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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.
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.
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.
Artificial neural networks, taking inspiration from biological neurons, have become an invaluable tool for machine learning applications. Recent studies have developed techniques to effectively tune the connectivity of sparsely-connected artificial neural networks, which have the potential to be more computationally efficient than their fully-connected counterparts and more closely resemble the architectures of biological systems. We here present a normalisation, based on the biophysical behaviour of neuronal dendrites receiving distributed synaptic inputs, that divides the weight of an artificial neuron’s afferent contacts by their number. We apply this dendritic normalisation to various sparsely-connected feedforward network architectures, as well as simple recurrent and self-organised networks with spatially extended units. The learning performance is significantly increased, providing an improvement over other widely-used normalisations in sparse networks. The results are two-fold, being both a practical advance in machine learning and an insight into how the structure of neuronal dendritic arbours may contribute to computation.
Achieving functional neuronal dendrite structure through sequential stochastic growth and retraction
(2020)
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.
Neurogenesis of hippocampal granule cells (GCs) persists throughout mammalian life and is important for learning and memory. How newborn GCs differentiate and mature into an existing circuit during this time period is not yet fully understood. We established a method to visualize postnatally generated GCs in organotypic entorhino-hippocampal slice cultures (OTCs) using retroviral (RV) GFP-labeling and performed time-lapse imaging to study their morphological development in vitro. Using anterograde tracing we could, furthermore, demonstrate that the postnatally generated GCs in OTCs, similar to adult born GCs, grow into an existing entorhino-dentate circuitry. RV-labeled GCs were identified and individual cells were followed for up to four weeks post injection. Postnatally born GCs exhibited highly dynamic structural changes, including dendritic growth spurts but also retraction of dendrites and phases of dendritic stabilization. In contrast, older, presumably prenatally born GCs labeled with an adeno-associated virus (AAV), were far less dynamic. We propose that the high degree of structural flexibility seen in our preparations is necessary for the integration of newborn granule cells into an already existing neuronal circuit of the dentate gyrus in which they have to compete for entorhinal input with cells generated and integrated earlier.
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.
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.
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.
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.