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Article  |   August 2011
Forward–backward masking of contrast patterns: The role of transients
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Journal of Vision August 2011, Vol.11, 15. doi:10.1167/11.9.15
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      John M. Foley; Forward–backward masking of contrast patterns: The role of transients. Journal of Vision 2011;11(9):15. doi: 10.1167/11.9.15.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Graham and Wolfson have shown that, after adaptation to an array of Gabor patterns, observers discriminate poorly between spatially identical Gabor arrays in which the contrasts are symmetrically above and below the adapt contrast. Ten experiments are reported that show that the effect occurs in simple contrast discrimination tasks and examine its properties using a two-alternative spatial forced-choice paradigm. The experiments show that the effect is quite robust and manifests itself in several different experimental paradigms. It occurs with masks as short as 50 ms, but it disappears when an interval as short as 30 ms intervenes between the test patterns and the masks. It occurs at 4 and 8 c/deg. The decrease in discrimination for straddle contrasts is accompanied by an increase in discrimination for contrasts above the mask contrast. A model is presented that describes and predicts the results. The key idea of the model is that patterns which suddenly change contrast produce two responses in the visual system, a V-response that is a function of the absolute value of the difference between the test contrast and the mask contrast and an S-response that is an increasing function of contrast.

Introduction
It has long been known that, if any visual stimulus is imaged on our retinas, that stimulus and stimuli similar to it become less visible. The general name of these phenomena is sensory or perceptual adaptation. This loss of sensitivity manifests itself both in an increase in absolute thresholds, the minimum stimulus magnitudes that we can detect, and a decrease in brightness, contrast, and color. These effects can be very large. For example, after looking at a bright surface for a minute, our threshold for seeing light increases by a factor of more than 100,000 (Hood & Finkelstein, 1986). Since it is generally to our advantage to be able to see things, this would seem to be more maladaptive than adaptive, although this loss of sensitivity to persisting stimuli may be a side effect of a system that has evolved to be acutely sensitive to change. 
If, instead of absolute threshold or perceptual magnitude, we examine our ability to see differences, we find that there is something directly advantageous about adaptation. Consider sensitivity to light first. When we have been in the dark for half an hour, we are exquisitely good at seeing very dim lights and can detect a few quanta. However, we are blind to the difference between lights of high intensity, even when they are very different. When we adapt to high intensity lights, dim lights become invisible, but we can now see differences among the high intensity lights (Hood & Finkelstein, 1986). Thus, the adaptive effect of light adaptation is that we see differences well around the adaptation level. A similar phenomenon occurs when we adapt to a contrast pattern. We become less sensitive to contrast, but when the adaptation level is sufficiently high, we become more sensitive to the differences in contrast over a high contrast range (Foley & Chen, 1997). Improved sensitivity to differences near the adaptation level has come to be recognized as the principal advantage produced by adaptation. 
Graham and Wolfson have recently discovered a phenomenon that appears to be completely inconsistent with this (Graham & Wolfson, 2007; Wolfson & Graham, 2007, 2009). In their paradigm, the observer adapts for 1 s to a grid of identical Gabor patterns all at the same contrast, for example, 50%. This is immediately followed by a brief test stimulus (for 94 ms). The test stimulus is composed of a spatially identical array of Gabor patches with two different contrasts in alternating rows (or columns), producing contrast-defined stripes that are either horizontal or vertical. The test array is immediately followed by a re-presentation of the adapt array again for 1 s. The observer's task is to identify the orientation of the contrast-defined stripes in the test stimulus. Note that in their task an observer can identify the orientation of the modulation without having to indicate which stripes are higher in contrast. 
In most of Wolfson and Graham's experiments, the independent variables were the adapt contrast and the two test contrasts, with the constraint that the difference between the test contrasts was held constant, typically at 10 or 20%. The dependent variable was the proportion of correct orientation identification responses. The proportion correct was often high, but surprisingly, when one of the two contrasts was above the adapt contrast and one below, performance got worse. When the contrasts were symmetrically above and below the adapt contrast, performance was much worse, often close to chance. Far from seeing well around the adapt contrast, their observers were almost blind to contrasts that symmetrically straddled the adapt contrast. They also showed that the phenomenon occurs when there are only four patterns in the array. I will call this phenomenon the Graham–Wolfson phenomenon. Although Wolfson and Graham focus on the decrement in discrimination in straddle conditions, they also showed that, relative to performance with no adapter, discrimination performance improves for contrast pairs in which both contrasts are just above or just below the mask contrast (Wolfson & Graham, 2009). 
Graham and Wolfson refer to their paradigm as an adaptation paradigm and to the phenomenon as adaptation. They pointed out that this phenomenon is not accounted for by existing models of contrast adaptation. They described it as “a new kind of adaptation,” which they associated with second-order processing. With context patterns that are only 1 s in duration and come immediately before and after the test stimuli, the paradigm would seem to be closer to what is usually called forward and backward masking. Consequently, I will refer to the context patterns as masks and the paradigm as forward–backward masking. Although the phenomenon has temporal properties like those of masking, we will see that it is associated with enhanced discrimination for many contrast pairs. 
The first question that I asked was does the phenomenon occur in a standard contrast discrimination task, that is, a task in which just two patterns are presented and the observer's task is to indicate which contrast is higher. Finding that it does occur in such tasks, I did a series of ten experiments to determine the conditions under which it occurs and to measure performance. The experiments will be referred to as Experiments 1 to 10. I then developed a model that describes and predicts performance in these tasks. 
Methods
Apparatus
A display system consisting of a computer, a Cambridge Research Systems graphics board, model VSG 2/5 with 15 bits of intensity resolution, and a Clinton monochrome display model DS2190P with a P45 phosphor, a resolution of 1184 × 848 pixels, and a frame rate of 100 Hz was used. 
Stimuli
Many features of the stimuli and the procedure were the same in most of the experiments. I will describe the common features here and the exceptions under the descriptions of the individual experiments. 
The background luminance was 50 cd/m2. The fixation mark was a plus sign with a luminance just enough higher than the background so that it could be clearly seen. The test stimuli were Gabor patterns in sine phase with reference to the fixation point so that the space average luminance of the patterns was equal to the background luminance. In all except one experiment, they had a center frequency of 4 c/deg and a standard deviation of 0.25 deg (1 cycle). Two test patterns with different contrasts were presented on each trial, with the exception that one of the patterns sometimes had a contrast of 0. The patterns were presented simultaneously, centered 0.8 deg above and below the fixation mark. 
On every trial, there were both forward and backward masks. The masks were identical to the test patterns, except for contrast, and they were presented in the same locations as the test patterns. In most of the experiments, the masks were presented for 1 s. They were immediately followed by the test patterns, which were presented simultaneously for 100 ms with a rectangular temporal waveform, and were immediately followed by the masks for 1 s. This temporal regime is the same as that employed by Graham and Wolfson, except that their test duration was 94 ms. In some experiments, mask duration, time between masks and tests (ISI), spatial frequency, and mask orientation were varied, as will be described. 
Procedure
The observer sat 115 cm from the display and a chin rest was used to keep the observer's head in this position. Viewing was binocular with natural pupils. Eye level was approximately at the center of the screen where the fixation mark was located. A two-alternative spatial forced-choice paradigm was used. On each trial, the higher contrast test pattern was presented randomly above or below the fixation mark and the lower test contrast in the other position. The observer responded by moving a lever up or down indicating whether the higher contrast was above or below the fixation mark. A tone marked the interval when the test patterns were presented. In most of the experiments, a second tone indicated whether the response was correct or incorrect, but experiments in which there was no error feedback were also performed. 
Four paradigms were used to measure the effects of the masks: (1) a threshold paradigm in which the lower of two contrasts (pedestal) was fixed and the higher contrast varied to find the contrast increment at which the responses were correct on 81% of the trials (Experiments 1 and 10); (2) a fixed contrast difference paradigm in which the difference between the two contrasts was fixed; the average contrast was varied over a wide range from below the mask contrast to above the mask contrast, and the proportion correct responses determined at each average contrast (Experiments 2, 3, 6, and 7); (3) a psychometric function paradigm in which the lower contrast was fixed and the higher contrast was varied over a wide range (Experiments 4, 5, and 8); (4) a symmetrical straddle paradigm in which the average test contrast equaled the mask contrast and the difference between the two test contrasts varied (Experiment 9). In the last three paradigms, the dependent variable was proportion correct. In all the experiments, trials were blocked by the mask contrast and the lower of the two test contrasts (pedestal contrast). The reason for blocking by pedestal contrast is that in several of the experiments different response rules were used for different pedestal contrasts. The number of trials per condition varied from 200 to 300 depending on the experiment. Observers had one or more practice sessions in each task prior to the experimental sessions. 
The observers were university students between the ages of 20 and 26. All had acuity of 20/20 with or without optical correction. They were not knowledgeable about the phenomenon or the hypotheses being tested. There were five observers: AJA, ALI, ATB, ATM, and PRR. ATB did nine of the ten experiments. Symbols used in this article are given in Table 1. Data from all the experiments are provided in a Supplementary Data Table
Table 1
 
Symbols used in this article.
Table 1
 
Symbols used in this article.
Symbol Quantity
Stimulus variables
C t Contrast of test pattern
C m Contrast of mask pattern
C p Contrast of pedestal (the lower of two test contrasts presented on a trial)
C ave The average of two test contrasts presented on a trial
C diff The difference between two test contrasts presented on a trial
ISI Temporal interval between masks and test pattern in seconds
Θ Orientation of the masks relative to the tests in degrees (Experiment 10)
 
Dependent variables
C tt Contrast of test pattern at threshold
P Proportion correct
 
Model parameters
S Et Excitatory sensitivity to test pattern t
S It Inhibitory sensitivity to test pattern t
S Im Inhibitory sensitivity to masks
p Exponent of excitatory signal
q Exponent of inhibitory signal
Z Value of maintained inhibition in the absence of a pattern
σ Standard deviation, Gaussian random variable added to each response
K Parameter of exponential decrease in masking with relative orientation (Experiment 10)
C tt0 Contrast of test pattern at threshold when test and mask have the same orientation (Experiment 10)
C ttA Asymptote of test pattern threshold as relative orientation increases (Experiment 10)
Experiments
Experiment 1: TvC functions for contrast discrimination at four levels of mask contrast
The TvC function for contrast discrimination refers to the function relating the increment threshold to the lower of the two contrasts (pedestal). This experiment measured the TvC functions without masks and at three levels of mask contrast. Here, the pedestal contrast was fixed in each condition and the contrast increment was varied using the Quest algorithm (Watson & Pelli, 1983) to find the threshold for discriminating pedestal plus increment from pedestal alone. The threshold was defined as the contrast increment that produced 81% correct performance. The pedestal contrasts spanned the range from 0 to 0.4. There were usually 8 values, but sometimes 9 were required to determine the low end of the function well. The mask contrasts were 0, 0.04, 0.08, and 0.16. The order of conditions was blocked by mask contrast, and the mask contrasts were presented in counterbalanced order. For each mask contrast, conditions with different pedestal contrasts were presented in random order. Within each condition, there was one mask contrast and one pedestal contrast. Tones provided error feedback. After one or two practice sessions, 6 measurements were made for each condition. 
The mean thresholds for each of the three observers are shown in Figure 1. The arrows indicate the mask contrasts. The mean standard errors were roughly constant over conditions. Mean standard errors as a proportion of mean threshold were AJA, 0.08; ATB, 0.12; and ALI, 0.09. When there are no masks, as pedestal contrast increases, threshold decreases and then increases, going well above the absolute threshold, a form that is often referred to as a “dipper function.” When there are forward–backward masks, thresholds are higher at low pedestal contrasts and then decrease, going below the no-mask thresholds and then increasing at high pedestal contrasts. So here we see a substantial advantage produced by the forward–backward mask; it reduces contrast discrimination thresholds for contrasts above the mask contrast by a factor of up to 0.5. As mask contrast increases, TvC functions are displaced upward and change shape. Note that the minimum value in each function occurs at a pedestal contrast above the mask contrast by a factor of approximately 1.5. In their general form, these functions look like those found by Foley (1994) for the effect of a fixed orthogonal superimposed mask on contrast discrimination and by Foley and Chen (1997) for the effect of a contrast adapter on contrast discrimination. However, it will be shown that these data require a different model. The smooth curves correspond to the model that will be described in the Model section. Although, when masks are presented, the lowest thresholds occur for contrast pairs above the mask contrast, there is no hint of a decrement in performance around the mask contrast. However, an analysis of these data showed that when the test increment was at threshold, the average of the pedestal and the pedestal plus increment was always above the mask contrast, so the experiment did not measure performance in straddle conditions where Graham and Wolfson found the worst performance. 
Figure 1
 
