The responses of V1 neurons can be modulated by stimulation outside the neurons’ classical receptive fields (e.g., Hubel & Wiesel, 1965; Nelson & Frost, 1985; Knierim & Van Essen, 1992; Kapadia, Westheimer, & Gilbert, 2000; etc). This surround or contextual modulation indicates that V1 neurons are not simply isolated bar and edge detectors. Instead they interact with each other with a potential for serving more complex visual functions. Among various issues of surround modulation studied by researchers in both neurophysiology and psychophysics are the roles of relative surround orientation, ranging from collinear (iso) to orthogonal (cross). In neurophysiological studies, surround facilitation and suppression have been reported at both iso and cross surround orientations (Hubel & Wiesel, 1965; Nelson & Frost, 1985; Knierim & Van Essen, 1992; DeAngelis, Freeman, & Ohzawa, 1994; Kapadia, Ito, Gilbert, & Westheimer, 1995; Sillito, Grieve, Jones, Cudeiro, & Davis, 1995; Toth, Rao, Kim, Somers, & Sur, 1996; Das & Gilbert, 1999; Hupe, James, Girard, & Bullier, 2001; Jones, Wang, & Sillito, 2002). The results are not always consistent, but the relative stimulus contrast has been cited as one important factor to account for some of the controversies about whether surrounds produce facilitation or suppression (Kapadia et al., 2000).

At the psychophysical (visual system) level, the influences of surround stimuli on contrast detection and discrimination may also provide insights into the surround modulation issue (e.g., Dresp, 1993; Polat & Sagi, 1993; Zenger & Sagi, 1996; Snowden & Hammett, 1998; Solomon, Watson, & Morgan, 1999; Yu & Levi, 2000). Collinear or iso-surround modulation has been studied most often (e.g., Polat & Sagi, 1993; Zenger & Sagi, 1996; Yu & Levi, 1997; Snowden & Hammett, 1998; Solomon et al., 1999), but only a small number of studies have investigated psychophysical surround modulation by cross-surround stimuli (Raasch, 1988; Polat & Sagi, 1993; Yu & Levi, 1998, 2000; Chen & Tyler, 2002; Yu, Klein, & Levi, 2002).

Strong cross-surround facilitation of contrast detection was demonstrated by Yu, Klein and Levi (2002), both with annular surround stimuli (replotted in this paper, Figure 3, top left panel) and with Gabor flankers similar to those used by Polat and Sagi (1993). Similar effects were first reported in a thesis by Raasch (1988). Yu, et al. (2002) found that cross-surround facilitation of contrast detection is surround-contrast dependent. It mainly manifests at low surround contrasts but not at high surround contrasts, which explains why no cross facilitation was evident in Polat and Sagi (1993). Control measurements by Yu, et al. (2002) showed that cross-surrounds facilitate contrast detection mainly by improving the internal signal/noise ratio rather than by reducing stimulus uncertainty.

For contrast discrimination, Yu and Levi (2000) reported two distinct types of surround effects, one narrowly tuned to iso-orientation, and the other broadly tuned to cross-orientation. Surround modulation of contrast discrimination at cross- and iso-orientations for a fixed pedestal contrast (0.40) is generally facilitative, but with distinct contrast dependencies. Cross-surrounds at contrasts higher than the pedestal contrast produce stronger facilitation that could completely eliminate masking. In contrast, iso-surround facilitation is diminished when the surround contrast is higher than the pedestal contrast.

In the current study we expanded our investigation to systematically study cross- and iso-surround modulation over a broad range of pedestal and surround contrasts by measuring complete threshold versus contrast (TvC) functions at multiple surround contrasts. We also fit the data quantitatively using a variant of the standard contrast-response function (which we call the Stromeyer-Foley function) to examine how the underlying contrast response function is modulated by cross- and iso-surrounds. Our experiments and data fitting enable us to test conclusions of previous researchers based on data collected from limited stimulus conditions, and to obtain a better understanding of psychophysical cross- and iso-surround modulation.

Three adult observers with normal or corrected-to-normal vision served in the study. BJ and ND were new to psychophysical observation and received several sessions of training. YC, one of the authors, was highly experienced.

