Commuting to work on a busy road, a driver must process a number of signals, such as the color of the coming light and the presence (or absence) of pedestrians. Researchers often employ behavioral measures, such as response times (RTs), in an attempt to determine whether several sources of information are processed at the same time (in parallel) or one after the other (serially). Such questions were the foundation of perception and elementary cognition in the late 19th century. After such questions had lain dormant for several decades, the 1960s witnessed a renaissance of interest in human information processing, including the issue of serial versus parallel search (also referred to as the “architecture issue”) in memory (Sternberg, 1966) and visual displays (Egeth, 1966).

Visual-search tasks have played an important role in the architecture issue. In such tasks, observers are required to indicate whether a target item is present within an array of distractors. Typically, the target item will be present on half of the trials. The number of items in the display—the set size—is systematically varied, and researchers often plot the time that it takes the observer to detect the target as a function of set size. Mean RTs typically increase with set size in a linear fashion, which suggested to researchers that search was serial in nature. However, it was soon demonstrated that parallel models could perfectly mimic these serial models, and indeed, serial models could also mimic parallel models in search paradigms (e.g., Atkinson, Holmgren, & Juola, 1969; Townsend, 1972; see the reviews in Townsend, 1990; Townsend & Ashby, 1983; Townsend & Wenger, 2004). Over the years, new experimental paradigms and related quantitative methodologies have been developed that can test parallel versus serial processing in ways that avoid the model-mimicking dilemma (e.g., Ashby & Townsend, 1980; Dzhafarov & Schweickert, 1995; Schweickert, 1978; Schweickert & Townsend, 1989; Townsend, 1984; Townsend & Wenger, 2004). These methodologies commonly use signatures in the entire RT distribution to distinguish between serial and parallel processing.

An unexplored signature of RT distributions with regard to the architecture issue is that of multimodality (a distribution that has two or more distinct maxima). In visual search, Cousineau and Shiffrin (2004) employed target and distractor stimuli that were difficult to distinguish and provided clear empirical evidence for multimodality. On target-present trials, for example, they found that several participants showed bimodal RT distributions for a set size of two (see Fig. 2 below). To capture these results, they formulated a standard serial model, as well as a modified standard serial model, both of which were shown to evince multimodality effects. The modified serial model provided the best fits. Although the parallel-versus-serial architecture issue was not the one they were trying to settle (their focus was on the termination issue), it is a logical step to question whether parallel models can evoke multimodality. Because serial models are in some ways more general than parallel models (see, e.g., Townsend, 1976b; cf. Marley & Colonius, 1992; Townsend, 1976a), it is unknown whether parallel models can mimic that type of behavior, or whether multimodal distributions are indeed diagnostic of serial processing.

The question of whether any parallel model can predict multimodality can be quickly cleared up. Since the first model employed by Cousineau and Shiffrin (2004) was a standard serial model, we can appeal to the conditions found by Townsend (1976b) to allow perfect parallel mimicking of a serial model. These conditions demonstrate that whenever a standard serial model predicts multimodality, so does its perfect parallel mimicker (see Townsend, 1976b). However, parallel models that perfectly mimic standard serial models call for the reallocation of resources across processing stages (e.g., Townsend & Ashby, 1983, pp. 83–91), a property that is incommensurate with parallel models that assume independent (i.e., noninteracting) channels. This raises our question of interest: Can a current independent parallel model from the literature plausibly produce multimodality?

We begin by elaborating on the seminal example of serial–parallel mimicry in visual-search tasks, where observing an increase in mean RTs as set size increases is typically taken to suggest a serial search strategy. We then move on to consider the limitations of this interpretation, recapping the work of Townsend and others. Next, we outline the computer simulations used to show another case of parallel–serial mimicry, in which a plausible independent-channel self-terminating parallel system can produce bimodal RT distributions. We describe the basic setup of the simulations and the assumptions of the model, which utilize a common model of decision-making, the linear ballistic accumulator (LBA; Brown & Heathcote, 2008). Finally, we show that a commonly available statistical test can detect this bimodality with as few as 1,000 simulated trials—a reasonable experimental sample.Footnote 1 We conclude that empirically observing a bimodal RT distribution is insufficient to reject accounts based on parallel processing, and even certain instances of independent parallel processing.