Experiment 1. Test contrast threshold vs. pedestal contrast functions at four mask contrasts for three observers. Error feedback was provided. Each point is the mean of six measurements. Arrows indicate mask contrasts. The smooth curves correspond to the V-response model that will be described below.
Figure 1
 
Experiment 1. Test contrast threshold vs. pedestal contrast functions at four mask contrasts for three observers. Error feedback was provided. Each point is the mean of six measurements. Arrows indicate mask contrasts. The smooth curves correspond to the V-response model that will be described below.
Experiment 2: Proportion correct for a constant contrast difference as a function of average contrast
This experiment was designed to be closer to Wolfson and Graham's (2007, 2009) experiments in stimuli and design, although the task was contrast discrimination rather than orientation discrimination. In this experiment, pedestal contrast and mask contrast were again the independent variables, but the difference between the pedestal contrast and the higher contrast was fixed and proportion correct was measured. The mask contrasts were 0.04, 0.08, 0.16, and 0.32. I found that, in order to prevent performance from going to 50 or 100%, I needed to use a small contrast difference with low contrast masks and increase the difference with the mask contrast. There were pairs in which both contrasts were below the mask contrast, pairs with both above, and pairs symmetrically straddling the mask contrast, as in Graham and Wolfson's experiments. 
There was an important difference in experimental design. Instead of presenting the test pairs in random order as Wolfson and Graham did, I presented them in blocks of 50 trials in which the same pair of contrasts was presented, randomly assigned to the two positions. Three easier trials proceeded each block. Responses to these were not included in the analysis. Tones provided error feedback. For each test pair and mask contrast, there were 300 trials for ATB and 250 trials for AJA. 
In the first condition with the first observer, it happened that both test contrasts were below the mask contrast. On the first 25 trials, the observer got 23/25 wrong; on the second 25, she got 22/25 right. When asked to explain, she said, “When I indicated the contrast that appeared higher, I got it wrong, so I reversed the response rule, and indicated the contrast that appeared lower.” She proceeded to use the first three trials of each block to determine the response rule and then to apply that rule throughout the block. She used the reverse rule in every condition in which both test contrasts were less than the mask contrast and the normal rule in the other conditions. She was consistent in her selection of a rule, except in the straddle conditions. When the second observer showed the same pattern of results, I instructed him to indicate the lower of the two contrasts (reverse response rule) whenever the average contrast was below the mask contrast and to indicate the higher contrast when the average contrast was equal to or greater than the mask contrast. The observer was instructed as to which rule to apply in each condition. I will refer to this instruction as the reverse/normal response rule. Without such an instruction, observers sometimes will do many trials before settling on a rule, especially when the discrimination is difficult. 
The results are shown for the two observers in Figure 2. Here, proportion correct is plotted against the average of the two contrasts (PvCave), as in Graham and Wolfson's papers. In every case, there is a substantial decrement in performance when the average contrast equals the mask contrast (the test contrasts are symmetrically above and below the mask contrast). Performance increases rapidly on either side of this point and then decreases for contrasts further away from the mask contrast. The data look remarkably like the data that Graham and Wolfson obtained in their contrast modulation discrimination task. They refer to this function form as a “butterfly.” On a linear contrast axis, the functions are approximately symmetrical about the mask contrast. So the phenomenon occurs for contrast discrimination, a finding that Graham and Wolfson have recently reported using a temporal forced-choice paradigm and a same–different task (Graham, Wolfson, Kwok, & Grinshpun, 2010). The smooth curves in the graphs correspond to a model that will be described below. The mean standard deviations of the proportion correct across blocks of 50 trials were AJA, 0.060 and ATB, 0.072. 
Figure 2
 
Experiment 2. Fixed contrast difference; reverse/normal rule; feedback. Proportion correct as a function of the average of the two test contrasts at four mask contrasts. (Left) AJA. (Right) ATB. Note that the contrast difference was increased with mask contrast to keep performance at approximately the same level. In each case, the minimum performance occurs when the average test contrast equals the mask contrast so that the test contrasts symmetrically straddle the mask contrast. The smooth curves show the predictions of the V-response model that will be described below. AJA, N = 250; ATB, N = 300.
Figure 2
 
Experiment 2. Fixed contrast difference; reverse/normal rule; feedback. Proportion correct as a function of the average of the two test contrasts at four mask contrasts. (Left) AJA. (Right) ATB. Note that the contrast difference was increased with mask contrast to keep performance at approximately the same level. In each case, the minimum performance occurs when the average test contrast equals the mask contrast so that the test contrasts symmetrically straddle the mask contrast. The smooth curves show the predictions of the V-response model that will be described below. AJA, N = 250; ATB, N = 300.
Experiment 3: Proportion correct for a constant contrast difference as a function of average contrast using the normal response rule
Here, the question is: what happens when there is no feedback and the observers use the normal response rule throughout? This experiment was identical to Experiment 2, except there was only one mask contrast, 0.32, the contrast difference was 0.08, observers were instructed to use the normal response rule (indicate the higher of the two contrasts) throughout, and there was no error feedback. There were 5 blocks of 50 trials, 250 trials per condition. 
The PvCave functions are shown in Figure 3. When both test contrasts are below the mask contrast, performance is below chance. When the test contrasts symmetrically straddle the mask contrast, performance is slightly above chance, and when both test contrasts are above the mask contrast, performance increases to a maximum and then decreases. For ATB, performance here is almost as good as in the C m = 0.32 condition in Experiment 2 even though the contrast difference is smaller. This may be an effect of practice. 
Figure 3
 
Experiment 3. Fixed contrast difference; normal response rule; no feedback. Proportion correct as a function of average contrast. The arrow indicates the mask contrast. The smooth curves correspond to the V-response model with parameters determined from Experiments 3 and 5 for PRR and an overall fit to Experiments 2, 3, 4, and 5 for ATB. (Left) PRR. (Right) ATB. N = 250.
Figure 3
 
Experiment 3. Fixed contrast difference; normal response rule; no feedback. Proportion correct as a function of average contrast. The arrow indicates the mask contrast. The smooth curves correspond to the V-response model with parameters determined from Experiments 3 and 5 for PRR and an overall fit to Experiments 2, 3, 4, and 5 for ATB. (Left) PRR. (Right) ATB. N = 250.
Experiment 4: Psychometric functions for contrast detection
What is the form of psychometric functions in the forward–backward masking paradigm? In the absence of masks, psychometric functions for detection are almost always sigmoidal. In this experiment, the lower contrast was fixed at 0 and the higher contrast varied over a large range, so the average contrast was half of the higher contrast. There were two mask contrasts, 0.04 and 0.16 for AJA and 0.04 and 0.32 for ATB. Thus, the experiment measured psychometric functions for contrast detection when the test patterns are masked. The trials were presented in blocks of 50 with the higher test contrast fixed over each block. For each mask and test contrast pair, there were 5 blocks of 50 trials each. Observers were instructed to use either the normal rule or the reverse rule before the start of each block. They used the reverse rule when the average contrast was below the mask contrast and the normal rule when it was equal to or above the mask contrast (reverse/normal rule). Error feedback was provided. Proportion correct is plotted against the higher contrast in Figure 4. As contrast increases, proportion correct increases and then drops sharply at the symmetrical straddle contrast. Here, the symmetrical straddle contrast is two times the mask contrast since the lower contrast is always zero. (If proportion correct were plotted against average contrast as in Figure 2, the drop would be at the point where the average contrast equals the mask contrast, as in that figure.) For the higher mask contrasts, performance in the symmetrical straddle conditions is better here than in most other experiments, a result that will be addressed in Experiment 9. The form of these psychometric functions is very unusual. However, similar functions have recently been found in other kinds of experiments (Baker, Meese, & Georgeson, 2010; Serrano-Pedraza & Derrington, 2008; Serrano-Pedraza, Goddard, & Derrington, 2007). 
Figure 4
 
Experiment 4. Psychometric function for contrast detection; reverse/normal rule; feedback. Proportion correct as a function of contrast in a contrast detection task with mask contrasts of 0.04 and 0.16 (AJA) or 0.32 (ATB) indicated by the arrows. The lower contrast is 0. The contrasts that symmetrically straddle the mask contrast are 0 and 2C m. The smooth curves correspond to the V-response model with parameters determined from Experiment 1 for AJA and an overall fit to Experiments 2, 3, 4, and 5 for ATB. N = 250.
Figure 4
 
Experiment 4. Psychometric function for contrast detection; reverse/normal rule; feedback. Proportion correct as a function of contrast in a contrast detection task with mask contrasts of 0.04 and 0.16 (AJA) or 0.32 (ATB) indicated by the arrows. The lower contrast is 0. The contrasts that symmetrically straddle the mask contrast are 0 and 2C m. The smooth curves correspond to the V-response model with parameters determined from Experiment 1 for AJA and an overall fit to Experiments 2, 3, 4, and 5 for ATB. N = 250.
Experiment 5: Psychometric functions for contrast discrimination
What is the form of psychometric function for contrast discrimination when the test contrasts are masked? In this experiment, the lower contrast was fixed at 0.1 and the mask contrast at 0.2. The higher test contrast was varied and the proportion correct was measured. Trials were blocked by the higher test contrast. As in Experiment 4, prior to each block the observer was instructed as to which response rule to use, normal or reverse. Here, the normal rule was used for all the test contrast pairs, and in separate blocks of trials, the reverse rule was used for average contrasts less than the mask contrast. ATB received error feedback; PRR did not. There were 300 trials in each condition. 
Figure 5 shows the proportion correct as a function of the higher test contrast. In the conditions in which the reverse rule was used, the functions are like those of Figure 4, except that they rise from the pedestal contrast (0.1) and they decrease near 0.3, the symmetrical straddle contrast for a mask of 0.2 and a pedestal of 0.1. In conditions in which the normal rule was used over the whole range of contrasts, performance goes below chance when the average contrast is below the symmetrical straddle contrast. In this experiment, there were two contrast pairs that straddle the mask contrast nonsymmetrically, with the average test contrast below the mask contrast. Proportion correct is relatively high in these conditions but not as high as for the most discriminable pairs. Note that in the symmetrical straddle conditions where the contrast difference is 0.2, performance is well above chance for both observers, but there is a decrement relative to performance in other conditions. 
Figure 5
 
Experiment 5. Psychometric function for contrast discrimination; ATB: feedback, PRR: no feedback. Proportion correct as a function of the higher contrast. Pedestal contrast = 0.1. Mask contrast = 0.2. (Left) PRR. (Right) ATB. The contrasts that symmetrically straddle the mask contrast are 0.1 and 0.3. The blue symbols and lines correspond to the conditions in which the normal response rule was used, and the red symbols and lines correspond to the conditions in which the reverse rule was used. N = 300. For PRR, the parameter values are the best values from a joint fit to Experiments 3 and 5, and for ATB, the parameter values are the best overall fit to Experiments 2, 3, 4, and 5.
Figure 5
 
