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Article  |   December 2013
Attentional dwell times for targets and masks
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Journal of Vision December 2013, Vol.13, 34. doi:10.1167/13.3.34
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      Anders Petersen, Søren Kyllingsbæk, Claus Bundesen; Attentional dwell times for targets and masks. Journal of Vision 2013;13(3):34. doi: 10.1167/13.3.34.

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Abstract
Abstract
Abstract:

Abstract  Studies on the temporal dynamics of attention have shown that the report of a masked target (T2) is severely impaired when the target is presented with a delay (stimulus onset asynchrony) of less than 500 ms after a spatially separate masked target (T1). This is known as the attentional dwell time. Recently, we have proposed a computational model of this effect building on the idea that a stimulus retained in visual short-term memory (VSTM) takes up visual processing resources that otherwise could have been used to encode subsequent stimuli into VSTM. The resources are locked until the stimulus in VSTM has been recoded, which explains the long dwell time. Challenges for this model and others are findings by Moore, Egeth, Berglan, and Luck (1996) suggesting that the dwell time is substantially reduced when the mask of T1 is removed. Here we suggest that the mask of T1 modulates performance not by noticeably affecting the dwell time but instead by acting as a distractor drawing processing resources away from T2. This is consistent with our proposed model in which targets and masks compete for attentional resources and attention dwells on both. We tested the model by replicating the study by Moore et al., including a new condition in which T1 is omitted but the mask of T1 is retained. Results from this and the original study by Moore et al. are modeled with great precision.

Introduction
Changes in our visual environment occur across both space and time. To detect and react on these changes, human are equipped with an attentional system that in most daily situations is sufficient to keep us out of trouble. Put under pressure, however, the system exhibits severe limitations. One such limitation is found when changes occur in close temporal proximity, as is the case in the attentional dwell time paradigm (Duncan, Ward, & Shapiro, 1994; Ward, Duncan, & Shapiro, 1996). Here, subjects are required to identify two backward masked targets presented on different spatial locations and with a systematically varied stimulus onset asynchrony (SOA). Results show a clear deficit in correctly reporting the second target (T2), when T2 is presented less than 500 ms after the first target (T1), and have been argued to reflect an attentional dwell on the first target. 
Recently, we presented the first quantitative computational model of the attentional dwell time, a theory of temporal visual attention (TTVA; Petersen, Kyllingsbæk, & Bundesen, 2012). The model is based on the neural theory of visual attention (Bundesen, Habekost, & Kyllingsbæk, 2005) and introduces the novel assumption that a stimulus retained in visual short-term memory (VSTM) temporarily takes up visual processing resources that otherwise could have been used to encode subsequent stimuli into VSTM. Besides explaining the dwell time effect per se, the model also explains performance in the special condition in which T1 and T2 are presented simultaneously at different locations (i.e., whole report condition) and performance when the exposure duration of the targets is varied. That is, the model is not only capable of explaining the dwell time effect but also may be used as a model for a wide range of findings in the spatial attention literature (for a review, see Bundesen, 1990). 
However, challenges for our model and others are findings by Moore, Egeth, Berglan, and Luck (1996) suggesting that the dwell time is substantially reduced when the mask of T1 is removed. Moore et al. (1996) ran an attentional dwell time paradigm containing two conditions: a Difficult condition in which T1 was immediately followed by a mask and an Easy condition in which the mask of T1 was delayed. Results showed a faster recovery of T2 in the Easy condition compared with the Difficult condition (see Figure 1), and Moore et al. suggested that this was because of a shorter dwell time (i.e., 100–200 ms) in the Easy condition. 
Figure 1
 
Least squares fits of the model to the data from Moore et al. (1996, experiment 1). The observed and predicted probabilities of correctly reporting T1 (pT1, squares and dashed lines) and T2 (pT2, circles and solid lines) are plotted as functions of SOA for the Difficult (gray) and Easy (black) conditions. The asterisks indicate where the difference in pT2 between the two conditions was significant (p < 0.05).
Figure 1
 
