Application of network analysis to the study of the branching patterns of dendritic fields

The technique of network analysis has been used to define the connectivity and growth of networks generated by monochotomous, dichotomous, and trichotomous branching. The number of distinct topologic branching patterns exhibited by networks with a given number of pendant arcs is defined mathematically; when all types are represented, a complete pendant arc series is formed. The frequency of occurrence of topologic types in these series is unique for a given hypothesis of growth. The growth of the small dendritic arrays such as the basal dendritic fields of neocortical pyramids may be studied by comparing the actual frequency of topologic types with those computed according to given hypotheses. For larger dendritic networks such as those of Purkinje cells in the cerebellum it is only practicable to use the topologic types formed by the peripheral parts of the tree as a basis for comparison. Individual dendritic segments can be ordered sequentially to define their hierarchical arrangement; the frequency of orders in a given network always forms an inverse geometric series. The ratio between orders is called the "bifurcation ratio," and the relationship in a given large series between adjacent orders becomes stabilized to a fixed or "established" bifurcation ratio at the periphery of the tree only. This "established ratio" characterizes the pattern of growth of the network. In the proximal part of the tree the ratio between adjacent orders is unstable and accounts for the variability of the overall bifurcation ratio exhibited by different networks with the same fundamental growth pattern and for the deviation of the overall from the established bifurcation ratio. For a given size of network the overall bifurcation ratio may be similar regardless of the mode of growth. It is concluded that the precise definition of branching structures afforded by network analysis makes this technique well suited for the study of the connectivity, growth, and morphology of dendritic trees.