Experiment 5. Psychometric function for contrast discrimination; ATB: feedback, PRR: no feedback. Proportion correct as a function of the higher contrast. Pedestal contrast = 0.1. Mask contrast = 0.2. (Left) PRR. (Right) ATB. The contrasts that symmetrically straddle the mask contrast are 0.1 and 0.3. The blue symbols and lines correspond to the conditions in which the normal response rule was used, and the red symbols and lines correspond to the conditions in which the reverse rule was used. N = 300. For PRR, the parameter values are the best values from a joint fit to Experiments 3 and 5, and for ATB, the parameter values are the best overall fit to Experiments 2, 3, 4, and 5.
Experiment 6: Fixed contrast difference—Effect of interstimulus interval on the relation between proportion correct and average contrast
What is the effect of inserting temporal intervals between the masks and the test patterns? In the experiments presented thus far, there was no time between the offset of the first mask and the onset of the tests or between the offset of the tests and the onset of the second mask. With this temporal regime, the Graham–Wolfson phenomenon appears to be quite robust. Experiment 6 examined the effect of introducing a time interval between the tests and the forward and backward masks. During these intervals, no pattern was presented; the luminance of the screen was uniform at the background luminance of 50 cd/m2, except for the fixation mark. It is characteristic of adaptation phenomena that, once the adapter is turned off, its effect dissipates over time, and performance returns to what it would be without an adapter. The effects of contrast adaptation have been shown to decrease as a negative power function of time after adapter offset and to persist for at least as long as the adapter was on, often several minutes (Greenlee, Georgeson, Magnussen, & Harris, 1991). 
Wolfson and Graham (2007) propose that, when the contrast changes, the adaptation level shifts from the first contrast to the second. If adaptation level determines performance, we would expect that when contrast changes, the contrast at which performance drops toward chance will shift over time from the first contrast to the second contrast. Here, we present a mask for 1 s, replace it by the 0 contrast background for a variable ISI, and then present our test patterns. The same ISI is inserted between the test patterns and the re-presentation of the mask. Unless the shift in adaptation level is extremely fast, we should be able to track it by measuring the proportion correct as a function of average contrast (PvCave) function at different interstimulus intervals (ISIs). Since we do not know in advance what contrast will correspond to the adaptation level for any ISI, we cannot instruct observers when to use the reverse rule. Consequently, in this experiment, the normal response rule was used throughout with no feedback. In this case, we know that at ISI = 0 the PvCave function will look like the functions in Figure 3 and performance will be below chance when C ave is below the mask contrast, C m. If, after the contrast changes to 0, the adaptation level shifts toward 0, we would expect to see a similar function shifted to the left. To test this hypothesis, we need a large range of average contrasts below the adapt contrast. The experiment was like Experiment 3 except that there was an interstimulus interval of 30, 50, or 150 ms both before and after the test stimuli. There was a fourth condition with no mask. The normal response rule was used throughout and there was no feedback. 
Figure 6 shows the results. With an interstimulus interval as short as 30 ms, the function is quite different than at ISI = 0. At this ISI, proportion correct goes to chance at a value of C ave below the mask contrast and then increases at the lowest average contrasts. At ISI = 50 ms, performance is slightly better, and at ISI = 150 ms, it is substantially better, but the functions have the same general form. When there is no mask, performance decreases close to monotonically as average contrast increases. These data show that the change in the PvCave function as ISI increases does not resemble a leftward shift of the ISI = 0 function. For ISI > 0, the data points are joined by line segments; no model is proposed here for these data. 
Figure 6
 
Experiment 6. Fixed contrast difference; normal response rule; no feedback. Proportion correct as a function of average contrast for a contrast difference of 0.08 and mask contrast of 0.32. The different symbols correspond to different interstimulus intervals in milliseconds. The ISI = 0 data (blue) are replotted from Experiment 3 and the smooth curve through them corresponds to the V-response model. (Left) PRR. (Right) ATB. N = 250.
Figure 6
 
Experiment 6. Fixed contrast difference; normal response rule; no feedback. Proportion correct as a function of average contrast for a contrast difference of 0.08 and mask contrast of 0.32. The different symbols correspond to different interstimulus intervals in milliseconds. The ISI = 0 data (blue) are replotted from Experiment 3 and the smooth curve through them corresponds to the V-response model. (Left) PRR. (Right) ATB. N = 250.
Experiment 7: Fixed contrast difference—Effect of brief masks on the relation between proportion correct and average contrast
Will the effect occur for very brief masks? Graham and Wolfson used mask duration of 1 s. Here, we reduced the mask duration to 50, 100, and 120 ms and measured PvCave for mask contrasts of 0.04 and 0.32. Otherwise, the methods were the same as in Experiment 2, reverse/normal rule with feedback. The results are shown in Figure 7. The functions have the same form as in Experiment 2, showing that the Graham–Wolfson phenomenon occurs with these brief masks. Note that AJA performs worse with 100-ms masks than when the masks are on for 1 s and that, for ATB, when mask duration is 50 ms, the performance curves are shifted slightly to the left. As will be explained in the Model section, changes in the model accommodate these changes in performance. The smooth curves correspond to this modified model. 
Figure 7
 
Experiment 7. Short duration masks. Fixed contrast difference; reverse/normal rule; feedback. Proportion correct as a function of average contrast for mask contrasts of 0.4 and 0.32. Mask durations: AJA, 100 ms; ATB, 50 and 120 ms. N = 250. The smooth curves correspond to the V-response model. Slight modifications to the model were made to accommodate the effects of these very brief masks.
Figure 7
 
Experiment 7. Short duration masks. Fixed contrast difference; reverse/normal rule; feedback. Proportion correct as a function of average contrast for mask contrasts of 0.4 and 0.32. Mask durations: AJA, 100 ms; ATB, 50 and 120 ms. N = 250. The smooth curves correspond to the V-response model. Slight modifications to the model were made to accommodate the effects of these very brief masks.
Experiment 8: Psychometric function for contrast discrimination—Effect of increasing spatial frequency to 8 c/deg
Does the phenomenon occur at higher spatial frequencies? Graham and Wolfson used 2 c/deg in their experiments. In Experiments 17, I used 4 c/deg. Here, I increased the spatial frequency to 8 c/deg, by doubling the viewing distance to 230 cm. All visual angles were thus halved. Otherwise, the experiment was identical to Experiment 5, except that for ATB two sets of measurements were made for average contrasts at and above the straddle contrasts (red and blue symbols). No error feedback was provided. The results are shown in Figure 8. They are very much like the results of Experiment 5, except that discrimination (deviation from chance performance) is not quite as good here. The smooth curves correspond to the model with only one parameter changed to account for lower performance. For PRR, no decrement in performance is apparent in the straddle condition. 
Figure 8
 
Experiment 8. Spatial frequency: 8 c/deg. Psychometric function for contrast discrimination; no feedback. Proportion correct as a function of the higher contrast. Mask contrast = 0.2. Pedestal contrast = 0.1. Blue: normal rule; red: reverse rule for contrasts below the straddle contrast, normal rule, above. Arrows indicate the mask contrast. The symmetrical straddle contrasts are the pedestal contrast, 0.1 and 0.3. N = 200. Note that for PRR there is no decrement in performance in the straddle condition.
Figure 8
 
Experiment 8. Spatial frequency: 8 c/deg. Psychometric function for contrast discrimination; no feedback. Proportion correct as a function of the higher contrast. Mask contrast = 0.2. Pedestal contrast = 0.1. Blue: normal rule; red: reverse rule for contrasts below the straddle contrast, normal rule, above. Arrows indicate the mask contrast. The symmetrical straddle contrasts are the pedestal contrast, 0.1 and 0.3. N = 200. Note that for PRR there is no decrement in performance in the straddle condition.
Experiment 9: Symmetrical straddle paradigm—Effect of the contrast difference
In Graham and Wolfson's description, the critical feature of the phenomenon is a decrement in discrimination performance in conditions in which the test contrasts straddle the mask contrast. We have found such a decrement in almost every data set, and in many of them, performance is near chance in the symmetrical straddle conditions. In this experiment, we presented only symmetrical straddle conditions. We varied the contrast difference over a large range at two mask contrasts to determine if performance depends on these variables. The two mask contrasts were 0.08 and 0.32. Mask duration was 0.5 s. Error feedback was provided. The proportion correct as a function of test contrast difference (PvCdiff) functions for three observers is shown in Figure 9. With the exception of one condition, proportion correct increases monotonically with contrast difference and goes above 0.95 for all three observers. Performance is better with the lower mask contrast. Clearly, our observers are not blind to contrast differences that symmetrically straddle the mask; they are just less sensitive to them. In the Model section, I will present a model that explains this effect. I cannot explain ATB's performance at the two highest contrast differences with C m = 0.08. Since she had done eight experiments before this one, it cannot be attributed to a lack of practice. An additional 200 trials at each of the aberrant data points did not improve performance. 
Figure 9
 
Experiment 9. Symmetrical straddle paradigm; normal rule; feedback. Proportion correct as a function of contrast difference for three observers. N = 300. The smooth curves correspond to the S-response model. Parameters were the same as for the V-response model, except for S m, which is higher for the S-response.
Figure 9
 
Experiment 9. Symmetrical straddle paradigm; normal rule; feedback. Proportion correct as a function of contrast difference for three observers. N = 300. The smooth curves correspond to the S-response model. Parameters were the same as for the V-response model, except for S m, which is higher for the S-response.
Experiment 10: Effect of the orientation of the masks relative to the tests on the contrast detection threshold
What happens to the detection threshold when the orientation of the masks is varied relative to the test patterns? On the basis of what is known about other masking phenomena, we would expect thresholds to decrease as the relative orientation of mask to test increases. Here, the mask contrast was 0.16. The mask duration was 1 s and the test duration was 100 ms. Here, test and mask were in cosine phase with the fixation point. However, for patterns of this size phase has no effect on detectability (Foley, Varadharajan, Koh, & Farias, 2007). Observers were instructed to use the normal response rule and error feedback was provided. The Quest algorithm was used to seek out the contrast that corresponds to 81% correct. The results are shown in Figure 10. This experiment measures the absolute contrast threshold in the context of a forward–backward mask. The points plotted here correspond to the leftmost points (C p = 0) on the TvC functions for masks of different orientations. They are also single points (81% correct) on the psychometric functions for detection in the context of masks of different orientations. They do not show whether or not the straddle effect occurs for masks that differ in orientation from the test or how the straddle effect depends on mask orientation. The threshold is greatest when mask and test have the same orientation. It falls sharply on either side of the maximum with a width at half-maximum of about ±7 deg for AJA and ±14 deg for ALI. These orientation bandwidths are much narrower than those typically found for superimposed masking (Foley, 1994). The threshold elevation functions have the same sharply peaked form as those typically found for contrast adaptation (Bradley, Switkes, & De Valois, 1988). Snowden (1992) showed that tuning depends on both spatial and temporal frequencies. He fitted Gaussian functions to his data. At 4 c/deg, the standard deviation of the Gaussians was about 20 deg, except at high temporal frequency. When test and masks have the same orientation, we would expect the threshold to be the same as the no pedestal threshold for this mask contrast in Experiment 1. For ALI, it is almost the same; for AJA, it is lower here. Since feedback was used here, it is possible that AJA picked up the reverse rule and the threshold corresponds to the middle contrast in Figure 4 at which performance is 81% correct. That contrast agrees with AJA's same orientation threshold in Experiment 10
Figure 10
 
Experiment 10. Threshold as a function of mask orientation with reference to test; normal rule; feedback; no pedestal. Mask contrast = 0.16. The smooth curve is the best fitting negative exponential function. (Left) AJA, N = 300. (Right) ALI, N = 200.
Figure 10
 
