The detectability and appearance of visual targets can be modulated by surround stimuli. In this study we asked how cross- and iso-oriented surrounds modulate contrast detection and discrimination in foveal vision. We systematically measured the Threshold-versus-Contrast (TvC) functions over a wide range of pedestal and surround contrasts. Our results show that cross-oriented surrounds lower the contrast threshold over the entire range of pedestal and surround contrasts, but iso-surround modulation of the TvC function is dependant on the relative contrast, being facilitative when the surround/pedestal contrast ratio C_{sur}/C_{ped} < 1 and suppressive when C_{sur}/C_{ped} > 1. Data fitting indicates that cross-surround modulation (facilitation) is mainly due to improved gain, except at very low and high surround contrasts. Iso-surround modulation on the other hand is more complicated, probably reflecting more than one surround process as determined by the relative contrast.

^{2}mean luminance, and 3.8° × 3.0° screen size at the 5.64-meter viewing distance). Luminance of the monitor was made linear by means of a 15-bit look-up table. Experiments were run in a dimly lit room.

*d*= 18 arcmin), which would maximize masking at high pedestal contrasts (Yu & Levi, 1997, 2000) because of a desensitization effect (Westheimer, 1967), but would change masking very little at low pedestal contrasts because desensitization at low pedestal contrasts is nearly negligible (Westheimer, 1967). The surround (S in Figure 1b) was a sinusoidal grating annulus abutting the pedestal with contrast varying from 0 to 0.80. The outer and inner diameters of the surround were 45 and 18 arcmin, respectively. Contrast thresholds were measured with a successive two-alternative forced-choice staircase procedure. The pedestal was presented in each of the two stimulus intervals (300 msec each) separated by a 400 msec inter-stimulus interval. Each stimulus interval was accompanied with an audio tone of the same duration. The target was randomly presented in one of the two stimulus intervals with the same onset and offset as the pedestal. The observers’ task was to judge which stimulus interval contained the target. Each trial was preceded by a 6.3′ × 6.3′ fixation cross which disappeared 100 msec before the beginning of the trial. Audio feedback was given on incorrect responses. Each staircase consisted of four preliminary reversals and eight experimental reversals. The step size of contrast change in preliminary reversals was 7.5% of the previous contrast and in experimental reversals it was 2.5%. Each correct response lowered the target contrast by one step and each incorrect response raised the target contrast by three steps, which resulted in a 75% convergence level of the staircase. The mean of the eight experimental reversals was taken as the contrast threshold. Each datum represents the mean of 5–6 replications, and the error bars represent +/−1 standard error of the mean.

*C*= 0 to 0.40 in a random order). TvC functions at different surround orientations (cross and iso) and contrasts (

_{ped}*C*= 0 to 0.80) were measured in a balanced order.

_{sur}*C*), being facilitative when

_{sur}/C_{ped}*C*< 1 and suppressive when

_{sur}/C_{ped}*C*> 1. These effects are evident in Figures 2–4.

_{sur}/C_{ped}*C*= 0). Individual data and data fitting outputs can be retrieved by clicking http://journalofvision.org/3/8/1/fig02-here.gif.

_{sur}*C*= 0.05 to 0.40), especially at

_{sur}*C*= 0.10 & 0.40, facilitation is mainly due to a downshift of the TvC function. At the lowest surround contrast (

_{sur}*C*= 0.025), facilitation is minimal at detection (

_{sur}*C*= 0), most evident at the dipper (

_{ped}*C*= 0.025 ∼ 0.10), and weakens as the pedestal contrast increases (which results in a steeper slope of the TvC function at high pedestal contrasts). At the highest surround contrast (

_{ped}*C*= 0.80), facilitation is mainly evident at high pedestal contrast with little effect at low pedestal contrasts, and the TvC function is flatter at high pedestal contrasts.