Visual search and parallel–serial mimicry

In a typical visual-search task (e.g., Treisman & Gelade, 1980), observers are requested to detect a target item among distractors as quickly as possible. The target may differ from the distractor item(s) by one feature (say, color—a blue target letter among red distractors) or require a combination of features to distinguish it from the distractor items (say, color and form—a blue X among red Xs, red Os, and blue Os). Another important manipulation is set size; trials may contain different numbers of items in the display, sometimes including a target item (positive trials) and sometimes not (negative trials). The inclusion of negative trials is important, so that participants cannot repeatedly press the “yes” button. Example displays for a single-feature search and a two-feature search are presented in Fig. 1a and b, respectively.

Fig. 1
figure 1

Example displays in a visual-search task. Target detection in panel a requires a single-feature search, whereas that in panel b requires search for a conjunction of two features. Panel c shows typical results

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Hypothetical results from such an experiment (e.g., Treisman & Gelade, 1980) are presented in Fig. 1c. For the single-feature condition—in this instance, finding a blue letter among red letters—typical search latencies are roughly the same regardless of display size. That is, the function relating mean RT as a function of the number of items in the display is flat, with a slope of zero. This finding had led to the notion that single-feature search is carried out in parallel (Treisman 1988, 1992). Search for a unique combination of features (conjunction), such as a blue X among distractors that can share with the target either their form or color, results in a monotonically increasing function. This result has been interpreted as the outcome of a slower, serial search that requires attention.

However, this parallel–serial interpretation is not altogether complete. As Townsend and others have shown, under some conditions, the parallel and serial models are mathematically equivalent (e.g., Dzhafarov & Schweickert, 1995; Schweickert, 1978; Townsend, 1990; Townsend & Ashby, 1983). More bluntly put, a parallel model can predict a monotonically increasing mean RT function, and even the straight-line prediction of standard serial models,Footnote 2 as set size increases (Algom, Eidels, Hawkins, Jefferson, & Townsend, in press). Empirically observing a monotonically increasing RT function in a search task, then, even a linear function, is insufficient to determine whether the processing of multiple items is being carried out in serial or in parallel. To overcome this mimicking problem, researchers have had to develop new methodologies (e.g., Townsend & Nozawa’s, 1995, systems factorial technology) or look for other “empirical signatures” that are unique to certain models. One such empirical signature that might be considered a typical prediction of serial processes is the bimodal (or multimodal) RT distribution.

The case for bimodal RT distributions in serial and parallel processes

A serial mode of processing makes the following testable prediction: The number of items that must be scanned before finding the target and responding should vary across trials, depending on where observers start their search and the position of the target. The consequence is a mixture of fast responses, when the target is detected early in the search, and slow responses, when it is detected later on. Combining together distributions of faster or slower responses, each having a different mode, results in a single mixture distribution that may be bimodal (or multimodal, if more than two positions are involved; for simplicity, we will focus on two processes from now on). The question then concerns the interpretation of bimodality when it is observed: Does a bimodal RT distribution necessarily mean that the underlying processes are serial?

As we discussed earlier, Cousineau and Shiffrin (2004) provided one of the most compelling examples of multimodality in the empirical data. They reported data from three participants engaged in a visual-search task, in which the target and distractor stimuli were constructed to encourage serial search (the items were wheels with spokes, and the target was defined by a conjunction of specific spokes). Reexamining their data, Donkin and Shiffrin (2011) noted that “positive responses [in the Cousineau & Shiffrin data] exhibited multimodal response time distributions, with the modes roughly corresponding to the serial position in which the target happened to be compared” (p. 2831, emphasis added). Although Cousineau and Shiffrin were primarily interested in the termination rule (whether participants scan the entire visual field or stop at some earlier point, such as upon identifying a target), their data provide a clear demonstration of multimodality in RT distributions from a visual-search task. Example data from three participants, with display sizes (DSs) of one, two, and four items, are presented in the left panel of Fig. 2.