The stimuli (Figure 1) were generated by a VisionWorks computer graphics system (Vision Research Graphics, Inc., Duham, NH) and presented on a U.S. Pixel Px19 monochrome monitor (1024 x 512 resolution, 0.28 mm (H) x 0.41 mm (V) pixel size, 117 Hz frame rate, 62 cd/m² mean luminance, and 3.8^o x 3.0^o 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.

The target (Figure 1) was a 10 arcmin long spatially localized D6 grating (the sixth derivative of a Gaussian, T in Figure 1b) centered in-phase on a circularly windowed sinusoidal grating pedestal (P in Figure 1b) of the same spatial frequency (8.0 cpd) and orientation (vertical) with contrast varying from 0 to 0.40. The D6 target (similar to a Gabor function with σ = 4.1 arcmin) was multiplied by a Gaussian window along its long axis (σ = 4.2 arcmin) and truncated at the target length. The size of the circular pedestal was larger than the target (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' x 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.

Each experimental session typically consisted of three segments lasting for approximately two hours. Each segment measured one TvC function (six staircases for six pedestal contrasts at Cped = 0 to 0.40 in a random order). TvC functions at different surround orientations (cross and iso) and contrasts (Csur = 0 to 0.80) were measured in a balanced order.

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 surround/pedestal contrast ratio (Csur/Cped ), being facilitative when Csur/Cped < 1 and suppressive when Csur/Cped > 1. These effects are evident in Figures 2 -4.

Figure 2 shows the TvC functions of the mean data from three observers and data fitting outputs (see later Data Fitting section) (cross-surround data in the left column and iso-surround data in the right column). The error bar for each datum represents either the average of the three corresponding individual error bars, or the standard errors of individual thresholds, whichever is larger. Each row shows cross- and iso-surround TvC functions at one surround contrast (in ascending order from 0.025 to 0.80), as well as the baseline TvC function (Csur = 0). Individual data and data fitting outputs can be retrieved by clicking here.

The mean and individual baseline functions (i.e., the TvC function with no surround shown by black asterisks and dashed lines replicated in each panel) resemble a typical TvC function (e.g.,Legge & Foley, 1980). As the pedestal contrast increases, the contrast threshold first decreases and then increases, forming a dipper near the detection threshold.

Cross-surrounds facilitate the entire TvC function at all surround contrasts (red circles). At low and moderate surround contrasts (Csur = 0.05 to 0.40), especially at Csur = 0.10 & 0.40, facilitation is mainly due to a downshift of the TvC function. At the lowest surround contrast (Csur = 0.025), facilitation is minimal at detection (Cped = 0), most evident at the dipper (Cped = 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 (Csur = 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.

Iso-surrounds facilitate the entire TvC function at the lowest surround contrast (Csur = 0.025) except for detection (Cped = 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 Csur = 0.80. The transition from suppression to facilitation appears to be determined by the relative contrast of the surround and pedestal (considered below). At Csur = 0.40, suppression changes to facilitation dramatically, producing a kink in the TvC function at high pedestal contrasts. At the highest surround contrast (Csur = 0.80), suppression is evident at low pedestal contrasts (Cped < = 0.1) with thresholds equal to the baseline at Cped = 0.20 and 0.40.

Several interesting properties emerge when the mean cross and iso data are plotted together (replotted from Figure 2) at each pedestal contrast as a function of the surround contrast (Figure 3 – lower abscissa). First, at higher pedestal contrasts (Cped = 0.10 ~ 0.40), cross- and iso-surrounds produce nearly identical facilitation when Cped > Csur (values less than 1 on the top abscissa, which shows the ratio of surround to pedestal contrast). Similar data reported previously at Cped = 0.40 (Yu & Levi, 2000) are also included (Figure 3, bottom right). However, iso facilitation diminishes when Cped < Csur, suggesting the influence of the relative contrast of pedestal and surround stimuli (see below).

Second, at lower pedestal contrasts, although cross facilitation tends to be strong, iso surrounds do not have much effect on contrast thresholds, except for slight facilitation at the lowest contrasts (Csur = 0.025) and some suppression when the surround contrast is much higher than the pedestal contrast (Csur = 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.

We hypothesize that cross and iso surround effects may share some common mechanisms. When Cped > Csur, these common mechanisms produce similar iso and cross surround facilitation. However, when Cped < Csur, 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 Csur/Cped > 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).

