The density recovery profile: a method for the analysis of points in the plane applicable to retinal studies

The density recovery profile is a plot of the spatial density of a set of points as a function of the distance of each of those points from all the others. It is based upon a two-dimensional point autocorrelogram. If the points are randomly distributed, then the profile is flat, with a value equal to the mean spatial density. Thus, any deviation from this value indicates that the presence of the object represented by the point alters the probability of encountering nearby objects of the same set. Increased value near an object indicates clustering, decreased value near an object indicates anticlustering. The method appears to be unique in its ability to provide quantitative measures of the anticlustered state. Two examples are presented. The first is based upon a sample of the distribution of the somata of starburst amacrine cells in the macaque retina; the second is based upon the distribution of the terminal enlargements on the dendrites of a single macaque ganglion cell that projects to the superior colliculus. In both cases, the density recovery profile is initially lower than the mean density, and increases up to the plateau at the value of the mean density. Two useful measures can be derived from this profile: an intensive parameter termed the effective radius, which quantifies the extent of the region of decreased probability and is insensitive to random undersampling of the underlying distribution, and an extensive parameter termed the packing factor, which quantifies the degree of packing possible for a given effective radius, and is insensitive to scaling. An extension of this method, applicable to correlations between two superimposed distributions, and based upon a two-dimensional point cross-correlogram, is also described.