Fig. 2
figure 2

Example data exhibiting bimodality in visual search (left) and arm reaching (right). The left panel presents an analysis of data collected by Cousineau and Shiffrin (2004), in a figure from “Visual Search as a Combination of Automatic and Attentive Processes,” by C. Donkin and R. M. Shiffrin, 2011, in L. Carlson, C. Hölscher, & T. F. Shipley (Eds.), Expanding the space of cognitive science: Proceedings of the 33rd Annual Conference of the Cognitive Science Society, pp. 2830–2835. Copyright 2011 by the Cognitive Science Society. Reprinted with permission. The right panel is from “Assessing Bimodality to Detect the Presence of a Dual Cognitive Process,” by J. B. Freeman and R. Dale, 2013, Behavior Research Methods, 45, pp. 83–97. Copyright 2013 by the Psychonomic Society. Reprinted with permission

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Freeman and Dale (2013) extended the scrutiny of unimodal versus bimodal distributions to another dependent variable, arm-reaching trajectories. They stipulated that “two processes that work on a different temporal scale . . . predict that the distribution of behavioral measures based on these responses will exhibit bimodality” (p. 84). The right panel of Fig. 2 illustrates trials from a hypothetical experiment in which responses were given by moving a computer mouse to the right or the left side. A mixture of trials of two different kinds (e.g., straight vs. convex trajectories) results in a bimodal distribution of the dependent variable (distance along the x-axis, in this example). It is crucial to note at this point that the two processes need not occur sequentially, one after the other, in order to exhibit bimodality. Rather, they can start at the same time, with one being slower than the other. This situation can readily be accomplished by an independent parallel system that processes one item quickly and efficiently, and the other more slowly. This idea forms the rationale for our simulations.

Constructing a parallel-model simulation that produces a bimodal RT distribution

The observable RT density function of a two-state model may be bimodal as it is a probability mixture of two unimodal distributions. A serial model with two stages of processing is one instance of such a model. In visual search, the stages may reflect the order of processing, with position a being processed first and position b being processed second. However, a parallel model with fast and slow processes for positions a and b, respectively, also qualifies as “mixing two distinct types of response” (Townsend & Ashby, 1983, p. 263). Next, we will describe an example of such a model in the context of visual search, and demonstrate that it can reasonably generate a bimodal RT distribution.

Our simulation was set up to mimic a visual-search experiment with a fixed set size of two. Thus, on each simulated trial, there were two visual positions (e.g., two horizontally aligned items, one to the left and the other to the right of a centrally positioned fixation point). Although we were only interested in the results of trials in which the target was present, our simulation worked like a typical visual-search experiment. Thus, on half of the trials the target was present, whereas for the other half the target was absent. When present, the target was positioned randomly (with equal overall numbers for each position).

The model: Parallel processing with attention gradient

We tested a simple parallel LBA model with “present” and “absent” accumulators for both the left and right positions. Each accumulator collected evidence toward some prescribed threshold, at some rate (as described below). If either target-present accumulator reached threshold before its corresponding target-absent accumulator, a target-present response was triggered immediately. This represented self-terminating search. For an “absent” response to be triggered, both target-absent accumulators had to reach threshold before their corresponding target-present accumulators. This represents exhaustive search. As soon as a response was triggered, the trial was terminated.

Visual attention may affect processing efficiency by way of improving the quality of evidence collection (i.e., increasing the rate of accumulation) in the attended region, by reducing the criterion in that area, or both. Regardless of the exact mechanism, it is said to have salutary effects on performance, as was suggested, for example, by Downing and Pinker (1985), and more recently by Müller, Mollenhauer, Rösler, and Kleinschmidt (2005). Downing and Pinker proposed a Gaussian attention gradient that enhances the processing of visual stimuli within a circumscribed region of space. Alternatively, Müller et al. proposed a Mexican hat function of attention modulation (see also Carrasco, 2011; Hopf et al., 2006, for reviews). The exact form of the modulation function is not important for our efforts, and may actually vary depending on the task difficulty. The critical point is the advantage in processing for items that are closer to the center of attention.

