Paradoxical Psychometric Functions (“Swan Functions”) are Explained by Dilution Masking in Four Stimulus Dimensions

Abstract

The visual system dissects the retinal image into millions of local analyses along numerous visual dimensions. However, our perceptions of the world are not fragmentary, so further processes must be involved in stitching it all back together. Simply summing up the responses would not work because this would convey an increase in image contrast with an increase in the number of mechanisms stimulated. Here, we consider a generic model of signal combination and counter-suppression designed to address this problem. The model is derived and tested for simple stimulus pairings (e.g. A + B), but is readily extended over multiple analysers. The model can account for nonlinear contrast transduction, dilution masking, and signal combination at threshold and above. It also predicts nonmonotonic psychometric functions where sensitivity to signal A in the presence of pedestal B first declines with increasing signal strength (paradoxically dropping below 50% correct in two-interval forced choice), but then rises back up again, producing a contour that follows the wings and neck of a swan. We looked for and found these “swan” functions in four different stimulus dimensions (ocularity, space, orientation, and time), providing some support for our proposal.

A Common Rule for Integration and Suppression of Luminance Contrast across Eyes, Space, Time, and Pattern

“…To formalize the relationships in our data, we used a simple functional model (derived from Meese & Summers, 2007; see also Baker et al, 2013) where the overall contrast response (“ resp ”) was given by where A and B are the Michelson contrasts of the A and B components, respectively, the exponents p and q were set to standard values of 2.4 and 2, respectively (from Legge & Foley, 1980), and the constant z was set to 2 for the purpose of illustration (e.g. Figure 4f).…”

“…Although there is variability amongst observers, the model does a fairly good job in predicting the high levels of average summation (black curves) that were found across the entire contrast range (we will consider differences across stimulus conditions and observers in Section 4). Because of the general success of this model (Equation (1)) across the four stimulus domains (Figure 5) here and elsewhere (Baker et al, 2013), we refer to this as the generic contrast integration model .…”

“…This arises from excitatory integration of inappropriate pedestal contrast and enhances the predicted level of summation in the model (for the A on AB versus AB on AB comparison). Dilution masking is distinct from more well-known forms of masking such as within-channel masking (Legge & Foley, 1980) and cross-channel masking (Foley, 1994; see Baker et al, 2013, for further comment). However, if LP and ASB were able to avoid this inappropriate integration by isolating the excitatory A response in the time and orientation domains (for some trials or pedestal contrasts at least), then this would diminish the level of summation, consistent with their results (see Figure 5).…”

“…At contrast detection threshold (both with and without an AB pedestal), observers were unable to identify whether the target was A or AB, suggesting that they were unable to achieve the benefit available by restricting signal integration to the target region (see Figure A1b in Appendix A). Blanket integration over the stimulus region results in a theoretical phenomenon that we have referred to elsewhere as dilution masking (Meese & Baker, 2011a; Meese & Summers, 2007; Baker et al, 2013; see also Appendix A). This arises from excitatory integration of inappropriate pedestal contrast and enhances the predicted level of summation in the model (for the A on AB versus AB on AB comparison).…”

“…Experiments that have measured contrast increment sensitivity at and above detection threshold (so-called “dipper functions”) suggest that the problem outlined above is overcome using an elaborate network of contrast interactions (Meese & Baker, 2011a; Meese & Summers, 2007) involving summation and counter-suppression: the gain control giveth with one hand and taketh away with the other (Baker, Meese, & Georgeson, 2013). Put simply, the contrast code is normalized (Albrecht & Geisler, 1993; Heeger, 1992) by the population of filter elements from which the contrast signal has been integrated.…”