Experiment 10. Threshold as a function of mask orientation with reference to test; normal rule; feedback; no pedestal. Mask contrast = 0.16. The smooth curve is the best fitting negative exponential function. (Left) AJA, N = 300. (Right) ALI, N = 200.
Summary of experimental results
These experiments show that the Graham–Wolfson phenomenon occurs in contrast discrimination tasks and that it is robust. They also show several properties of the phenomenon: 
  1.  
    When the normal response rule is used, TvC functions decrease and then increase as pedestal contrast increases (Experiment 1). The masks increase thresholds at low pedestal contrasts and decrease them at high pedestal contrasts, so that the TvC functions cross the function for the no-mask case. Masks substantially improve contrast discrimination for contrasts above the mask contrast.
  2.  
    When the normal response rule is used throughout, the PvCave function is below chance at low values of C ave; it decreases to a minimum and then rises steeply when C ave is close to C m. When C ave is above C m, performance is above chance; it rises to a maximum and then decreases (Experiment 3).
  3.  
    When the reverse response rule is used for conditions in which C ave is less than C m, as C ave increases, proportion correct increases, then decreases to a minimum when C ave = C m, increases again, and then decreases at high values of C ave. The maximum is at a C ave about 1.3 to 2 times the mask contrast; this factor is larger at lower mask contrasts (Experiment 2).
  4.  
    Psychometric functions show the same dependence of proportion correct on the relation between C ave and C m as is found in the constant difference experiments, and there are two symmetrical forms of each function depending on the response rule (Experiments 4 and 5).
  5.  
    In conditions in which C ave = C m (symmetrical straddle conditions), the normal response rule produces above chance performance. Proportion correct increases as C diff increases and decreases as C m increases (Experiment 9).
  6.  
    As mask contrast increases, for a constant contrast difference, the PvCave function becomes closer to chance (P = 0.5). To maintain discrimination performance, C diff must increase with C m, as it did in these experiments (Experiment 2). Wolfson and Graham did not find a decrease in discrimination with an increase in mask contrast. They concluded that “changing the adapt contrast simply shifts the curves along the contrast axis” (Wolfson & Graham, 2009). In spite of this general decrease in discrimination as C m increases, at high values of C ave, discrimination improves with C m. This happens because the PvCave function also shifts upward along the C ave axis with C m. This is evident in Figure 1 and is the principal advantage produced by the masks.
  7.  
    The phenomenon is very sensitive to a temporal interval between masks and test and is not seen when an interval of 30 ms or more intervenes between the masks and the test patterns (Experiment 6).
  8.  
    The phenomenon occurs with masks as short as 50 ms, but with masks this brief, PvCave functions are shifted slightly toward lower contrasts (Experiment 7).
  9.  
    The phenomenon occurs at a spatial frequency of 8 c/deg (Experiment 8) as well as 4 c/deg.
  10.  
    Forward–backward masking of pattern detection is sharply tuned to the relative orientation of mask to target (Experiment 10).
Wolfson and Graham (2007) refer to the phenomenon as a “new kind of contrast adaptation.” “Masking” and “adaptation” can refer to experimental paradigms, to phenomena, and to underlying processes, but there is not a sharp distinction between them in any of these meanings. A stimulus presented before a test is usually called an adapter when it is on for several seconds or more and a forward mask when it is brief. The effects of an adapter (adaptation) increase with adapter duration for many minutes and persist for at least as long (Magnussen & Greenlee, 1985). The effects of a brief forward mask generally last less than a second (Foley & Boynton, 1993). The Graham–Wolfson phenomenon can be produced with 50-ms pre and post context patterns (Experiment 7). It appears not to persist for even 30 ms after context pattern offset (Experiment 6). Thus, in terms of its temporal properties, the phenomenon resembles masking more than adaptation. However, like other context patterns, a forward–backward “mask” produces substantial improvements in contrast discrimination in some conditions. 
Model
How can we understand this set of results? Graham and Wolfson (2007) proposed a model to account for their early results. The central element of the model is a new kind of complex channel (Buffy channel). The channel has three stages of spatial filtering with different point transforms between the first and second stages and the second and third stages. The response of the second point transform is a function of the difference between the output of the second filter and the average output of this filter over the preceding interval. The response function is piecewise linear. Although this model fits some of their data well (Wolfson & Graham, 2007, 2009), it remains to be seen how wide a scope this model will have. In their more recent papers, Wolfson and Graham have not fitted this model. Instead, they have emphasized three explanatory ideas. The first is that the response of the detecting mechanism is V-shaped with a minimum at the mask contrast. They point out that this would account for the straddle effect and the fact that discrimination performance is better for pairs on either side of the mask contrast. A V-shaped response function also accounts for our finding that, when C ave is below the mask contrast, performance goes below chance, unless the observer applies the reverse response rule. The second idea is that there is a “Weber-like process” that influences performance when the test contrasts are far from the mask contrast. They describe this process as a contrast gain control process of the normalization type. The third idea is that there are at least two channels with different asymmetric response functions to account for the above chance performance in the symmetrical straddle conditions. Although these ideas account for the principal phenomena qualitatively, they do not account for them quantitatively. 1  
Background: The S-response model accounts for many phenomena of contrast discrimination
I will start with what is known about contrast discrimination, masking, and adaptation and build on this to develop a model of the Graham–Wolfson phenomenon. When the threshold for discriminating the contrast of two gratings or Gabor patterns is measured as a function of the lower of the two contrasts (pedestal contrast), the result resembles that shown in Figure 11 (left). Nachmias and Sansbury (1974) showed that contrast discrimination functions have this form and proposed that contrast discrimination could be explained by a simple model: The response of the visual system to contrast increases as an accelerating function of contrast when contrast is low; the variance of the responses is constant, and a contrast increment that produces a constant increase in response will be at the contrast discrimination threshold. Nachmias and Sansbury did not specify the form of this function, but they showed that the lower part of the function can be approximated by a power function with a power between 2 and 3. That same year Stromeyer and Klein (1974) also reported that discrimination improves at low pedestal contrasts and proposed a slightly different model in which variance increases with mean response. The fundamental idea here (that we can use thresholds to infer the response of the system) was proposed by Fechner in 1859 (Fechner, 1966). He identified the response with the magnitude of the percept; in recent models, it is identified with the response magnitude in the mechanism that determines discrimination performance. 
Figure 11
 
(Left) Threshold vs. contrast (TvC) function for contrast discrimination. (Right) Corresponding response vs. contrast function. At any pedestal contrast, the contrast difference threshold is the increase in contrast that produces a constant increase in response. Functions are computed using the best parameters for ATB from the current experiments.
Figure 11
 
(Left) Threshold vs. contrast (TvC) function for contrast discrimination. (Right) Corresponding response vs. contrast function. At any pedestal contrast, the contrast difference threshold is the increase in contrast that produces a constant increase in response. Functions are computed using the best parameters for ATB from the current experiments.
Legge and Foley (1980) used this idea to create a model of the threshold vs. contrast function for a grating that is masked by a superimposed grating that is different from the test grating. More specifically, it was a model that described the family of S-shaped threshold vs. mask contrast functions for masks that differ in spatial frequency from the test pattern. When the patterns are narrow, 1.5 cycles, the TvC functions have approximately the same form on log–log coordinates; the effect of mask spatial frequency is to shift the functions along the log mask contrast axis. This family of TvC functions is accounted for by a family of response versus contrast (RvC) functions that are the same except for a multiplicative change in sensitivity to mask contrast. 
Foley (1994) showed that this model fails when the mask varies in orientation and when the TvC function for contrast discrimination is measured in the presence of a fixed mask with a different orientation than the test. He proposed a model that retains the idea that discrimination and superimposed masking depend on an S-shaped response function but allows the form of this function to change with the mask contrast. Foley showed that when the stimuli are narrow Gabor patterns, contrast discrimination can be accounted for by the responses of a single pattern vision mechanism. The mechanism has a linear excitatory receptive field, which responds to net excitation with an accelerating nonlinearity. The mechanism also receives a divisive inhibitory input that has a constant component as well as a component that increases as an accelerating function of the pattern contrast. The mechanism response is equal to the ratio of excitation to inhibition and it is perturbed by random Gaussian noise. What distinguishes this 1994 model from the earlier ones is that the response function changes shape when the context in which the pattern is viewed changes. Ross and Speed (1991) had earlier obtained similar results and proposed a similar explanation, although their model of the effects is quite different. Foley's model is a general model that has been shown to provide a basis for models of performance in a variety of tasks, including contrast discrimination, superimposed masking (Foley, 1994; Holmes & Meese, 2004; Meese, 2004; Meese & Hess, 2004; Meese & Holmes, 2002; Watson & Solomon, 1997), forward and backward masking (Foley & Chen, 1999), masking with temporally modulated patterns (Boynton & Foley, 1999), masking with concentric patterns (Chen & Foley, 2004), lateral masking (Chen & Tyler, 2001, 2002, 2008; Meese, Challinor, Summers, & Baker, 2009), interocular masking (Meese, Challinor, & Summers, 2008; Meese, Georgeson, & Baker, 2006; Maehara, Huang, & Hess, 2010), masking by chromatic stimuli (Chen, Foley, & Brainard, 2000a, 2000b), contrast adaptation (Foley & Chen, 1997), contrast matching (Meese & Hess, 2004; Snowden & Hammett, 1998; Xing & Heeger, 2001), and selective attention (Foley & Schwarz, 1998). Foley's function for the response of a pattern vision mechanism to a single pattern can be written as follows: 
R = ( C t S E t ) p / ( ( C t S I t ) q + Z ) + e ,
(1)
where C t is the contrast of the test pattern; S Et, S It, p, q, and Z are parameters of the model, and e is a sample of a Gaussian random variable with mean 0 and constant standard deviation, σ. The numerator is referred to as the excitatory term and the denominator as the divisive inhibitory term. For contrast discrimination in the absence of any context pattern, this response function is very similar to the Legge and Foley function. The functions shown in Figure 11 are based on Foley's model and the best parameters for ATB derived from the current experiments. I will call this response an S-response. Model functions are plotted on linear contrast axes because these make it easier to see the relation between the TvC functions and the response vs. contrast (RvC) functions. 
A context pattern can produce changes in inhibition, excitation, or both. This is reflected in the model by adding a term to the denominator or to both the numerator and the denominator of the response function. Foley (1994) measured TvC functions for contrast discrimination of Gabor patterns in the presence of masks of different orientations superimposed on the test patterns. An orthogonal mask increased thresholds at low pedestal contrasts, but as pedestal contrast increased, threshold decreased, went below the threshold in the absence of the mask, and then rose at high pedestal contrasts. A TvC function for contrast discrimination in the presence of an orthogonal mask is illustrated in Figure 12 (left) for a case with typical parameter values. The no-mask function from Figure 11 is shown for comparison. The model accounts for the effect of a fixed orthogonal mask by adding a term corresponding to inhibition from the mask to the denominator of the response function. The function becomes 2  
R = ( C t S E t ) p / ( ( C t S I t + C m S Im ) q + Z ) + e ,
(2)
where C m is the contrast of the fixed mask and S Im is the inhibitory sensitivity to the mask. The other symbols are the same as in Equation 1. The response function corresponding to the TvC function in the left panel of Figure 12 is shown in the right panel. When test patterns and masks are similar in orientation, the masks also excite, but Equation 2 will suffice for our current purpose. In a contrast discrimination task, the mechanism responses to the two contrasts and the behavioral response on any trial are related by the following assumption: When instructed to use the normal response rule, the observer indicates the stimulus for which the mechanism response is greatest; when instructed to use the reverse rule, the observer indicates the interval in which the mechanism response is least. 
Figure 12
 