We assumed that attention is always directed to the left position, so items on that side would fall within focal attention, whereas items on the opposite side would get less attention. This assumption could be relaxed without loss of generality, as long as the location of attention and the location of the target item remained unrelated. For simplicity, we kept attention at a fixed location. The attentional advantage could be modeled by a number of model parameters, depending on the accumulator model chosen for the task. Our model of choice was the LBA, which we describe next.

The linear ballistic accumulator

The LBA (Brown & Heathcote, 2008) is a model of rapid choice that has five parameters: base time (nondecision time), maximum possible start point, mean drift rate, drift-rate variance, and response threshold. A response in the LBA model begins with a random amount of evidence, a, sampled from a uniform distribution between 0 and A. The evidence accumulates at a linear and fixed (i.e., ballistic) rate that is sampled from a normal distribution with mean v and standard deviation s, until a threshold amount, b, is collected. The observed RT is the sum of the time taken for evidence to reach threshold in the accumulator associated with the chosen response, plus the time taken for nondecision aspects of RT, such as the time taken to encode stimuli and the time taken to execute the motor response, t 0. Formally, the RT on each trial can be written as T = t 0 + (b – a)/v.

Because evidence is accumulated linearly and at a fixed rate, there is no within-trial variability (unlike in other successful models of choice and RT). This simplification allowed Brown and Heathcote (2008) to derive closed-form solutions for the probability density function f(t) and the cumulative distribution function F(t), which makes the LBA easy to simulate.

The setup for the simulation is illustrated at the bottom of Fig. 3. There were four parallel, independent linear and ballistic accumulators—the target-present and -absent accumulators for each position. As we mentioned before, the response on each trial was determined by the outcomes in each position. If either target-present accumulator reached threshold before its corresponding target-absent accumulator, a “target-present” response was triggered immediately. In contrast, a “target-absent” response was triggered only if both target-absent accumulators reached threshold before their corresponding target-present accumulators. Figure 3 illustrates a two-item display, with distractor O on the left and target X on the right. Since we assume that the left position is within focal attention (illustrated by the yellow circle), the accumulators corresponding to the left item should collect evidence at a higher rate and/or require less evidence (lower threshold).

Fig. 3
figure 3

Illustration of the four parallel, independent accumulators set up for the simulation. The top box illustrates a possible display in a hypothetical visual-search task. The bottom illustrates the corresponding model. The more-attended region on the left of the display is marked by the yellow circle. Since the target item, X, is on the less-attended side of the display, its corresponding correct accumulator (target present) has a lower drift rate than does the correct accumulator on the left (target absent). The threshold is also displayed as being lower for the more-attended accumulators. For each accumulator, the Y-axis represents evidence accumulated, and the X-axis represents time. See the text for details concerning the LBA parameters and their stochastic nature

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The simulation

We used two accumulators for the left, more-attended position, and two for the right, less-attended position. For the correct accumulator in the more-attended position (in Fig. 3, the target-absent accumulator on the left), we fixed the values of all five parameters, as follows: s = 1, t 0 = 0.1, A = 0.5, b = 1, and v = 6. These values are approximately based on estimates that we have obtained from other studies using the LBA to fit empirical data (e.g., Eidels, 2012; Eidels, Donkin, Brown, & Heathcote, 2010). For the less-attended correct accumulator (in Fig. 3, the target-present accumulator on the right), we fixed the values of s, t 0, and A to be the same as those of the attended accumulator. Critically, we systematically varied the values of the rate and threshold parameters of the less-attended correct accumulator. We allowed the drift rate, v, to vary from 6 to 0 (in steps of –0.05). We similarly varied the values of the threshold parameter, b, from 1 to 2 (in steps of 0.01). For both levels of attention, the only parameter to differ between correct accumulators and their incorrect counterparts was the drift rate, which was reduced by a factor of 1/2.

