To study spatial interactions corresponding to the non-classical receptive field organization for human vision, we used a dual-masking paradigm to measure how target contrast discrimination can be affected by the relative location of the flankers. The observers' task was to detect a 4 cycle/deg vertical Gabor superimposed on a matching Gabor pedestal in the presence of vertical Gabor flankers. The flankers were either (i) collinear with the target and varying in distance or (ii) at a fixed distance from the target but with varying in location relative to the vertical axis. Compared with the no-flanker condition, the collinear flankers *decreased* target threshold at low pedestal contrasts (facilitation) and *increased* threshold at high contrasts (suppression). The low contrast facilitation increased with distance up to 4 wavelengths and decreased beyond that. Both facilitative and suppressive flanker effects were greatest at the collinear location and decreased monotonically as flanker location deviated from the collinear axis. These flanker effects are modeled with our sensitivity modulation model, which suggests that the flanker effects are multiplicative terms applied to both the excitatory and inhibitory terms of a divisive inhibition response function. The model parameters show that the facilitative flanker effect is narrowly tuned in space. The data are not compatible with a model of additive normalization by the pedestal contrast or with the uncertainty model.

*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, and

*u*

_{ x}and

*u*

_{ y}were the horizontal and the vertical displacements of the pattern, respectively. 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 vertically oriented. The contrasts of the flankers (c) were −6 dB or 0.5. The target and the pedestal were centered at the fixation point; hence, their displacement

*u*

_{ y}was zero. Figure 1 illustrates the arrangement of flankers in different test conditions.

*u*

_{ y}) of 0.35°, 0.71°, 1.06°, 1.41°, and 2.12°, corresponding to 1.4, 2.8, 4.2, 5.6, and 8.4 times target carrier wavelength. In the azimuth condition, the center of the flankers was always 0.71° away from the center of the target but were placed at orientations of 11° (for observer CC only), 22°, 30°, 45°, and 90° away from the vertical axis. Hence,

*u*

_{ x}was parameterized as 0.71°*sin(

*θ*) and

*u*

_{ y}as 0.71°*cos(

*θ*),with

*θ*being the azimuth or the angular deviation from the vertical axis. All stimuli were presented concurrently. The temporal waveform of the stimuli was a 90-ms rectangular pulse.

*Ψ*threshold-seeking algorithm (Kontsevich & Tyler, 1999b) to measure the threshold at 75% correct response level. There were 40 trials for each threshold measurement. Each datum point reported was an average of 4 to 8 repeated measures. We randomized the sequence in which pedestal contrast and flanker distance and location were presented in each threshold measurement.

^{2}.

*λ*in Figure 4 are the same as the collinear TvC functions in Figure 2 for the respective observers. This and the no-flanker conditions are plotted here again for reference. The target threshold at the 1.4

*λ*flankers was apparently dominated by the flankers acting as a pedestal and was practically constant for the measured pedestal contrasts. The masking effect at high pedestal contrast was observed at the 2.8

*λ*to 4.2

*λ*conditions. However, there it was unclear whether the flanker effect persisted at high pedestal contrasts for long distance (>5.6

*λ*) flankers.

*λ,*the flankers elevated the target detection threshold in the zero pedestal condition by 6–9 dB (a 2- to 3-fold increase). Based on the proximity between the flankers and the targets, this pronounced masking effect may be due to the overlap between the receptive field of the target detector and the flankers. As a result, the flankers also acted as a pedestal to cause strong threshold elevation. Conversely, the 2.8

*λ*to 5.6

*λ*flankers produced pronounced facilitation for target detection at no pedestal. Two of the observers (SS and SW) even showed a robust

*facilitation*for the 8.4

*λ*flanker condition. The distance effect on target detection was consistent with previous reports (Polat & Sagi, 1993).

*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

*f*

_{ j}(

*x, y*) is defined by a Gabor function (see Methods section). (It is assumed that we are concerned with the output of the neural mechanism located for maximal activation by the image, i.e., that

*E*

_{ j}′ = max [

*E*

_{ j}′(

*x*′,

*y*′)] across the visual field.) If the image

*g*(

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

*c,*as was used in our experiment, Equation 2 can be simplified to

*S*

_{ ej}is a constant defining the excitatory sensitivity of the mechanism. The detailed derivation of Equation 2′ from Equation 2 has been discussed elsewhere (Chen, Foley, & Brainard, 2000).

*Si*

_{ j}= Σ

_{ n}(

*w*

_{ n}

*Se*

_{ j}

*) is the sensitivity of the*

^{q}*j*th mechanism to the divisive inhibition input.

*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 as 1 in our model fitting. When the flanker is presented, we simply replace

*R*

_{j}(Equation 4) by

*R*

_{j}′ (Equation 6) in Equation 7.

*p, q,*and

*z,*were the same for all conditions in each observer. (2) The sensitivity parameter (

*S*

_{e}) to the target was set at 100. (3) The contribution of the flanker to the sensitivity parameters (

*S*

_{e}and

*S*

_{i}) was set to 0 for all flankers that were more than one wavelength unit away from the target; and (4) the lateral effect parameters Ke and Ki were set to 1 in the pedestal only conditions.