(Left) TvC function when an orthogonal mask of fixed contrast is present in addition to the pedestal (red). (Right) Corresponding response vs. contrast function. In each case, functions for the no-mask case (Figure 11) are shown in blue for comparison. The functions are derived from Foley (1994, model 2, here called the S-response model) using typical parameter values. Note that with an orthogonal mask, the minimum of the TvC function is generally below the mask contrast.
Figure 12
 
(Left) TvC function when an orthogonal mask of fixed contrast is present in addition to the pedestal (red). (Right) Corresponding response vs. contrast function. In each case, functions for the no-mask case (Figure 11) are shown in blue for comparison. The functions are derived from Foley (1994, model 2, here called the S-response model) using typical parameter values. Note that with an orthogonal mask, the minimum of the TvC function is generally below the mask contrast.
To describe and predict thresholds, one more assumption is needed. In the two-alternative spatial forced-choice paradigm used here, this assumption is that at threshold there is a constant difference between the mean response to the pedestal alone and the mean response to the pedestal plus test increment: 
Δ R = R p t R p ,
(3)
where Δ
R
is the difference between the mean responses. The value of Δ
R
can be set arbitrarily without affecting the fit of the model to data, although it will affect the values of the parameters. Here, it is set to 1, a value used in previous applications of the model. 
Models in which the standard deviation varies with contrast can account for TvC functions equally well, provided that there is a corresponding change in the mean response function (Klein & Levi, 2009; Legge, Kersten, & Burgess, 1987). There are an infinite number of such pairs of functions. In principle, it is possible to determine the functions separately using psychophysical methods. In practice, the data are inconsistent (Georgeson & Meese, 2006; Klein, 2006). TvC functions alone are not sufficient to independently specify the two functions. It has been proposed that the decrease in threshold as pedestal contrast increases is due to stimulus uncertainty (Pelli, 1985). However, that would seem to be an unlikely explanation of the threshold decrease of more than a factor of 10 when there is a high contrast forward–backward mask that carries information about the time and spatial form of the test. Consequently, I use the simpler constant standard deviation model here. 
This model accounts well for contrast discrimination, superimposed masking, and other phenomena. The phenomenon that is most relevant here is pattern adaptation. Greenlee and Heitger (1988) showed that pattern adaptation increases contrast discrimination thresholds at low contrasts and decreases thresholds at high contrasts. Foley and Chen (1997) and Ross, Speed, and Morgan (1993) measured TvC functions for contrast discrimination with and without pattern adaptation and showed that the TvC function with adaptation is qualitatively the same as that produced by an orthogonal mask. The two sets of results are different in that for Foley and Chen the TvC function after adaptation crosses the TvC function for no adaptation; Ross and Speed's results do not clearly show this. Wilson and Humanski (1993) found TvC functions that were linear on log–log coordinates both before and after adaptation and proposed a different gain control model to account for them. The form of the Foley and Chen TvC function is illustrated in the left panel of Figure 12 with the corresponding response function in the right panel. Thus, adapting to a Gabor pattern that has the same orientation as the test produces an effect of the same form as a superimposed orthogonal mask. That effect is modeled by an increase in the divisive inhibitory input to the detecting mechanism. This inhibitory input persists when the adapter is turned off and gradually decreases over time. An adapter with the same orientation as the test will excite the detecting mechanism, but that excitation ends before the test patterns are presented (Foley & Chen, 1997; Ross & Speed, 1991). The effect of the added inhibition produced by the mask or adapter is both to suppress the response and to change the shape of the response function. This change is not a simple multiplication of the response or of the contrast; it is a nonlinear transformation of the response function that suppresses the response and shifts the steepest region of the function along the contrast axis. 
I first fitted the S-response model to the results of Experiment 1. Although the model produces dipper-shaped curves that cross at high pedestal contrast, it cannot produce good fits to these data. More specifically, it fails by underpredicting the decrease in threshold as pedestal contrast increases in the conditions where a forward–backward mask is presented. I did not attempt to fit the S-response model to the data of Experiments 29, because it is obvious that it cannot predict the local decrease in performance at the mask contrast. We need a new model to account for the Graham–Wolfson phenomenon found in forward–backward masking. 
V-response model accounts for many properties of the Graham–Wolfson phenomenon
To account for the results of the forward–backward masking experiments described above, I assume that when there is little or no time between the masks and the tests, a second response is produced in the visual system. This response is an increasing function of the absolute value of the difference between the test contrast and the mask contrast. It is a response to the change or transient between the mask and the test. This response is given by the following equation: 
R = ( a b s ( C t C m ) S E t ) p ( a b s ( C t C m ) S I t + C m S Im ) q + Z + e ,
(4)
where the symbols have the same meaning as in the earlier equations. Note that the equation has the same form as Equation 2, except that abs(C tC m) takes the place of C t. This response is 0 when C t = C m. It is illustrated in Figure 13 (right). The response increases on either side of 0 are symmetrical and have the same general form as the S-response. The response is a function of the magnitude of the change in contrast (transient) from the mask to the test and does not depend on the sign of the change. I will refer to it as a V-response. As in the S-response model, I assume that at threshold the difference between the mean responses to the pedestal plus increment and pedestal alone will be 1. Figure 13 also shows the TvC functions corresponding to this response function. As illustrated in Figures 4 and 5, there are sometimes three different values of the contrast increment at which performance corresponds to the threshold value (0.81 correct here). The red curve in the left panel is the TvC function for the highest of the three thresholds. This is the threshold that is found when the normal response rule is used, as in Experiment 1. This threshold is very high at low pedestal contrasts but is substantially below the no-mask threshold (blue) at high pedestal contrasts. This improvement in contrast discrimination at high pedestal contrasts increases with mask contrast and reaches at maximum at C m = 0.3–0.4 for typical parameters. Here, as can be seen, the increase is about a factor of 2. The other two TvC functions in Figure 13 correspond to the middle and low thresholds that the model predicts when the reverse response rule is used. These thresholds only exist when the pedestal contrast is below the mask contrast by at least the threshold increment. Overall, the V-response produces good contrast discrimination for all pedestal contrasts except those in a small range just below the mask contrast. 
Figure 13
 
(Left) TvC function with no mask (blue) and the three TvC functions for mask contrast of 0.32. The functions with masks correspond to the three contrasts at which performance is at threshold (see Figures 4 and 5). Red: Threshold in high contrast range where normal response rule is used as in Experiment 1. When the reverse rule is used, there are two thresholds for low contrast pedestals: green, middle threshold and black, lowest threshold. The functions are derived using Equation 4 and the best parameters for ATB in the current experiments. Far from masking, the forward–backward masks improve discrimination at high contrasts and produce good discrimination at low contrasts, but there is a small range of pedestal contrasts just below the mask contrast at which contrast discrimination is poor. (Right) The corresponding response functions for C m = 0 and 0.32.
Figure 13
 
(Left) TvC function with no mask (blue) and the three TvC functions for mask contrast of 0.32. The functions with masks correspond to the three contrasts at which performance is at threshold (see Figures 4 and 5). Red: Threshold in high contrast range where normal response rule is used as in Experiment 1. When the reverse rule is used, there are two thresholds for low contrast pedestals: green, middle threshold and black, lowest threshold. The functions are derived using Equation 4 and the best parameters for ATB in the current experiments. Far from masking, the forward–backward masks improve discrimination at high contrasts and produce good discrimination at low contrasts, but there is a small range of pedestal contrasts just below the mask contrast at which contrast discrimination is poor. (Right) The corresponding response functions for C m = 0 and 0.32.
In most of the experiments, the dependent variable was proportion correct. To predict proportion correct, we have to consider the variability of the responses. Since the responses are assumed to be perturbed by Gaussian noise with a constant standard deviation, the proportion correct in a two-alternative discrimination task, P, depends on the distribution of the difference ΔR = R ptR p, which has mean ΔR and standard deviation
2
σ. I assume that the proportion correct is given by 
P c = P ( Δ R > 0 ) = 0 P ( Δ R ) d Δ R ,
(5)
where P is the probability distribution of ΔR. Thus, I assume that the proportion correct corresponds to the area under the Gaussian response distribution for ΔR > 0. We might call this cumulative normal distribution the internal psychometric function, since it relates the mean mechanism response to the proportion correct. Note that because the model assumes that the relation between contrast and mechanism response is nonlinear, the model implies that the function relating proportion correct to the contrast difference is not a cumulative normal distribution as experiments have shown (Bird, Henning, & Wichmann, 2002; Foley & Legge, 1981; Garcia-Perez & Alcala-Quintana, 2007). The value of the standard deviation is arbitrary here. I set it at 1.11, so that when ΔR is 1 the proportion correct is 0.81, corresponding to performance at threshold in Experiment 1. This allows us to use the same parameter values for all the experiments. For a specific pair of contrasts, given the mean response to each of them and this value of the standard deviation, we can use the internal psychometric function, which corresponds to the cumulative normal distribution (Equation 5), to determine the proportion correct. 
When performance is determined by the V-response, and C ave < C m, the higher contrast will produce the lower mechanism response on average, and if the normal response rule is applied, the probability of being correct will be below 0.5. If the reverse rule is applied in these conditions, probability correct will be above 0.5 and will be symmetrical to the normal rule data about the mask contrast. The V-response accounts for the form of normal rule psychometric functions as follows: When both test contrasts are below the mask contrast, the response difference (Equation 3) will be negative, and by Equation 5, the proportion correct will be less than 0.5. As the contrast difference increases, proportion correct will decrease until the higher contrast equals the mask contrast. As the higher test contrast increases above the mask contrast, the proportion correct will increase until the higher contrast is 2 times the mask contrast. That is the straddle point at which the V-response does not discriminate between the two test patterns. As the higher contrast increases above the straddle point, the response difference will become increasingly positive and proportion correct will increase with an asymptote at 1. Use of the reverse rule reflects this function about the P = 0.5 line. I will show that the V-response accounts for the experimental results described above, except for results in the symmetrical straddle condition; they are accounted for by the S-response. 
Fitting the model and predicting experimental results
The V-response model was fitted to the data of Experiment 1 by finding the values of the parameters that minimized the root mean squared log error (RMSE). Since multiplying the numerator and the denominator by any constant does not change the response and is equivalent to changes in S Et, S It, S Im, and Z, it is possible to fix the value of one of these parameters without loss of goodness of fit. I fixed S Et to 100, so that there were five free parameters in these fits. The standard deviation of the responses, σ, does not affect the predictions here. The model was fitted to the data using a program that sought the parameters that minimized the root mean squared error (RMSE). Since there are many local minima in the error space, 50 fits were made starting from different initial values, and the fit with the lowest RMSE was taken as the best fit. The values of the best fitting parameters for AJA and ALI and the corresponding RMSE values are given in Table 2. The curves in Figure 1 for AJA and ALI correspond to the best fitting model. The curves for ATB are a prediction from fits to other data. The fits to the TvC functions are good, with RMSE between 0.11 and 0.21 when expressed as contrasts, comparable to other TvC function fits. The reason that there is no local decrease in performance near the mask contrast is that, when the normal response rule is used, the threshold is always above C m. Consequently, at threshold, C ave is always sufficiently greater than C m that the contrast difference at which the dip would occur is always below the threshold. 
Table 2
 
Summary of model fits. The first four columns show the primary fits for four observers. For three observers, these fits were used to predict performance in the other experiments specified in the row headed “Predicted.” Other columns show fits to individual experiments. Some of these required a different value of S Im (in italics).
Table 2
 