Overall, our systematic changes resulted in 12,220 combinations of rate and threshold (in the less-attended accumulator). We ran two simulations. In the first, we generated 100,000 target-present trials (of 200,000 total trials) for each rate and threshold combination. This simulation demonstrated that bimodality is achievable with plausible LBA parameter estimates. In the second, we generated only 1,000 target-present trials for each rate and threshold combination—the same amount reported by Cousineau and Shiffrin (2004). We assessed the correct responses for multimodality using Hartigan’s dip test (Hartigan & Hartigan, 1985), a commonly available statistical test.Footnote 3 This simulation demonstrated that bimodality can be achieved with a plausible number of experimental trials.

Results

Figure 4 shows the results from our first simulation for a selected subset of parameter values. The RT distributions displayed are for correct responses to target-present trials. Each column in the figure corresponds to a different combination of parameter values. The drift rate for the more-attended accumulator was fixed at 6, but the rates for the less-attended accumulator were 5 (left column), 4.5 (second from left), 4 (third from left), and 3.5 (rightmost column). The thresholds for the attended and unattended correct accumulators in this example were 1 and 1.4, respectively. The rows correspond to the sources of the distribution. The top row presents RT distributions for the attended accumulator, the middle row presents RT distributions for the less attended accumulator, and the bottom row presents the distribution of the mixture of the two previous distributions. Note that the first two rows of distributions are not available for the scientist to view in empirical data, since trials from the two positions are intermixed. Only the empirical distribution, at the bottom of the figure, is available for the researcher to inspect. In particular, the bottom row shows how bimodality builds up in the observed distribution as we decrease the rate of evidence accumulation in the less-attended channel, from left to right. Error rates also increase from left to right (5.18 %, 6.04 %, 7.17 %, and 8.35 %, respectively).

Fig. 4
figure 4

Simulation results for correct target-present decisions for a selected subset of parameter values: Response time distributions for targets in the attended position (a) and the less-attended position (b). The bottom row (c) shows the (empirical) mixture distributions. See the text for details

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The results depicted in Fig. 4 demonstrate that an independent (self-terminating) parallel system can produce bimodal distributions, given plausible parameter estimates. Importantly, Fig. 4 also shows that the RTs in our simulations are realistic. In our simulation, the target item appeared in the less-attended region on half of the trials. For this half of the trials, then, the unattended accumulator determined RTs. Since lowering drift rate and/or increasing threshold leads to slower RTs, it is possible that bimodality could be detected only by mixing very slow trials from this unattended accumulator—slower than are typically observed in empirical data—with faster RTs being determined by the attended accumulator. Figure 4 suggests that this is not the case.

Figure 5 shows the results of Hartigan’s dip tests on each combination within the entire parameter space for both simulations. To recount, the simulations differ only in the number of trials (100,000 vs. 1,000 target-present trials). The horizontal x- and y-axes of the figure thus represent the systematic changes in the threshold (b) and drift rate (v) parameters of the less-attended accumulator. The vertical axis shows the p values from Hartigan’s dip test, with small values (p < .05) rejecting the null hypothesis that the observed distribution is unimodal. The larger simulation (left panel) shows a clear diagonal divide highlighting the space in which bimodality can be detected, whereas the smaller simulation (right panel) shows a noisier pattern. This is commensurate with the drop in power that would be expected, given the reduction in trial numbers by a factor of 100. Crucially, it is evident from both panels that bimodal RT distributions are not a unique prediction of serial processes, or even of serial-mimicking parallel systems; an independent self-terminating parallel system with the justifiable addition of an attention gradient can lead to bimodal RT distributions—even for a substantial part of the parameter space using only 1,000 trials.

Fig. 5
figure 5

Space of p values from Hartigan’s dip test for bimodality as a function of the rate (v) and threshold (b) in the less-attended accumulator. The left panel presents the simulated results for 100,000 target-present trials, whereas the right panel displays the (noisier) results for 1,000 target-present trials. Small p values suggest rejecting the null hypothesis of a unimodal response time distribution

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Discussion

Our simulation results, summarized in Fig. 5, suggest that an independent parallel model with the addition of an attention gradient can generate a bimodal RT distribution that can be detected by Hartigan’s dip test with as few as 1,000 trials. Bimodality can be achieved by increasing the decision threshold in less-attended region(s), by lowering the evidence accumulation rate in the less-attended region, or by various combinations of the two. Bimodality is therefore insufficient to rule out self-terminating parallel models, even those with independent channels.

