The influence of time structure on number prediction motion

Abstract

Time structure refers to the ratio of time spent moving during visible segments and occluded segments in position prediction motion (PM) tasks. Recent research has found that an equal time structure can improve performance in position prediction motion tasks. However, there is no research to explore the influence of time structure on the number PM task. In three behavioural experiments, this study explored whether an equal time structure improved number prediction motion performance, as it did in position tasks. The results demonstrated that an equal time structure could improve participant performance in the number prediction motion task (Experiment 1). However, equal time structures did not improve task performance when the numbers before the transition number were presented regularly one by one (Experiment 2). Experiment 3 showed that participants could actively take advantage of equal time information when the numbers before the transition number were presented irregularly. These findings demonstrated that when the number sequence was not presented in order, people could use the time structure heuristics to estimate completion time estimates (CTEs). People could sub-vocally count through mental number space when the number sequence was presented in order.

Keywords

Time structure number prediction motion prediction motion

Humans often need to judge a time interval by some cues in daily life. For example, people usually estimate how long the moving car arrives at us in the busy street. In this case, they need to take advantage of physical cues of the car (such as speed, distance, acceleration, or size) to judge whether they can pass the street safely (Aguilar-Lleyda et al., 2018; Battaglini & Ghiani, 2021; Baurès et al., 2018; Bennett & Benguigui, 2013; Keshavarz et al., 2017; Levulis et al., 2015). Researchers have applied prediction motion (PM) tasks to investigate motion processing (Baurès et al., 2017, 2018; Makin & Poliakoff, 2011).

In the laboratory, PM tasks are usually conducted by asking participants to press a button when the moving occluded object reaches a particular point, such as the end of the occluder (Baurès et al., 2021). These are regarded as position PM tasks in physical space. However, motion occurs not only in physical space (as assessed in position prediction motion tasks and interruption tasks), but also in feature space (Blaser et al., 2000; Blaser & Sperling, 2008; Sheth et al., 2000). Researchers can more clearly understand the mechanistic processes underlying PM task performance based on different task dimensions. People often interact with digital instruments in daily life. For example, a driver often needs to estimate when a red light will end. A pilot often needs to determine when an aircraft will reach its designated height based on an altimeter. Recently, Makin and Chauhan (2014) used a novel PM task paradigm, the so-called number PM task, in which people were asked to press a button as soon as the number presented on a screen reached zero during a rapid countdown that disappeared before reaching zero (Makin & Chauhan, 2014). This study showed the close relationship between performance in position and number PM tasks. People who responded earlier in the position PM task tended to do so in the number PM task as well (Makin & Chauhan, 2014). Consequently, they proposed that a common rate control mechanism was used during the occlusion period, which could update the mental imagery of moving objects in the occluded segment (Makin, 2018; Makin & Bertamini, 2014).

Recently, Chang and Jazayeri (2018) found that the time structure can significantly influence the position PM task (Qin et al., 2022). The time structure (T) refers to the ratio of travel time between the visible segment (first segment) and occluded segment (second segment) in position PM tasks (Chang & Jazayeri, 2018). When T = 1.0, indicating that the time spent moving is the same across the two segments, this condition is called an equal time structure. Other conditions are called unequal tine structures, and T ≠ 1.0. Time structure was found to be a useful cue for PM tasks (Qin et al., 2022). Imagining catching an approaching bouncing ball from ground, we usually use the velocity and position cues to estimate when it will contact our hand (Nijhawan, 1994). However, if the factors affecting the judgement of velocity and location are unreliable, for example, when the surrounding is lit diffusely to be able to see the ball, people might estimate the time from the temporal structure of the sound when the ball bounces off the ground (Chang & Jazayeri, 2018). In such a scenario, we can solely rely on time cue to ascertain when the object reaches the target position. Chang and Jazayeri (2018) found that performance was better with equal time structures than with unequal time structures in visual modality. Researchers also found that people performed better when the time structure was equal in the auditory modality (Qin et al., 2022). However, there is no research to explore the role of time structure in number PM task. According to the finding in the position tasks, we assume that people will perform better at the equal time structure condition than under the unequal time structure condition.