Least squares fits of the model to the data from Moore et al. (1996, experiment 1). The observed and predicted probabilities of correctly reporting T1 (pT1, squares and dashed lines) and T2 (pT2, circles and solid lines) are plotted as functions of SOA for the Difficult (gray) and Easy (black) conditions. The asterisks indicate where the difference in pT2 between the two conditions was significant (p < 0.05).
In this article, we present theoretical and empirical evidence suggesting that masking of T1 modulates performance in the attentional dwell time paradigm but does so without noticeably affecting the attentional dwell time. Instead, we show that the faster recovery of T2 found by Moore et al. (1996) when no mask was presented after T1 can be explained by assuming that masks not only stop processing of a target but also act as distractors: If T1 is masked, T2 will have to compete with both T1 and the mask of T1 for processing resources. However, if T1 is not masked, T2 will not have to compete with other objects for processing resources after the presentation (and decay) of T1, resulting in a faster recovery of correctly reporting T2. We use our computational model of the attentional dwell time (TTVA; Petersen et al., 2012) to substantiate our reasoning. Moreover, we conducted an experiment replicating the findings by Moore et al. and additionally revealing that the presentation of the mask of T1 without T1 itself resulted in a transient impairment of correctly reporting T2. This is direct evidence that the mask not only stops processing of the target but also acts as a distractor and that attention dwells on a mask in the same way as attention dwells on a target before moving to the next objects. The results from Moore et al. (1996) and our experiment were modeled with great precision by TTVA. 
Theory of temporal visual attention
The mathematical model is based on Bundesen's theory of visual attention (TVA; Bundesen, 1990) and the later neural interpretation of TVA (NTVA; Bundesen et al., 2005). In this section, we first give a short introduction to TVA and NTVA followed by a description of how these models of attention are extended into the temporal domain in the theory of temporal visual attention (TTVA). 
The theory of visual attention
The TVA proposes that an object x in the visual field is encoded into VSTM by encoding a categorization of the object into VSTM. A categorization has the form “object x belongs to category i” (or equivalently, “object x has feature i”), where i is a perceptual category (e.g., a certain color, shape, movement, or spatial position). The speed or processing rate at which the categorization that object x belongs to category i is encoded into VSTM is given by the rate equation of TVA,  where η(x,i) ∈ ℝ+ ∪ {0} is the strength of the sensory evidence that object x belongs to category i, βi ∈ [0,1] is the perceptual bias associated with category i, and wx ∈ ℝ+ ∪ {0} is the attentional weight of object x, which is divided by the sum of attentional weights across all objects in the visual field, S
In TTVA, a special case of TVA known as the fixed-capacity independent race model (FIRM; Shibuya & Bundesen, 1988) is used to describe how objects are encoded into VSTM. In FIRM, the total processing rate of an object x is defined as  where C is the total processing capacity. That is, the visual system is assumed to have a limited processing capacity, and objects in the visual field compete for these limited processing resources by the attribution of attentional weights. Objects with high weights are allocated more processing resources than objects with low weights. Objects will then race against each other to access VSTM. The more processing resources allocated to an object, the higher is the probability that the object will be encoded into VSTM. 
When storage limitations of VSTM can be neglected, the probability of encoding object x into VSTM before the end of the presentation of object x is given by  where vx is the processing rate of object x, τ is the exposure duration of object x, and t0 is the longest ineffective exposure duration (a.k.a. the threshold for visual perception). That is, if the exposure duration of an object is shorter than t0, the probability of encoding the object into VSTM is zero. However, if the exposure duration of the object is longer than t0, the probability of encoding the object into VSTM increases exponentially as a function of τt0. More specifically, t0 is defined as t1t2, where t1 is the time from when the stimulus display is presented until the race toward VSTM is initiated, and t2 is the time from when the postmask is presented until the race is interrupted by the mask (Petersen et al., 2012; Shibuya & Bundesen, 1988). 
Dyrholm, Kyllingsbæk, Espeseth, and Bundesen (2011) found evidence that t0 varies across trials. This variation can be modeled by assuming that t0 is approximately normally distributed with mean μ0 and a small standard deviation σ0 such that the time it takes for an object to be encoded into VSTM is modeled as coming from an ex-Gaussian distribution (i.e., a convolution of a normal distribution and an exponential distribution; Luce, 1986). The probability of encoding an object into VSTM is then approximated by  where ϕ(x) and Φ(x) are the probability density function and the cumulative distribution function of the standard normal distribution, respectively. 
VSTM and feedback loops
The neural interpretation of TVA (NTVA; Bundesen et al., 2005) implements the equations of TVA at the level of cortical neurons. In NTVA, the number of cortical neurons representing an object x is proportional to the relative attentional weight of the object, wx/∑zS wz, and the level of activity in the neurons representing object x corresponds to the multiplicative scaling of the relative attentional weight by η(x,i)βi, such that object x races toward VSTM with a processing rate given by the rate equation of TVA. 
Furthermore, NTVA suggests an implementation of VSTM building on the Hebbian notion that short-term memory is based on retaining information in positive feedback loops that sustain activity in the processing neurons representing the information (Hebb, 1949). In this way, visual representations can outlast the original sensory stimulation, leaving time for recoding into a more permanent storage. Thus, in NTVA, the VSTM map does not in itself contain representations of features of objects but serves as a topographic map of objects feeding back impulses to the feature neurons from which they originated. Figure 2 illustrates a unit in the VSTM map gating two feedback loops. A categorization of an object x becomes encoded in VSTM by becoming embedded in a positive feedback loop, which is closed when, and only when, a unit representing object x in the VSTM map is activated. If the VSTM unit is inactive, impulses from the feature units are not fed back to the feature units, and no feedback loop is established. 
Figure 2
 
Two feedback loops gated by the same unit in the VSTM map of objects. The logical AND units (marked by ampersands) indicate that to sustain a positive feedback loop, both impulses from a feature unit and a unit in the VSTM map are needed. Copyright © 2005 by the American Psychological Association. Reproduced with permission. The official citation that should be used in referencing this material is Bundesen, C., Habekost, T., & Kyllingsbæk, S. (2005). A neural theory of visual attention: Bridging cognition and neurophysiology. Psychological Review, 112, 291–328. The use of APA information does not imply endorsement by APA.
Figure 2
 
