To clarify the role of lateral interaction and uncertainty reduction in collinear flanker facilitation of contrast discrimination, we investigated the effect of flankers and spatial cueing on the contrast discrimination threshold. The task of the observers was to detect a 1.3 cyc/deg vertical Gabor target superimposed on one of the Gabor pedestals presented at 3° to the left and right of the fixation point. We measured the target threshold vs. pedestal contrast (TvC) functions in four (2 × 2) conditions: With or without the presence of collinear flankers and with or without a spatial cue that indicated the location of the target. The presence of the flankers lowered the target threshold at low pedestal contrasts, but increased the threshold at high contrasts. The presence of spatial cues lowered the target threshold at all pedestal contrasts regardless of the presence of flankers. The flanker effect was similar in cued and non-cued conditions, suggesting that the effect was similar regardless of cues. The TvC functions are well fit by a divisive inhibition model in which the presence of collinear flankers increases both the excitatory and inhibitory terms of a divisive inhibition response function, while cueing increases only the excitatory term.

_{c}

^{2}+ E

_{A}

^{2})

^{1/2}where E

_{c}is the effect of the cue and E

_{A}is the effect of the factor A. Thus, given that the cue alone can account for 65% of the total facilitation effect in the cue-plus-flanker condition, the other factor A, when presented alone, should be able to account for 76% of total facilitation. Since neither the nonlinearity nor the effect of other factors was assessed in Petrov et al. (2006), it remains inconclusive whether the collinear flanker effect is a cueing effect. Third, their evidence for uncertainty reduction came from the cueing effect. While it is common to explain the cueing effect in terms of uncertainty reduction (Foley & Schwarz, 1998), this notion is at odds with some recent studies which consider the cueing effect as a change in sensitivity of the system (Lu & Dosher, 1998). Hence, the nature of the cueing effect in collinear facilitation requires further scrutiny.

*per se*. The cueing effect provided by the flanker would be negligible compared to that of the cue itself. Thus, it allows us to estimate the effect of factors other than cueing in flanker facilitation.

^{2}.

*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 vertical displacements of the pattern respectively. All patterns had a spatial frequency (

*f*) of 1.3 cycles per degree and a scale parameter (

*σ*) of 0.35 deg. The pedestal and the target were placed 3 deg to the left and/or right of the central fixation point, with the displacement (

*u*

_{ y}) of zero. The displacement of flankers (

*u*

_{ y}) above and below the target is 2.31 deg, corresponding to 3 times target carrier wavelength. The contrast of the flankers (

*c*) was −6 dB or 50%.

*f*

_{j}(

*x, y*) defined by a Gabor function. The excitation of this linear operator to an image

*g*(

*x, y*) is given as

*g*(

*x, y*) is a periodic pattern with a contrast c, as we used in our experiment, Equation 2 can be simplified to

_{j}is a constant called the excitatory sensitivity of the j-th channel. Detailed derivation from Equations 2 to 2′ has been discussed elsewhere (Chen, Foley, & Brainard, 2000).

_{j}plus an additive constant z. That is,

_{j}is the summation of a non-linear combination of the rectified excitations of all relevant channels within the same hypercolumn feeds back to channel j. This divisive inhibition term I

_{j}is

_{j}is a positive value serving as an inhibitory sensitivity. If the uncertainty level throughout the whole experiment is the same, in a 2AFC experiment as ours, the observer's performance can be determined by one channel that gives the greatest response difference between the two intervals (Tyler & Chen, 2000). The difference in response is given as

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Baseline | Cue | Flankers | Cue plus flankers | Baseline | Cue | Flankers | Cue plus flankers | Baseline | Cue | Flankers | Cue plus flankers | Baseline | Cue | Flankers | Cue plus flankers | Baseline | Cue | Flankers | Cue plus flankers | |

Se | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |

Si | 122.10 | 122.10 | 122.10 | 122.10 | 128.82 | 128.82 | 128.82 | 128.82 | 126.04 | 126.04 | 126.04 | 126.04 | 129.42 | 129.42 | 129.42 | 129.42 | 132.65 | 132.65 | 132.65 | 132.65 |

