We used a dual-masking paradigm to study how contrast discrimination is influenced by the presence of adjacent stimuli differing in orientation. The task of the observer was to detect a vertical Gabor target superimposed on a vertical Gabor pedestal in the presence of flankers. The Gabor flankers had orientations ranging from 0° (parallel to the target) to 90° (orthogonal). The flankers had two different facilitatory effects: (a) Threshold facilitation. The flankers facilitated target detection at low pedestal contrasts. This facilitation was narrowly tuned to flanker orientation. (b) Pedestal enhancement. The flankers at high contrast enhanced the masking effectiveness of the pedestal. This pedestal enhancement changed little with flanker orientation. We fitted the data with a sensitivity modulation model in which the flanker effects were implemented as multiplicative factors modulating the sensitivity of the target mechanism to both excitatory and inhibitory inputs. The model parameters showed that, (a) pedestal enhancement occurs when flanker facilitation to the pedestal is greater than to the target; (b) while the sensitivity modulation was tuned sharply with flanker orientation, the ratio between the excitatory and the inhibitory factors remained constant. The explanation of the flanker orientation effect requires the both the values of each factor and the ratio between them.

*B*was the mean luminance,

*C*was the contrast of the pattern ranging from 0 to 1,

*f*was the spatial frequency,

*σ*was the scale parameter (standard deviation) of the Gaussian envelope,

*u*was the vertical displacement of the pattern, and

*ϑ*was the orientation of the Gabor patch. All patterns had a spatial frequency (

*f*) of 4 cycles per degree and a scale parameter (

*σ*) of 0.1768°. The target and the pedestal were centered at the fixation point; hence the displacement

*u*

_{y}was zero. The two flankers were placed above and below the target with a displacement (

*u*

_{y}) of 0.75°. The target and the pedestal were vertically oriented with

*ϑ*= 0°. The flankers had orientations ranged from 0° (vertical) to 90°. The flanker orientation deviated from the target by values of 11°, 23°, 30°, 45°, 60°, and 90°. The contrasts of the flankers (

*C*) were −6dB or 0.5. All stimuli were presented concurrently with the temporal waveform of the stimuli was a 90 ms pulse.

^{2}.

*t*(6.31)=2.5046,

*p*=0.023 < 0.05), and from 2.9 to 0.2 dB for CCC (

*t*(3)=4.79,

*p*=0.0086 < 0.05). Polat and Sagi (1993) also reported a reduction of facilitation in similar conditions. At high pedestal contrasts, however, the orthogonal flankers showed similar effects to the parallel flankers. The TvC function was shifted to the left with little, if any, difference from the effect of the parallel flankers.

*t*(5.25)=1.2292,

*p*= 0.1368 >0.1; for SAS,

*t*(6.84)=0.1063,

*p*=0.4594 > 0.1). Every flanker produced a similar masking effect. Thus, the flanker masking either has no orientation tuning or is very broadly tuned to orientation.

*Between*hypercolumns (or other local subdivisions), the interaction is in the form of a lateral sensitivity modulation (shown outside the dotted box in Figure 4).

*Within*each hypercolumn, the mechanism response is influenced by other mechanisms in the same hypercolumn through a subsequent process of contrast normalization or divisive inhibition (shown within the dotted box). The original version of this model was developed to explain the variety of flanker effects on response functions of striate cortical cells (Chen & Kasamatsu, 1998; Chen et al. 2001) and the same mathematical form was later discovered to explain the psychophysical data as well (Chen & Tyler, 2001). Xing & Heeger (2001) also proposed a model of a similar form to account for lateral effects.

*j*is a linear operator within a spatial sensitivity profile

*f*

_{j}(

*x,y*). The excitation of this linear operator to an image

*g*(

*x,y*) is given as where the linear filter

*f*

_{j}(

*x,y*) is defined by a Gabor function (see Methods section). If the image

*g*(

*x,y*) is a periodic pattern with contrast

*C*, as was used in our experiment, Equation 1 can be simplified to where

*Se*

_{j}is a constant defining the excitatory sensitivity of the mechanism. Detailed derivation of Equation 1′ from Equation 1 has been discussed elsewhere (Chen, Foley & Brainard, 2000).

*j*-th mechanism to the divisive inhibition input.