Summary of model fits. The first four columns show the primary fits for four observers. For three observers, these fits were used to predict performance in the other experiments specified in the row headed “Predicted.” Other columns show fits to individual experiments. Some of these required a different value of S Im (in italics).
4 c/deg 50-ms mask 8 c/deg Symmetrical straddle
Observer AJA ALI ATB PRR AJA ATB ATB PRR PRR ATB ATM
Experiments 1 1 2, 3, 4, 5 3, 5 7 7 8 8 9 9 9
Data sets 4 4 9 3 2 3 2 2 2 2 2
Data points 32 33 69 22 12 18 16 12 8 8 8
Points fitted 32 33 60 19 10 17 14 10 8 8 8
Parameters
    S Et 100 100 100 100 100 100 100 100 100 100 100
    S It 63.78 70.26 64 62 63.78 64 64 62 62 64 80
    S Im 23.15 38.56 37 12.6 54 37 56 17.58 83 180 45
    p 2.54 2.05 2.6 2.6 2.54 2.6 2.6 2.6 2.6 2.6 2.6
    q 2.15 1.60 2.26 2.32 2.15 2.26 2.26 2.32 2.32 2.26 2.36
    Z 6.22 2.20 3.5 12.9 6.22 3.5 3.5 12.9 12.9 3.5 16
    σ 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11
    C m factor 0.83
SSE 117.96 82.38 0.446 0.152 0.022 0.046 0.033 0.210 0.009 0.073 0.024
RMSE 1.91 1.58 0.086 0.089 0.047 0.052 0.049 0.145 0.033 0.078 0.055
Predicted 2, 4, 7 1, 7, 9, 10 9, 10
 

Notes: S Et and σ were fixed throughout at 100 and 1.11, respectively. In these fits, sometimes there is a valley in the error space along which RMSE varies less than 1%. In those cases, p was fixed at 2.6. For Experiment 1, SSE and RMSE are given in decibels, since error in decibels was minimized in these fits as is common for TvC data. For Experiment 9 (8 c/deg), S Im is higher than for 4 c/deg. For Experiment 10 (straddle condition), fit is to S-response model. Most of the data sets for each observer are fitted with the same five parameters. The C m factor for ATB (Experiment 7) is the factor multiplied by the mask contrast to give effective mask contrast.

The V-response model was also fitted to the data of Experiments 29. Points corresponding to the symmetrical straddle conditions were excluded from these fits; they will be dealt with separately. The model fits most of the data well. In making these fits, the parameter space was scanned at increasingly higher resolutions to find the parameters that provided the lowest RMSE. S Et was again fixed at 100 and σ was fixed at 1.11. In these fits, there is often a valley in the error space resulting in different parameter sets that yield essentially equally good fits (RMSE within 1%). In these cases, I fixed the value of p to 2.6, a typical value. RMSE averaged less than 0.05. The parameters that produce the best fit vary across observers and, to a lesser extent, across experiments. We expect large differences to occur across observers sometimes, since every visual system is different. On the other hand, when the stimuli and the visual system remain the same, we would expect the parameters to remain the same except for measurement errors. In principle, it is possible to estimate a single set of parameters by fitting the data from all the experiments in which the stimuli and the observers were the same in a single fit. I did something different. I used the first experiment or set of experiments done by an observer to determine a set of parameters for that observer. I then used these parameters to predict the results of the other experiments done by that observer. Table 2 contains the experiments on which the fits were based, the parameters that produced the lowest RMSE for those experiments, the SSE and RMSE of the fit, and the experiments whose results were predicted using these parameters. For AJA only the data of Experiment 1 were fitted and the resulting parameters were used to predict the results of the three other experiments that he participated in. All the smooth curves in the graphs correspond to fits of the model or to predictions based on fits of the model to other data. So the V-response model with five free parameters for each observer fits most of the data. There are four exceptions: Experiments 7, 8, 9, and 10
1. In Experiment 7, where the mask duration was much shorter than in the other experiments, different changes in the model were required for the two observers. AJA's performance with the 100-ms masks was worse than his performance with the same test contrast pairs with the 1-s masks (Experiment 2). This surprising result was accounted for in the model by allowing inhibitory sensitivity to the mask to be higher for the 100-ms masks than the 1-s masks. It was 54 here and 23 in Experiment 2. ATB did not show this overall decrease in performance. However, for the 50-ms masks, her performance functions were shifted slightly to the left. This was modeled by substituting an effective mask contrast for the mask contrast in the model. An effective mask contrast 0.83 times mask contrast gave the best fit to these data. This suggests that the response to these brief masks has a smaller amplitude than masks of 100 ms or longer. The smooth curves in the figure correspond to this slightly modified, six free parameter model (Figure 7, ATB, red and blue curves). Note that the mask duration was shortened from 1 s to 120 ms for ATB without affecting any parameters (green curve). Graham and Wolfson (2007) used adapting patterns whose contrast varied sinusoidally in time and addressed the question of how the adaptation level (my effective mask contrast) depends on the preceding contrast waveform. They conclude that the contrast is integrated over about 250 ms to determine the momentary adaptation level. Our results suggest an even shorter integration time but are far too limited to be definitive. 
2. In Experiment 8, where the patterns had a spatial frequency of 8 c/deg, performance was worse than in the corresponding experiment at 4 c/deg (Experiment 5). This implies that one or more of the parameters must be different here. I fitted the Experiment 8 data by allowing S Im to vary and setting the other four parameters to their best values when spatial frequency was 4 c/deg. This value of S Im is given in Table 2 and the curves in Figure 8 correspond to this model. Although the model produces reasonably good fits to the Experiment 8 data that are not improved appreciably by allowing other parameters to vary as well, this model will not be a generally satisfactory model of the effect of spatial frequency. The reason is that S Im affects only sensitivity to the masks, while it is well established that sensitivity to test patterns is also affected by their spatial frequency. There is not sufficient data in this study to adequately model the effects of spatial frequency. 
3. In Experiment 9, the V-response model predicts chance performance in all symmetrical straddle conditions and, therefore, is a completely inadequate model of these data. So the Experiment 9 results present more of a challenge than the others. Wolfson and Graham (2007, 2009) noted that performance in symmetrical straddle conditions in their experiments tends to be greater than chance and that this cannot be explained by a symmetrical V-response that goes to 0 at the mask contrast. They proposed that there may be two or more V-response units that have different slopes above and below the mask contrast and are mirror images of each other about the mask contrast. I am not able to exclude this possibility, but I propose an alternative explanation for performance in these conditions. I assume that an S-response, which explains some pattern adaptation results when there is a time interval between the offset of the adapter and the onset of the test (Foley & Chen, 1997), is produced in conditions in which the V-response is also produced (short ISIs). For any contrast pair, performance is determined by the response that is most sensitive to the contrast difference. For most contrast pairs, the V-response to the test contrasts is more sensitive than the S-response and discrimination performance is determined by it. However, in symmetrical straddle conditions, the difference between the V-responses produced by the two contrasts is zero, and therefore, the V-response is blind to these contrast differences, no matter how large. The S-response, on the other hand, is monotonic increasing over the whole range of contrast, will be different for any pair of straddle contrasts, and will increase with the difference between those contrasts. I fitted the S-response model to the data of Experiment 9. It provides reasonably good fits and it does so using the same parameters as the best parameters in the V-response model, except for S Im, which is much higher here. A summary of these fits is given in Table 2. The smooth curves in Figure 9 correspond to these fits. Thus, a model in which the visual system produces both a V-response and an S-response, with the most sensitive response determining performance in any condition, accounts for both the reordering of contrast magnitudes below the mask contrast and the limited ability to discriminate in the straddle conditions. 
Both the V-response and the S-response are shown in Figure 14 for the best parameters of PRR and ATB for the two models and a mask contrast of 0.32. Although both responses are suppressed as mask contrast increases, the S-response is suppressed more. As a consequence, the difference in responses for all pairs of contrasts, except straddle pairs, is greater for the V-response. For symmetrical straddle pairs, the difference in S-responses is greater. Here, we see another advantage produced by the V-response. It enables much better discrimination of contrast than the S-response does over most of the contrast range. However, it is unable to discriminate symmetrical straddle pairs, and we discriminate them poorly using the less sensitive S-response. Unfortunately, the better discrimination afforded by the V-response is accompanied by consistent errors in judging which contrast is higher when both are below the mask contrast. That might not be a great disadvantage because the most common function of our pattern vision system is the identification of patterns and that is often maintained over contrast reversals. However, it is not clear that patterns with contrasts below the mask contrast are perceived as patterns (see the Phenomenology section). 
Figure 14
 
Comparison of the V-response and the S-response when mask contrast is 0.32. Curves are based on the best parameter values from each model for each observer. (Left) PRR. (Right) ATB. Both responses are suppressed as the mask contrast increases, but the S-response is suppressed more. Note that except for contrast pairs that straddle the mask contrast, the response difference is always greater for the V-response.
Figure 14
 