Two feedback loops gated by the same unit in the VSTM map of objects. The logical AND units (marked by ampersands) indicate that to sustain a positive feedback loop, both impulses from a feature unit and a unit in the VSTM map are needed. Copyright © 2005 by the American Psychological Association. Reproduced with permission. The official citation that should be used in referencing this material is Bundesen, C., Habekost, T., & Kyllingsbæk, S. (2005). A neural theory of visual attention: Bridging cognition and neurophysiology. Psychological Review, 112, 291–328. The use of APA information does not imply endorsement by APA.
Extensions to the temporal domain
TVA has convincingly predicted results from numerous behavioral paradigms in which a group of backward masked stimuli are presented simultaneously (e.g., whole report and partial report). To model behavior in these paradigms, a single race among the stimuli (in most cases letters) is initiated. For ease of exposition, it may be assumed that calculation of attentional weights begins when the letters are presented (Time 0) and finishes such that the race based on the weights can begin at time t0. The race is interrupted as soon as the mask is presented, at time τ, yielding a race duration of τt0 (see Shibuya & Bundesen, 1988, and Petersen et al., 2012, for a more elaborate description of the race). 
However, when a temporal gap is introduced between the presentation of two backward masked letters, as is the case in the attentional dwell time paradigm, Petersen et al. (2012) suggested that multiple calculations of attentional weights occur such that a new race based on the current attentional weights is initiated every time a letter is presented. Thus, when two letters (T1 and T2) are presented with a temporal gap, two calculations of attentional weights will be initiated: one when T1 is presented and one when T2 is presented. This implies that the calculations will finish at separate time points, resulting in two redistributions of the attentional resources: one at t01 (i.e., t0 for T1) and one at t02 (i.e., t0 for T2). 
Moreover, Petersen et al. (2012) suggested that neurons engaged in a feedback loop are prevented from being reallocated to process other objects in a later race without first being disengaged from the loop. As in NTVA, neurons are locked in a feedback loop to retain a representation of the encoded object in VSTM. The time it takes to lock a neuron in a feedback loop is assumed to be exponentially distributed with a rate parameter λl and is referred to as the lock time. The exponential distribution may also be described by its mean μl = 1/λl, which we refer to as the mean lock time. After a neuron has been locked to retain a representation of an encoded object in VSTM, the neuron will dwell on the object such that the representation can be recoded into a longer-term store, such as the speech loop, for later report. This is referred to as the dwell time of the locked neurons. As for the lock time, it is assumed that the dwell time is exponentially distributed with a rate parameter λd and with a mean of μd = 1/λd (the mean dwell time). After the dwell time has passed, a neuron is released (i.e., the feedback loop is broken) and can be reallocated to process other objects. 
Modeling data from Moore et al. (1996, experiment 1)
Petersen et al. (2012) showed that the attentional dwell time effect observed by Duncan et al. (1994) can be explained by locking and releasing visual processing neurons. When T1 and T2 are presented simultaneously (i.e., SOA = 0), the processing rate of either letter is (1/2)C, and we predict equal probabilities of encoding T1 and T2 (see Figure 2). However, if T1 has a head start (i.e., SOA > 0), it is most likely that t01t02. This will give T1 an advantage over T2 because all neurons, in the interval between t01 and t02, will be allocated to T1, resulting in a processing rate of C. Furthermore, neurons allocated to process T1 (or the mask for T1) in the interval between t01 and t02 may be locked to retain the representation of T1 (or the mask of T1) in VSTM. Consequently, T2 will lack processing resources when it is presented approximately 100 to 200 ms after T1. When the interval between t01 and t02 increases above 200 to 300 ms, T1 gradually loses its advantage because neurons will begin to be released from T1 and again become available for the processing of T2. Thus, when the interval between t01 and t02 is long (e.g., 900 ms), almost all of the visual processing neurons will have been locked to and released from T1 and will now be available for the processing of T2, resulting in both letters having a processing rate of C (i.e., the probabilities of correctly reporting T1 and T2 will be the same). 
As touched on in the Introduction, Moore et al. (1996) ran an attentional dwell time paradigm containing two conditions. In the Difficult condition, T1 was immediately followed by a mask that lasted until the response, whereas in the Easy condition, the mask of T1 was delayed and presented at the same time as the mask of T2. Results showed that the temporary impairment in correctly reporting T2 lasted for a shorter time in the Easy condition compared with the Difficult condition (see Figure 1), and Moore et al. suggested that this was because of a shorter dwell time (i.e., 100–200 ms) in the Easy condition. 
TTVA provides a different explanation for the data observed by Moore et al. (1996) given the following special assumptions on the computation of attentional weights in the attentional dwell time paradigm: (a) When a mask follows a target at the same location without any delay, the mask inherits the attentional weight of the target. (b) When a stimulus appears at a new location, the processing of any previously presented stimuli is halted and replaced by a new computation of attentional weights encompassing all stimuli in the visual field. (c) The time it takes to compute and implement attentional weights is approximately normally distributed with a mean of μ0 and a standard deviation of σ0. (d) Once the attentional weights have been computed and implemented, a new competition (race) between stimuli to become encoded into VSTM is started. 
The Difficult condition in Moore et al. (1996, experiment 1) is similar to the experiment of Duncan et al. (1994), and TTVA can account for the data in this condition without additional assumptions. In the Easy condition, however, T1 is no longer followed immediately by a mask (except when SOA = 0), but its retinal representation decays gradually until the mask is presented. We approximate the effect of the decaying trace of T1 by adding an equivalent additional exposure duration (see Loftus, Johnson, & Shimamura, 1985), μdecay, to the exposure duration of T1, τ1, such that the effective exposure duration of T1 becomes  Furthermore, in the Easy condition, we assume that the attentional weight of T1, wT1, equals the attentional weight of T2 at any time t below or equal to the effective exposure duration of T1, whereas wT1 equals 0 at any time t greater than the effective exposure duration of T1. Thus, in the Easy condition,  As a result, the relative attentional weight of T2 in the Easy condition increases when the time of the redistribution of attentional weights initiated by the presentation of T2, wT2, t02, occurs after T1 has decayed (i.e., T1 no longer competes for attentional resources). Thus,  In contrast, we assume that the relative attentional weight of T2 in the Difficult condition equals 1/2 (i.e., wT1 = wT2) both when t02τ1 and when t02 > τ1. That is, we assume that the mask of T1 in the Difficult condition acts as a distractor that inherits its attentional weight from T1 and competes with T2 for attentional resources. In the Easy condition, T2 will not have to compete with the mask of T1. This explains why the probability of correctly reporting T2 follows a different time course in the Easy compared with the Difficult condition. For a more elaborate description of the model, see the supplemental material
The final model has six free parameters: μl (mean lock time), μd (mean dwell time), C (total processing capacity), μ0 (mean of the longest ineffective exposure duration, t0), σ0 (standard deviation of t0), and μdecay (additional exposure duration of T1). Figure 1 shows a least squares fit of the model to the data by Moore et al. (1996, experiment 1), and Table 1 lists the estimated parameters together with the goodness of the fits measured by the square root of the mean squared deviation (RMSD). The model makes a good prediction of the data, which is reflected in the low RMSD value (but see the section “Modeling our Data” for a discussion on the tradeoff between goodness-of-fit and the complexity of the model). The estimated parameters are reasonable and with roughly the same order of magnitude as previously estimated values (cf. Table 2). The only exception is a mean of t0 (μ0) of 1 ms, which is lower than normally estimated. This result may reflect the lack of a data point at SOA = 0, which would have helped to constrain μ0. Thus, a better estimate of μ0 could be provided if this data point was included (see the “Experiment” section). 
Table 1
 
