2.6.1 Stimuli used in area summation experiments
Traditionally, stimuli used to investigate area summation in vision were luminance increments. These were either presented as bright discs or Gaussian blobs of various diameters (Graham et al., 1939; Barlow, 1958; Bijl, Koenderink, & Koenderink, 1993). However, following the recognition that stimuli are processed by the visual system in separate channels correspond- ing to different spatial scales (Section 2.3.2), experiments were then conducted using spatially bandpass stimuli such as windowed sine-wave gratings and Gabor patches (see Section 3.4). Gratings are superior to luminance increments due to the fact that varying the size of a lu- minance stimulus will shift its energy to lower spatial frequencies. This makes the area sum- mation properties of data from those stimuli more difficult to interpret, as changes in the size of the stimulus are confounded with changes in its spectral properties. On the other hand, increasing the area of a contrast-defined stimulus such as a grating will cause it to activate a greater number of spatially distributed detectors tuned to the same spatial frequency. Further support for the use of gratings comes from experiments conducted using periodic stimuli that show a dependence of the summation effect on the number of cycles of the stimulus shown, rather than a reliance on absolute aperture size (Hoekstra, van der Groot, van den Brink, & Bilsen, 1974; Howell & Hess, 1978).
2.6.2 Previous results from luminance-defined stimuli
For luminance-increment stimuli, psychophysical data conform to Ricco’s law when the stim- ulus’s size is varied within an area which is relatively small. In the fovea, linear summation is found up to three minutes of arc (Graham et al., 1939), in the periphery this behaviour extends
over larger areas (0.4 - 0.5 deg2in Graham et al., 1939; Barlow, 1958). These data are con-
sistent with linear summation within the smallest available receptive field of the observer at that eccentricity. Increasing the size of a luminance stimulus beyond this typically uncovers a range of sizes over which summation has a slope of −1
2 (following Piper’s law; Barlow, 1958;
Bijl et al., 1993). Ideal summation of this kind would be predicted if the observer had access to equally sensitive receptive fields at every spatial frequency and was certain about the stimulus being shown, as the entire stimulus could then be detected by a single receptive field matched to its extent. For larger areas summation typically continues at progressively lower rates until thresholds become entirely independent of stimulus size (Bijl et al., 1993).
2.6.3 Previous results from contrast-defined stimuli
Linear summation has been found for grating stimuli when their size is varied within the range expected to be detected by the receptive field of a single simple cell (Legge, 1978; Kersten, 1984; Polat & Tyler, 1999; Foley et al., 2007). This level of summation is also seen between a “full” grating stimulus and a contrast modulated version (“Swiss cheese” or “Battenberg”) which has been multiplied by a checkerboard plaid to introduce small “checks” of full and zero contrast (see Section 3.4.3 and 3.4.6; Meese & Summers, 2007; Meese, 2010; Baker & Meese, 2011). These linear short-range summation effects are predicted by any model which includes a linear filtering stage (for example convolution by Gabor approximations to simple cell recep- tive fields). As the stimulus is increased in size the summation slope transitions from −1 (linear summation) to a shallower gradient. This usually involves a region with a slope of −1
2, though
this may be an artefact of a transition between a slope of −1 and a slope of −1
4, rather than
representing the operation of summation processes that produce a slope with this gradient. In the fovea, investigations of summation confound effects of stimulus size with those of the inhomogeneity in contrast sensitivity across the visual field. This will lead to studies tending to underestimate the level of summation between the outputs of the detectors. Nevertheless, Kersten (1984) found summation slopes of −1
2 over four grating cycles in the fovea. Other
studies have explained foveal data by incorporating a model of the sensitivity inhomogene- ity into their summation models, these studies typically find summation consistent with that which would produce slopes of −1
4, if the inhomogeneity were not present (Robson & Gra-
ham, 1981; Watson & Ahumada, 2005; Meese & Summers, 2007, 2012). Originally these data were presented as evidence for probability summation (Robson & Graham, 1981), however the contemporary “noisy energy” model (which combines square law transduction with a tem- plate matched to the stimulus extent) also accounts for these data (Meese & Summers, 2012). The effects of the inhomogeneity in contrast sensitivity across the visual field can be avoided by presenting stimuli in the periphery, as there are areas where its effect is very small (Robson & Graham, 1981). Results from summation experiments conducted in the periphery are in- consistent however. Mayer and Tyler (1986) found linear summation (slope of −1) over more
than eight stimulus cycles in the periphery (much larger than the estimated size of simple cell receptive fields), however the results of this study have not been replicated elsewhere in the literature. Some investigations have found spatially extensive summation with a slope of −1 2
(Manahilov et al., 2001; Meese & Hess, 2007), whereas other studies have found only fourth- root summation when testing over a wider area (Robson & Graham, 1981). It is not clear why these differences in summation slope have been found. The steeper slopes for smaller stimuli in the periphery could be explained in the combination model if one of the two components shallowing the slope became inactive under those conditions, either by linearisation of the transducer or by a breakdown in the template stage (resulting in mandatory pooling over an inefficiently large area).
2.6.4 Conclusions from previous contrast detection threshold studies
From previous data, the two models that provide good accounts of the data are the probability summation model and the noisy energy model that combines square-law transduction with a template stage (with both models including initial linear filtering stages). Additional evidence from the empirical slopes of the psychometric functions compared against those predicted by the models favours the noisy energy combination model (Meese & Summers, 2009, 2012). Models of extended summation in the fovea have however always relied on their accounts of the inhomogeneity in contrast sensitivity as a major determinant of the predictions that they make. In addition, the inconsistent results from summation experiments conducted in the periphery make assessments of behaviour here difficult to conduct conclusively without further studies (as data could be selected from the literature to fit any viable model). These outstanding issues are addressed in Chapter 6 and Chapter 7 of this thesis.
2.6.5 Area summation in contrast discrimination studies
Early studies investigating the summation of suprathreshold contrast using the contrast dis- crimination method found a lack of area summation above threshold when a target of increas- ing size was placed on a pedestal matched to its extent (Legge & Foley, 1980). These find- ings are compatible with the probability summation hypothesis, provided noise becomes cor- related between detectors above threshold. By manipulating target and pedestal extent sepa- rately however, studies have found greater levels of summation (Bonneh & Sagi, 1999; Meese, Hess, & Williams, 2005). By measuring contrast discrimination thresholds for Swiss cheese gratings (see Section 3.4.3) added to a full grating pedestal, Meese and Summers (2007) found substantial summation inconsistent with probability summation, but consistent with the phys- iological combination of transduced signals (combined with a suppressive contrast gain con- trol; see Heeger, 1992; Foley, 1994).