Many studies have shown close connections between numbers and space (Hubbard et al., 2005). Furthermore, relative to position PM tasks, number PM tasks can avoid tracking (Barnes, 2008; DeLucia & Liddell, 1998; Makin & Poliakoff, 2011) and tau strategies (Bootsma & Oudejans, 1993; DeLucia, 2013; Lee et al., 1983; Savelsbergh, 1991). Therefore, in the present study, we applied the number PM task paradigm (Makin & Chauhan, 2014) and designed three behavioural experiments to explore the influences of equal time structures in improving number PM task performance, similar to improvements observed in the position task. First, we investigated the influence of time structure on number PM task performance in Experiment 1. Only the initial number and transition number can be observed in Experiment 1, and therefore, participants could only make completion time estimates (CTEs) in the number PM task based on time structure. In Experiment 2, these numbers between the initial number and transition number were presented regularly to explore whether participants took advantage of the time structure or the regular rhythm context to make CTEs. In Experiment 3, we explored whether the participants actively applied equal time intervals to make CTEs when these numbers between the initial number and transition number were presented irregularly.

Experiment 1

This experiment explored the influence of the time structure on performance in the number PM task. Based on previous studies on position PM tasks, we supposed that an equal time structure could improve performance in the number PM task.

Methods

Participants

A priori power analysis using G* Power (Faul et al., 2007) determined the appropriate sample size. A power analysis, in which, by setting α = .05, 22 participants provided .80 power, determined a medium-size effect (f = .25; Cohen, 1992). It was feasible to exceed this minimum sample size recommendation, so we collected 31 (four males; age range: 18–22 years; M_age: 19.58 years) undergraduate students from Shaanxi Normal University to participate in the experiment in exchange for course credits. All participants had normal or corrected-to-normal vision and were naïve to the study’s purpose. The study was conducted according to the guidelines of the Declaration of Helsinki and approved by the Institutional Review Board (or Ethics Committee) of Shaanxi Normal University (protocol code IRB201703121 and date of approval on 10/1/2021). The data and experimental codes for this study are available for download through https://osf.io/62ytb/.

Apparatus

We used the experiment codes written by Makin (2018). The experimental procedure was conducted using PsychoPy 3.0. Stimuli were presented using a Lenovo ThinkVision L1900PA. The screen resolution was 1,920 × 1,280 pixels (horizontal by vertical), and the monitor refresh rate was 60 Hz. The participants were required to sit at a table in a dimly lit room, approximately 60 cm from the cathode-ray tube (CRT) monitor. The centre of the monitor screen was positioned between the two eyes.

Stimuli and design

The number PM task is depicted in Figure 1a. In Experiment 1, only the initial and transition numbers were seen in each trial. Every trial began with “10.00,” “8.00,” or “6.00” in the centre of the screen, which was called the initial number. The blue numbers, whose font was Times New Roman and whose height was 2.0, were presented on the black screen. Then, the number disappeared and counted down to “0.” It was visible again when the number reached 4.00, which was called the transition number. The participants needed to press the space key as soon as they thought the number reached 0. After this, the feedback was presented on the screen, which reminded participants of how long they deviated from the real time that the number reached zero. To create different scenarios so that participants would not exhibit stereotyped responses, we designed two different speeds as a control variable within each level of the time structure. As the velocity was the same in each trial, the time ratio was equal to the number distance ratio before and after the transition number. Thus, the time ratios were 1.5 (for 10.00), 1.0 (for 8.00), and 0.5 (for 6.00). The condition in which the time ratio = 1.0 was called an equal time structure, while the other two conditions were called unequal time structures. Table 1 summarises the durations before and after the transition numbers for different combinations across speeds and time structures. Different levels of speed and number distances were chosen because, like most previous position PM tasks, the actual temporal interval from the transition point to the target point was between 0.5 and 4 s (Chang & Jazayeri, 2018; Makin & Bertamini, 2014; Makin & Chauhan, 2014).

