We investigated how two co-aligned adjacent stimuli (flankers) influence threshold versus pedestal contrast (TvC) functions in binocular, monocular, and dichoptic presentations. Targets were presented to the two eyes or to only one eye. Pedestals and flankers were presented to the same eye to which the target was presented (binocular or monocular presentations) or to the other eye (dichoptic presentation). In the binocular presentation of targets and pedestals, the binocular flankers lowered thresholds at low pedestal contrasts. The monocular flankers had a similar effect to the binocular flanker, although the threshold reduction was smaller. In the dichoptic presentation of a target and a pedestal, flankers lowered thresholds when flankers were presented to the eye where targets were presented. In contrast, dichoptic flankers elevated thresholds at intermediate pedestal contrasts when a pedestal was also dichoptically presented. We fitted binocular contrast gain control models to the data. It follows from the fitting results that flankers modulate outputs from spatial filters in the monocular processing stage of contrast gain control.

*threshold versus pedestal contrast*(TvC) function. Although several researchers have reported that binocular flankers elevate contrast discrimination thresholds (Chen & Tyler, 2001; Zenger-Landolt & Koch, 2001), it is still unclear how they change in the presence of monocular or dichoptic flankers. These experiments addresses three issues that need to be resolved to better understand the mechanisms underlying flank facilitation. First, is there a difference in threshold reduction between the binocular and monocular flankers? Second, how does dichoptic presentation modulate sensitivity for discrimination? Third, using the two current models (Maehara & Goryo, 2005; Meese, Georgeson, & Baker, 2006) of binocular combination, are the facilitative effects of flankers, additive or multiplicative? The answer to these questions bears upon the nature and site of interactions between spatial filters that occur in early visual processing.

^{2}. The viewing distance was 57 cm.

^{2}. Vertical Gabor patterns were used for GM and PCH. Because it was possible that the interocular alignment was not perfect for vertical Gabor patterns, horizontal Gabor patterns were used for JB. Horizontal patterns are less affected by misalignment issues.

*dichoptic flankers*when a target and flankers were presented to different eyes. The no-flanker condition was tested for each presentation conditions of targets and pedestals. There were 12 levels of pedestal contrast (−∞, −52.8, −48.7, −44.5, −40.7, −36.4, −32.4, −28.3, −24.2, −20.1, −16.0, and −12.0 dB). That is, the experimental design was defined by a combination of the pedestal presentation (3), the flanker presentation (2 or 3), and the pedestal contrast (12).

*F*(11, 22) = 3.54,

*p*= 0.006. As a whole, thresholds were not significantly different between the no-flanker and binocular-flanker conditions,

*p*> 0.05. The TvC functions had the typical dipper shape, as reflected in the significant quadratic main effect of Pedestal Contrast,

*F*(1, 2) = 3837,

*p*< 0.001. Although Chen and Tyler (2001) also reported the threshold reduction in the presence of binocular flankers, their results were different from ours in that flankers elevated thresholds throughout the high pedestal contrast range (−30 to −10 dB).

*F*(2, 4) = 12.4,

*p*= 0.019. We can see from the center panels of Figure 2 that the flankers lowered thresholds. On the other hand, data points almost overlap between the no-flanker and dichoptic-flanker conditions (red circles and cyan diamonds). That is, the dichoptic flankers were not effective. The Flanker main effect was qualified by the significant Flanker × Pedestal Contrast interaction that the monocular flankers lowered thresholds at low pedestal contrasts but not at high pedestal contrasts,

*F*(22, 44) = 3.68,

*p*< 0.001. The threshold reduction looks smaller for the monocular flankers than for the binocular flankers. Actually, the monocular flankers lowered the mean detection thresholds by 2.1 dB while the binocular flankers lowered it by 3.4 dB, although the

*t*-test showed no significant difference between them,

*p*> 0.05. The TvC functions had the dipper shape in the monocular presentation, as reflected in the significant quadratic main effect of Pedestal Contrast,

*F*(1, 2) = 581,

*p*= 0.002, as well as in the binocular presentation, as reported previously.

*F*(2, 4) = 15.5,

*p*= 0.013. This threshold reduction was limited to low pedestal contrasts, producing a significant Flanker × Pedestal Contrast interaction,

*F*(22, 44) = 4.29,

*p*< 0.001. The right panels of Figure 2 show that the dichoptic flankers elevated thresholds at middle pedestal contrasts. This would also contribute to the significant interaction. The TvC functions showed threshold elevation at high pedestal contrasts with a very small dip, as reflected in the significant quadratic main effect of Pedestal Contrast,

*F*(1, 2) = 55.1,

*p*= 0.018.

