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. [Display omitted] •Neuronal cell types can be distinguished by the topologies of their dendritic arbors•For many neurons, the distribution of subtree sizes is scale invariant•Postsynaptic spines and branchlets cause deviations from scale invariance•The subtree-size distribution reflects the underlying branching processes Liao et al. found that the dendritic arbors of many neurons have topological scale invariance: the distributions of subtree sizes follow power laws. Simulations indicate that scale invariance arises from repetitive application of branching rules, including those that optimally balance material costs and conduction times.