In this study, we investigated the functional mechanism by which spatial attention excludes unwanted information, a consequence of attention that has been consistently demonstrated at the neuronal level, the neural population level, and the overall behavioral level. The effect of spatial attention was measured using a temporal cuing paradigm. External noise, whose spatial frequency characteristics were systematically manipulated, was added to the signal stimulus. Contrast thresholds were measured as functions of the pass-band of the external noise to reveal the spatial frequency characteristics of the perceptual template in both the attended and unattended conditions. We found that spatial attention excludes external noise uniformly across all the spatial frequencies without changing the spatial frequency selectivity of the perceptual template.

^{2}(background = 27 cd/m

^{2}). Observers viewed the displays binocularly with natural pupil at a viewing distance of approximately 72 cm.

*only*one T, indicated by the arrow cue in the center of the display. In the attended condition, the cue occurred 167 ms before the target onset (Figure 2b); in the unattended condition, the cue occurred 75 ms after (Figure 2c). Across trials, the cue pointed to each of the four locations with equal probability. Two independent external noise frames, each lasting 33 ms, were shown, one immediately before and one immediately after the presentation of the Ts in each spatial region. Twelve filters, six low-pass and six high-pass, were used to generate filtered external noise (Figure 1c and 1d). The method of constant stimuli (Woodworth & Schlosberg, 1954) with seven signal contrast levels was used to estimate threshold contrasts in all the external noise and attention conditions.

*f*

_{0}= 0,

*f*

_{1}= 0.34,

*f*

_{2}= 0.68,

*f*

_{3}= 1.36,

*f*

_{4}= 2.72,

*f*

_{5}= 5.44, and

*f*

_{6}= 10.88 c/deg (Figure 1c and 1d). The gains of the

*i*

^{th}low- and the

*i*

^{th}high-pass filters are where

*σ*

_{ext}= 0.33. The contrast range of the display system was from −1.0 to +1.0. Pixel contrast distribution with a standard deviation of 0.33 conformed reasonably to Gaussian. (2) The Fourier transformation of the noise image was computed. (3) One of the 12 digital filters (Equation 1) was applied to the output form (2). (4) An inverse Fourier transformation was performed on the filtered image to produce an external noise image in real space. The filtered external noise values were then sampled at 256 equally spaced linear contrast levels from −100% to +100%.

*M*

_{i}(

*x,y*) with normalized total energy

*S*

_{j}(

*x,y*) The noisy response of the

*i*

^{th}template to the

*j*

^{th}signal is where

*ɛ*is assumed to be Gaussian distributed due to internal and/or external noise (for discussion, see The Perceptual Template Model).

*M*

_{i}(

*x,y*), the maximum sensitivity for identifying the

*i*

^{th}stimulus is obtained when the expected total difference between its output to the matched signal and that to the nonmatched signals is maximized (Duda, Hart, & Stork, 2001).

*λ*is a Lagrange multiplier. For an optimal template, the first-order derivative of Equation 7 is zero1:

*d*

_{j}(

*M*

_{i}) =

*R*

_{i}(

*M*

_{i}) −

*R*

_{j}(

*M*

_{i}) is the same for ∀

*i*and all

*j*≠

*i*. In other words, the expected “responses” of a template to all the nonmatching Ts are identical. We calculated the Fourier magnitude spectra of the optimal templates as a function of spatial frequency. The resulting functions were identical for the four optimal templates and are plotted as a single function in Figure 3d.

*T*(

*j*) in each spatial frequency range

*f*

_{j}≤

*f*<

*f*

_{j+1}(

*j*= 0, …, 5), (2) a nonlinear transducer function (∥·∥

^{γ}) whose output is a power function of its input, (3) a multiplicative internal noise whose amplitude is proportional to the total energy in the stimulus (×

*N*

_{m}), (4) an additive internal noise with mean amplitude 0 and standard deviation

*N*

_{a}, and (5) a task-dependent decision process based on the

*noisy*output. Although four templates are required to identify the orientations of the Ts, the spatial frequency characteristics of the four templates were assumed to be the same (see Optimal templates). The functional form of this model is briefed in the equations below, and is illustrated in the pattern of predictions in Figure 4b.