*E*

_{ ij}″ and monocular inhibitory signal

*I*

_{ ij}″. The monocular excitation produced by pattern component

*i,*which is presented to eye

*j,*is

*C*

_{ ij}is component contrast and

*S*

_{ E}is the excitatory sensitivity of the mechanism either to target or pedestal. Since excitation is a linear process, monocular excitation produced by target plus pedestal is the sum of their individual excitations:

*E*

_{L}and

*E*

_{R}originate in the left and right eyes, respectively. These monocular excitations are raised to power

*m*(nonlinear transducer) and then summed to yield binocular excitation

*E*:

*S*

_{ I}is an inhibitory sensitivity of mechanism. Then, the monocular inhibitory signals are half-wave rectified. The binocular inhibitory signal

*I*is a sum of the rectified signals raised to power

*n*:

*p*or

*q*after the summation of monocular signals. Then, the mechanism response

*R*is computed as the binocular excitation divided by the binocular inhibitory signal plus a semi-saturation constant

*z*. These calculations are expressed as

*R*

_{ t+ p}) exceeds the response to the pedestal alone (

*R*

_{ p}) by a constant value. Stated more specifically, behavioral thresholds depend on the value of the decision variable

*D*:

*D*= 1.

*K*

_{Ej}and

*K*

_{Ij}are modulation factors for monocular excitations and monocular inhibitory signals, respectively. The additive modulation factors have positive values when flankers are presented. Although there are different factors for the left and right eyes, their values are the same between two eyes (

*K*

_{EL}=

*K*

_{ER};

*K*

_{IL}=

*K*

_{IR}) for the binocular flankers. For the no-flanker condition, all the modulation factors are set to be 0, being ineffective. In the case that flankers are presented to only one eye (monocular or dichoptic flanker), modulation factors are effective only for the eye where flankers are presented whereas they are ineffective for the other eye.

*S*

_{ I},

*m, n, p, q,*and

*z*were free parameters that were not fixed in advance.

*S*

_{ E}was a fixed parameter. The red smooth curves in the top panels of Figure 2 correspond to the best fit for the mean data. The root mean squared errors (RMSEs) of the fits were 0.655 dB for the mean data, 0.810 dB for GM, 1.13 dB for PCH, and 1.25 dB for JB. These errors were close to the mean SEs of thresholds for the no-flanker condition (1.82 dB for the mean, 0.701 dB for GM, 0.938 dB of PCH, and 1.17 dB for JB) and similar to fitting errors reported in the previous studies (Maehara & Goryo, 2005; Meese et al., 2006). After the first fitting, the model was fitted to the data for the conditions with flankers. The modulation factors were set to be free for the second fit. Other parameters were fixed to the values estimated by the first fit. The blue and cyan curves in the middle row panels of Figure 2 correspond to the best fit to the mean data (see Figure S1 in the supplementary materials for individual fits). The RMSEs of the second fits were 1.01 dB for the mean data, 1.25 dB for GM, 1.33 dB for PCH, and 1.75 dB for JB (Table 2). These errors were close to the mean SEs of thresholds for conditions with flankers (1.50 dB for the mean, 0.773 dB for GM, 1.01 dB for PCH, and 0.925 dB for JB). The fits were reasonably good.

