primary variable than wire length or temporal delays, and moreover, the new spine economy principle may be important for brain evolutionary design in a broader context.

Cerebral cortex is an outer layer of the brain in mammals, and it plays a critical part in various cognitive processes such as learning, memory, attention, language, and consciousness. The cerebral cortex contains a number of neuroanatomical parameters whose values are essentially conserved across species and brain sizes, which suggests that these particular parameters are somehow important for brain efficient functioning. This study shows that the fractional volumes of five major cortical components both neuronal and non-neuronal (axons, dendrites, spines, glia/astrocytes, capillaries) are also approximately conserved across mammals, and neural wire (axons and dendrites) occupies the most of cortical space. Moreover, the fractional volumes form a special hierarchy of dependencies, being approximately equal to integer powers of 1/3. Is there any evolutionary principle of cortical organization that would explain these properties? This study finds that there are two different theoretical principles that can provide answers: one standard related to minimization of neural wire fractional volume, and a new proposition associated with economical maximization of spine content. However, the latter principle produces more robust results, which suggests that spine economical maximization is potentially an alternative to the more common “wire minimization” in explaining the cortical layout. Therefore, the current study can become an important contribution to our understanding (or debating) of the main factors influencing the evolution of local cortical circuits in the brain.

Development of dendritic form and function

The nervous system is populated by numerous types of neurons, each bearing a dendritic arbor with a characteristic morphology. These type-specific features influence many aspects of a neuron's function, including the number and identity of presynaptic inputs and how inputs are integrated to determine firing properties. Here, we review the mechanisms that regulate the construction of cell type-specific dendrite patterns during development. We focus on four aspects of dendrite patterning that are particularly important in determining the function of the mature neuron: (a) dendrite shape, including branching pattern and geometry of the arbor; (b) dendritic arbor size;

Topological characterization of neuronal arbor morphology via sequence representation: I--motif analysis

BACKGROUND

The morphology of neurons offers many insights into developmental processes and signal processing. Numerous reports have focused on metrics at the level of individual branches or whole arbors; however, no studies have attempted to quantify repeated morphological patterns within neuronal trees. We introduce a novel sequential encoding of neurite branching suitable to explore topological patterns.

RESULTS

Using all possible branching topologies for comparison we show that the relative abundance of short patterns of up to three bifurcations, together with overall tree size, effectively capture the local branching patterns of neurons. Dendrites and axons display broadly similar topological motifs (over-represented patterns) and anti-motifs (under-represented patterns), differing most in their proportions of bifurcations with one terminal branch and in select sub-sequences of three bifurcations. In addition, pyramidal apical dendrites reveal a distinct motif profile.

CONCLUSIONS

The quantitative characterization of topological motifs in neuronal arbors provides a thorough description of local features and detailed boundaries for growth mechanisms and hypothesized computational functions.

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.

Quantifying the Number of Discriminable Coincident Dendritic Input Patterns through Dendritic Tree Morphology

Current developments in neuronal physiology are unveiling novel roles for dendrites. Experiments have shown mechanisms of non-linear synaptic NMDA dependent activations, able to discriminate input patterns through the waveforms of the excitatory postsynaptic potentials. Contextually, the synaptic clustering of inputs is the principal cellular strategy to separate groups of common correlated inputs. Dendritic branches appear to work as independent discriminating units of inputs potentially reflecting an extraordinary repertoire of pattern memories. However, it is unclear how these observations could impact our comprehension of the structural correlates of memory at the cellular level. This work investigates the discrimination capabilities of neurons through computational biophysical models to extract a predicting law for the dendritic input discrimination capability (M). By this rule we compared neurons from a neuron reconstruction repository (neuromorpho.org). Comparisons showed that primate neurons were not supported by an equivalent M preeminence and that M is not uniformly distributed among neuron types. Remarkably, neocortical neurons had substantially less memory capacity in comparison to those from non-cortical regions. In conclusion, the proposed rule predicts the inherent neuronal spatial memory gathering potentially relevant anatomical and evolutionary considerations about the brain cytoarchitecture.

Statistical analysis and data mining of digital reconstructions of dendritic morphologies

Neuronal morphology is diverse among animal species, developmental stages, brain regions, and cell types. The geometry of individual neurons also varies substantially even within the same cell class. Moreover, specific histological, imaging, and reconstruction methodologies can differentially affect morphometric measures. The quantitative characterization of neuronal arbors is necessary for in-depth understanding of the structure-function relationship in nervous systems. The large collection of community-contributed digitally reconstructed neurons available at NeuroMorpho.Org constitutes a “big data” research opportunity for neuroscience discovery beyond the approaches typically pursued in single laboratories. To illustrate these potential and related challenges, we present a database-wide statistical analysis of dendritic arbors enabling the quantification of major morphological similarities and differences across broadly adopted metadata categories. Furthermore, we adopt a complementary unsupervised approach based on clustering and dimensionality reduction to identify the main morphological parameters leading to the most statistically informative structural classification. We find that specific combinations of measures related to branching density, overall size, tortuosity, bifurcation angles, arbor flatness, and topological asymmetry can capture anatomically and functionally relevant features of dendritic trees. The reported results only represent a small fraction of the relationships available for data exploration and hypothesis testing enabled by sharing of digital morphological reconstructions.