_{sur}*C*= 0.025) except for detection (

_{sur}*C*= 0) (blue circles). As the surround contrast increases, facilitation is limited to higher pedestal contrasts, and iso-surround modulation at lower pedestal contrast becomes suppressive, raising thresholds above the baseline when

_{ped}*C*= 0.80. The transition from suppression to facilitation appears to be determined by the relative contrast of the surround and pedestal (considered below). At

_{sur}*C*= 0.40, suppression changes to facilitation dramatically, producing a kink in the TvC function at high pedestal contrasts. At the highest surround contrast (

_{sur}*C*= 0.80), suppression is evident at low pedestal contrasts (

_{sur}*C*< = 0.1) with thresholds equal to the baseline at

_{ped}*C*= 0.20 and 0.40.

_{ped}*C*= 0.10 ∼ 0.40), cross- and iso-surrounds produce nearly identical facilitation when

_{ped}*C*>

_{ped}*C*(values less than 1 on the top abscissa, which shows the ratio of surround to pedestal contrast). Similar data reported previously at

_{sur}*C*= 0.40 (Yu & Levi, 2000) are also included (Figure 3, bottom right). However, iso facilitation diminishes when

_{ped}*C*<

_{ped}*C*, suggesting the influence of the relative contrast of pedestal and surround stimuli (see below).

_{sur}*C*= 0.025) and some suppression when the surround contrast is much higher than the pedestal contrast (

_{sur}*C*= 0.80). Lack of surround modulation at lower pedestal contrast, especially at detection, is specific to the annular iso-surround stimuli we used, as significant iso-facilitation is evident when collinear flankers restricted to near the ends of a target stimuli are used (e.g., Polat & Sagi, 1993). Solomon & Morgan (2000) and Yu, et al. (2002) showed that collinear iso facilitation can be suppressed by non-collinear stimulus components in annular iso surround stimuli.

_{sur}*C*>

_{ped}*C*, these common mechanisms produce similar iso and cross surround facilitation. However, when

_{sur}*C*<

_{ped}*C*, iso surround facilitation is diminished and thresholds approach the baseline. In later data fitting, we will show that a combination of baseline fitting and cross data fitting nicely describes many of the iso-surround effects; however, it fails to account for the iso-surround inhibition when the ratio

_{sur}*C*> 1 (top abscissa values greater than 1 in Figure 3, lower abscissa values > 1 in Figure 4, and easily seen in the lower right panel of Figure 2).

_{sur}/C_{ped}*C*) (Figure 4). Clearly, cross-surrounds facilitate regardless of

_{sur}/C_{ped}*C*, though facilitation tends to be stronger at a lower

_{sur}/C_{ped}*C*. On the other hand, iso-surround modulation is dependent on

_{sur}/C_{ped}*C*. As

_{sur}/C_{ped}*C*increases, iso-surround modulation changes from facilitation to suppression. Iso-surrounds mainly produce facilitation when

_{sur}/C_{ped}*C*≤ 1 and suppression when

_{sur}/C_{ped}*C*> 1.

_{sur}/C_{ped}*d*’function, Stromeyer & Klein, 1974; Legge & Foley, 1980; Foley, 1994; Boynton, Demb, Glover, & Heeger, 1999; Chen & Tyler, 2001) to fit TvC data. This function can be written as: We call this

*d*′ contrast response function the Stromeyer-Foley function, in which

*C*is the stimulus contrast,

*p*is the log-log slope at low contrast,

*w*is the log-log slope at high contrast,

*C*is the contrast at the kink point where lines drawn through the high and low asymptotes intersect as seen in Figure 6a, and

_{k}*K*controls the height of the function. A deeper exploration of the role each of the parameters plays in controlling the shape of the

*d*′ function and the TvC function is taken up in the Appendix.

*d*′ function can be determined from contrast discrimination data such as shown in Figure 2. The connection between the

*d*′ function and the TvC function is given by: where

*C*is the pedestal contrast and

_{ped}*C*is the test threshold in a contrast discrimination task.