Table 1.

Durations before and after the transition number (ms).

Initial number	Before transition number		After transition number
Initial number	Speed 1	Speed 2	Speed 1	Speed 2
6.00	500	1,000	1,000	2,000
8.00	1,000	2,000	1,000	2,000
10.00	1,500	3,000	1,000	2,000

Figure 1.

Number prediction motion task. (a) Experiment 1, (b) Experiment 2, and (c) Experiment 3.

Procedure

In a silent and dimly illuminated room, the participants were asked to press a key with their right index finger as soon as they thought the number in the screen centre reached zero. After this, feedback was presented on the screen that informed participants about how early or late they pressed the button. The participants initially completed 12 practice trials. Then, the formal experiment started, which comprised 120 trials in total. The feedback exists in both practice trials and formal trials. The two factors of time structure and speed were fully counterbalanced. The trials were presented in a randomised order for every subject. Each of the six possible combinations comprised 20 repetitions; therefore, all participants completed 120 trials.

Analysis

The actual completion time estimates (CTEa) refer to the actual time from the disappearance of the transition number 4.00–0.00. The estimated CTE (CTEe) was defined as the duration from disappearance to the key-pressing time. CTE performance was evaluated by constant error (CE) and variance error (VE), which are defined as follows

CE = {\bar{C T E e}}_{j} - C T E a_{j}

(1)

VE = \sum_{i = 1}^{n} \frac{{(C T E e_{i} - \bar{C T E e})}^{2}}{n}

(2)

where i and j represent the ith trial and jth experimental treatment, respectively. The CE reflects a tendency in the participants’ judgement. A positive CE value indicates CTE underestimation, while a negative CE value indicates CTE overestimation (Braly & DeLucia, 2020). VE represents the CE variance over the repetitions of the same trial. The root mean squared error (RMSE) can also be used to assess participant performance in the PM task, which is defined using CE and VE (Jazayeri & Shadlen, 2010) as follows

RMSE = \sqrt{C E^{2} + V E^{2}}

(3)

The RMSE can be used to compare performance in different conditions. Due to its high effectiveness (Chang & Jazayeri, 2018), the RMSE value is primarily used to represent each participant’s performance in the following analysis.

An analysis of variance (ANOVA) of one-factor three-level repeated measurement was conducted in this experiment. All data were examined for normality and sphericity using Shapiro–Wilk’s and Mauchly’s Sphericity tests, respectively. All of the independent variables followed a normal distribution in each level of time structure, permitting the use of parametric statistical analyses. When the sphericity was not assumed, the Greenhouse–Geisser correction was utilised.

Results and discussion

We excluded the outliers that absolute values of CE were >3.0 SDs of each condition (1.88%). Then we calculated the mean CE, VE, and RMSE of participants in each condition. Figure 2a shows the CE in the three different time structures. The ANOVA showed a significant effect of time structure, F(1.37, 40.69) = 105.62, p < .001, $η_{p}^{2} = . 78$ . Post hoc analyses showed that CE values were significantly lower when T = 0.5 than when T = 1.0 and T = 1.5, both ps < .001. The CE was much lower when T = 1.0 than when T = 1.5, p < .001. The results illustrated an earlier response in the smaller time structure conditions and a later response in the large time structure condition, which was consistent with previous results for the position PM task (Li et al., 2015).

Figure 2.

Results of Experiments 1–3. (a, d, g) The first column shows the CE results. (b, e, h) The second column shows the VE results. (c, f, I) The third column shows the RMSE results.

