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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.
Much is known about the computation in individual neurons in the cortical column. Also, the selective connectivity between many cortical neuron types has been studied in great detail. However, due to the complexity of this microcircuitry its functional role within the cortical column remains a mystery. Some of the wiring behavior between neurons can be interpreted directly from their particular dendritic and axonal shapes. Here, I describe the dendritic density field (DDF) as one key element that remains to be better understood. I sketch an approach to relate DDFs in general to their underlying potential connectivity schemes. As an example, I show how the characteristic shape of a cortical pyramidal cell appears as a direct consequence of connecting inputs arranged in two separate parallel layers.