Topological scaling law

Branching allows neurons to make synaptic contacts with large numbers of other neurons, facilitating the high connectivity of nervous systems. Neurons have geometric properties such as branch lengths and diameters that minimize construction costs and optimize signal transmission and intracellular transport. In this work, we asked whether neuronal arbors also have topological properties that reflect their growth and/or functional properties. We discovered that the subtree-size distribution, the average number of branches that support dendrite tips, follows a power law whose slope varies among different neuronal types from a wide range of invertebrate and vertebrate species. Through simulation, we show that the slope depends on the symmetry and density of the arbor. Furthermore, inclusion of postsynaptic spines and other terminal processes on branches causes a deviation of the subtree-size distribution from the power law. Thus, the subtree-size distribution is a topological property that reflects the underlying functional morphology of dendrites.