Universal features of dendrites through centripetal branch ordering

Abstract

Dendrites form predominantly binary trees that are exquisitely embedded in the networks of the brain. While neuronal computation is known to depend on the morphology of dendrites, their underlying topological blueprint remains unknown. Here, we used a centripetal branch ordering scheme originally developed to describe river networks—the Horton-Strahler order (SO)–to examine hierarchical relationships of branching statistics in reconstructed and model dendritic trees. We report on a number of universal topological relationships with SO that are true for all binary trees and distinguish those from SO-sorted metric measures that appear to be cell type-specific. The latter are therefore potential new candidates for categorising dendritic tree structures. Interestingly, we find a faithful correlation of branch diameters with centripetal branch orders, indicating a possible functional importance of SO for dendritic morphology and growth. Also, simulated local voltage responses to synaptic inputs are strongly correlated with SO. In summary, our study identifies important SO-dependent measures in dendritic morphology that are relevant for neural function while at the same time it describes other relationships that are universal for all dendrites.

Distinct Relations of Microtubules and Actin Filaments with Dendritic Architecture

“…Class IV WT neurons also had the most complex dendritic arbors as measured by the number of branch terminals (683 ± 44.7), followed by class IV Form3OE (365 ± 29.9) and the least complex class I (26 ± 4.8). Correspondingly, the maximum Strahler (centripetal) orders ( Vormberg et al., 2017 ; see Glossary in Supplemental Information for definition) for the three groups were 7, 6, and 4, respectively ( Figure 2 D). These results are consistent with previous analyses of da neurons ( Grueber et al., 2002 ; Nanda et al., 2018b ).…”

A developmental stretch-and-fill process that optimises dendritic wiring

“…A large palette of branching statistics were collected for each dendrite reconstruction separately using simple combinations of TREES toolbox functions: (1) Total dendrite length L was the sum of all internode lengths (len tree), (2) number of branch points B were calculated as the sum of all nodes that are branch points (B tree), (3) covered surface area S was calculated with a novel custom written TREES toolbox function span tree that counts 1 − bins after morphological closing with a disc of 4µm radius of a binary matrix with 1 − bins at tree node locations after resampling the trees to 1µm internode distances -similarly to previous work on fly lobula plate tangential cells (Cuntz et al, 2008), (4) Sholl analysis (Bird and Cuntz, 2019) that calculates the number of intersections of a tree with a growing circle centred on the dendrite root was done by using the dedicated TREES toolbox function sholl tree, (5) normalised Sholl analysis was done by normalising the Sholl radii to 6× the mean Euclidean distance in the tree (eucl tree) and normalising the number of intersections with the integral over the Sholl intersections diagram, (6) branch order distributions were calculated by identifying how many nodes and the overall dendritic cable length of their connected segments in the tree had a given branch order (BO tree), (7) normalised branch orders were normalised to the next integer of 2.5× the mean branch order in any given tree and the cable distribution was normalised to the total length of dendrite, (8) branch length distributions were obtained by combining two TREES toolbox functions that calculate the path length from each node to the root (Pvec tree) and that dissect the trees into their individual branches (dissect tree), (9) normalised branch lengths were normalised to 3× the mean branch length in the tree and branch length frequency was normalised to the integral over the distribution, (10) compression value distributions were calculated as the ratio between Euclidean distances (eucl tree) and the path distances (Pvec tree) to the root for all nodes, (11) the Strahler order distributions were calculated as the percentages of dendrite length per given Strahler order (strahler tree) (Vormberg et al, 2017), (12) branch angle distributions were obtained by calculating the angles in the branching plane of each branch point (angleB tree).…”

Modelling brain-wide neuronal morphology via rooted Cayley trees

“…Neurons have been widely viewed as binary trees (e.g. 27 ), here we first examine this by looking into the degree (i.e. the number of branches each node links to) of each node.…”