_{test}*C*is small compared to

_{k}*C*as for the present data, Equation 1 becomes: Thus,

*K*is approximately the

*d*′ value at 100% contrast. The connection of

*K*to the high contrast Weber fraction is derived by keeping the leading terms of the Taylor’s expansion of Equations 2 and 3, based on the assumption that

*C*<<

_{test}*C*. So the Weber fraction is given by:

_{ped}*C*= 1 the Weber fraction is simply 1/

_{ped}*Kw*. Equations 4 and 5 also illustrate that the log-log slope of the TvC function (

*C*as a function of

_{test}*C*) is 1-

_{ped}*w*. Thus the high contrast portion of the TvC function pins down the parameters

*K*and

*w*. A full understanding of the connection between

*C*and

_{test}*C*in the TvC function depends on many factors including the nature of the underlying transducer function, the amount of uncertainty, early and late gain control, and the amount of additive and multiplicative noise. We present a simplified model of visual processing which includes each of these factors in the Appendix (Figure 7); for the present, we fit the data with the four parameter Stromeyer-Foley function and look for systematic changes in the parameters produced by the iso and cross surrounds. In particular, we are interested in whether the Stromeyer-Foley function can account for the surround effects on the basis of parameter changes, or whether additional factors need to be included.

_{ped}^{2}), was the lowest (χ

^{2}= 28.4,

*df*= 17) when

*p*was a single value. The χ

^{2}was reduced when all four parameters were allowed to float (χ

^{2}= 23.2), but the values of parameters became unstable due to the high correlation between

*p*and other parameters. In addition, the reduction of chi square from 28.4 to 23.2 was insufficient to justify the five extra parameters.

*K, C*, and

_{k}*w*are plotted against the surround contrast in Figure 5 (

*p*has a single value across the six surround contrasts), as are values of 1/

*wK*(approximately the Weber fraction at

*C*=1.0 as shown in Equation 5). The error bars on the parameters estimates are based on the variance output by the lsqnonlin program (without making the reduced chi square heterogeneity correction), except for error bars of 1/

*wK*that were calculated with a Monte Carlo simulation based on the standard errors of

*w*and

*K*. The

*d*′ functions associated with these parameter values are also shown (Figure 6).

*K*is nearly equally raised at all surround contrasts, indicating that a cross surround at any (visible) contrast improves the gain. A more detailed analysis in the Appendix will show that a raised

*K*vertically lifts the Stromeyer-Foley function. However, such a gain change could occur at different stages of visual processing (Node 3 or 7 of Figure 7), so the exact mechanism underlying gain change cannot be unequivocally determined. The gain improvement is clearly a dominant effect of cross surround modulation at

*C*= 0.1 ∼ 0.4. At very low and high contrasts (

_{sur}*C*=0.05 and 0.8), the cross surround also changes the kink contrast (

_{sur}*C*) and the high-contrast slope

_{k}*w*. A change of

*C*may indicate a change of pooled divisive inhibition (node 7 in Figure 7) as discussed in the Appendix. And a change of w may indicate a change of saturation or gain control of the transducer function (node 6 in Figure7), or stimulus dependent multiplicative noise (node 9 in Figure 7), or both. Finally, Figure 5 also indicates that the high contrast Weber fraction, 1/

_{k}*wK*, is a smoothly decreasing function of surround contrast. Because

*K*is fairly constant, this change is mainly contributed by

*w*that increases as a function of surround contrast.

*C*≥ 0.10, where the fits indicate more facilitation than is evident in the actual data. This discrepancy may be due to individual differences (it is only true for the two novice observers but not for the highly practiced author). However, if this is a genuine effect reflecting additional suppression near

_{sur}*C*=

*C*= 0.10 (possibly reducible with learning), it can be simulated by adding a subtractive component (−a*(1+0.125/

_{ped}*C*)

^{−2}) to the Stromeyer-Foley function (Figure 2, left column, dotted green lines). The parameter values in this new component were chosen to give a decent fit of suppression near

*C*= 0.1. They were chosen by a rough trial and error procedure, since there are not enough data points to constrain data fitting.