Cross and iso surround modulation can be summarized by plotting normalized mean contrast thresholds (i.e. threshold-with-surround/threshold-with-no-surround) against the surround/pedestal contrast ratio (Csur/Cped) (Figure 4). Clearly, cross-surrounds facilitate regardless of Csur/Cped, though facilitation tends to be stronger at a lower Csur/Cped. On the other hand, iso-surround modulation is dependent on Csur/Cped. As Csur/Cped increases, iso-surround modulation changes from facilitation to suppression. Iso-surrounds mainly produce facilitation when Csur/Cped ≤ 1 and suppression when Csur/Cped > 1.

In order to fit the TvC data we used a contrast response function (the 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:

The parameters of the 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:

When Ck is small compared to C as for the present data, Equation 1 becomes:

At Cped = 1 the Weber fraction is simply 1/Kw. Equations 4 and 5 also illustrate that the log-log slope of the TvC function (Ctest as a function of Cped ) is 1-w. Thus the high contrast portion of the TvC function pins down the parameters K and w. A full understanding of the connection between Ctest and Cped 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.

We first used Equation 1 to fit cross-surround effects. The cross TvC functions at all six surround contrasts were fitted simultaneously via a nonlinear least square method (the Matlab lsqnonlin function). To reduce the number of parameters, we experimented by letting one parameter be a single value (the same for all surround contrast conditions) and the other three parameters be vectors (different values for different surround contrast conditions). The total number of parameters was 3*6+1 = 19. The number of data points being fit was 36. The goodness of the fit, chi square (χ²), was the lowest (χ² = 28.4, df = 17) when p was a single value. The χ² was reduced when all four parameters were allowed to float (χ² = 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.

The fitted values of K, Ck, and 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).

The Stromeyer-Foley function generally captures the properties of cross surround modulation (Figure 2, left column, solid red lines) quite well. As shown in Figure 5, 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 Csur = 0.1~ 0.4. At very low and high contrasts (Csur =0.05 and 0.8), the cross surround also changes the kink contrast (Ck) and the high-contrast slope w. A change of Ck 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/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.

The fitted curves often miss the data points at the 0.10 pedestal contrast when Csur ≥ 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 C = Cped = 0.10 (possibly reducible with learning), it can be simulated by adding a subtractive component (–a*(1+0.125/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 Cped = 0.1. They were chosen by a rough trial and error procedure, since there are not enough data points to constrain data fitting.

Unlike cross-surround data fitting, iso-surround data fitting is more qualitative than quantitative. First we attempted to use Equation 1 to fit the iso-surround data, and the same fitting procedure as for cross-surround data was followed. Although fitting was reasonably good for iso TvC functions at Csur = 0.025, 0.05 as well as at Csur = 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 Csur = 0.40. Fitting for the iso TvC function at Csur = 0.10 also misses the dipper when the slope of the TvC function at high pedestal contrasts is satisfied.

Earlier we pointed out that iso-surround effects are nearly identical to cross-surround effects when Csur < Cped. When Csur > Cped 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 Cped < Csur and the cross fitting curve at Cped > Csur 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 Cped = Csur: 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 Csur < 0.10 the Cped = Csur point was grouped with Cped<Csur (the perceived contrast of the pedestal was reduced because of its smaller size). When Csur > 0.10, the Cped = Csur point was grouped with Cped>Csur. For example, at Csur= 0.05, we combine the baseline curve up to Cped = 0.05 and cross curve starting at Cped = 0.10, leaving a gap between Cped= 0.05 and Cped = 0.10. However at Csur = 0.40, we combine the baseline curve up to Cped = 0.20 and cross curve starting at Cped = 0.40, leaving a gap between Cped = 0.20 and Cped = 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 (Csur = 0.80), probably as a result of multiplicative noise induced by high-contrast surround stimuli (discussed below).

In this study we systematically measured cross- and iso-surround modulation of the TvC function over a wide range of surround contrasts. Cross-surround facilitation is evident across the entire TvC function. Iso-surround modulation of the TvC function is dependent on the relative contrast (Csur/Cped), being facilitative when Csur/Cped <1 and suppressive when Csur/Cped >1. Data fitting indicates that cross-surround modulation (facilitation) can be reasonably accounted for by changes in the Stromeyer-Foley 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.

