Coarse-graining model reveals universal exponential scaling in axonal length distributions

The exponential distance rule (EDR) is a well-documented phenomenon suggesting that the distribution of axonal lengths in the brain follows an exponential decay pattern. Nevertheless, individual-level axon data supporting this assertion is limited to Drosophila and mice, while inter-region connectome data is also accessible for macaques, marmosets, and humans. Although axon-level data in Drosophila and mice support the generality of the EDR, region-level data can significantly deviate from the exponential curve. In this study, we establish that the axon number-weighted length distribution of region-level connections converges onto a universal curve when rescaled to the mean axonal length, demonstrating similarities across different species. To explain these observations, we present a simple mathematical model that attributes the observed deviations from the EDR in the weighted length distribution of inter-regional connectomes to the inherent coarse-graining effect of translating from neuron-level to region-level connectomics. We demonstrate that the qualitative predictions of the model are robust with respect to various aspects of brain region-geometry, including dimensionality, resolution, and curvature. On the other hand, the performance of the model exhibits a monotonous dependence on the amount of region-geometry related detail incorporated into the model. The findings validate the universality of the EDR rule across various species, paving the way for further in-depth exploration of this remarkably simple principle.