_{ped}*C*= 0.025, 0.05 as well as at

_{sur}*C*= 0.80 (Figure 2, right column, dotted blue lines), Equations 1 and 2 do not capture the suppression-to-facilitation transition at high pedestal contrasts of the iso TvC function at

_{sur}*C*= 0.40. Fitting for the iso TvC function at

_{sur}*C*= 0.10 also misses the dipper when the slope of the TvC function at high pedestal contrasts is satisfied.

_{sur}*C*<

_{sur}*C*. When

_{ped}*C*>

_{sur}*C*there is not much iso modulation except some facilitation at the lowest surround contrast and suppression at the highest surround contrast. That is, iso-surround effects appear to be primarily a two-state function, and which state they are in is determined by the relative contrast. To demonstrate this, we combined the segments of the baseline fitting curve at

_{ped}*C*<

_{ped}*C*and the cross fitting curve at

_{sur}*C*>

_{ped}*C*and plotted them together with the iso TvC data in Figure 2. The specific rules for the combination require an assumption about what to do near the transition point when

_{sur}*C*=

_{ped}*C*: The perceptual matching point of the center and surround will depend on their overall contrast. At low contrasts the surround’s perceived contrast is expected to be larger than the pedestal’s because of the larger size of the surround. We measured the detection thresholds for the pedestal and for the surround for subject YC and found the pedestal threshold to be 1.3 times the surround threshold. At high contrasts the perceived contrast of the center and surround are expected to be equal. Because of this effect of perceived contrast, in Figure 2 when

_{sur}*C*< 0.10 the

_{sur}*C*=

_{ped}*C*point was grouped with

_{sur}*C*<

_{ped}*C*(the perceived contrast of the pedestal was reduced because of its smaller size). When

_{sur}*C*> 0.10, the

_{sur}*C*=

_{ped}*C*point was grouped with

_{sur}*C*>

_{ped}*C*. For example, at

_{sur}*C*= 0.05, we combine the baseline curve up to

_{sur}*C*= 0.05 and cross curve starting at

_{ped}*C*= 0.10, leaving a gap between

_{ped}*C*= 0.05 and

_{ped}*C*= 0.10. However at

_{ped}*C*= 0.40, we combine the baseline curve up to

_{sur}*C*= 0.20 and cross curve starting at

_{ped}*C*= 0.40, leaving a gap between

_{ped}*C*= 0.20 and

_{ped}*C*= 0.40. Figure 2 right column shows that a combination of baseline fits (thick black lines) and cross fits (thick red lines) nicely account for most of the iso data. The exception is the extra inhibition seen at the highest surround contrast (

_{ped}*C*= 0.80), probably as a result of multiplicative noise induced by high-contrast surround stimuli (discussed below).

_{sur}*C*), being facilitative when

_{sur}/C_{ped}*C*<1 and suppressive when

_{sur}/C_{ped}*C*>1. Data fitting indicates that cross-surround modulation (facilitation) can be reasonably accounted for by changes in the Stromeyer-Foley

_{sur}/C_{ped}*d*′ function, mainly a raised gain, except at very low and high surround contrasts. Iso-surround modulation on the other hand is more complicated, probably reflecting more than one process and is affected by the relative contrast.

*d*′ function. For example, a raised gain due to cross surround modulation could equally possibly occur at different stages of visual processing, even within the frame of our simple model. Similarly, a reduced high-contrast slope of the TvC function (1-

*w*) could indicate either a change of saturation or gain control of the transducer, or increased stimulus dependent multiplicative noise. These multiple possibilities suggest that data fitting does not provide sufficient power to fully constrain the psychophysical mechanisms of surround modulation.

*p*(the low-contrast slope) across surround contrasts. A change of

*p*would indicate a change of uncertainty due to cross surround modulation. Indeed, the effects of cross-surrounds seen in Figure 2 do not resemble the pattern seen in Figure 8b which simulates the TvC function changes that would occur if

*p*varies. Thus it is unlikely that the surround produces facilitation through an uncertainty reduction mechanism. Elsewhere, we (Yu, Klein & Levi, 2002) provided experimental evidence that the cross facilitation of detection cannot be fully explained by uncertainty reduction. One of these arguments is that the bottom of the dipper of the unflanked TvC curve can be strongly facilitated by the cross-oriented surrounds (see Figure 2). Uncertainty reduction is unlikely to account for the surround’s facilitation of the dipper regime since the presence of the pedestal should have minimized the uncertainty so there wouldn’t be any uncertainty for the surround to minimize.