Parameter estimates by least squares fits of the model to the group data reported by Moore et al. (1996) and individual data from the 9 participants in the current experiment.
Table 1
 
Parameter estimates by least squares fits of the model to the group data reported by Moore et al. (1996) and individual data from the 9 participants in the current experiment.
μl (ms) μd (ms) μ0 (ms) σ0 (ms) C (Hz) α μdecay (ms) RMSD
Moore et al. 403 211 1 24 23 136 0.02
Participant 1 63 215 17 19 26 0.30 125 0.05
Participant 2 204 315 24 24 44 0.41 212 0.06
Participant 3 84 219 29 16 50 0.02 235 0.08
Participant 4 79 655 19 40 73 0.79 67 0.05
Participant 5 48 221 12 9 21 0.38 261 0.04
Participant 6 57 254 26 12 34 0.17 300 0.07
Participant 7 150 494 25 12 58 0.71 225 0.06
Participant 8 790 194 30 11 66 0.14 200 0.04
Participant 9 128 508 13 15 88 0.2 252 0.05
Table 2
 
Range of previous parameter estimates from Petersen et al. (2012), Dyrholm et al. (2011), Habekost, Petersen, and Vangkilde (2013), and Bundesen (1990; Table 1). Note: N denotes the number of subjects in the studies.
Table 2
 
Range of previous parameter estimates from Petersen et al. (2012), Dyrholm et al. (2011), Habekost, Petersen, and Vangkilde (2013), and Bundesen (1990; Table 1). Note: N denotes the number of subjects in the studies.
μl (ms) μd (ms) μ0 (ms) σ0 (ms) C (Hz) α μdecay (ms) N
Petersen et al. 32–144 234–611 9–30 8–38 14–93 6
Dyrholm et al. −30–79 0–26 5–217 0.12–2.7 347
Habekost et al. 0–39 29–106 0.17–1.6 68
Bundesen 6–30 197–983 5
For comparison with previous dwell time estimates, we calculated the mean time from the moment when processing neurons were first allocated to T1 until they were released. This time is given by 1/C + μl + μd and resulted in a value of 614 ms. A comparable estimate was calculated by Moore et al. (1996) by fitting ogive-normal functions to individual data and using the point of inflection as a measure of the time at which the presentation of T1 no longer interfered with the encoding of T2 (i.e., the dwell time of attention on T1). The mean point of inflection for the Difficult condition was found to be 625 ms, which is remarkably close to the estimate from our model. 
Experiment
The presented model predicts that both targets and masks encoded into VSTM will take up attentional resources and that the observed impairment of correctly reporting T2 is caused not only by the encoding of T1 but also by the encoding of the mask of T1. Our experiment tested this prediction by presenting T1 without the mask of T1 in one condition and the mask of T1 without T1 in a different condition. The paradigm was constructed as a further development of the paradigm by Moore et al. (1996) by adding a third condition in which T1 was deleted but the stimulus formerly masking T1 was retained (Mask condition). Whereas the combined effect of presenting T1 and the mask of T1 is examined in the Difficult condition, the simple effects of presenting only T1 or only the mask of T1 are investigated in the Easy and Mask conditions, respectively. 
Method
Participants
Nine female psychology or anthropology students (mean age = 27 years) from the University of Copenhagen were paid a standard fee by the hour for participating in the experiment. All had normal or corrected-to-normal vision. 
Targets
The targets were all 26 uppercase letters of the English alphabet constructed from 27 unique line segments. The letters were white with a width and height of 1.10° and 1.65°, respectively. 
Masks
The masks were constructed from the same 27 unique line segments that were used for constructing the stimulus letters (see Figure 3). No two masks were identical (see Petersen & Kyllingsbæk, 2013, for results that show the importance of varying the mask in the attentional dwell time paradigm). Each mask was made by randomly choosing 14 of the 27 unique line segments and shifting the 14 segments independently of each other either 0.55° to the left (probability 0.2), 0.55° to the right (probability 0.2), 0.55° up (probability 0.2), 0.55° down (probability 0.2), or not at all (probability 0.2). This procedure made the size of the masks slightly larger than the size of the letters. 
Figure 3
 
Trial outline for each of the three conditions in our experiment. T1 and T2 are represented by the letters K and L, respectively. The Difficult and Easy conditions are similar to the conditions in Moore et al. (1996, experiment 1), except that in their study, the initial presentation of the boxes and fixation cross was 500 ms, the average exposure duration of the targets was τaverage = 60 ms, and the SOAs were 0, 200, 350, 500, 650, and 1100 ms.
Figure 3
 
