Definitions

MST_rule

A greedy algorithm can be implemented which optimizes locally total wiring and path length to the root inspired by Cajal’s laws of conservation of material and conduction time. This represents an extension to the minimum spanning tree (MST) algorithm. One parameter, the balancing factor bf, determines the formants of potential trees

adapted from
PLoS supp. material by Hermann Cuntz

← resampling_a_tree
→ quadratic_diameter_taper

The figure above exemplifies the general approach to obtain a locally optimized graph. In the process, unconnected carrier points (red dots) connect one by one to the nodes of a tree (black dots). At each step, the unconnected carrier point, which is closest to the tree according to some cost function, connects to the node in the tree to which it is closest. The distance cost in this case is composed of two components inspired by Ramón y Cajal’s laws of neuronal branching: 1. the wiring cost corresponding to the Euclidean distance to the node in the tree (red dashed lines show three sample segment distances for carrier point P); 2. the conduction time cost, corresponding to the path length from the root (large black node) to the carrier point P. In the example here, even though P is closer to node 5 in Euclidean terms, the additional cost of path length (adding distance between node 4 and node 5) might tip the balance in favour of node 4. A balancing factor bf weighs these two cost functions against each other (see “MST_tree”).

This approach produces realistic neuronal branching structures in all cases. The balancing factor between the two costs determines the electrotonic compartmentalization of the neuronal tree. At one extreme, one finds the pure minimum spanning tree, at the other, the entirely compartmentalized stellate structure, which connects each carrier point directly to the root.