The S-F function is characterized by four parameters: p, w, K and Ck in Equation 1 or p, w, th and ck in Equation 6. The parameters 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 Cped = 1.0 (see Equation 5) under the assumption that Ck <<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 (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.

The parameter Ck can be calculated from the parameters th, p, K and w using Equation 7. Ck can also be determined graphically by drawing two straight lines on the log-log d' function that match the high and low contrast asymptotic regions as shown in Figure 6a. Ck is the pedestal contrast where the two asymptotic lines intersect. It specifies the region where the d' slope changes from p to w.

Figure 7 shows a simplified model of visual processing that consists of three stages: early processing stage (Nodes 1 &2), intermediate processing stage including an excitatory branch (Nodes 3 & 4) and a divisive inhibitory branch (Nodes 5, 6 & 7), and an output stage (Nodes 8 & 9). Each of the nine nodes indicates where a nonlinearity or gain control process could be influenced by the surround. A detailed discussion of each node will be presented in conjunction with Figure 8; here we present an overview of the nodes. Nodes 1, 3, 5 and 8 represent gain control mechanisms where the signal gets multiplied by a constant, i.e. output = constant* input. Node 1 is in early stage and affects threshold, th. Nodes 3 (excitatory branch of intermediate processing) and 8 (output), control the output scale factor, 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.

In the lower left corner of each node in Figure 7 is one of four symbols representing the action performed at that node: (a) An X (nodes 1, 3, 5 and 8) represents a gain control stage where the signal is multiplied by a constant. (b) An accelerating curve (nodes 2, 4 and 6) represents a nonlinearity where the output is a power function of the input (y = xa). (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 d' function at high contrasts.

The right side of each node shows how the S-F function is altered at each node. The S-F function is represented by a broken line where the break is at Ck with the slopes of the two line segments given by 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.

We now discuss how a surround can modify specific parameters of the S-F function. To illustrate these effects Figure 8 shows separate nodes from Figure 7 (left panel) and associated plots of the S-F function (middle panels) and TvC function (right panels). The middle of the three curves in each S-F or TvC panel is the S-F or TvC function with parameters that best fit our unflanked averaged data: p=2.27, w = 0.57 (or q = p-w =1.70), K=18.4, Ck=0.040 (corresponding to th = 0.031 and ck = 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.

Figure 8a shows the effect of a gain control modulation at node 1, producing a change in threshold, th, in Equation 6, while fixing p, w, and ck. This manipulation of parameters shifts the 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).

Figure 8b shows the effect of a modification in node 2. The main effect is to change p, the log-log slope at low contrast. The semi-saturation contrast, Ck is also altered. It is interesting that the high contrast portions of the S-F curves including parameters 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:

A comparison of the blue and black curves in Figure 8b provides a good example of the seemingly paradoxical finding that inhibition can produce facilitation. In the 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.

Figure 8c shows the effect of a modification at either node 3 or 8 of Figure 7, producing a change in K in Equation 1 without changing p, w, and Ck. This manipulation of parameters simply shifts the log-log 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.

Figure 8d shows the effect of altering the gain control at node 5. This manipulation shifts the log-log 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.

Figure 8e shows the effect of a change in either node 6 or node 9. The effect here is on w, the log-log slope at high contrast while leaving unchanged th and p, the low contrast parameters of Equation 6, and also ck, 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-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.

Figure 8f shows the effect of varying the saturation point, Ck in Equation 1, by a modification at node 7. This manipulation shifts the log-log 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 Ck 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 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.

As our final example, Figure 8g shows that by combining a shift downward plus an equal shift leftward one can produce an approximately leftward shift of the TvC function. This type of shift has been reported by Chen & Tyler (2001), but it is not seen in our data. There is an ambiguity as to which nodes are involved in producing the shifts seen in Figures 8g as well as 8a, c and d, since there are four nodes (1, 3, 5 and 8) that produce translations in different directions, but a general translation can be represented by just two parameters.

A similar ambiguity is present for the slope parameters, 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.

This research is supported by National Institute of Health grants R01EY01728 and R01EY04776. We thank Alex Tauras for his help with constructing Figure 7. Commercial relationships: none.