Trial outline for each of the three conditions in our experiment. T1 and T2 are represented by the letters K and L, respectively. The Difficult and Easy conditions are similar to the conditions in Moore et al. (1996, experiment 1), except that in their study, the initial presentation of the boxes and fixation cross was 500 ms, the average exposure duration of the targets was τaverage = 60 ms, and the SOAs were 0, 200, 350, 500, 650, and 1100 ms.
Design
The paradigm included three mixed conditions (cf. Figure 3): the Difficult and Easy conditions replicated the experiment by Moore et al. (1996). In the Difficult condition, T1 was postmasked until response, whereas in the Easy condition, the mask of T1 was delayed until the presentation of the mask of T2. In the Mask condition, T1 was omitted and only the mask of T1 was presented. In all three conditions, seven SOAs between T1 and T2 were used (0, 100, 200, 300, 400, 600, and 900 ms), and letters were chosen randomly without replacement so that each letter was used once and only once as the first and second target for each SOA to avoid variation caused by different salience of the letters. Furthermore, letters were chosen so that within a trial, the two target letters were always different. With three conditions, seven SOAs and 26 letters, one block of the experiment consisted of 546 trials. 
Procedure
Stimuli were presented on a 19-in. CRT monitor at 100 Hz using in-house custom-made software written in C++. A white fixation cross (0.55° × 0.55°) was displayed on a black background together with four white boxes serving as place holders (1.65° × 2.20°). The boxes were placed at the corners of an imaginary square 5.5° from fixation (cf. Figure 3). Participants initiated every trial themselves by pressing the space bar on the keyboard, which was followed by a short delay of 200 ms. In the Difficult condition, a first target letter (T1) was then presented for an individually calibrated exposure duration of τ ms (40 ≤ τ ≤ 90, M = 54.4 ms, SD = 14.2 ms) followed by a mask that stayed on the screen until the end of the trial. In the Easy condition, however, T1 was presented without a mask, and in the Mask condition, the mask was presented without T1. In all three conditions, a second target letter (T2) was presented in one of the three remaining boxes after either an SOA between T1 and T2 (SOA) of 0, 100, 200, 300, 400, 600, or 900 ms (Difficult and Easy conditions) or an SOA between the mask of T1 and T2 of SOAMT = SOA − τ. (Mask condition). T2 was presented for the same exposure duration as T1 and was masked for 240 ms. Following the presentation of the target(s) and masks, participants responded by typing the letters on a keyboard in any order they preferred. A forced-choice procedure was applied such that participants were required to type in two letters in response to all trials even if this meant they had to guess. This procedure also applied to trials in the Mask condition, even though only one target was presented. Participants were instructed to maintain central fixation during trials but were allowed to move their eyes between trials. Eye movements were measured using a head-mounted eye tracker (EyeLink II). 
All participants were tested in two sessions on separate days. Each session comprised 50 practice trials and two experimental blocks. Before the first block of the first session, participants performed an adaptive psychophysical calibration procedure (accelerated stochastic approximation; Kesten, 1958), which adjusted the exposure duration of the letters, τ, such that on average one of the two letters (i.e., about 50% of the presented letters) were correctly reported in the Difficult condition with simultaneous presentation of the two letters (i.e., SOA = 0). The procedure comprised 50 trials in which the exposure duration on the current trial was adjusted on the basis of the exposure duration and the number of correctly reported targets on the previous trial. Each participant performed three calibrations. The first calibration was initiated using an exposure duration of 100 ms and the following two calibrations used the outcome of the previous calibration as the starting point. 
Results
Eye movements were analyzed to ensure that participants maintained central fixation during the trials (see Petersen & Kyllingsbæk, 2013, for dramatic effects of eye movements in the attentional dwell time paradigm). A trial was categorized as a trial with eye movements if the gaze deviated more than 2.75° from the fixation cross (i.e., half the distance from the fixation cross to the target locations) at any time during the trial before the onset of the second mask. We did not measure eye movements after the onset of the second mask, assuming that they would not influence the identification of T2. If no eye movements were registered outside the 2.75° boundary, the trial was categorized as a trial without eye movements. The analysis showed that across subjects, the percentage of trials with eye movements was low (M = 0.95%, SD = 1.00); nevertheless, these trials were removed from further analysis of the data. 
Figure 4 shows the mean probability of correctly reporting T1 (pT1) and T2 (pT2) as functions of SOA for each of the three conditions. (For the Easy and Difficult conditions, the mean probability of pT1 and pT2 is presented at SOA = 0.) In accordance with Moore et al. (1996), the accuracy data for the Difficult and Easy conditions were analyzed by conducting two 2-way repeated-measures analyses of variance (ANOVAs) for T1 and T2, respectively, with SOA and condition (Difficult vs. Easy) as within-subject factors, excluding trials with SOA = 0. Not surprisingly, a significant main effect of condition was found for T1, F(1, 8) = 56.131, p < 0.001, Display FormulaImage not available = 0.875, whereas no main effect of SOA was found, F(5, 40) = 1.435, p = 0.152, ns. The results also showed a significant Condition × SOA interaction for T1, F(5, 40) = 2.873, p = 0.026, Display FormulaImage not available = 0.264. 
Figure 4
 
Observed and predicted probabilities of correctly reporting T1 (pT1, squares and dashed lines) and T2 (pT2, circles and solid lines) as functions of SOA for the Difficult (gray), Easy (black), and Mask (white) conditions of our experiment. Error bars show the standard error of the mean, and the asterisk indicates the significant difference between pT2 in the Difficult and Easy conditions at SOA = 300 ms (p < 0.05).
Figure 4
 
