One Rule to Grow Them All: A General Theory of Neuronal Branching and Its Practical Application
(A) The tree consists of cylinders or frusta (red) connecting each two nodes along the directed edges (away from the root node, arrows). Branch points and termination points represent the topology (topological points). A branch is a set of continuation points between two topological points. The labelling of the nodes is unique following three principles: hierarchical sorting, continuous labelling preserving sub-tree consistency and topological sorting (see text). (B) Rearrangement of node locations on a sample tree. Examples of equidistant node redistribution resulting in 10 or 20 µm resampling and a 20 µm resampling including length conservation (see text and “resampling” section of Methods). (C–E) Unique representations of topology and electrotonic properties from sample tree from (B). (C) Applying topological sorting, a unique electrotonic equivalent tree can be constructed by mapping node label hierarchy on the branch angle (equivalent tree). (D) The adjacency matrix depicts the connectivity between the nodes of a tree. The corresponding electrotonic signature (current transfer from a node to another, i.e. the potential difference measured in one node as a result of a current injection into another) describes the dendritic compartmentalization (see text). The electrotonic signature corresponding to the 20 µm resampled tree preserves the compartmentalization of the original tree. (E) A one-dimensional string fully describes the topology once the nodes of a tree are sorted topologically. Green pieces represent branches ending with a branch point while black pieces end with a termination point. Branch lengths correspond to real metric length and their order follows the node label sorting. Because all representations observe the same continuous labelling, they preserve the sub-tree structure (a red transparent patch highlights one such sub-tree throughout all representations in (C–E)).