*C*in Equation 1 and shifts the log-log

_{k}*d*′ curve diagonally approximately in the direction of the high contrast slope (see Figure 8f). If cross surrounds modulate contrast responses by solely manipulating the gain control pool, it would effectively change the value of

*C*in the denominator of Equation 1. However, our data fitting indicates that a significant change of the value of

_{k}*C*only occurs at very low and high surround contrasts, discounting the single gain control pool idea as a full explanation for cross-surround modulation.

_{k}*C*= 0.48. They reported that this high contrast iso-surround suppresses low contrast (

_{sur}*C*<= 0.10) discrimination but does not affect detection (

_{ped}*C*= 0) or high contrast discrimination (

_{ped}*C*> 0.10). These data are similar to ours at

_{ped}*C*= 0.40 (Figure 2, row d), except that significant facilitation is shown in our data at high pedestal contrasts. A key difference is that Snowden and Hammett’s target and pedestal were matched in size, while we used a larger pedestal that approaches the limits of the underlying perceptive field. Under their stimulus conditions, the abutting iso-surround actually covers part of the perceptive field center as well as the antagonistic surround. The resulting mutual cancellation would diminish facilitation. Snowden and Hammett (1998) made the assumption that (iso) surround modulation is a variation of regular masking, and high contrast surrounds would act as low contrast pedestals because of the separation. This assumption would not predict the iso-surround facilitation shown in our experiments, since the presence of iso-surrounds would be equivalent to a small increase of the supra-threshold pedestal contrast, which would lead to slightly elevated contrast thresholds, rather than significant facilitation.

_{sur}*d*′ function can be written as: Besides Equation 1, the function can also be written in terms of the detection threshold: Here

*c*and

*c*are the contrasts

_{k}*C*and

*C*from Equation 1 expressed in threshold units (

_{k}*c*=

*C/th*;

*c*=

_{k}*C*).

_{k}/th*c*is the contrast at which the denominator doubles. The parameters

_{k}*p*and

*w*are the same as in Equation 1. The detection threshold,

*th*, is defined to be the contrast giving

*d*′=1, as can be seen by setting

*c*= 1 (

*C*=

*th*) in

*K*. The connection between

*th*and

*th*, obtained by equating 1 and 6, is:

*p, w, K*and

*C*in Equation 1 or

_{k}*p, w, th*and

*c*in Equation 6. The parameters

_{k}*th*and

*p*are determined by the low pedestal contrast region of the TvC curve, with

*th*being the threshold (

*d*′ = 1) for zero pedestal contrast and

*p*being the slope of the log-log

*d*′ function at low pedestal contrasts. The parameters

*K*and

*w*are determined by the large pedestal contrast region of the TvC curve with 1-

*w*being the log-log slope of the TvC function (see Equation 5) and 1/

*Kw*being the test contrast for

*C*= 1.0 (see Equation 5) under the assumption that

_{ped}*C*<<1.0, as is typical for spatial frequencies below about 15 c/deg. For the parameters of the unflanked condition (see Figure 6), the jnd at a 100% pedestal (

_{k}*C*= 1.0) is 1/

*Kw*= 1/(18.6*0.57) = 0.094. A test contrast of 9.5% at a 100% pedestal gives a Weber fraction of 0.094, a reasonable value at high pedestal contrasts. Another way of looking at these numbers is that a 9.4% test contrast together with the high contrast TvC slope of

*w*= 0.57 gives

*K*= 1/(0.094*0.57) = 18.6.

*K*is approximately the

*d*′ at

*C*= 1.0. This means that there are approximately

*K*= 19 jnds from zero contrast to 100% contrast.

*C*can be calculated from the parameters

_{k}*th, p, K*and

*w*using Equation 7.