Observed and predicted probabilities of correctly reporting T1 (pT1, squares and dashed lines) and T2 (pT2, circles and solid lines) as functions of SOA for the Difficult (gray), Easy (black), and Mask (white) conditions of our experiment. Error bars show the standard error of the mean, and the asterisk indicates the significant difference between pT2 in the Difficult and Easy conditions at SOA = 300 ms (p < 0.05).
The ANOVA for T2 showed no main effect of condition, F(1, 8) = 0.35, p = 0.571, ns, but a significant main effect of SOA, F(5, 40) = 50.35, p < 0.001, Display FormulaImage not available = 0.863, reflecting a higher accuracy of correctly reporting T2 at long as compared with short SOAs. The ANOVA also showed a marginally significant Condition × SOA interaction, F(5, 40) = 2.20, p = 0.073, Display FormulaImage not available = 0.216. Multiple comparisons between pT2 in the Difficult and Easy conditions for each SOA indicated a significant difference at SOA = 300 ms, t(8) = 4.44, p < 0.05, d = 1.49 (Bonferroni-corrected paired t test), whereas no significant differences were found at other SOAs, all t(8) ≤ 1.56, ns. In sum, this analysis indicated that with increasing SOA, performance began to improve sooner in the Easy compared with the Difficult condition, consistent with the findings by Moore et al. (1996). 
Finally, the results from the Mask condition were analyzed by conducting a one-way repeated-measures ANOVA with SOA as a within-subject factor. The ANOVA revealed a significant effect of SOA, F(5, 40) = 11.356, p < 0.001, Display FormulaImage not available = 0.587, suggesting that much like the presentation of T1 in the other conditions, the presentation of the mask of T1 in the Mask condition resulted in a lower probability of correctly reporting T2 at SOAs in the range between roughly 100 and 400 ms. This supports our hypothesis that a mask acts as a distractor and attention dwells on a mask just like it dwells on a target. 
Modeling our data
Whereas the data reported by Moore et al. (1996) were modeled at the group level, the data from our own experiment were modeled separately for each participant. The Difficult and Easy conditions were modeled in the same way as before, but a few additional assumptions had to be made to model the Mask condition. First, we assumed that in the Mask condition, t01 was normally distributed with mean τ1 + μ0 and standard deviation σ0. Second, we assumed that the attentional weight of a mask, wM, was different from the attentional weight of a target. Thus, the relative attentional weight of T2 in the Mask condition is given by  where α = wM1/wT2 is the relative attentional weight of a mask and is included as an additional free parameter in the model (see the supplemental material for a more elaborate description of the model). Figure 4 shows the mean of the least squares fits of the model to the individual data from our experiment. The estimated parameters are listed in Table 1 together with the goodness of the fits. 
The model made good predictions of the data from our experiment, even though the RMSD values were a bit higher than for the fit to the data from Moore et al. (1996). The difference in goodness of fit should be expected because the data from Moore et al. were fitted on the group level, whereas the data from our own experiment were fitted for each individual participant. To address the issue about tradeoff between goodness-of-fit and the complexity of a model, we tested whether the two extra parameters (wM and μdecay) introduced to the model in order to fit the data from Moore et al. and the data from our own experiment were really needed: An F test comparing a model in which wM was introduced as a free parameter with a nested model in which the weight of a mask was equal to the weight of a target revealed that the more complex model provided a significantly better fit than the nested model when fitted to the data from our own experiment, F(9, 261) = 3.11, p < 0.01. In addition, an F test comparing a model in which wM and μdecay were included as free parameters with the nested model in which only wM was included as a free parameter and no decay was assumed revealed again that the more complex model provided a significantly better fit than the nested model when fitted to the data from our own experiment, F(9, 252) = 2.37, p < 0.05. Correspondingly, we found that the model in which μdecay was included as a free parameter significantly improved the fit to the data from Moore et al. compared with a nested model in which no decay was assumed, F(1, 18) = 14.35, p < 0.01. Taken together, we see this as evidence that the most complex model (introducing wM and μdecay as free parameters) is indeed needed to obtain the most accurate fits to the data.1 
Turning to the estimated values of the parameters, they seem quite plausible and with roughly the same order of magnitude as previously estimated values (cf. Table 2), except that the estimate for the mean lock time, μl, of Participant 8 is very high, reflecting a very small deficit in correctly reporting T2 compared with the deficits shown by the other participants. (Similar exceptions have been reported in the literature on the attentional blink; see Martens, Munneke, Smid, & Johnson, 2006; Martens & Valchev, 2009. For more general work on the attentional blink, see Chun & Potter, 1995; Olivers & Meeter, 2008; Shapiro, Raymond, & Arnell, 1994.) Apparently, the behavior of the model (i.e., faster recovery of T2 in the Easy condition compared with the Difficult condition) is not linked to a special combination of parameter values. As illustrated in Figure 5, changing the values of C and μdecay moderately does not change the overall behavior of the model dramatically. Rather, the general shape of the predicted curves is inherent to the model (see Supplementary Figure S1 for similar illustrations related to the other parameters in the model). 
Figure 5
 
Predicted probabilities of correctly reporting T1 (pT1, dashed lines) and T2 (pT2, solid lines) as functions of SOA in the Difficult (gray) and Easy (black) conditions for three different values of C (A): 20 Hz (thin lines), 40 Hz (medium size lines), and 80 Hz (thick lines), and three different values of μdecay (B): 100 ms (thin lines), 200 ms (medium size lines), and 300 ms (thick lines) The remaining parameters were fixed at μl = 180 ms, μd = 300 ms, μ0 = 20 ms, and σ0 = 20 ms (see Supplementary Figure S1 for similar illustrations related to these parameters).
Figure 5
 