These results are commensurate with earlier studies on the effects of visual attention using signal detection theory. Farah (1989, Exp. 4) used signal detection measures that separate perceptual and decisional factors, and showed that allocating visual attention to a particular region in space reduced the criterion (beta) in that region. More recent studies, using a different method, have led to a debate whether attention affects the decision threshold or the quality of the percept (i.e., evidence accumulation rate). For example, Carrasco and colleagues (e.g., Anton-Erxleben, Abrams, & Carrasco, 2010; Carrasco & McElree, 2001) argued that attention enhances the rate of evidence accumulation and the apparent contrast, but does not affect bias, whereas Schneider (e.g., Schneider, 2011; Schneider & Komlos, 2008) has stressed that attention alters the decision threshold but not appearance. Our study assumes no side in this debate and makes no claim about how attention enhances performance. Rather, it simply aims to show that bimodality could result from the effects of such an attention gradient, whether the improvement is in rate, criterion, or both.

Close inspection of Fig. 5 may suggest that the attentional effect of a threshold change on bimodality is more robust than that of rate: Increasing the threshold by 60 %, from 1 to 1.6 (while rate was held fixed), resulted in the detection of bimodality by the dip test (at p < .05), whereas a similar result based on rate changes required that rate decrease by half an order of magnitude, from 6 to about 1.2 (while the threshold was fixed). This outcome, however, is likely to depend on the parameters of the simulated model. In particular, the maximum value of the start-point distribution, A, was initially set to 0.5. Recall that the amount of evidence to accumulate is given by subtracting the start point, a [sampled from a uniform distribution ~U(0, A)], from the threshold, b. With an initial threshold of 1, relatively small changes to b will result in proportionally dramatic changes of the expression b – a. However, with a smaller value of A, the same changes in the decision threshold should have less impact. We tested this by repeating the same simulations for various values of A, and found when A was set to 0.2, the effects of rate and threshold were roughly equal (i.e., similar proportional changes in either rate or threshold were required to obtain confidence in the bimodality of the RT distribution).

As always, the present arguments have limitations. For instance, we do not profess that parallel processing is likely under every experimental design. For instance, by making the targets difficult to identify, Cousineau and Shiffrin (2004) increased the likelihood of serial processing. One can even envisage an experimental design that would employ the use of eyetracking so that the target would appear only at a fixation. This would make parallel processing impossible—even after unlimited practice. Finally, it is important to note that we do not profess that the underlying architecture of our independent, self-terminating parallel model naturally gives rise to bimodal distributions. Rather, the justifiable addition of an attention gradient is what makes bimodality possible.

Conclusions

Multimodality might seem like a trademark of serial systems. Although no analytic results exist for general or parameterized serial models as to the conditions that suffice to elicit bimodality, the models constructed elsewhere and successfully applied to multimodal data have been based on serial architectures—which forms a type of existence proof. The situation for parallel systems has been even murkier. Not only has no analytic work been carried out on such systems to date, it has not been made manifest that they are even capable of producing multimodal distributions (though Cousineau & Shiffrin, 2004, allowed for that possibility).

We therefore began by recapping the general serial–parallel equivalence theorem that demonstrated that parallel models can, a fortiori, mimic standard serial models. However, these parallel models that mimic standard serial models are a special case. Unlike so-called standard parallel models, they allow for some form of channel interaction, by reallocating resources or capacity across processing completion stages (see, e.g., Townsend & Ashby, 1983, pp. 48, 88, and 138). Standard parallel models, on the other hand, assume independent channels. Thus, we were interested as to whether self-terminating independent parallel models could be capable of predicting multimodal processing time distributions. On the basis of our results, we concluded that observing a bimodal RT distribution in an experiment (such as, but not restricted to, visual search) is not sufficient to reject accounts that are based on either serial-mimicking parallel system, or even standard, self-terminating, independent-channels parallel processing.