*C*can also be determined graphically by drawing two straight lines on the log-log

_{k}*d*′ function that match the high and low contrast asymptotic regions as shown in Figure 6a.

*C*is the pedestal contrast where the two asymptotic lines intersect. It specifies the region where the

_{k}*d*′ slope changes from

*p*to

*w*.

*K*. Node 5, in the divisive inhibitory branch of intermediate processing will be seen to affect the high contrast region of the

*d*′ function. Nodes 2, 4 and 6 correspond to power law nonlinearities in their respective branches of the model, i.e. output = input

^{power}. Node 7 corresponds to an additive gain control that affects the saturation point of the divisive inhibition branch of the model. This node corresponds to the pooled gain control found in many modfels of cortical processing, as will be discussed. Node 9 is multiplicative output noise.

*y*=

*x*). (c) A decelerating curve (node 7) represents a saturating stage. For example, the surround could contribute to a gain control pool that adds a constant to the denominator of the S-F function. Even without a surround there would be a semi-saturation constant contributing to the denominator. (d) A letter “N” (node 9) represents multiplicative noise at the output. The multiplicative nature of this noise will reduce the log-log slope of the

_{a}*d*′ function at high contrasts.

*C*with the slopes of the two line segments given by

_{k}*p*and

*w*, and the height of the lines given by

*K*or

*th*. A pair of broken lines are shown in each node representing how the surround can alter the S-F function by modulating that node. The black lines are identical across all nodes and set the baseline, and red lines show how the S-F function is changed by surround influence. For a multiplicative node the surround would modify the gain at that stage. For a nonlinear node the surround would change the power exponent. For the additive gain control pooling node 7 the presence of the surround would give an additive contribution at that node. For the output noise node 9, the surround could suppress or enhance the noise. The division sign between nodes 4 and 8 represents the divisive inhibition typical of feed-forward gain control models.

*p*=2.27,

*w*= 0.57 (or

*q*=

*p-w*=1.70),

*K*=18.4,

*C*=0.040 (corresponding to

_{k}*th*= 0.031 and

*c*= 1.29). The other two curves shown in red and blue correspond to the S-F and TvC functions with specific model parameters decreased and increased by a factor of √2 except as discussed below.

_{k}*th*, in Equation 6, while fixing

*p, w*, and

*c*. This manipulation of parameters shifts the

_{k}*d*′ curve horizontally. The TvC curve (right panel) shows a downward shift which is greater at low contrast than at high, similar to the effect of a low contrast cross surround as seen in the second and third panels of Figure 2 (also see Figure 6 for

*d*′ function change).

*p*, the log-log slope at low contrast. The semi-saturation contrast,

*C*is also altered. It is interesting that the high contrast portions of the S-F curves including parameters

_{k}*K*are unchanged. This is because the signal is altered in both the numerator and denominator of the S-F function. There has been a long-standing debate over whether the low contrast facilitation associated with

*p*> 1 is caused by uncertainty reduction when a pedestal is present or by an accelerating transducer function (Legge, Kersten, & Burgess, 1987). An increase of stimulus uncertainty depresses the

*d*′ function at low contrast while leaving it unchanged at high contrast, precisely as is seen in Figure 8b. A similar effect is obtained by a change of exponent at node 2 in Figure 7 giving a

*d*′ function whose p dependence is: This form of the

*d*′ function, in which a fixed

*b*has replaced

*c*of Equation 1, would have

_{k}^{p-w}*C*change as

_{k}*p*is varied. The effects of cross-surrounds seen in Figure 2 do not resemble the pattern seen in Figure 8b so it is unlikely that the surround produces facilitation through an uncertainty reduction mechanism or through a node 2 mechanism.

*d*′ function the red curve is reduced compared to the baseline black curve, indicating higher uncertainty or suppression at low contrast. Indeed this suppression raises the detection threshold substantially, as seen at the leftmost portion of the TvC function (the right panel). However, there is a crossover whereby for pedestal contrasts above 6% the

*d*′ suppression results in contrast discrimination facilitation, such that the red curve is below the black curve. This result occurs because the suppression at low

*d*′ steepens the

*d*′ function at medium contrasts as occurs with iso surround inhibition. Another mechanism for crossover will be discussed in connection with Figure 8g.