Predicted probabilities of correctly reporting T1 (pT1, dashed lines) and T2 (pT2, solid lines) as functions of SOA in the Difficult (gray) and Easy (black) conditions for three different values of C (A): 20 Hz (thin lines), 40 Hz (medium size lines), and 80 Hz (thick lines), and three different values of μdecay (B): 100 ms (thin lines), 200 ms (medium size lines), and 300 ms (thick lines) The remaining parameters were fixed at μl = 180 ms, μd = 300 ms, μ0 = 20 ms, and σ0 = 20 ms (see Supplementary Figure S1 for similar illustrations related to these parameters).
For comparison with previous estimates of the attentional dwell time, we calculated the mean time from the moment when processing neurons were first allocated to T1 until they were released. This time is given by 1/C + μl + μd and resulted in a mean value across the participants in our experiment of 520 ms, SD = 251 ms, which is not very different from the 450 ms estimated by Duncan et al. (1994). 
Finally, we wanted to test the interpretation by Moore et al. (1996) that the attentional dwell time was shorter in the Easy as compared with the Difficult condition. Thus, we fitted an alternative model to the data from Moore et al. (1996, experiment 1) and the data from our own experiment, a model in which the mean dwell time, μd, in the Easy condition was allowed to vary freely, independently of the mean dwell time in the remaining conditions. F tests comparing the alternative model with the nested model without the extra parameter revealed that the alternative model did not provide a significantly better fit than the nested model neither when fitted to the data from Moore et al., F(1, 17) = 0.11, ns, nor when fitted to the data from our own experiment, F(9, 243) = 0.34, ns. This indicates that competition between masks and target for processing resources is more likely to explain the observed data than a reduction of the dwell time in the Easy condition. 
Our results are not the first evidence that the item following immediately after T1 has a pronounced effect on the probability of correctly reporting T2. Raymond, Shapiro, and Arnell (1992) showed that removing the post-T1 distractor in the attentional blink (AB) paradigm attenuated the AB, whereas removing one or more distractors after the post-T1 distractor did not affect the AB significantly. The study was later replicated by Chun and Potter (1995) with a consistent pattern of results. The posttarget distractor also plays a key role in the interference theory by Shapiro et al. (1994). Here, the AB is a result of both T1 and the post-T1 distractor entering VSTM such that T2 receives a diminished weight and thus is more open to interference from the other items in VSTM. Finally, the Boost and Bounce model by Olivers and Meeter (2008) also assigns special meaning to the post-T1 distractor. Whereas T1 initiates a transient excitatory feedback that boosts the sensory representation of T1 and the following items, the post-T1 distractor triggers a strong transient inhibitory feedback (the bounce), reducing the sensory representation of the post-T1 distractor and the following items. Thus, if T2 is presented immediately after the post-T1 distractor (i.e., at lag 2), it is less likely to be perceived correctly. 
General discussion
Attentional dwell time defines our inability to perceive spatially separate events when they occur in rapid succession. Duncan et al. (1994) estimated the dwell time to be about 500 ms using a paradigm in which two backward masked targets were presented. Moore et al. (1996) challenged this estimate by showing that the dwell time is substantially reduced when the mask of the first target (T1) is removed and suggested an estimate in the range of 100 to 200 ms. 
In this article, we presented evidence suggesting that the mask of T1 modulates performance, not by noticeably increasing the time attention remains focused on T1 as suggested by Moore et al., but instead by acting as a distractor drawing processing resources away from T2. Building on this idea, we presented a detailed quantitative model of performance in experiment 1 of Moore et al. (1996), replicated their data including a new condition in which T1 was omitted but the mask of T1 was retained. Results from the new condition revealed that attention dwells on a mask in the same way as attention dwells on a target before moving to the next objects. 
Our modeling is based on the TTVA proposed by Petersen et al. (2012), which is an extension of the NTVA (Bundesen et al., 2005) into the temporal domain. In TTVA, retaining a stimulus in VSTM takes up visual processing resources that otherwise could have been used for analyzing the current stimuli. Thus, resources are locked and cannot be used for processing subsequent stimuli until the stimulus in VSTM has been reported or recoded into a more durable format, which causes the long attentional dwell times. Petersen et al. (2012) used the model to provide a detailed account of performance in both a standard attentional dwell time experiment and an extension with varied exposure durations of the target stimuli. In the current study, we showed how the data of Moore et al. (1996) and our own replication and extension of their data can be explained with great precision by TTVA assuming that the retinal representation of an unmasked target decays and that a mask has an attentional weight and a processing rate of its own. These additional assumptions allowed TTVA to account for the data in all of the three conditions described in this article: one in which T1 and T2 were masked (Difficult condition), one in which T1 was unmasked and T2 was masked (Easy condition), and one in which the mask of T1 was presented without T1, and T2 was masked (Mask condition). The estimates we obtained for attentional dwell times were similar to those reported by Duncan et al. (1994). Thus, our results suggest an attentional dwell time in the range of 500 ms regardless of whether T1 is masked or not (but see also Theeuwes, Godijn, & Pratt, 2004; Ward, 2001). 
Supplementary Materials
Acknowledgments
The research was supported by the University of Copenhagen Programme of Excellence, the Danish Council for Strategic Research, the Sapere Aude program of the Danish Council for Independent Research, and Inge Lehmanns Legat. 
Commercial relationships: none. 
Corresponding author: Anders Petersen. 
Email: anders.petersen@psy.ku.dk. 
Address: Department of Psychology, University of Copenhagen, Copenhagen, Denmark. 
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Footnotes
1  The F ratio tests used here are equivalent to generalized log likelihood ratio tests based on chi-square statistics. Further, because we are comparing two nested models, the selection procedure using the log likelihood ratio test yields the same results as a selection procedure that identifies the model that generalizes best based on the method of minimum description length (cf. Pitt, Myung, & Zhang, 2002, footnote 2).
Figure 1
 
Least squares fits of the model to the data from Moore et al. (1996, experiment 1). The observed and predicted probabilities of correctly reporting T1 (pT1, squares and dashed lines) and T2 (pT2, circles and solid lines) are plotted as functions of SOA for the Difficult (gray) and Easy (black) conditions. The asterisks indicate where the difference in pT2 between the two conditions was significant (p < 0.05).
Figure 1
 
Least squares fits of the model to the data from Moore et al. (1996, experiment 1). The observed and predicted probabilities of correctly reporting T1 (pT1, squares and dashed lines) and T2 (pT2, circles and solid lines) are plotted as functions of SOA for the Difficult (gray) and Easy (black) conditions. The asterisks indicate where the difference in pT2 between the two conditions was significant (p < 0.05).
Figure 2
 
