Mathematical modeling of dendritic growth in vitro

The dendritic branching pattern of cultured hippocampal neurons was analyzed to obtain mathematical parameters that fit the time-dependent growth of dendrites under limited extrinsic influence. Cultured neurons were stained with a non-toxic carbocyanine dye (diO) and pyramidal-shaped neurons that were physically separated from one another were analyzed at post-plating days 1, 2, 3, 4, 6 and 7. The geometric branching pattern of the dendrites was analyzed using a mathematical model that incorporates random effects in the form of a Galton-Watson branching process where splitting of one branch is statistically independent of the splitting of all other branches, and deterministic effects in the form of a parameter that measures the extent to which dense patterns (clusters) or sparse patterns (elongated trees) are formed. The geometric branching pattern of the dendrites was analyzed using a mathematical model that incorporates random and deterministic effects. The model parameters were estimated via the method of maximum likelihood. The data suggest that in vitro basal dendrites grow according to a purely random branching process without pronounced dense or sparse patterns, while apical dendrites tend to form elongated trees with fewer secondary bifurcations. This trend is quantified, and it depends on the culture conditions in which the neurons are grown. The quantitative assessment of various influences on dendritic growth patterns are discussed.