*K*in Equation 1 without changing

*p, w*, and

*C*. This manipulation of parameters simply shifts the log-log

_{k}*d*′ curve vertically. Figure 5 shows that one of the clearest effects of a cross surround of any contrast is to produce an increase in

*K*, which is a dominant change in the middle region of cross surround contrasts (10%–40%). However, at very low and high surround contrasts

*K*is not the only parameter that is affected by the surround.

*d*′ curve diagonally in the direction of the low contrast slope. Node 5 is on the branch of the model that affects the denominator of Equation 1 and Equation 6. Chen and Tyler (2001) use this as one of their sites of gain control. By attenuating the signal portion of that branch there is minimal effect at very small signal strength as shown in Figure 8d. This is the type of shift that is seen in the iso surround data of Figure 2 at low surround contrasts.

*w*, the log-log slope at high contrast while leaving unchanged

*th*and

*p*, the low contrast parameters of Equation 6, and also

*c*, the log-log break point in threshold units. The TvC function, shown in the right panel of Figure 8e has a high contrast log-log slope of 1-

_{k}*w*(as discussed following Equation 3), thus the slope of the right hand side of the TvC changes, similar to the effect of an 0.8 contrast cross surround (Figure 2 bottom panel, also see Figure 6). There are two general explanations of factors controlling

*w*: transducer saturation (node 6) or stimulus dependent (multiplicative) noise (node 9). The approach taken in this paper is largely agnostic on this topic since one could argue that the denominator of the Stromeyer-Foley function could come either from saturation (or gain control) of the transducer function or it could come from multiplicative noise. Foley (1994) associates the threshold elevation of the TvC function at high contrasts with a divisive gain control of the contrast response function. Stromeyer and Klein (1974), on the other hand, fit the increasing contrast discrimination threshold at high pedestal contrasts using multiplicative noise. Kontsevich, Chen, & Tyler (2002) present data and arguments in favor of the multiplicative noise explanation. Based on unpublished data, we conclude that both factors (a saturating transducer function plus multiplicative noise) strongly contribute to the shape of the

*d*′ function at large pedestal contrasts. For simplicity, in the present article we do not introduce multiplicative noise (it would slightly modify the S-F function with a Pythagorean sum of the various noise sources in the S-F denominator), but it is important to keep in mind that such noise could well play an important role in suprathreshold discrimination. The connection of this discussion to the effect of the surround on the parameters shown in Figure 5 is that the presence of a low contrast surround could suppress the multiplicative noise, thereby reducing

*w*.

*C*in Equation 1, by a modification at node 7. This manipulation shifts the log-log

_{k}*d*′ curve diagonally approximately in the direction of the high contrast slope. This is the model of pooled divisive inhibition proposed by Malik & Perona (1990), Heeger (1992), Albrecht & Geisler (1991) and Foley (1994). In this proposal the surround would add to the gain pool at node 7, effectively increasing the value of

*C*in the denominator of Equation 1. Figure 8f shows that this manipulation has a sizeable effect on the low contrast region but minimal effect at high pedestals of both the

_{k}*d*′ and the TvC functions. Since this is not what is seen in our cross-surround data, we can discount the single gain control pool idea as a full explanation for the effect of cross-oriented surrounds. However, in the iso-surround case we do see strong examples of extra inhibition at low pedestal contrasts when the surround contrast is high. This is precisely what is expected for the standard gain control (or noise intrusion) shown in Figure 8f.

*p*and

*w*. Node 4 just affects the slope of the branch in the numerator, thereby altering both the low and the high contrast slopes (not shown in Figure 8). Since there are four nodes (2, 4, 6, 9) that affect the two slope parameters, the present experiments are unable to pin down the nodes where the surround modifies the slope. As was discussed earlier in connection with Figure 5, the surround had minimal effect on

*p*and only a small effect on

*w*.