The Sholl analysis of neuronal cell images: semi-log or log-log method?

Although the Sholl analysis is a quantitative method for morphometric neuronal studies and its application provides many benefits to neurobiology since it is obvious, common and meaningful, there are many unresolved theoretical issues that need to be addressed. Nevertheless, it can be practiced without much background or sophistication. The two different methods of the Sholl analysis--log-log and semi-log--have been applied previously without a clear basis as to what to use. To make an adequate choice of the method, one should try and accept that one which gives a better result. We consider that some of the underlying principles, assumptions and limitations for the Sholl analysis can be found in basic mathematics. In order to compare the results of applications of the semi-log and log-log methods to the same neuron, we introduce the concept of determination ratio as the ratio of the coefficient of determination for the semi-log method and that for the log-log method. If the semi-log method is better as related to the log-log method, the value of this parameter is larger than 1. Having in mind that dendrites exhibit enormously diverse forms, we point out that the semi-log Sholl method is more frequently utilizable in practice. Only the neurons, whose dendritic trees have one or a few sparsely ramified dendrites being much longer than the rest ones, could be successfully and exactly analysed using the log-log method. We also compare the Sholl analysis with fractal analysis for the characterization of neuronal arborization patterns and found that between the Sholl and fractal analysis exist various and important analogies.