Comparison of the topology and growth rules of motoneuronal dendrites.

The complexity, shape, and branching modes of the dendrites of spinal motoneurons were compared in cat, rat, and frog using topological analysis and growth models. The complexity of motoneuronal dendrites, measured as the mean number of terminal segments, varied significantly among samples and was related to contractile properties of innervated motor units. Despite this variation, all mature motoneurons having a mean number of terminal segments per dendrite greater than ten (up to 24.3) exhibited a narrow range of values of coefficients describing the symmetry of tree shapes (0.42-0.47). This implies low variability in the topological shape of motoneuronal dendrites of different animals. This similarity of tree shapes proved to be a result of the similarity of growth rules. The growth of the dendrites could be described to a first approximation by a two-parameter (Q and S) model called the QS model and by a multitype Markovian model. The estimation of parameters of the QS model, in which parameter Q is related to the probability of branching of intermediate segments, revealed that Q was equal or close to 0, implying that branching of dendrites is restricted to terminal segments. The estimates of the parameter S, which describes whether the probability of branching increases (S < 0) or decreases (S > 0) exponentially with segment order, were positive. This was in agreement with the results of estimation of probabilities of branching provided by the Markovian model, which showed that the branching probabilities decreased with segment order in an exponential manner in most of the neurons studied. The QS and Markovian models involve different assumptions about the sequence and timing of branching events, and selection of the best model can provide insight into details of dendritic outgrowth. Extensive simulation of tree outgrowth using a Markovian model revealed significant differences between stimulated trees and real dendrites, particularly with regard to variability of the number of terminals and to symmetry. In contrast, the QS model provided a good fit to the mean values and standard deviations of basic topological parameters. This model is adequate to describe the shape of mature motoneuronal dendrites. It implies that dendritic branches have many opportunities to bifurcate during the whole time of development and that bifurcating potency of a branch is a function of the number and position of other branches of that dendrite. Combined with analysis of metrical properties such as lengths of segments, the QS model can assist in a quantitative analysis of development and plasticity.