**Abstract**:

**Abstract**
**What are the temporal dynamics of perceptual sampling during visual search tasks, and how do they differ between a difficult (or inefficient) and an easy (or efficient) task? Does attention focus intermittently on the stimuli, or are the stimuli processed continuously over time? We addressed these questions by way of a new paradigm using periodic fluctuations of stimulus information during a difficult (color-orientation conjunction) and an easy (+ among Ls) search task. On each stimulus, we applied a dynamic visual noise that oscillated at a given frequency (2–20 Hz, 2-Hz steps) and phase (four cardinal phase angles) for 500 ms. We estimated the dynamics of attentional sampling by computing an inverse Fourier transform on subjects' d-primes. In both tasks, the sampling function presented a significant peak at 2 Hz; we showed that this peak could be explained by nonperiodic search strategies such as increased sensitivity to stimulus onset and offset. Specifically in the difficult task, however, a second, higher-frequency peak was observed at 9 to 10 Hz, with a similar phase for all subjects; this isolated frequency component necessarily entails oscillatory attentional dynamics. In a second experiment, we presented difficult search arrays with dynamic noise that was modulated by the previously obtained grand-average attention sampling function or by its converse function (in both cases omitting the 2 Hz component to focus on genuine oscillatory dynamics). We verified that performance was higher in the latter than in the former case, even for subjects who had not participated in the first experiment. This study supports the idea of a periodic sampling of attention during a difficult search task. Although further experiments will be needed to extend these findings to other search tasks, the present report validates the usefulness of this novel paradigm for measuring the temporal dynamics of attention.**

*sequentially*on the stimuli (Treisman & Gelade, 1980; Wolfe, 1998; Wolfe et al., 1989), acting as a “spotlight” that switches from one stimulus (or group of stimuli) to another (Vanrullen, Carlson, & Cavanagh, 2007), or whether it processes them all at the same time in a continuous or

*parallel*manner (Eckstein, Thomas, Palmer, & Shimozaki, 2000; Palmer, Ames, & Lindsey, 1993). The first hypothesis—but a priori not the second—should predict that difficult visual search involves a periodic temporal dynamic of attentional sampling (Vanrullen & Dubois, 2011).

*A(t)*that we aim to measure over the interval of 0 to 0.5 s. Our working definition of attention entails that stimulus information

*S(t)*at different moments in time will weigh more or less on the perceptual decision as a function of the value

*A(t)*at that time; in other words, the subjects' perception can be approximated (potentially with an additive or multiplicative constant) by a linear combination of stimulus information and attentional sampling: For a given temporal frequency

*ω*, we call P

_{cosω}, P

_{sinω}, P

_{-cosω}, and P

_{-sinω}can be directly estimated by presenting a visual stimulus whose temporal profile is modulated with a cosine, sine, –cosine, or –sine function (respectively) and measuring the resulting perceptual performance (e.g., d-primes). This is the purpose of Experiment 1 (see Figure 1). In theory, some of these measures should be expected to provide redundant results: because P

_{cosω}= −P

_{-cosω}and P

_{sinω}= −P

_{-sinω}, only one measurement for each pair should (in theory) be sufficient. In practice, however, one can imagine that modulating the visual stimulus at a particular frequency could induce performance changes independently of the exact phase (for example, if faster modulation frequencies cause an increase of arousal or vigilance). Therefore, we systematically measured the effect of each phase modulation (cosine, sine) and their inverse (−cosine, −sine) on performance. Any unwanted factor that would equally affect all modulation phases at a given frequency would then be eliminated in the following equations.

*as C*

_{ω}*= (P*

_{ω}_{cosω}− P

_{-cosω})/2 − i. (P

_{sinω}− P

_{-sinω})/2. In other words, Fourier series analysis implies that the attentional sampling function should be proportional to In other words, measuring the complex coefficients C

*every 2 Hz (from 2 to 20 Hz) will give us an estimate of the attentional function (note that values greater than 20 Hz are not considered here for practical purposes, although of course they may be relevant for attention).*

_{ω}*n*= 10,000), and the amplitude spectra of surrogate attentional functions (either based on grand-average attentional functions or on individual functions) were used to estimate significance. To achieve this, the 10,000 surrogate amplitude spectra were ranked in ascending order, separately for each frequency. The 9,501th, 9,901th, 9,991th, and 10,000th values were considered as the respective limits of four different confidence intervals (95%, 99%, 99.9%, and 99.99%), which are represented with different colors in the background of the four corresponding graphs. An experimentally observed spectral amplitude value was considered significantly different from the corresponding null distribution with

*p*< 0.05 if it exceeded the 95% confidence threshold (and with

*p*< 0.01 above the 99% confidence threshold, and so on). To take into account the possible problem of multiple statistical comparisons across the 10 frequencies used (2–20 Hz), we adopted a conservative statistical threshold of

*p*< 0.001. For both tasks, we also ran the same analysis, discarding the signals from stimulus onset (0–83 ms) and stimulus offset (417–500 ms). This was aimed at verifying that any periodicity highlighted by the previous analysis would not be due merely to an increased sensitivity to the onset and/or the offset of the stimuli.

