The Flatness of Bifurcations in 3D Dendritic Trees: An Optimal Design

The geometry of natural branching systems generally reflects functional optimization. A common property is that their bifurcations are planar and that daughter segments do not turn back in the direction of the parent segment. The present study investigates whether this also applies to bifurcations in 3D dendritic arborizations. This question was earlier addressed in a first study of flatness of 3D dendritic bifurcations by Uylings and Smit (1975), who used the apex angle of the right circular cone as flatness measure. The present study was inspired by recent renewed interest in this measure. Because we encountered ourselves shortcomings of this cone angle measure, the search for an optimal measure for flatness of 3D bifurcation was the second aim of our study. Therefore, a number of measures has been developed in order to quantify flatness and orientation properties of spatial bifurcations. All these measures have been expressed mathematically in terms of the three bifurcation angles between the three pairs of segments in the bifurcation. The flatness measures have been applied and evaluated to bifurcations in rat cortical pyramidal cell basal and apical dendritic trees, and to random spatial bifurcations. Dendritic and random bifurcations show significant different flatness measure distributions, supporting the conclusion that dendritic bifurcations are significantly more flat than random bifurcations. Basal dendritic bifurcations also show the property that their parent segments are generally aligned oppositely to the bisector of the angle between their daughter segments, resulting in "symmetrical" configurations. Such geometries may arise when during neuronal development the segments at a newly formed bifurcation are subjected to elastic tensions, which force the bifurcation into an equilibrium planar shape. Apical bifurcations, however, have parent segments oppositely aligned with one of the daughter segments. These geometries arise in the case of side branching from an existing apical main stem. The aligned "apical" parent and "apical" daughter segment form together with the side branch daughter segment already geometrically a flat configuration. These properties are clearly reflected in the flatness measure distributions. Comparison of the different flatness measures made clear that they all capture flatness properties in a different way. Selection of the most appropriate measure thus depends on the question of research. For our purpose of quantifying flatness and orientation of the segments, the dihedral angle β was found to be the most discriminative and applicable single measure. Alternatively, the parent elevation and azimuth angle formed an orthogonal pair of measures most clearly demonstrating the dendritic bifurcation "symmetry" properties.