Neuronal convergence in early contrast vision: Binocular summation is followed by response nonlinearity and area summation

Abstract

We assessed summation of contrast across eyes and area at detection threshold (C(t)). Stimuli were sine-wave gratings (2.5 c/deg) spatially modulated by cosine- and anticosine-phase raised plaids (0.5 c/deg components oriented at +/-45 degrees ). When presented dichoptically the signal regions were interdigitated across eyes but produced a smooth continuous grating following their linear binocular sum. The average summation ratio (C(t1)/([C(t1+2)]) for this stimulus pair was 1.64 (4.3 dB). This was only slightly less than the binocular summation found for the same patch type presented to both eyes, and the area summation found for the two different patch types presented to the same eye. We considered 192 model architectures containing each of the following four elements in all possible orders: (i) linear summation or a MAX operator across eyes, (ii) linear summation or a MAX operator across area, (iii) linear or accelerating contrast transduction, and (iv) additive Gaussian, stochastic noise. Formal equivalences reduced this to 62 different models. The most successful four-element model was: linear summation across eyes followed by nonlinear contrast transduction, linear summation across area, and late noise. Model performance was enhanced when additional nonlinearities were placed before binocular summation and after area summation. The implications for models of probability summation and uncertainty are discussed.

Contrast integration over area is extensive: A three-stage model of spatial summation

“…Previous detailed analyses (Meese, 2010;Meese & Summers, 2007, 2009) have consistently shown that spatial probability summation across the elements of our standard size filter (spatial frequency bandwidth of 1.6 octaves, orientation bandwidth of T25-, 4-lobe sinephase receptive field) cannot account for our results. This was confirmed again here as follows.…”

“…The Swiss cheese stimulus was designed to combat this problem (Meese & Summers, 2007), at least in part. For example, if the check size is small compared to the rate at which sensitivity declines with eccentricity, then each type of cheese ("black" and "white") are spread fairly equally over the inhomogeneity, with the net result that sensitivity is fairly equal for the two (Meese & Baker, 2011;Meese & Summers, 2007, 2009. However, if the check size is large compared to the rate of decline, then the size of the integration aperture might be underestimated because the loss of sensitivity in the peripheral hole regions would mean that there is little benefit from filling them in.…”

“…However, recent psychophysical evidence (Meese, 2010;Meese & Baker, 2011;Meese & Summers, 2007, 2009) is inconsistent with this. A more successful model is one in which contrast integration (linear spatial summation of contrast) follows a nonlinear (square-law) contrast transducer (Foley, Varadharajan, Koh, & Farias, 2007;Meese & Summers, 2007, 2009) and additive noise (Meese, 2010).…”

“…However, recent psychophysical evidence (Meese, 2010;Meese & Baker, 2011;Meese & Summers, 2007, 2009) is inconsistent with this. A more successful model is one in which contrast integration (linear spatial summation of contrast) follows a nonlinear (square-law) contrast transducer (Foley, Varadharajan, Koh, & Farias, 2007;Meese & Summers, 2007, 2009) and additive noise (Meese, 2010). The cascade of the two quadratic effects produced by square-law transduction (Legge, 1984) and summation of signal and noise (Campbell & Green, 1965;Tyler & Chen, 2000) means that sensitivity improves with the fourth root of area, similar to the conventional probability summation model and several empirical reports (see Meese & Baker, 2011 for a review).…”

Binocular functional architecture for detection of contrast-modulated gratings

“…This lack of consensus probably springs, in part, from the smallness of these effects -just a few dB -coupled with the considerable difficulty of getting reliable, unbiassed threshold estimates to the required accuracy. Over the last 30 years, computer-controlled, forced-choice psychophysical methods with better display technology, and much greater numbers of trials, have improved the reliability of threshold estimation, and with these methods estimates of the binocular advantage tend to cluster around 4-5dB (Legge, 1984;Maehara & Goryo, 2005;Meese & Baker, 2011;Meese et al, 2006;Meese & Summers, 2009;Simmons & Kingdom, 1998;Simmons, 2005) -as if forming a compromise between the theoretically salient values of 3 and 6 dB (which represent an ideal observer with independent noise in the two eyes versus linear summation with fixed noise). In a retrospective analysis, however, we found that the variation between studies may not be random.…”