*t*tests, according to the prediction that a search array presented with an SNR following the estimated attentional function should lead to better search performance than the opposite signal.

*n*= 14 for both tasks), we verified that subjects used the appropriate search strategy for the two tasks that we intended to use in Experiment 1: finding a + among Ls or a conjunction of color (red or green) and orientation (30° or 330° from upright). We presented the stimuli for unlimited durations with a set size randomly drawn between four and eight elements and computed the RT × set size slopes. As expected, the slopes were near zero for the L versus + task: 7.7 ms ± 7.9 ms per element,

*t*(13) = 3.1,

*p*= 0.01, for target present, and 11.6 ms ± 6.7 ms per element,

*t*(13) = 5.5,

*p*< 0.01, for target absent—

*t*tests performed under the null hypothesis that the slopes are equal to zero (i.e., the task involved minimal attention). Slopes were strongly positive for the conjunction task: 70 ms ± 7.7 ms per element,

*t*(13) = 25.6,

*p*< 0.0001, for target present, and 184.6 ms ± 27.4 ms per element,

*t*(13) = 19.1,

*p*< 0.0001, for target absent (i.e., the task involved significant attentional resources).

*p*< 10

^{−3}, a conservative statistical threshold due to multiple comparisons across frequencies). In Figure 4B, the amplitude spectrum was computed first on each individual sampling function and subsequently averaged. In this case, the spectrum presented an overall decreasing shape, significantly different from chance at frequencies of 2, 4, and 6 Hz. The existence of similar low-frequency components (2, 4, 6 Hz) in the amplitude spectra from both the individual and grand-average sampling functions implies that these components not only present increased amplitude for each subject but also are strongly phase locked between subjects. As we will see, these low-frequency components mostly reflect increased sensitivity to the onset and offset of the stimulus sequence.

*t*(6) = 2.5,

*p*= 0.04, for target present and 149.5 ms ± 35 ms per element,

*t*(6) = 4.7,

*p*= 0.0034, for target absent.

*F*(1, 12) = 8.2,

*p*= 0.014). Moreover, no significant difference was found between subject groups (between-subject factor “initial/naïve,”

*F*(1, 12) = 0.9,

*p*= 0.37), nor any significant interaction between the two factors (“test/control” × “initial/naive”:

*F*(1, 12) = 0.6,

*p*= 0.471).

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*η*, exceeds a predefined criterion c, that is, The template-matching and decision processes in Equation 2 are identical to those described by Solomon (2002, equation 2, p. 106). In summary, the observer depicted in Supplementary Figure S1 will respond yes when and will respond no otherwise.

_{i}and a time-invariant external noise pattern noise

_{i}, both of them amplitude-modulated in time by a sinusoidal function of the same frequency

*ω*

_{i}but opposite phase (let us call

*φ*

_{i}the phase applied to the signal). Within the context of our experiment, Equation 3 can therefore be rewritten as Because the signal

_{i}and noise

_{i}patterns are time-invariant, the equation becomes We further simplify this equation by assuming that the (positive) attentional sampling function

*A*(

*t*) is scaled such that its integral over the display interval is equal to 1, thus, The remaining integral in Equation 6 directly corresponds to the perceptual variable P used in the main text (more precisely, P

_{cosω}, P

_{sinω}, P

_{-cosω}, and P

_{-sinω}are obtained by setting the phase

*φ*

_{i}to 0, −

*π*/2,

*π*, and

*π*/2, respectively). This variable P fluctuates between −1 and 1. When P = 1, reflecting an ideal match between the attentional sampling function and the sinusoidal stimulation for that trial, the external noise terms in Equation 6 are canceled and the response depends solely on the input signal pattern. On the other hand, when P = −1, implying that the attentional sampling function oscillates in phase opposition with the sinusoidal stimulation for that trial, the signal terms in Equation 6 disappear and the response solely depends on the noise.

*η*

_{i}was taken from a Gaussian distribution with mean of 0 and standard deviation of 1 (this value was chosen to produce a range of d′ values between 0 and 2, compatible with those observed experimentally). We assumed that the signal pattern was identical to the target template on target-present trials (that is, w

^{t}signal

_{i}= 1) and was orthogonal to the target template on target-absent trials (that is, w

^{t}signal

_{i}= 0). The noise pattern was also assumed to be orthogonal

*on average*to the target template but with trial-by-trial deviations in either direction; that is, the dot product w

^{t}noise

_{i}was taken from a Gaussian distribution with a mean of 0 and standard deviation of 0.5 (this value implies that a fully random noise pattern will match the target template

*better*than the target itself on 2.28% of trials—a rather liberal estimate). Finally, the criterion c was chosen (based on the observed false alarm rates during the experiment) to be c = 0.5. Supplementary Figure S2 illustrates the d′ values of this model observer obtained for various values of the variable P. Based on this near-linear relationship, we can conclude that in our study, d′ was indeed an appropriate experimental approximation of the variable P.