*θ*° is the same as the flanker at

*θ*+ 180° and is assumed to have the same effect as the flanker at −

*θ*° (symmetry about collinear axis) and

*θ*+ 180° (symmetry about orthogonal axis). That is,

*a*

_{m}and

*b*

_{m}are free parameters, and

*m*is either

*e*or

*i*to denote excitatory and inhibitory modulation parameters. As shown below, the value of parameter

*b*

_{m}′ also depends on the distance of the flankers The fit of this equation is shown in Figure 6 as smooth curves. The bandwidth of the tuning function, measured as the half height point, was at 46°–48° for both Ke and Ki, all observers. This result is comparable with the estimation of perceptual field by Ledgeway, Hess, and Geisler (2005) with a contour integration paradigm. There is very little, if any, individual difference on the azimuth tunings.

*t*denotes the distance between the target and the flanker, and

*b*

_{m}″,

*c*

_{m}, and

*d*

_{m}are free parameters. The excitatory parameter Ke peaks between 2.1 and 3.5

*λ*(where

*λ*is the target wavelength). Previous lateral-masking studies reported that flankers produced the maximum facilitation effect on target detection when the distance between the target and the flankers was about 3

*λ*(Polat & Sagi, 1993; Solomon et al. 1999). Since the lateral effect on detection is dominated by the excitatory parameter (Chen & Tyler, 2001), our result explains such distance effects in lateral masking. The inhibitory parameter Ki peaks between 1.9 and 2.8

*λ,*closer than Ke. In contrast with the

*azimuth*tuning, which also shows a great consistency among observers, the bandwidth of the

*distance*tuning function showed notable individual differences. The half-height distance was about 1.2

*λ*for Ke and Ki across all observers at the near end (closer to the target) but ranged from 4 to 8

*λ*at the far end.

*a*

_{m}=

*a*

_{m}′/

*b*

_{m}′. We may then plug the parameter values from Equation 11 back into Equation 6 to estimate how the response of the target detector changes with the spatial location of the flanker, which is the “non-classical receptive field” (NCRF) of the target detector. As depicted in the pseudocolor plots of Figure 9, the NCRF is contrast dependent. The low contrast NCRF showed an extended excitatory region near the collinear axis of the target detector while the high contrast NCRF showed an inhibitory region along the same axis.

*σ*is the scale parameter,

*u*

_{ y}is the distance between the flanker and the target as discussed in the Methods section, and

*s*is the scale parameter of the underlying receptive field and was the only extra free parameter to be estimated. It is obvious that the best TvC prediction fits the data poorly for all three observers ( Figure 10, red curves). It predicts a much narrower range of facilitation (1–2 wavelength units) than the data shown (extending over ∼6 wavelength units).

_{ t}′ and Se

_{ m}′ are the sensitivities of the off-peak linear filter to the target and to the nearby flankers, respectively, and

*u*

_{ t}and

*u*

_{ m}are the distances between the center of the off-peak linear filter to the target and the nearby flanker respectively. The sum of

*u*

_{ t}and

*u*

_{ m}is the distance between the flanker and the target. We then fit this off-peak model to the data (green dashed curves in Figure 10). Even with two extra parameters,

*u*

_{ t}and

*s,*there is no improvement in the qualitative fit to the data and is still a gross mismatch with the form of the flanker distance function. It is important to stress that any attempt to account for the spatial interaction function in terms of the TvC function needs to be assessed by such a quantitative modeling procedure, and that any such explanation is likely to fail if the range of spatial interactions are very different for the masking and facilitatory regions of the function.

*λ*. (Conversely, the results showed continuous summation up to 6

*λ*for peripheral detection sites, but that performance is not relevant to this study.) Thus, a multiple size model cannot explain the data in Figure 10 either.

*sensitivity*of the target mechanism. The effect of the flanker is to multiply the response function by a factor (which will be an additive constant in logarithmic coordinates). When this effect is played through the generation of the TvC function, the flanker effect is to shift the high contrast portion of the TvC function horizontally to the left on a log–log plot. A detail analysis of the flanker effect of the shape of the TvC function was provided in Chen and Tyler (2001).

*increased*threshold for high contrast pedestals. Third, the target was always placed halfway between the flankers. Hence, flankers in all the azimuths should provide the same cue about the location of the target and hence have the same uncertainty reduction. Yet, the detection threshold was modulated significantly (Figure 3) with flanker azimuth. In summary, for both theoretical and empirical reasons, our results cannot be explained by a reduction in uncertainty.

*decreased*target threshold at low pedestal contrasts (facilitation) and

*increased*threshold at high contrasts (suppression). The low contrast facilitation increased with distance up to 4 wavelengths and decreased beyond that. Both facilitative and suppressive flanker effects were greatest at the collinear location and decreased monotonically as flanker location deviated from the collinear axis. This result reveals a spatial interaction field that is strongly dependent on the contrast at the target location. At low contrasts, the field of interaction on contrast discrimination is spatially extensive and is inhibitory for aligned flankers and facilitatory elsewhere. As contrast increases, the field of interactions shrinks and the configuration effects invert, becoming facilitatory for collinear flankers and inhibitory elsewhere. These effects cannot be explained in terms of the response properties of the classical receptive field, as represented by the transducer model based on the contrast discrimination function, but seem to be true spatial interactions corresponding to the non-classical receptive field recorded in cortical cells.