Topology recapitulates ontogeny of dendritic arbors

Branching of dendrites and axons allows neurons to make synaptic contacts with large numbers of other neurons, facilitating the high connectivity of the nervous system. Neurons have geometric properties, such as the lengths and diameters of their branches, that change systematically throughout the arbor in ways that are thought to minimize construction costs and to optimize the transmission of electrical signals and the intracellular transport of materials. In this work, we investigated whether neuronal arbors also have topological properties that reflect the growth and/or functional properties of their dendritic arbors. In our efforts to uncover possible topological rules, we discovered a function that depends only on the topology of bifurcating trees such as dendritic arbors: the tip-support distribution, which is the average number of branches that support n dendrite tips. We found that for many, but not all, neurons from a wide range of invertebrate and vertebrate species, the tip-support distribution follows a power law with slopes ranging from -1.4 and -1.8 on a log-log plot. The slope is invariant under iterative trimming of terminal branches and under random ablation of internal branches. We found that power laws with similar slopes emerge from a variety of iterative growth processes including the Galton-Watson (GW) process, where the power-law behavior occurs after the percolation threshold. Through simulation, we show the slope of the power-law increases with the branching probability of a GW process, which corresponds to a more regular tree. Furthermore, the inclusion of postsynaptic spines and other terminal processes on branches causes a characteristic deviation of the tip-support distribution from a power law. Therefore, the tip-support function is a topological property that reflects the underlying branching morphogenesis of dendritic trees.