*Ke*and

*Ki*denote the sensitivity modulation factors to the excitatory and the inhibitory inputs respectively. Therefore, the response function with the presence of flankers becomes

*Ke*and

*Ki*are functions of flanker contrast. However, in the experiment reported in this paper, only two flanker contrasts (0% and 50%) were used. Therefore, we simply take

*Ke*and

*Ki*to have a value of 1 when the flanker contrast is 0 (thus reducing Equation 5 to Equation 3) and as free parameters to be estimated when the flanker contrast is 50%. As shown below (see sec. 4.2), both

*Ke*and

*Ki*are required in order to account for different aspects of the flanker effect. In our experiment, we measured the target threshold on a pedestal using a 2AFC paradigm in which the observer has to discriminate a target superimposed on a pedestal from the pedestal alone. Suppose the observer’s performance is determined by the local mechanism that gives the greatest response difference between the two intervals. When there are no flankers, the difference in response is given as where

*j*is the mechanism that gives the greatest response difference,

*b*denotes the pedestal contrast and

*b*+

*t*denotes the target-plus-pedestal contrast. The target reaches the threshold when its contrast increases by a certain amount (Legge & Foley, 1980), designated 1 in our model fitting. When the flanker is presented, we simply replace

*R*

_{j}(from Equation 3) by

*R′*

_{j}(from Equation 5) in Equation 6.

CCC | SAS | |||
---|---|---|---|---|

TvC function | Se | 100* | 100* | |

parameters | Si | 106 | 95 | |

P | 2.98 | 2.80 | ||

Q | 2.28 | 2.15 | ||

σ | 35.79 | 10.18 | ||

Lateral | Ke** | 0° | 2.58 | 2.72 |

modulation | 11° | 1.66 | ||

factors | 22° | 1.42 | ||

Ke** | 30° | 2.06 | ||

45° | 1.33 | |||

60° | 2.11 | |||

90° | 1.29 | 2.11 | ||

Ki** | 0° | 3.28 | 3.67 | |

11° | 2.24 | |||

22° | 1.96 | |||

30° | 2.58 | |||

45° | 1.83 | |||

60° | 2.56 | |||

90° | 1.66 | 1.99 |

*Ke*and

*Ki*represent the strength of the lateral effects received by the target mechanism. The parameter

*Ke*is required to account for the facilitation that occurs at zero or low pedestal contrasts (Polat & Sagi, 1993, 1994). Given the parameter values, when the pedestal is not presented and the target is near threshold, the magnitude of the divisive inhibition term

*I*(Equation 5) is negligible compared with the additive constant

*σ*. Thus, in this scenario, Equation 6 can be simplified to

*Ke*raised to a power of 1/

*p*. If

*Ke*is larger than 1, the target threshold decreases. This result explains the lateral masking effect found by Polat and Sagi (1993, 1994), the in-phase flanker effect of Solomon et al. (1999) and the initial flanker facilitation at lower end of the TvC functions.

*Ke*and

*Ki*. When the pedestal contrast is sufficiently high, the additive constant (

*σ*) is negligible compared with the inhibition term (

*I*) in the response function (Equaton 5). Thus, we can simplify the response without the flankers as (

*E*

^{p}/

*I*) and the response with flankers as (

*Ke*/

*Ki*)*(

*E*

^{p}/

*I*). That is, the response function with flankers is the ratio between

*Ke*and

*Ki*multiplied by the no-flanker response function. Translating the responses to thresholds, the threshold difference between the flanker and the no-flanker conditions is proportional to the ratio

*Ki*/

*Ke*. Since

*Ke*and

*Ki*are independent of pedestal contrast, this ratio gives a parallel shift of TvC functions horizontally on log-log coordinates. The presence of the flankers reduces the responses and increases the thresholds, consistent with

*Ki*being greater than

*Ke*.

*equivalent contrast*. The flankers have a facilitatory effect on both the target and the pedestal. When there is no pedestal or the pedestal is weak, one only needs to consider the flanker effect on the target. Due to the flanker facilitation, a target with a particular contrast in the flanker condition produces the same response in the system as a target with a higher contrast in the no-flanker condition. Hence, the threshold in the flanker condition is lower than in the no-flanker condition. When the pedestal contrast is high, in addition to the facilitation on the target, which pushes the TvC functions down, it becomes necessary to consider the flanker effect on the pedestal. Empirically, we find that the pedestal is effectively facilitated by the flankers, producing the same effect as a pedestal with a higher contrast in the no-flanker condition. This facilitatory effect is essentially the same as pushing the TvC function leftward in logarithmic coordinates. Since the target threshold at this part of TvC functions increases with pedestal contrast, a leftward shift means that the target threshold increases in the flanker condition at the same pedestal contrast relative to the no-flanker condition. The net result of these two processes seen in the data (Figure 1 and 2) is the sum of a leftward movement produced by the facilitation of the pedestal and a downward movement by the facilitation of the target.