Two feedback loops gated by the same unit in the VSTM map of objects. The logical AND units (marked by ampersands) indicate that to sustain a positive feedback loop, both impulses from a feature unit and a unit in the VSTM map are needed. Copyright © 2005 by the American Psychological Association. Reproduced with permission. The official citation that should be used in referencing this material is Bundesen, C., Habekost, T., & Kyllingsbæk, S. (2005). A neural theory of visual attention: Bridging cognition and neurophysiology. Psychological Review, 112, 291–328. The use of APA information does not imply endorsement by APA.
Figure 2
 
Two feedback loops gated by the same unit in the VSTM map of objects. The logical AND units (marked by ampersands) indicate that to sustain a positive feedback loop, both impulses from a feature unit and a unit in the VSTM map are needed. Copyright © 2005 by the American Psychological Association. Reproduced with permission. The official citation that should be used in referencing this material is Bundesen, C., Habekost, T., & Kyllingsbæk, S. (2005). A neural theory of visual attention: Bridging cognition and neurophysiology. Psychological Review, 112, 291–328. The use of APA information does not imply endorsement by APA.
Figure 3
 
Trial outline for each of the three conditions in our experiment. T1 and T2 are represented by the letters K and L, respectively. The Difficult and Easy conditions are similar to the conditions in Moore et al. (1996, experiment 1), except that in their study, the initial presentation of the boxes and fixation cross was 500 ms, the average exposure duration of the targets was τaverage = 60 ms, and the SOAs were 0, 200, 350, 500, 650, and 1100 ms.
Figure 3
 
Trial outline for each of the three conditions in our experiment. T1 and T2 are represented by the letters K and L, respectively. The Difficult and Easy conditions are similar to the conditions in Moore et al. (1996, experiment 1), except that in their study, the initial presentation of the boxes and fixation cross was 500 ms, the average exposure duration of the targets was τaverage = 60 ms, and the SOAs were 0, 200, 350, 500, 650, and 1100 ms.
Figure 4
 
Observed and predicted probabilities of correctly reporting T1 (pT1, squares and dashed lines) and T2 (pT2, circles and solid lines) as functions of SOA for the Difficult (gray), Easy (black), and Mask (white) conditions of our experiment. Error bars show the standard error of the mean, and the asterisk indicates the significant difference between pT2 in the Difficult and Easy conditions at SOA = 300 ms (p < 0.05).
Figure 4
 
Observed and predicted probabilities of correctly reporting T1 (pT1, squares and dashed lines) and T2 (pT2, circles and solid lines) as functions of SOA for the Difficult (gray), Easy (black), and Mask (white) conditions of our experiment. Error bars show the standard error of the mean, and the asterisk indicates the significant difference between pT2 in the Difficult and Easy conditions at SOA = 300 ms (p < 0.05).
Figure 5
 
Predicted probabilities of correctly reporting T1 (pT1, dashed lines) and T2 (pT2, solid lines) as functions of SOA in the Difficult (gray) and Easy (black) conditions for three different values of C (A): 20 Hz (thin lines), 40 Hz (medium size lines), and 80 Hz (thick lines), and three different values of μdecay (B): 100 ms (thin lines), 200 ms (medium size lines), and 300 ms (thick lines) The remaining parameters were fixed at μl = 180 ms, μd = 300 ms, μ0 = 20 ms, and σ0 = 20 ms (see Supplementary Figure S1 for similar illustrations related to these parameters).
Figure 5
 
Predicted probabilities of correctly reporting T1 (pT1, dashed lines) and T2 (pT2, solid lines) as functions of SOA in the Difficult (gray) and Easy (black) conditions for three different values of C (A): 20 Hz (thin lines), 40 Hz (medium size lines), and 80 Hz (thick lines), and three different values of μdecay (B): 100 ms (thin lines), 200 ms (medium size lines), and 300 ms (thick lines) The remaining parameters were fixed at μl = 180 ms, μd = 300 ms, μ0 = 20 ms, and σ0 = 20 ms (see Supplementary Figure S1 for similar illustrations related to these parameters).
Table 1
 
Parameter estimates by least squares fits of the model to the group data reported by Moore et al. (1996) and individual data from the 9 participants in the current experiment.
Table 1
 
Parameter estimates by least squares fits of the model to the group data reported by Moore et al. (1996) and individual data from the 9 participants in the current experiment.
μl (ms) μd (ms) μ0 (ms) σ0 (ms) C (Hz) α μdecay (ms) RMSD
Moore et al. 403 211 1 24 23 136 0.02
Participant 1 63 215 17 19 26 0.30 125 0.05
Participant 2 204 315 24 24 44 0.41 212 0.06
Participant 3 84 219 29 16 50 0.02 235 0.08
Participant 4 79 655 19 40 73 0.79 67 0.05
Participant 5 48 221 12 9 21 0.38 261 0.04
Participant 6 57 254 26 12 34 0.17 300 0.07
Participant 7 150 494 25 12 58 0.71 225 0.06
Participant 8 790 194 30 11 66 0.14 200 0.04
Participant 9 128 508 13 15 88 0.2 252 0.05
Table 2
 
Range of previous parameter estimates from Petersen et al. (2012), Dyrholm et al. (2011), Habekost, Petersen, and Vangkilde (2013), and Bundesen (1990; Table 1). Note: N denotes the number of subjects in the studies.
Table 2
 
Range of previous parameter estimates from Petersen et al. (2012), Dyrholm et al. (2011), Habekost, Petersen, and Vangkilde (2013), and Bundesen (1990; Table 1). Note: N denotes the number of subjects in the studies.
μl (ms) μd (ms) μ0 (ms) σ0 (ms) C (Hz) α μdecay (ms) N
Petersen et al. 32–144 234–611 9–30 8–38 14–93 6
Dyrholm et al. −30–79 0–26 5–217 0.12–2.7 347
Habekost et al. 0–39 29–106 0.17–1.6 68
Bundesen 6–30 197–983 5
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