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Fig. 13.9. Edge cooccurrence in nature and long-range lateral connections in PGLISSOM. The distributions of excitatory lateral connections in the model are compared with the edge-cooccurrence statistics in nature to see how well they match perceptual requirements. (a) The Bayesian edge-cooccurrence statistics in natural images (Geisler et al. 2001; reprinted with permission, copyright 2001 by Elsevier). Each location in polar coordinates (&phi, &delta) contains a small round disk, representing the likelihood ratios of all possible orientations &theta at direction &phi and distance &delta by color coding; the &theta with the highest ratio is shown in the foreground (&theta, &phi, and &delta are defined as in Figure 13.8). Each likelihood ratio represents the conditional probability that a pair of edge elements in configuration (&theta, &phi, &delta) belongs to the same physical contour vs. different physical contours in natural images. The conditional probabilities were determined through manual labeling of contours in real world images. The most likely elements are aligned along cocircular paths emanating from the center. (b) The distributions of &theta, &phi, and &delta for the lateral excitatory connections in GMAP (Choe and Miikkulainen 2004; reprinted with permission, copyright 2004 by Springer). Each location (&phi, &delta) displays two values: (1) The color scale in the background shows the relative log-probability of finding a target receptive field at that location, and (2) the black oriented bars represent the most probable orientation &theta of the target receptive field at that location (not plotted for the weakest connections). The figure shows that neurons with receptive fields aligned on a common smooth contour are most likely to be connected with lateral excitatory connections. This distribution corresponds closely to the edge cooccurrence patterns in nature, suggesting that the model is well suited for encoding grouping relations in natural images.