*Ke*and

*Ki*change with flanker orientation. Both parameters drop quickly as the flanker orientation deviates from the target orientation. This effect is more obvious with the parameters for CCC’s data (Panel b), which has more sample values at small orientation differences. The change of parameters can be characterized by a linear combination of two Gaussian functions of flanker orientation (smooth curves). One Gaussian is narrowly tuned with a scale parameter (“standard deviation”) for Ke of 2.55° (SAS) or 4.49° (CCC), and for

*Ki*of 2.56° (SAS), or 4.48° (CCC); and the other is broadly tuned with scale parameters for

*Ke*of 72.43° (SAS) or 77.47° (CCC), and for

*Ki*of 63.88° (SAS), or 72.21° (CCC). The similarities in the tuning functions for

*Ke*and

*Ki*are consistent with the idea that the excitatory and inhibitory lateral modulation effects are from the same source and act on different agents in the target mechanism. We acknowledge that the Gaussian parameters for SAS are less constrained due to the limited number of samples, but they are nevertheless of similar magnitudes to those for CCC. One Gaussian function of flanker orientation cannot capture the behavior of

*Ke*and

*Ki*as it provides a much poorer fit to the data (

*F*(8,10) = 37.87,

*p*< 0.0001). Thus, it is clear that there must be two components for both the excitatory and inhibitory lateral modulations: one narrowly tuned to flanker orientation and the other broadly tuned.

*Ke*and

*Ki*drop rapidly with flanker orientation. Since the target threshold at low pedestal contrast is determined by the value of

*Ke*, this dramatic change of Ke reflects the narrow flanker orientation tuning in target thresholds in low pedestal contrast.

*Ki*is greater than that of

*Ke*, it decreases with flanker orientation at about the same rate as does Ke. As a result, the ratio

*Ki*/

*Ke*(green open triangles in Figure 5) is roughly constant for all flanker orientations. This constant

*Ki*/

*Ke*ratio is reflected in the data as the flanker suppression that is broadly tuned in orientation.

*magnitude*of the response. Contrast discrimination experiments, which measure the increment threshold from a base contrast as shown in section 4.1, concern the

*slope*of the contrast response function in relation to the prevailing noise. Hence, it is meaningless to compare directly the discrimination and matching data. It is possible that, in the same experimental setup, discrimination threshold increases (slope of the response function is flatter) while the apparent contrast also increases (the magnitude of the response to base contrast increases). Yu, Klein, and & Levi (2001) actually reported in the same study that the surround showed different effects on contrast discrimination and contrast matching, and were puzzled by that difference. Nevertheless, it is possible to derive the response function from the discrimination performance, as shown in section 4.1 (Equations 5 & 6). The magnitude of the response is proportional to

*Ke*/

*Ki*(see sec. 4.2). From Table 1, it is easy to determine that, on average, the ratio

*Ke*/

*Ki*between 0° and 90° changes from 0.76 to 0.83 or a 9% increase (for SAS, 0.74 to 0.88; and for CCC, 0.79 to 0.78). This change, though close to zero, is comparable with the “slight facilitation” reported by Yu, Klein, and Levi. It is evident, therefore, that the apparently contradictory prior results are in fact compatible with our model.

*λ*) of the measured flanker effects reported by Polat and Sagi.

*E*and

*I*in Equation 5) terms in the response function. Its contribution to the mechanism response is added to that of the target. Suppose that the flanker contrast is constant in the flanker conditions as in our experiments, then equating the flankers to a weak pedestal is equivalent to increasing

*E*and

*I*in Equation 5 by a constant. On the other hand, the contribution of the pedestal to

*E*and

*I*, and in turn the response, increases with pedestal contrast. Thus, the TvC function in the presence of the flankers will converge to the TvC function without any flankers as pedestal contrast increases. Snowden & Hammett (1998) derived the same prediction for contrast discrimination in the presence of a patterned surround.

*multiplicative*factor that modulates the responses of the target mechanism.