Ke | 1 | 1 | 1 | 1 | 1 | 1 | 1.33 | 1.33 | 1 | 1 | 1.34 | 1.34 | 1 | 1 | 1.32 | 1.32 | 1 | 1 | 1.35 | 1.35 |

Ki | 1 | 1 | 1 | 1 | 1 | 1 | 2.85 | 2.85 | 1 | 1 | 2.87 | 2.87 | 1 | 1 | 2.63 | 2.63 | 1 | 1 | 2.93 | 2.93 |

Ka | 1 | 1 | 1 | 1 | 1 | 2.02 | 1 | 2.02 | 1 | 1 | 1 | 1 | 1 | 1.31 | 1 | 1.31 | 1 | 1 | 1 | 1 |

Kb | 1 | 1 | 1 | 1 | 1 | 1.59 | 1 | 1.59 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.78 | 1 | 0.78 |

z | 129.82 | 129.82 | 129.82 | 129.82 | 615.17 | 615.17 | 615.17 | 615.17 | 388.76 | 388.76 | 388.76 | 388.76 | 334.64 | 334.64 | 334.64 | 334.64 | 438.88 | 438.88 | 438.88 | 438.88 |

p | 2.68 | 2.68 | 2.68 | 2.68 | 3.13 | 3.13 | 3.13 | 3.13 | 3.07 | 3.07 | 3.07 | 3.07 | 2.95 | 2.95 | 2.95 | 2.95 | 3.17 | 3.17 | 3.17 | 3.17 |

q | 2.20 | 2.20 | 2.20 | 2.20 | 2.60 | 2.60 | 2.60 | 2.60 | 2.51 | 2.51 | 2.51 | 2.51 | 2.44 | 2.44 | 2.44 | 2.44 | 2.63 | 2.63 | 2.63 | 2.63 |

m | 1.03 | 0.68 | 2.40 | 1 | 1 | 1 | 1 | 1 | 1.92 | 1 | 1.92 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

n | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

SSE | 64.49 | 26.88 | 58.12 | 33.40 | 41.92 | |||||||||||||||

Free parameters | 7 | 8 | 7 | 7 | 7 |

*d*′, and in turn the threshold changes with the ratio of the monitored channel, m, and the relevant channel, n (Tyler & Chen, 2000) but not the number of the monitored channel itself. Hence, any change of m will be balanced out by the proportional change in n. Hence, set n = 1 would not reduce the goodness-of-fit of the model as long as m is a free parameter.

*F*(1,24) = 4.69,

*p*= .04). In this model, the effect of the flankers is similar to Model 2. The cue increases the excitatory term by 31% (Ka = 1.31) and thus shifts TvC functions downward relative to the non-cue condition.

*F*(1,24) = 8.61,

*p*=.007).

*p*> .01). Hence, we do not favor Model 2, on the principle of parsimony. Similarly, Model 5 is not favored because it fits the data worse than Model 4 even though the number of the free parameters are the same. The RMSE of Model 4, 1.02 dB, is close to the mean standard error of the data (0.97 dB). The model accounts for 94.5% of the variance of the averaged data and gives an excellent description of the data (

*X*

^{2}(31,

*N*= 5) 10.24,

*p*= .9999).

*X*

^{2}measurement for goodness-of-fit reached unity. This parameter increment and decrement defined the 68% confidence interval. Under the Gaussian noise assumption, the range of this confidence interval was multiplied by 1.96 to get the 95% confidence interval. Table 2 shows the results. In general, the fit parameter values for this model were quite stable.

Parameter | Estimated value | 95% confidence interval |
---|---|---|

Si | 129.42 | 127.22 ∼ 131.54 |

Ke | 1.32 | 1.11 ∼ 1.57 |

Ki | 2.63 | 1.36 ∼ 3.97 |

Ka | 1.31 | 1.19 ∼ 1.50 |

z | 334.64 | 234.22 ∼ 386.27 |

p | 2.95 | 2.28 ∼ 4.13 |

q | 2.44 | 1.93 ∼ 3.13 |

*t*(4) = 3.14,

*p*= 0.034), it was only about 1/3 of the cueing effect measured in the main experiment. Therefore the cueing effect we measured in the main experiment can hardly be due to forward masking.