Morphometric study of the development of Purkinje cell dendritic trees in the mouse using vertex analysis

Vertices are the points in an arborescence which interconnect segments and comprise terminal or pendant vertices (Vp), nodal or branching points and the root point. Branching points may be dichotomous (Vd) or trichomtomous (Vt), etc., and are subdivided into distinct two-dimensional topological entities according to the number of terminal vertices they drain, i.e. Vds comprise primary vertices (Va), connecting two Vps; secondary vertices (Vb), connecting one Vp and one Vd or one Vt; and tertiary vertices (Vc), connecting either two Vds, two Vts or one Vt and one Vd. The four types of Vt (Va', Vb', Vc', Vd') similarly connect three, two, one and zero Vps respectively. Each Vt may be transformed into two Vds thus, Va' = Va + Vb; Vb' = Va/3 + 4Vb/3 + Vc/3; Vc' = Vb + Vc and Vd' = 2Vc. Analysis proceeds by transforming mixed trees containing varying proportions of Vds and Vts into entirely dichotomous branching structures. The topology is then defined by the Va/Vb ratio which has a unique value according to the mode of growth and the frequency of Vts. Vertices are ordered by a centrifugal technique. The frequency distribution of vertices of different order allow the changes in growth characteristics and in remodelling to be detected within particular regions of the tree. Metrical parameters are readily incorporated into the analysis since all vertices are interconnected by segments of finite length and are given the same order magnitude as the vertex they drain. The analytical capabilities of the method are exemplified by its application to the study of growth and plasticity in the dendritic trees of Purkinje cells in the mouse. Growth is defined in metrical and topological terms and sites of reorganization within the mature tree are identified.