*S*

_{0}(

*j*) in each spatial frequency range

*f*

_{j}≤

*f*<

*f*

_{j+1}(Figure 1b) and contrast

*c*, and filtered external noise with expected Fourier power spectrum

*σ*

^{2}

_{ext}

*H*

^{passband}(

*i*). The signal in the stimulus is processed through the perceptual template and the nonlinear transducer to yield a signal output,

*S*

_{1}: where

*α*is the relative efficiency of the signal stimulus. The value of alpha scales the degree of match between the template and the signal in other domains that are not explicitly measured in the experiment (e.g., time) (Lu & Dosher, 2001).

*α*can be regarded as a parameter of the PTM model.

*S*

_{1}with variance

*Var*

_{total}(across trials) is submitted to the decision process with signal discriminability,

*d*′, determined by the signal-to-noise ratio2:

*c*

_{τ}(

*i*) — signal contrast required for the observer to reach a particular performance criterion level

*d*′ — can be expressed as a function of the pass-band and the cutoff spatial-frequency by inverting Equation 13:

*c*

_{τ}) predictions of an example PTM model with a known template. Thresholds are shown for three criterion performance levels as functions of cutoff frequencies of the low-pass and high-pass filtered external noise — the so-called “TvF” (threshold versus frequency) functions. To derive the spatial frequency characteristics of the perceptual template

*T*(

*j*) for an observer, TvF at one or several performance criterion levels are measured and then fit using Equation 14 with

*α*,

*T*(

*j*),

*N*

_{a},

*N*

_{m}, and

*γ*as parameters (Lu & Dosher, 2001)Performance signatures of attention mechanisms

*N*

_{a}by

*A*

_{a}(≤ 1.0), (2) uniform external noise exclusion across all spatial-frequencies (multiplying

*σ*

_{ext}by

*A*

_{ext}(≤ 1.0)), (3) changing the spatial-frequency characteristics of the perceptual template (replacing

*T*(

*j*) with

*T*

_{att}(

*j*) in the attended condition), and (4) multiplicative internal noise reduction (multiplying

*N*

_{m}by

*A*

_{m}(≤ 1.0)). The effect of the four mechanisms of attention can be expressed in a

**single equation after modifying the corresponding terms in**Equation 14:

*d*′s of 0.84, 1.24, and 1.68 in four-alternative forced-identification, were estimated from the psychometric functions (Wichmann & Hill, 2001a; Wichmann & Hill, 2001b). Threshold contrasts at all three performance levels are displayed in Figure 6a as functions of the cutoff spatial frequency of the filtered external noise. The four TvF functions in each panel of Figure 6a correspond to the low-pass and high-pass external noise conditions in both attended and unattended conditions.

*r*

^{2}=0.9798, 0.9802, and 0.9811). For observers KY and QL, this model is superior to all its subset models (

*p*< .001), and the model that assumes all four mechanisms does not improve the fit to the data (

*p*> .15). For observer SM, this model is superior to all its subsets (

*p*< .001), and the model that assumes all four mechanisms provides only a marginally better fit to the data (

*p*> .07).

N_{a} | N_{m} | α | γ | A_{a} | A_{ext} | r^{2} | |
---|---|---|---|---|---|---|---|

KY | .0620 | .4153 | .1165 | 1.804 | .5564 | .8529 | .9798 |

QL | .0338 | .3840 | .1635 | 2.093 | .6102 | .8942 | .9802 |

SM | .0397 | .4043 | .1227 | 1.972 | .6465 | .9072 | .9812 |

*A*

_{ext}to its

*γ*th power because both the external noise and

*A*

_{ext}pass through the nonlinear transducer function. For KY, QL, and SM,

*A*

^{γ}

_{ext}= 0.7505, 0.7913, 0.8253, reflecting the magnitude of threshold reduction in the attended condition in high external noise.

^{1}The second-order derivative of Equation 7 is also required to be negative for the optimal template.