Figure 2b compares VE values across conditions. The ANOVA demonstrated that the effect of time structure was significant, F(2, 60) = 28.20, p < .001, $η_{p}^{2} = . 49$ . Post hoc analyses demonstrated that VE values were significantly lower when T = 1.0 than when T = 0.5 and T = 1.5, p = .001 and p < .001, respectively. In addition, the difference between T = 0.5 and T = 1.5 was significant, p < .01. The results indicated that people responded more stably in the equal condition.

Figure 2c compares RMSE values across conditions. The analyses revealed a significant effect of time structure on RMSE, F(2, 60) = 39.12, p < .001, $η_{p}^{2} = . 57$ . Post hoc analyses showed that RMSE values were significantly lower when T = 1.0 than when T = 0.5 and T = 1.5, both ps < .001, which indicated that participant performance was better when the time structure was equal. Moreover, there was a significant difference between T = 0.5 and T = 1.5, p < .001. The results suggested that an equal time structure could improve performance in the number PM task, which was consistent with the results in the position PM task (Chang & Jazayeri, 2018) and verified our hypothesis.

The results were consistent with those of position PM tasks: From the view of CE results, participants tended to overestimate CTEs when T = 0.5 and T = 1.0 (Qin et al., 2022). Participants tended to underestimate CTEs when T = 1.5 (Li et al., 2015). The VE results indicated that participants performed more stably at equal time structure condition than at unequal time structure conditions. In addition, the results of RMSE demonstrated that the participants’ performance was significantly higher at equal time structure condition than unequal time structure conditions (Chang & Jazayeri, 2018; Qin et al., 2022).

Experiment 2

In Experiment 1, we found that an equal time structure can improve participant performance in the number PM task. We presented these numbers between initial and transition numbers regularly in Experiment 2, which allowed the participants to observe the process of the initial number decreasing at a constant speed, such as 8.00 (initial number), 7.00, 6.00, 5.00, and 4.00 (transition number). The numbers disappeared after the transition number was presented. The participants were asked to respond when they judged that the number had reached zero (see Figure 1b). The participants could employ two possible strategies to judge the time at which the number reached zero. First, the participants could take advantage of the time structure heuristics to estimate the task, as in Experiment 1. Performance with an equal time structure would be better, as observed in Experiment 1. Second, the participants could use the regular rhythms to develop expectancies about the moment the number reached zero (Sanabria et al., 2011). The performance across the three conditions would not be different when they used rhythmic information to estimate the CTEs since regular rhythmic cues existed in all conditions.

Methods

Participants

Analogously to Experiment 1, in Experiment 2, we tested 30 (6 males; age range: 18–22 years; M_age: 19.68 years) undergraduate students from Shaanxi Normal University. All participants had normal or corrected-to-normal vision and were naïve to the study’s purpose.

Apparatus, stimuli, procedure, and design

The settings were the same as those in Experiment 1, except that the numbers between the initial and transition numbers were also presented (such as “8.00,” “7.00,” “6.00,” “5.00,” and “4.00”). Thus, these numbers could form a regular digital sequence before they disappeared. The influence of time structure on performance in the number PM task was tested when there were regular rhythmic cues. After 12 practice trials, the participants were asked to complete 120 formal trials in total.

Results and discussion

We excluded the outliers that absolute values of CE were >3.0 SDs of each condition (1.05%). Then we calculated the mean CE, VE, and RMSE of participants in each condition. Figure 2d depicts the comparison of CE values. The ANOVA revealed that the effect of time structure was not significant, F(1.36, 39.54) = 2.59, p > .05. Figure 2e compares VE values, and ANOVA showed a significant effect of time structure, F(2, 58) = 5.10, p = .009, $η_{p}^{2} = . 15$ . Post hoc analyses demonstrated that VE values were lower when T = 0.5 than when T = 1.5, p = .011. The differences between T = 0.5 and T = 1.0 and between T = 1.0 and T = 1.5 were not significant, both ps > .05. Figure 2f compares RMSE values, and ANOVA revealed no significant effect of time structure, F(2, 58) < 1, p > .05.

