title: Vertex analysis of neural tree structures containing trichotomous nodes.
abstract: Vertex analysis defines dichotomous networks in terms of the relative frequencies of pendant (Vp) and nodal (Vd) vertices. Va, Vb and Vc comprise the 3 classes of Vd which connect 2, 1 and 0 Vp respectively. The vertex ratio (VR) of Va/Vb may be used to define topology since Va and Vc approach equality in large networks. The introduction of trichotomous nodes (Vt, comprising Va', Vb', Vc' and Vd', connecting 3, 2, 1 and 0 Vp) confounds vertex analysis in as much as the values of VR must be recalculated to maintain the definition of topology. Each Vt is equivalent to 2Vd. Va', Vc' and Vd' may be unequivocally transformed into Va + Vb; Vb + Vc; and 2Vc respectively. However, Vb' potentially contains Va, Vb and Vc in proportions which must be arbitrarily set, since the mode of growth only determines the frequencies of Vt and not their transformation into Vd pairs. We have assumed that all possible pairs of transformed Vb' are equally likely and, accordingly, Vb' is equivalent to Va/3 + 4Vb/3 + Vc/3. The values of VR for different modes of growth, featuring different frequencies of Vt, may thus be obtained by computer growth simulation. The mode of growth of observed networks may thereby be ascertained from VR, providing all Vt are detectable in the tree.
mesh_terms: D003712:Dendrites; D006801:Humans; D008433:Mathematics; D008959:Models, Neurological; D009420:Nervous System / Q000254:growth & development*