Twin summation model | |||||||||
---|---|---|---|---|---|---|---|---|---|

S _{ E} | S _{ I} | m | n | p | q | z | SSE | RMSE | |

Mean | 100 | 56.0 | 1.69 | 1.52 | 1.65 | 1.29 | 4.52 | 15.4 | 0.655 |

GM | 100 | 60.9 | 2.05 | 1.34 | 2.07 | 1.14 | 12.5 | 23.6 | 0.810 |

PCH | 100 | 46.6 | 1.70 | 1.37 | 1.43 | 1.36 | 2.25 | 45.7 | 1.13 |

JB | 100 (fixed) | 61.6 | 1.39 | 2.02 | 1.45 | 1.60 | 4.46 | 56.2 | 1.25 |

Two-stage model | |||||||||

S _{ E} | m | s | p | q | z | d | SSE | RMSE | |

Mean | 100 | 1.81 | 1.60 | 1.28 | 30.2 | 0.155 | 0.130 | 19.8 | 0.742 |

GM | 100 | 1.96 | 1.57 | 1.30 | 32.8 | 0.396 | 0.160 | 20.6 | 0.757 |

PCH | 100 | 2.20 | 1.35 | 1.06 | 4.6 | 1.20 | 0.187 | 61.3 | 1.31 |

JB | 100 (fixed) | 1.47 | 3.56 | 2.70 | 1.3 | 1.30 | 0.231 | 64.6 | 1.34 |

Twin summation model with additive modulation | ||||
---|---|---|---|---|

K _{ Ej} | K _{ Ij} | SSE | RMSE | |

Mean | 0.729 | 0.466 | 60.6 | 1.01 |

GM | 1.27 | 0.959 | 94.5 | 1.25 |

PCH | 0.502 | 0.205 | 107 | 1.33 |

JB | 0.600 | 0.529 | 183 | 1.75 |

Twin summation model with multiplicative modulation | ||||

K _{ Ej} | K _{ Ij} | SSE | RMSE | |

Mean | 1.74 | 1.97 | 54.2 | 0.951 |

GM | 2.22 | 2.70 | 129 | 1.47 |

PCH | 1.62 | 1.64 | 108 | 1.34 |

JB | 1.54 | 1.84 | 167 | 1.67 |

Two-stage model with additive modulation | ||||

K _{ Ej} | K _{ Ij} | SSE | RMSE | |

Mean | 0.612 | 0.000 | 61.7 | 1.01 |

GM | 1.06 | 0.000 | 96.7 | 1.27 |

PCH | 0.493 | 0.324 | 103 | 1.31 |

JB | 0.250 | 0.000 | 214 | 1.89 |

Two-stage model with multiplicative modulation | ||||

K _{ Ej} | K _{ Ij} | SSE | RMSE | |

Mean | 1.62 | 2.18 | 109 | 1.35 |

GM | 2.03 | 3.34 | 217 | 1.90 |

PCH | 2.27 | 2.40 | 112 | 1.37 |

JB | 2.52 | 2.75 | 200 | 1.83 |

*E*

_{L}and

*E*

_{R}are raised to power

*m*and divided by a sum of the two monocular excitations and a constant

*s,*yielding the first-stage outputs

*F*

_{L}and

*F*

_{R}:

*p,*and then subjected to the second-stage divisive inhibition. This calculation is expressed as

*R*is the mechanism response and

*z*is a constant. The inhibition from other mechanisms is also omitted here. A target contrast will be the threshold when the response to the target plus pedestal (

*R*

_{ t + p}) exceeds the response to the pedestal alone (

*R*

_{ p}) by a constant value

*d*. That is,

*D*=

*d*at the threshold. The constant

*d*is one of the free parameters, whereas it was fixed to be 1 in the twin summation model.

*z*was close to 0 (e.g., 0.0044) whereas a constant

*s*had an extremely large value (e.g., 907). Since this makes the two-stage model work in a different way from its design, we excluded fits where a constant

*z*was estimated to be smaller than 0.02.

*m, p, q, s, z,*and

*d*were free to vary. The RMSEs of the fits were 0.742 dB for the mean data, 0.757 dB for GM, 1.31 dB for PCH, and 1.34 dB for JB ( Table 1). These errors were comparable with RMSEs of the fits of the twin summation model.

*K*

_{ Ij}were estimated to be 0 for GM and JB ( Table 1). This means that only the excitatory modulation possibly explains the flanker effect.

*s*or

*z*. For example, if a semi-saturation constant

*z*was removed, the mechanism response of the twin summation model for the binocular presentation will become

*K*

_{ Ej}and

*K*

_{ Ij}cancel mutually. A similar cancellation occurs also in the two-stage model if the constant

*s*is removed from Equation 16. Therefore, the fits show that flankers are not effective at very high pedestal contrasts (bottom panels of Figure 2). The monocular flankers with the dichoptic pedestal shift TvC functions only downward because the dichoptic flankers do not affect the response to a pedestal presented to the other eye. In contrast, the dichoptic flankers with a monocular pedestal shift TvC functions leftward. In both cases, flankers induce no threshold change at very high pedestal contrasts due to the cancellation of modulation factors.

S _{ E} | Excitatory sensitivity parameters of spatial filters |

S _{ I} | Inhibitory sensitivity parameters of spatial filters |

m | Exponents for monocular excitations |

n | Exponents for monocular inhibitory signals |

p | Exponents for binocular excitations |

q | Exponents for binocular inhibitory signals |

z | Semi-saturation constants |

s | Semi-saturation constants for the first-stage contrast gain control |

d | Response differences necessary to detect a target |

K _{ Ej} | Excitatory modulation factors for eye j |

K _{ Ij} | Inhibitory modulation factors for eye j |