In Experiment 2, participants seemed to use a different strategy from those in Experiment 1. Participants always tended to underestimate the CTEs under the equal or unequal time structure conditions. Moreover, the differences of VE values were not significant at equal time structure condition than at unequal time structure conditions. Most importantly, the results of RMSE revealed that there was no significant difference between these three time structure conditions, which indicated that the number of rhythm intervals before the transition number had no impact on CTEs estimation. All the results of CE, VE, and RMSE suggested that the response pattern was different between Experiment 2 and Experiment 1. Thus, the results confirmed the second hypothesis, that the participants could use regular rhythmic cues to make CTEs.

Experiment 3

The performance improvement in the equal time structure condition was not replicated in Experiment 2. However, did this mean that the same temporal intervals before and after the transition number were useless when the first interval was divided? To test this hypothesis, we presented the irregular rhythmic cues before the transition number: that is, the numbers between the initial number and transition number appeared irregularly in this experiment (see Figure 1c). We predicted that participants’ performance would be better in the equal time interval condition if they had actively utilised the equal intervals.

Methods

Participants

As in Experiments 1 and 2, in Experiment 3 we tested 30 (6 males; age range: 18–22 years; M_age: 19.68 years) participants from Shaanxi Normal University. All participants had normal or corrected-to-normal vision and were naïve to the study’s purpose. One participant was excluded because of procedure error.

Apparatus, stimuli, procedure, and design

All conditions were the same as in Experiment 2 except for the stimulus presentation. In this experiment, the digital sequence decreased at a constant speed. However, the numbers between the initial and transition numbers appeared randomly to prevent participants from forming a regular rhythm that could be used to judge when the number decreased to zero.

Results and discussion

We excluded the outliers that absolute values of CE were >3.0 SDs of each condition (1.18%). Then we calculated the mean CE, VE, and RMSE of participants in each condition. Figure 2g shows the results for the CE values, and ANOVA revealed a significant influence of time structure on CE, F(1.27, 35.51) = 10.66, p = .001, $η_{p}^{2} = . 28$ . The post hoc analyses showed that CE values were significantly lower when T = 0.5 and T = 1.0 than when T = 1.5, both ps < .01. The difference between T = 0.5 and T = 1.0 was not significant, p > .05.

Figure 2h compares VE values and shows a significant effect of time structure on VE, F(2, 56) = 9.24, p < .001, $η_{p}^{2} = . 25$ . The post hoc analyses showed that VE values were significantly lower when T = 1.0 than when T = 0.5 and 1.5, p = .011 and p < .001, respectively. The difference between T = 0.5 and T = 1.5 was not significant, p > .05. These data indicated that, relative to unequal time structures, an equal time structure could improve CE stability.

Figure 2i compares RMSE values, and ANOVA revealed a significant influence of time structure on RMSE, F(2, 56) = 10.48, p < .001, $η_{p}^{2} = . 27$ . The post hoc analyses showed that RMSE values were lower when T = 1.0 than when T = 0.5 and T = 1.5, p < .001 and p = .002, respectively. The difference between T = 0.5 and T = 1.5 was not significant, p > .05. The results demonstrated that participants performed better when the temporal intervals were the same before and after the transition number.

The results of Experiment 3 were similar to those of Experiment 1: The CE results showed that participants tended to press button earlier when T = 0.5 and tended to press button later when T = 1.5. In addition, the results of VE revealed that participants’ response was more stable at equal time structure condition than at unequal time structure conditions. What’s more, participants judged more accurately at equal time structure condition than at unequal time structure conditions, which indicated that when the temporal interval of two segments was the same, the participants’ performance was better. The results suggested that participants might apply the same strategy as in Experiment 1.

Comparison of three experiments data

From the above analysis, participants seemed to use different strategies in the three experiments. In this section, we compared the three experiments data using a 3 (Experiment: 1, 2, 3) × 3 (T: 0.5, 1.0, 1.5) mixed ANOVA with Experiment as a between-subject factor and time structure as a within-subject factor. If participants used the different strategies in the three experiments, there would be a significant interaction between Experiment and time structure.

