A simple two-dimensional simulation of sampling the pattern of ocular-dominance strips with averaging-box voxels. (**a**) The pattern of ocular-dominance strips in the monkey in a drawn reconstruction by S. LeVay (http://hubel.med.harvard.edu/b27.htm; ). To simulate the spatial scale of the human ocular-dominance pattern, we assumed a strip width of 0.75 mm and superimposed square voxels (right) of up to 6-mm width. (**b**) For voxels of any given width placed in random locations on the pattern, we determined the probability density function of the voxel contrast, i.e. (L-R)/voxel area, where L and R are the areas of left and right ocular dominance within the voxel, respectively. When voxels are very small (width: 0.2 mm), almost all of them are located either entirely in the left or entirely in the right ocular-dominance domain; few voxels straddle the boundary between the domains. When voxels are large (width: 3-6 mm), each voxel samples both domains and most of their contrast, thus, cancels out. (**c**) The expected decoding information predicted by the simulation (ideal-observer t value, vertical axis) plotted as a function of voxel width for single voxels (dotted gray) and for a voxel population covering the entire pattern (solid black). As shown in (b) larger voxels suffer from contrast cancellation. However, larger voxels also benefit from greater noise reduction through averaging the signal within their boundaries. Consider the single voxel case (dotted gray line). As voxel size increases for very small voxels, noise reduction through averaging improves decoding (dotted gray line rises). Single-voxel decoding performance peaks when voxel width approximately matches domain width (i.e. 0.75 mm). As voxel size increases further, signal cancellation becomes an issue and single-voxel decoding suffers. However, for large voxels (width: 3-6 mm), the two effects end up balancing each other, for the following reason: When voxel width doubles, voxel area squares, and noise (i.e. standard error of the contrast estimate) is divided by two. What happens to the signal (i.e. the contrast estimate)? For intuition, we can imagine the strips as a vertical grating. Adjacent ocular dominance strips in the voxel cancel and any signal comes from at most a single strip at the edge of the voxel that ends up without a partner. When voxel width is doubled, the area of the non-cancelling strip relative to the area of the voxel (i.e. the contrast) is divided by 2. So as voxel width doubles, the contrast and its standard error are both divided by 2, thus the voxel's t value remains unaffected. This holds only for strip patterns, such as ocular dominance columns. For a checkerboard pattern, for example, the t value would roughly be divided by 2 each time voxel width doubles. However, there would still be some contrast, even for large voxels covering many squares of the checkerboard (unless the voxel's width is an exact multiple of the checkerboard period).

For the entire population of voxels (solid black line), the picture is different: decoding performance drops monotonically as the voxels get larger. Note, however, that the whole-volume performance using all voxels in the pattern area (black axis on the left) is much higher than the single-voxel performance (gray axis on the right). The ideal observer will optimally weight each voxel, thus benefiting from high contrast in single small voxels, while the overall factor by which the noise is reduced through combining the evidence across the pattern area is the same for all voxel sizes (square root of the total pattern area). In the plot, the decoding information is represented by the ideal-observer t value, which is the contrast t value obtained by optimally weighting the voxel input. Assuming Gaussian independent and identically distributed noise at each location of the neuronal pattern, the ideal-observer t value is monotonically related to d', decoding accuracy, and stimulus-response mutual information in bits. The units are arbitrary as they scale with the noise level in the neuronal input.

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