Comparison of the V-response and the S-response when mask contrast is 0.32. Curves are based on the best parameter values from each model for each observer. (Left) PRR. (Right) ATB. Both responses are suppressed as the mask contrast increases, but the S-response is suppressed more. Note that except for contrast pairs that straddle the mask contrast, the response difference is always greater for the V-response.
4. The models as described thus far imply nothing about the tuning of the mechanisms to orientation, so they do not describe the results of Experiment 10. Here, I use a simple exponential decay function to describe the decrease in threshold with an increasing difference between the orientation of the tests and the orientation of the masks. The smooth curves in the graphs correspond to the negative exponential function: 
C t t = C t t 0 e ( K | Θ | ) + C t t A ,
(6)
where C tt is the contrast of the test at threshold, C tt0 is the threshold when the mask and test have the same orientation, Θ is the relative orientation of test and mask in deg, K is a parameter that determines the decrease in threshold with increasing orientation difference, and C ttA is the threshold in the absence of a mask. This model was fitted to the data of Experiment 10 (K = −0.1 for AJA and −0.05 for ALI). 
Tuning is much narrower here than for superimposed masking. Why? Foley (1994) showed that, in superimposed masking, thresholds are jointly determined by the excitation and divisive inhibition produced by all the components of the stimulus. In superimposed masking, the mask both excites and inhibits the detecting mechanism, but excitation is much more narrowly tuned. This excitation by the mask has the effect of reducing the threshold for small orientation differences but not for large orientation differences, producing more rounded threshold elevation curves. In the model of forward–backward masking, excitation by the mask does not persist in the test interval. This has the effect of increasing the threshold for small orientation differences and thus narrowing the orientation tuning of the masking process. Similar threshold elevation functions have been reported for contrast adaptation using the conventional paradigm (Bradley et al., 1988). 
The absolute threshold with a mask in the same orientation as the test was also determined for AJA in Experiments 1 and 4 and for ALI in Experiment 1. For ALI, there is good agreement between the two measures. For AJA, the value here, 0.28, is lower than in the other two experiments, 0.38 in Experiment 1 and 0.37 in Experiment 4. Since there was feedback here, it is possible that AJA used the reverse rule and this threshold corresponds to the second highest contrast at which there is 0.81 correct (see Figure 4). 
In Figure 6 (Experiment 6), there are no smooth curves through the data for ISI > 0. Since the S-response model had fitted the data of Foley and Chen (1997) with an ISI of 133 ms, I expected that the same model would apply here for ISI values this long or longer. In fact, it does not. Although, it is possible to find a response function that will explain these data, the challenge is to find a response function that makes sense in terms of underlying processes. 
Phenomenology
In addition to measuring performance in these tasks, I asked the observers to describe their visual experiences. Initially, none of the observers reported seeing the contrast change that occurred when the test patterns were presented. Instead, they perceived a change in the mask that was generally different in the two test pattern positions. ATB described this as a “blink” and PRR described it as a “flicker.” ATB based her responses on the magnitude of the blink. PRR initially tried to base her responses on the number of cycles in the flicker but switched early to the magnitude of the flicker. When instructed to indicate the higher contrast, both reported that they indicated the percept of higher magnitude. AJA reported that he saw no difference in magnitude, but one of the test patterns appeared to come on earlier than the other, and when instructed to indicate the higher contrast, he indicated the one that appeared to come on first. When instructed to use the normal response rule, all three performed consistently, above chance when the average test contrast was greater than the mask contrast and below chance when the average was below. When they used the reverse rule, the low contrast side of these functions was inverted. 
As ATB and PRR continued to do experiments, they began to sometimes report seeing a contrast change during the test interval and the change was sometimes different in the two possible test pattern positions. This tended to occur reliably when both contrasts were above the mask contrast and especially when they were well above. The contrast changes reported were almost always increases. After this phenomenology became apparent, observers were instructed to base their responses on perceived contrast in the conditions in which this was possible. They reported that they were able to do this in some, but not all, conditions. However, this change in perceptual criteria had no discernable effect on discrimination performance. Thus, the same steeply varying V-response appears to be associated with qualitatively different percepts in different conditions. 
During the experiments, observers made reports only after completing a block of trials. After the experiments were completed, PRR and ATB spent a session just looking at the patterns and making phenomenological reports. In neither case were the reports completely consistent, either within or between observers, but we can make some descriptive statements: 
  1.  
    When ISI = 0, mask duration ≥500 ms, and both contrasts were below the mask contrast, contrast changes were not seen. When one contrast was above the mask contrast, a contrast increase was sometimes seen. This percept became more frequent as test contrast increased. PRR reported seeing a contrast change at lower test contrasts than ATB.
  2.  
    When ISI = 0 and mask duration is 50 ms (Experiment 7), reports were inconsistent; ATB reported contrast changes even in some conditions where both test contrasts were below the mask contrast. PRR reported only differences in flicker.
  3.  
    When ISI = 30 or 50 ms (Experiment 6) and the average test contrast was less than the mask contrast, flicker magnitude was again the dominant percept, and when the average test contrast was above the mask contrast, contrast changes were the dominant percept. For an ISI of 150 ms, contrast changes were reported in all conditions. So, when ISI > 0, we again have different percepts associated with contrast discrimination that depend on the relation between mask and test contrasts.
  4.  
    In symmetrical straddle conditions, the normal response rule usually produced proportions correct above 0.50. The rare exceptions were close to 0.50. The dominant percept on which observers thought they were basing their responses, flicker or contrast, was not consistent, with a slight tendency to report flicker more when the contrast difference was large. So test contrasts above the mask contrast do not reliably produce percepts of contrast increase. The dominant percept again had no discernable effect on performance.
Both conditions in which the model attributes performance to the V-response and those in which performance is attributed to the S-response produce inconsistent phenomenological reports of the appearance of the test patterns, sometimes flicker and sometimes spatial pattern. There are some conditions that reliably produce one or the other report. What are we to make of the inconsistent phenomenology? There is a widely used assumption in psychophysics that a response in one mechanism is associated with one percept. If this is the case, it seems plausible that the V-response produces the flicker percept and the S-response produces the contrast pattern percept. It is possible that both responses occur in all conditions, although one is perceptually dominant. Observers sometimes report seeing both a contrast change and flicker. According to the analysis presented above, one of the responses reliably determines performance, the one that produces the best discrimination performance. It may be that the other response sometimes produces the dominant percept and determines the phenomenological report. When there is an interval between the masks and the tests, this seems quite plausible. In these conditions, there will be large transients between the masks and the zero contrast during the ISI. They will be quite visible, but they will be identical at the two positions and will not contribute to discrimination. If the V-response is not associated with a percept of the stimulus pattern, it would not contribute to pattern identification, even though it contributes to contrast discrimination. 
Biological basis of the two responses
Although the model postulates two responses to the patterns used here, it does not require two mechanisms. It is possible that there is a single mechanism that produces a V-response to a contrast change and an S-response to a maintained contrast. The finding that the two responses are the same in the absence of a mask is consistent with this hypothesis. The transient V-response must be such that it is very sensitive to change but does not carry information about the direction of contrast change. Many simple and complex cells in the visual cortices of cats and monkeys respond to both the onset and the offset of an optimal grating pattern with a transient increase in firing rate. For complex cells, if the pattern is presented for 100 ms or longer, the magnitude of on and off responses is about the same (Liang, Shen, Sun, & Shou, 2008). If similar responses are produced by increases and decreases in contrast, they could correspond to the V-responses of the model. Cortical cells vary greatly in their response profiles; many continue to respond after the initial on response, others return to their baseline rate. Among those that continue to respond, the response profiles vary greatly (Albrecht, Geisler, Frazor, & Crane, 2002). These continuing responses could correspond to the S-responses. There is one aspect of the results that may be difficult to reconcile with the single mechanism idea. That is the finding that the S-response is more suppressed by the masks than the V-response. This suggests that the V-response and S-response occur in different pathways. These pathways could contain cells that are predominantly transient or predominantly sustained, the first carrying the V-response and the second the S-response. The hypothesis that there are distinct transient and sustained pathways, which has a long history, is also consistent with the hypothesis that each pathway evokes a distinct percept. However, as described above, here reported percepts are not closely related to performance. 
In the models presented here, the mechanism responses are single-valued, whereas even very brief stimuli produce neural responses that are extended in time. The mechanism responses may be understood as momentary values of responses that are extended in time. By varying ISI, Foley and Chen (1999) showed that for the S-response the model excitatory signal lasts for less than 40 ms and the inhibitory signal lasts for about 100 ms under their conditions. This produces a mechanism response not greatly different than the response of some visual cortical neurons. It is not clear whether a similar analysis can be made of the V-response, because it does not seem to mediate performance except at very short ISIs. 
Conclusion
The V-response model describes or predicts the results of most of the experiments well using five free parameters for each observer with minor changes to account for the effects of very brief masks or changes in spatial frequency. The S-response model describes performance in the symmetrical straddle conditions. Parameters are the same, except for sensitivity to divisive inhibition, which is greater for the S-response. 
The results show that forward–backward masking is closely related to superimposed masking and contrast adaptation. In each of these paradigms, the S-response function is both suppressed and changed in form. Both effects are produced by adding a constant to the divisive inhibitory signal. The changes in form are such as to steepen the response function near the mask contrast and thereby improve contrast discrimination for contrasts in that range. Forward–backward masking with a 0 ISI adds a complication to this story. It produces response transients that are steep (V-responses) and thereby enable very good discrimination in a range around the mask contrast but have the same sign for increases and decreases in contrast and are thereby blind to contrasts that symmetrically straddle the mask contrast and poor at discriminating other straddle contrast pairs. Here, the beneficial effect is produced by subtraction. Fortunately, S-responses enable us to discriminate contrast in the range where the V-response is blind. 
So pattern vision is complex. When there is a temporally close forward–backward mask, a contrast pattern produces two responses in the visual system. Both responses change in different, highly nonlinear ways when a mask or an adapter is presented. The changes in both responses are adaptive in that they enable better contrast discrimination near the mask contrast than would be possible without them. Our visual system has evolved to see many things well, but the evolutionary adaptations are subtle. 
Supplementary Materials
Supplementary PDF - Supplementary PDF 
Acknowledgments
Partial support for this project was provided by NIH Grant EY 12743. Some of these results were reported at the 2010 Annual Meeting of the Vision Sciences Society. I thank Jerome Tietz for technical assistance and the five observers who participated in the study. I thank Norma Graham and Sabina Wolfson for helpful discussions of this phenomenon and critical reading of an earlier version of the manuscript. 
Commercial relationships: none. 
Corresponding author: John M. Foley. 
Email: foley@psych.ucsb.edu. 
Address: Department of Psychological and Brain Sciences, University of California, Santa Barbara, CA 93106, USA. 
Footnotes
Footnotes
1  In a paper that I received after this paper was written, Graham (in press) provides a more complete statement of the Graham and Wolfson model and shows a fit to some more of their data.
Footnotes
2  Foley (1994) combined the inhibitory terms in a different way when the mask was of a different orientation than the test. Here, the inhibitory term is written for the case in which the mask has the same orientation as the test, because that is the condition in Experiments 19.
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Figure 1
 
Experiment 1. Test contrast threshold vs. pedestal contrast functions at four mask contrasts for three observers. Error feedback was provided. Each point is the mean of six measurements. Arrows indicate mask contrasts. The smooth curves correspond to the V-response model that will be described below.
Figure 1
 
Experiment 1. Test contrast threshold vs. pedestal contrast functions at four mask contrasts for three observers. Error feedback was provided. Each point is the mean of six measurements. Arrows indicate mask contrasts. The smooth curves correspond to the V-response model that will be described below.
Figure 2
 
Experiment 2. Fixed contrast difference; reverse/normal rule; feedback. Proportion correct as a function of the average of the two test contrasts at four mask contrasts. (Left) AJA. (Right) ATB. Note that the contrast difference was increased with mask contrast to keep performance at approximately the same level. In each case, the minimum performance occurs when the average test contrast equals the mask contrast so that the test contrasts symmetrically straddle the mask contrast. The smooth curves show the predictions of the V-response model that will be described below. AJA, N = 250; ATB, N = 300.
Figure 2
 
Experiment 2. Fixed contrast difference; reverse/normal rule; feedback. Proportion correct as a function of the average of the two test contrasts at four mask contrasts. (Left) AJA. (Right) ATB. Note that the contrast difference was increased with mask contrast to keep performance at approximately the same level. In each case, the minimum performance occurs when the average test contrast equals the mask contrast so that the test contrasts symmetrically straddle the mask contrast. The smooth curves show the predictions of the V-response model that will be described below. AJA, N = 250; ATB, N = 300.
Figure 3
 
Experiment 3. Fixed contrast difference; normal response rule; no feedback. Proportion correct as a function of average contrast. The arrow indicates the mask contrast. The smooth curves correspond to the V-response model with parameters determined from Experiments 3 and 5 for PRR and an overall fit to Experiments 2, 3, 4, and 5 for ATB. (Left) PRR. (Right) ATB. N = 250.
Figure 3
 
Experiment 3. Fixed contrast difference; normal response rule; no feedback. Proportion correct as a function of average contrast. The arrow indicates the mask contrast. The smooth curves correspond to the V-response model with parameters determined from Experiments 3 and 5 for PRR and an overall fit to Experiments 2, 3, 4, and 5 for ATB. (Left) PRR. (Right) ATB. N = 250.
Figure 4
 
Experiment 4. Psychometric function for contrast detection; reverse/normal rule; feedback. Proportion correct as a function of contrast in a contrast detection task with mask contrasts of 0.04 and 0.16 (AJA) or 0.32 (ATB) indicated by the arrows. The lower contrast is 0. The contrasts that symmetrically straddle the mask contrast are 0 and 2C m. The smooth curves correspond to the V-response model with parameters determined from Experiment 1 for AJA and an overall fit to Experiments 2, 3, 4, and 5 for ATB. N = 250.
Figure 4
 
Experiment 4. Psychometric function for contrast detection; reverse/normal rule; feedback. Proportion correct as a function of contrast in a contrast detection task with mask contrasts of 0.04 and 0.16 (AJA) or 0.32 (ATB) indicated by the arrows. The lower contrast is 0. The contrasts that symmetrically straddle the mask contrast are 0 and 2C m. The smooth curves correspond to the V-response model with parameters determined from Experiment 1 for AJA and an overall fit to Experiments 2, 3, 4, and 5 for ATB. N = 250.
Figure 5
 
Experiment 5. Psychometric function for contrast discrimination; ATB: feedback, PRR: no feedback. Proportion correct as a function of the higher contrast. Pedestal contrast = 0.1. Mask contrast = 0.2. (Left) PRR. (Right) ATB. The contrasts that symmetrically straddle the mask contrast are 0.1 and 0.3. The blue symbols and lines correspond to the conditions in which the normal response rule was used, and the red symbols and lines correspond to the conditions in which the reverse rule was used. N = 300. For PRR, the parameter values are the best values from a joint fit to Experiments 3 and 5, and for ATB, the parameter values are the best overall fit to Experiments 2, 3, 4, and 5.
Figure 5
 