Results and discussion

For CE, the results showed that the main effects of time structure, F(1.34, 116.86) = 65.88, p < .001, $η_{p}^{2} = . 43$ , and Experiment condition, F(2, 87) = 13.72, p < .001, $η_{p}^{2} = . 24$ , were significant. In addition, the interaction between the two factors was also significant, F(4, 174) = 35.26, p < .001, $η_{p}^{2} = . 45$ . The simple effect analyses showed that, in Experiment 1, the CE values were lower when T = 0.5 than when T = 1.0 and T = 1.5, both ps < .001. The value of CE was lower when T = 1.0 than when T = 1.5, p < .001. In Experiment 2, there was no significant difference in the CE values at three time structure conditions, all ps > .05. The results of Experiment 3 were the same as Experiment 1, CE was lower when T = 0.5 than when T = 1.0 and T = 1.5, p = .033 and p < .001, respectively. The value of CE was lower when T = 1.0 than when T = 1.5, p < .001.

For VE, the main effects of time structure, F(1.86, 160.59) = 30.94, p < .001, $η_{p}^{2} = . 26$ , and Experiment condition, F(2, 87) = 57.26, p < .001, $η_{p}^{2} = . 57$ , were significant. There was an interaction between time structure and Experiment, F(4, 174) = 5.62, p < .001, $η_{p}^{2} = . 11$ . The simple effect analyses indicated that VE was significantly lower when T = 1.0 than when T = 0.5 and T = 1.5 in Experiment 1, p = .001 and p < .001, respectively. The VE values were lower when T = 0.5 than when T = 1.5, p < .001. In Experiment 2, the differences between T = 1.0 and T = 0.5 and between T = 1.0 and T = 1.5 were not significant, both ps > .05. The difference between T = 0.5 and T = 1.5 was significant, p = .046. Again, the results of Experiment 3 were similar to Experiment 1, which demonstrated that VE was significantly lower when T = 1.0 than when T = 0.5 and T = 1.5 in Experiment 1, both ps < .001, respectively. The difference between T = 0.5 and T = 1.5 was not significant, p > .05.

For RMSE, the main effects of time structure, F(1.83, 159.74) = 38.48, p < .001, $η_{p}^{2} = . 31$ , and Experiment condition, F(2, 87) = 94.78, p < .001, $η_{p}^{2} = . 69$ , were significant. In addition, the interaction of the two factors was also significant, F(4, 174) = 12.80, p < .001, $η_{p}^{2} = . 23$ . The simple effect analyses demonstrated that, in Experiment 1, the RMSE values were significantly lower when T = 1.0 than when T = 0.5 and T = 1.5, both ps < .001. The difference between T = 0.5 and T = 1.5 was also significant, p < .001. In Experiment 2, there was no significant difference in the RMSE values at three time structure levels. In Experiment 3, the RMSE values were significantly lower when T = 1.0 than when T = 0.5 and T = 1.5, both ps < .001. The difference between T = 0.5 and T = 1.5 was not significant, p > .05.

From the analyses of CE, VE, and RMSE, we found that the interaction between the two factors was always significant, which indicated that participants had a different response pattern between the three experiments. The results revealed that participants always had a similar performance in Experiment 1 and Experiment 3, which indicated that people could take advantage of the same strategy in both experiments. However, the response pattern of participants had an apparent difference in Experiment 2 with Experiments 1 and 3. The results suggested that participants could use a different strategy in Experiment 2.

General discussion

The present set of experiments was designed to explore in detail the influence of time structure on number PM tasks. Experiment 1 found that the participants judged CTEs more accurately with equal time structure conditions than with unequal time structures, which was the same finding as that previously observed in position PM tasks. Experiment 2 showed no improvement in performance based on an equal time structure when regular rhythmic cues were available. In Experiment 3, similar to Experiment 1, the participants performed better with an equal time structure than with unequal time structures when numbers between the initial and transition numbers were presented irregularly.

