Theory and data for area summation of contrast with and without uncertainty:evidence for a noisy energy model

Contrast sensitivity improves with the area of a sine-wave grating, but why? Here we assess this phenomenon against contemporary models involving spatial summation, probability summation, uncertainty, and stochastic noise. Using a two-interval forced-choice procedure we measured contrast sensitivity for circular patches of sine-wave gratings with various diameters that were blocked or interleaved across trials to produce low and high extrinsic uncertainty, respectively. Summation curves were steep initially, becoming shallower thereafter. For the smaller stimuli, sensitivity was slightly worse for the interleaved design than for the blocked design. Neither area nor blocking affected the slope of the psychometric function. We derived model predictions for noisy mechanisms and extrinsic uncertainty that was either low or high. The contrast transducer was either linear (c1.0) or nonlinear (c2.0), and pooling was either linear or a MAX operation. There was either no intrinsic uncertainty, or it was fixed or proportional to stimulus size. Of these 10 canonical models, only the nonlinear transducer with linear pooling (the noisy energy model) described the main forms of the data for both experimental designs. We also show how a cross-correlator can be modified to fit our results and provide a contemporary presentation of the relation between summation and the slope of the psychometric function.

Fourth-root summation of contrast over area:no end in sight when spatially inhomogeneous sensitivity is compensated by a witch's hat

Measurements of area summation for luminance-modulated stimuli are typically confounded by variations in sensitivity across the retina. Recently we conducted a detailed analysis of sensitivity across the visual field (Baldwin et al, 2012) and found it to be well-described by a bilinear “witch’s hat” function: sensitivity declines rapidly over the first 8 cycles or so, more gently thereafter. Here we multiplied luminance-modulated stimuli (4 c/deg gratings and “Swiss cheeses”) by the inverse of the witch’s hat function to compensate for the inhomogeneity. This revealed summation functions that were straight lines (on double log axes) with a slope of -1/4 extending to ≥33 cycles, demonstrating fourth-root summation of contrast over a wider area than has previously been reported for the central retina. Fourth-root summation is typically attributed to probability summation, but recent studies have rejected that interpretation in favour of a noisy energy model that performs local square-law transduction of the signal, adds noise at each location of the target and then sums over signal area. Modelling shows our results to be consistent with a wide field application of such a contrast integrator. We reject a probability summation model, a quadratic model and a matched template model of our results under the assumptions of signal detection theory. We also reject the high threshold theory of contrast detection under the assumption of probability summation over area.