Linking macroscopic with microscopic neuroanatomy using synthetic neuronal populations

Dendritic morphology has been shown to have a dramatic impact on neuronal function. However, population features such as the inherent variability in dendritic morphology between cells belonging to the same neuronal type are often overlooked when studying computation in neural networks. While detailed models for morphology and electrophysiology exist for many types of single neurons, the role of detailed single cell morphology in the population has not been studied quantitatively or computationally. Here we use the structural context of the neural tissue in which dendritic trees exist to drive their generation in silico. We synthesize the entire population of dentate gyrus granule cells, the most numerous cell type in the hippocampus, by growing their dendritic trees within their characteristic dendritic fields bounded by the realistic structural context of (1) the granule cell layer that contains all somata and (2) the molecular layer that contains the dendritic forest. This process enables branching statistics to be linked to larger scale neuroanatomical features. We find large differences in dendritic total length and individual path length measures as a function of location in the dentate gyrus and of somatic depth in the granule cell layer. We also predict the number of unique granule cell dendrites invading a given volume in the molecular layer. This work enables the complete population-level study of morphological properties and provides a framework to develop complex and realistic neural network models.

Computational models of neurons and neural networks provide a valuable avenue to test our understanding of brain regions and to make predictions to guide future experimentation. Each neuron has a unique dendritic tree, features of which can vary depending on the location of the neuron within the particular brain region. In this study, we generated a complete population of dendritic trees for the most numerous type of neuron in the hippocampus, the dentate gyrus granule cell, using a realistic three-dimensional structural context to drive the generation process. Morphological properties can now be studied at the level of complete neuronal populations, and this work provides a foundation to build upon in the construction of large-scale, data-driven neuroanatomical and network models.

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.

Preserving neural function under extreme scaling

Important brain functions need to be conserved throughout organisms of extremely varying sizes. Here we study the scaling properties of an essential component of computation in the brain: the single neuron. We compare morphology and signal propagation of a uniquely identifiable interneuron, the HS cell, in the blowfly (Calliphora) with its exact counterpart in the fruit fly (Drosophila) which is about four times smaller in each dimension. Anatomical features of the HS cell scale isometrically and minimise wiring costs but, by themselves, do not scale to preserve the electrotonic behaviour. However, the membrane properties are set to conserve dendritic as well as axonal delays and attenuation as well as dendritic integration of visual information. In conclusion, the electrotonic structure of a neuron, the HS cell in this case, is surprisingly stable over a wide range of morphological scales.

Functional genomic analyses of two morphologically distinct classes of Drosophila sensory neurons: post-mitotic roles of transcription factors in dendritic patterning

Background

Neurons are one of the most structurally and functionally diverse cell types found in nature, owing in large part to their unique class specific dendritic architectures. Dendrites, being highly specialized in receiving and processing neuronal signals, play a key role in the formation of functional neural circuits. Hence, in order to understand the emergence and assembly of a complex nervous system, it is critical to understand the molecular mechanisms that direct class specific dendritogenesis.

Methodology/Principal Findings

We have used the Drosophila dendritic arborization (da) neurons to gain systems-level insight into dendritogenesis by a comparative study of the morphologically distinct Class-I (C-I) and Class-IV (C-IV) da neurons. We have used a combination of cell-type specific transcriptional expression profiling coupled to a targeted and systematic in vivo RNAi functional validation screen. Our comparative transcriptomic analyses have revealed a large number of differentially enriched/depleted gene-sets between C-I and C-IV neurons, including a broad range of molecular factors and biological processes such as proteolytic and metabolic pathways. Further, using this data, we have identified and validated the role of 37 transcription factors in regulating class specific dendrite development using in vivo class-specific RNAi knockdowns followed by rigorous and quantitative neurometric analysis.

Conclusions/Significance

This study reports the first global gene-expression profiles from purified Drosophila C-I and C-IV da neurons. We also report the first large-scale semi-automated reconstruction of over 4,900 da neurons, which were used to quantitatively validate the RNAi screen phenotypes. Overall, these analyses shed global and unbiased novel insights into the molecular differences that underlie the morphological diversity of distinct neuronal cell-types. Furthermore, our class-specific gene expression datasets should prove a valuable community resource in guiding further investigations designed to explore the molecular mechanisms underlying class specific neuronal patterning.

Central synapses release a resource-efficient amount of glutamate

Why synapses release a certain amount of neurotransmitter is poorly understood. We combined patch-clamp electrophysiology with computer simulations to estimate how much glutamate is discharged at two distinct central synapses of the rat. We found that, regardless of some uncertainty over synaptic microenvironment, synapses generate the maximal current per released glutamate molecule while maximizing signal information content. Our result suggests that synapses operate on a principle of resource optimization.