Experiment 5. Psychometric function for contrast discrimination; ATB: feedback, PRR: no feedback. Proportion correct as a function of the higher contrast. Pedestal contrast = 0.1. Mask contrast = 0.2. (Left) PRR. (Right) ATB. The contrasts that symmetrically straddle the mask contrast are 0.1 and 0.3. The blue symbols and lines correspond to the conditions in which the normal response rule was used, and the red symbols and lines correspond to the conditions in which the reverse rule was used. N = 300. For PRR, the parameter values are the best values from a joint fit to Experiments 3 and 5, and for ATB, the parameter values are the best overall fit to Experiments 2, 3, 4, and 5.
Figure 6
 
Experiment 6. Fixed contrast difference; normal response rule; no feedback. Proportion correct as a function of average contrast for a contrast difference of 0.08 and mask contrast of 0.32. The different symbols correspond to different interstimulus intervals in milliseconds. The ISI = 0 data (blue) are replotted from Experiment 3 and the smooth curve through them corresponds to the V-response model. (Left) PRR. (Right) ATB. N = 250.
Figure 6
 
Experiment 6. Fixed contrast difference; normal response rule; no feedback. Proportion correct as a function of average contrast for a contrast difference of 0.08 and mask contrast of 0.32. The different symbols correspond to different interstimulus intervals in milliseconds. The ISI = 0 data (blue) are replotted from Experiment 3 and the smooth curve through them corresponds to the V-response model. (Left) PRR. (Right) ATB. N = 250.
Figure 7
 
Experiment 7. Short duration masks. Fixed contrast difference; reverse/normal rule; feedback. Proportion correct as a function of average contrast for mask contrasts of 0.4 and 0.32. Mask durations: AJA, 100 ms; ATB, 50 and 120 ms. N = 250. The smooth curves correspond to the V-response model. Slight modifications to the model were made to accommodate the effects of these very brief masks.
Figure 7
 
Experiment 7. Short duration masks. Fixed contrast difference; reverse/normal rule; feedback. Proportion correct as a function of average contrast for mask contrasts of 0.4 and 0.32. Mask durations: AJA, 100 ms; ATB, 50 and 120 ms. N = 250. The smooth curves correspond to the V-response model. Slight modifications to the model were made to accommodate the effects of these very brief masks.
Figure 8
 
Experiment 8. Spatial frequency: 8 c/deg. Psychometric function for contrast discrimination; no feedback. Proportion correct as a function of the higher contrast. Mask contrast = 0.2. Pedestal contrast = 0.1. Blue: normal rule; red: reverse rule for contrasts below the straddle contrast, normal rule, above. Arrows indicate the mask contrast. The symmetrical straddle contrasts are the pedestal contrast, 0.1 and 0.3. N = 200. Note that for PRR there is no decrement in performance in the straddle condition.
Figure 8
 
Experiment 8. Spatial frequency: 8 c/deg. Psychometric function for contrast discrimination; no feedback. Proportion correct as a function of the higher contrast. Mask contrast = 0.2. Pedestal contrast = 0.1. Blue: normal rule; red: reverse rule for contrasts below the straddle contrast, normal rule, above. Arrows indicate the mask contrast. The symmetrical straddle contrasts are the pedestal contrast, 0.1 and 0.3. N = 200. Note that for PRR there is no decrement in performance in the straddle condition.
Figure 9
 
Experiment 9. Symmetrical straddle paradigm; normal rule; feedback. Proportion correct as a function of contrast difference for three observers. N = 300. The smooth curves correspond to the S-response model. Parameters were the same as for the V-response model, except for S m, which is higher for the S-response.
Figure 9
 
Experiment 9. Symmetrical straddle paradigm; normal rule; feedback. Proportion correct as a function of contrast difference for three observers. N = 300. The smooth curves correspond to the S-response model. Parameters were the same as for the V-response model, except for S m, which is higher for the S-response.
Figure 10
 
Experiment 10. Threshold as a function of mask orientation with reference to test; normal rule; feedback; no pedestal. Mask contrast = 0.16. The smooth curve is the best fitting negative exponential function. (Left) AJA, N = 300. (Right) ALI, N = 200.
Figure 10
 
Experiment 10. Threshold as a function of mask orientation with reference to test; normal rule; feedback; no pedestal. Mask contrast = 0.16. The smooth curve is the best fitting negative exponential function. (Left) AJA, N = 300. (Right) ALI, N = 200.
Figure 11
 
(Left) Threshold vs. contrast (TvC) function for contrast discrimination. (Right) Corresponding response vs. contrast function. At any pedestal contrast, the contrast difference threshold is the increase in contrast that produces a constant increase in response. Functions are computed using the best parameters for ATB from the current experiments.
Figure 11
 
(Left) Threshold vs. contrast (TvC) function for contrast discrimination. (Right) Corresponding response vs. contrast function. At any pedestal contrast, the contrast difference threshold is the increase in contrast that produces a constant increase in response. Functions are computed using the best parameters for ATB from the current experiments.
Figure 12
 
(Left) TvC function when an orthogonal mask of fixed contrast is present in addition to the pedestal (red). (Right) Corresponding response vs. contrast function. In each case, functions for the no-mask case (Figure 11) are shown in blue for comparison. The functions are derived from Foley (1994, model 2, here called the S-response model) using typical parameter values. Note that with an orthogonal mask, the minimum of the TvC function is generally below the mask contrast.
Figure 12
 
(Left) TvC function when an orthogonal mask of fixed contrast is present in addition to the pedestal (red). (Right) Corresponding response vs. contrast function. In each case, functions for the no-mask case (Figure 11) are shown in blue for comparison. The functions are derived from Foley (1994, model 2, here called the S-response model) using typical parameter values. Note that with an orthogonal mask, the minimum of the TvC function is generally below the mask contrast.
Figure 13
 
(Left) TvC function with no mask (blue) and the three TvC functions for mask contrast of 0.32. The functions with masks correspond to the three contrasts at which performance is at threshold (see Figures 4 and 5). Red: Threshold in high contrast range where normal response rule is used as in Experiment 1. When the reverse rule is used, there are two thresholds for low contrast pedestals: green, middle threshold and black, lowest threshold. The functions are derived using Equation 4 and the best parameters for ATB in the current experiments. Far from masking, the forward–backward masks improve discrimination at high contrasts and produce good discrimination at low contrasts, but there is a small range of pedestal contrasts just below the mask contrast at which contrast discrimination is poor. (Right) The corresponding response functions for C m = 0 and 0.32.
Figure 13
 
(Left) TvC function with no mask (blue) and the three TvC functions for mask contrast of 0.32. The functions with masks correspond to the three contrasts at which performance is at threshold (see Figures 4 and 5). Red: Threshold in high contrast range where normal response rule is used as in Experiment 1. When the reverse rule is used, there are two thresholds for low contrast pedestals: green, middle threshold and black, lowest threshold. The functions are derived using Equation 4 and the best parameters for ATB in the current experiments. Far from masking, the forward–backward masks improve discrimination at high contrasts and produce good discrimination at low contrasts, but there is a small range of pedestal contrasts just below the mask contrast at which contrast discrimination is poor. (Right) The corresponding response functions for C m = 0 and 0.32.
Figure 14
 
Comparison of the V-response and the S-response when mask contrast is 0.32. Curves are based on the best parameter values from each model for each observer. (Left) PRR. (Right) ATB. Both responses are suppressed as the mask contrast increases, but the S-response is suppressed more. Note that except for contrast pairs that straddle the mask contrast, the response difference is always greater for the V-response.
Figure 14
 
Comparison of the V-response and the S-response when mask contrast is 0.32. Curves are based on the best parameter values from each model for each observer. (Left) PRR. (Right) ATB. Both responses are suppressed as the mask contrast increases, but the S-response is suppressed more. Note that except for contrast pairs that straddle the mask contrast, the response difference is always greater for the V-response.
Table 1
 
Symbols used in this article.
Table 1
 
Symbols used in this article.
Symbol Quantity
Stimulus variables
C t Contrast of test pattern
C m Contrast of mask pattern
C p Contrast of pedestal (the lower of two test contrasts presented on a trial)
C ave The average of two test contrasts presented on a trial
C diff The difference between two test contrasts presented on a trial
ISI Temporal interval between masks and test pattern in seconds
Θ Orientation of the masks relative to the tests in degrees (Experiment 10)
 
Dependent variables
C tt Contrast of test pattern at threshold
P Proportion correct
 
Model parameters
S Et Excitatory sensitivity to test pattern t
S It Inhibitory sensitivity to test pattern t
S Im Inhibitory sensitivity to masks
p Exponent of excitatory signal
q Exponent of inhibitory signal
Z Value of maintained inhibition in the absence of a pattern
σ Standard deviation, Gaussian random variable added to each response
K Parameter of exponential decrease in masking with relative orientation (Experiment 10)
C tt0 Contrast of test pattern at threshold when test and mask have the same orientation (Experiment 10)
C ttA Asymptote of test pattern threshold as relative orientation increases (Experiment 10)
Table 2
 
Summary of model fits. The first four columns show the primary fits for four observers. For three observers, these fits were used to predict performance in the other experiments specified in the row headed “Predicted.” Other columns show fits to individual experiments. Some of these required a different value of S Im (in italics).
Table 2
 
Summary of model fits. The first four columns show the primary fits for four observers. For three observers, these fits were used to predict performance in the other experiments specified in the row headed “Predicted.” Other columns show fits to individual experiments. Some of these required a different value of S Im (in italics).
4 c/deg 50-ms mask 8 c/deg Symmetrical straddle
Observer AJA ALI ATB PRR AJA ATB ATB PRR PRR ATB ATM
Experiments 1 1 2, 3, 4, 5 3, 5 7 7 8 8 9 9 9
Data sets 4 4 9 3 2 3 2 2 2 2 2
Data points 32 33 69 22 12 18 16 12 8 8 8
Points fitted 32 33 60 19 10 17 14 10 8 8 8
Parameters
    S Et 100 100 100 100 100 100 100 100 100 100 100
    S It 63.78 70.26 64 62 63.78 64 64 62 62 64 80
    S Im 23.15 38.56 37 12.6 54 37 56 17.58 83 180 45
    p 2.54 2.05 2.6 2.6 2.54 2.6 2.6 2.6 2.6 2.6 2.6
    q 2.15 1.60 2.26 2.32 2.15 2.26 2.26 2.32 2.32 2.26 2.36
    Z 6.22 2.20 3.5 12.9 6.22 3.5 3.5 12.9 12.9 3.5 16
    σ 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11
    C m factor 0.83
SSE 117.96 82.38 0.446 0.152 0.022 0.046 0.033 0.210 0.009 0.073 0.024
RMSE 1.91 1.58 0.086 0.089 0.047 0.052 0.049 0.145 0.033 0.078 0.055
Predicted 2, 4, 7 1, 7, 9, 10 9, 10
 

Notes: S Et and σ were fixed throughout at 100 and 1.11, respectively. In these fits, sometimes there is a valley in the error space along which RMSE varies less than 1%. In those cases, p was fixed at 2.6. For Experiment 1, SSE and RMSE are given in decibels, since error in decibels was minimized in these fits as is common for TvC data. For Experiment 9 (8 c/deg), S Im is higher than for 4 c/deg. For Experiment 10 (straddle condition), fit is to S-response model. Most of the data sets for each observer are fitted with the same five parameters. The C m factor for ATB (Experiment 7) is the factor multiplied by the mask contrast to give effective mask contrast.

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