The vertebrate retina contains a large number of different types of neurons that can be distinguished by their morphological properties. Assuming that no location should be without a contribution from the circuitry and function linked to a specific type of neuron, it is expected that the dendritic trees of neurons belonging to a type will cover the retina in a regular manner. Thus, for most types of neurons, the contribution to visual processing is thought to be independent of the exact location of individual neurons across the retina. Here, we have investigated the distribution of AII amacrine cells in rat retina. The AII is a multifunctional amacrine cell found in mammals and involved in synaptic microcircuits that contribute to visual processing under both scotopic and photopic conditions. Previous investigations have suggested that AIIs are regularly distributed, with a nearest-neighbor distance regularity index of ~4. It has been argued, however, that this presumed regularity results from treating somas as points, without taking into account their actual spatial extent which constrains the location of other cells of the same type. When we simulated random distributions of cell bodies with size and density similar to real AIIs, we confirmed that the simulated distributions could not be distinguished from the distributions observed experimentally for AIIs in different regions and eccentricities of the retina. The developmental mechanisms that generate the observed distributions of AIIs remain to be investigated.

Grain boundaries (GBs) play an important role in material behavior, so considerable effort has gone into determining their structure and properties. Studies of GBs have revealed a correlation between the GB energy and expansion of the planes normal to the GB, or the so-called normal volume expansion. In this investigation, the volume expansion at several GBs was experimentally determined using extended energy-loss fine structure (EXELFS) analysis in a scanning/transmission electron microscope, allowing changes in the nearest-neighbor (n.n.) distances to be determined with nanometer spatial resolution. EXELFS performed on three-model GBs showed that the average n.n. distances at the GBs increased with increasing GB energy. Additionally, the total volume expansion at the GBs, calculated using complementary plasmon energy profiles, showed excellent agreement with volume expansions measured using other experimental techniques. Hence, this study demonstrates that EXELFS is a useful technique to measure the normal volume expansion at GBs. When combined with the results from complementary studies on the same GBs using valence electron energy-loss spectroscopy, this work further shows that the GB energy increases in relation to both the decrease in electron density at the GB and an accompanying increase in specific volume expansion at the GB.

Mosaics of neurons are usually quantified by methods based on nearest-neighbor distance (NND). The commonest indicator of regularity has been the ratio of the mean NND to the standard deviation, here termed the ‘conformity ratio.’ However, an accurate baseline value of this ratio for bounded random samples has never been determined; nor was its sampling distribution known, making it impossible to test its significance. Instead, significance was assessed from goodness-of-fit to a Rayleigh distribution, or from another ratio, that of the observed mean NND to an expected mean predicted by theory, termed the dispersion index. Neither approach allows for boundary effects that are common in experimental mosaics. Equally common are ‘missing’ neurons, whose effects on the statistics have not been studied. To address these deficiencies, random patterns and real neuronal mosaics were analyzed statistically. Ns independent random-point samples of size Np were generated for 13 Np values between 25 and 6400, where Ns × Np ≥ 144,000. Samples were generated with rectangular boundaries of aspect ratio 1:1, 1:5, and 1:10 to examine the influence of sample geometry. NND distributions, conformity ratios, and dispersion indices were computed for the resulting 45,997 independent random patterns. From these, empirical sampling distributions and critical values were determined. NND distributions for small-to-medium, bounded, random populations were shown to differ significantly from Rayleigh distributions. Thus, goodness-of-fit tests are invalid for most experimental mosaics. Charts are presented from which the significance of conformity ratios or dispersion indices can be read directly. The conformity ratio reacts conservatively to extremes of sample geometry, and provides a useful and safe test. The dispersion index is nonconservative, making its use problematic. Tests based on the theoretical distribution of the dispersion index are unreliable for all but the largest samples. Random deletions were also made from 33 real retinal ganglion cell mosaics. The mean NND, conformity ratio, and dispersion index were determined for each original mosaic and 36 independent samples at each of nine sampling levels, retaining between 90% and 10% of the original population. An exclusion radius, based on a spatial autocorrelogram, was also calculated for each of these 10,725 mosaic samples. The mean NND was moderately insensitive to undersampling, rising smoothly. The exclusion radius was remarkably insensitive. The conformity ratio and dispersion index fell steeply, sometimes failing to reach significance while half of the cells still remained. For the same 33 original mosaics, linear regression showed the exclusion radius to be 62 ± 3% of the mean NND.