There is no research on the role of time structure in number PM task. In this research, we found that time structure had a significant impact on the number task in Experiment 1, which indicated that participants judged more accurately at the equal time structure condition compared with the unequal conditions. Previous studies have shown a significant influence of such equal conditions on accuracy in the position PM task (Qin et al., 2022). In addition, the same results were reported not only in the visual modality but also in the auditory modality in previous research (Qin et al., 2022). In the position PM tasks, researchers put forward that people could use a time transformation strategy, which implies that people could derive an estimate of the duration before the transition point and scale it by the ratio of the distance before and after the transition point (Chang & Jazayeri, 2018). In the number PM task, we thought that people could use the same strategy as position PM task, which indicated that people might estimate the duration before the transition number and scale it by the ratio of the number distance before and after the transition number. Former researchers claimed that mental transformations of more complex cognition process engendered more noise (Pine et al., 1996; Schlicht & Schrater, 2007; Sober & Sabes, 2005). Therefore, the reason why people performed better in the equal time structure could be that the mental transformation was easier when time structure was equal condition compared with when the time structure was unequal conditions.

The RMSE results of Experiment 2 showed that there was no difference between equal and unequal time structure conditions when regular rhythmic information was available. We discussed the results from two aspects. First, the results of Experiment 2 were different from Experiments 1 and 3. In the latter two experiments, participants judged more accurately at equal time structure condition. The difference indicated that participants could use a different strategy in Experiment 2. Comparing the results of the three experiments, it showed that the interaction between experiment and time structure was always significant, which confirmed that the participants did adopt the different strategies to estimate CTEs. Based on this, we speculated that participants could sub-vocally count through mental number space during visible period based on the rhythmic context, and carry-on sub-vocally counting at the same rate during occlusion in Experiment 2. However, they could not do this in the other two experiments. Given that the preferred strategy was unavailable, participants relied more on time structure heuristics in Experiments 1 and 3.

What’s more, the results of Experiment 2 also indicated that the number of rhythm intervals did not influence the estimation of CTEs. Breska and Ivry (2020) found that more rhythmic intervals could improve participants’ performance in temporal anticipation, because rhythmic context could help participants to generate temporal expectancies. The more rhythmic context, the more stable the temporal expectation. However, we found that the number of rhythm intervals did not influence participants’ estimation of CTEs. In the number PM task, people didn’t have to judge the CTEs according to the temporal expectancy. They could sub-vocally count at a constant rate until the number reached zero. Humans are very sensitive to speech and number information. Some studies indicated that phonological loops in the working memory system could retain and retell accurately the rhythmic information of voice information, such as music and language (Patel, 2003). And the accuracy was not influenced by the number of prompting clues. Therefore, the number of rhythm intervals did not influence participants’ performance.

Although the rhythm context still existed in Experiment 3, it could not provide effective information for making CTEs because of the irregularity of this rhythmic information. Therefore, the temporal network needed to find a more effective way to make the CTEs. The results of Experiment 3 had a similar response pattern with Experiment 1, which demonstrated that people estimated the CTEs more accurately at equal time structure condition than at unequal time structure conditions. The results indicated that the participants might also apply mental transformation strategy in Experiment 3. In this scenario, these middle numbers could have been actively ignored, meaning that only the initial number, transition number, and interval between the two numbers were represented in working memory. In this way, the participants could conduct this task as in Experiment 1 with the exception that an active process of inhibiting or ignoring was needed. Neuroimaging techniques can be used to test this hypothesis. Significant activation in the dorsolateral prefrontal cortex (DLPFC) would be predicted if the process was actively inhibited by the temporal network (Friedman & Robbins, 2022; Menon & D’Esposito, 2022). It was concluded that when the middle rhythm context was not reliable, the participants could actively take advantage of time structure heuristics to make the CTEs in the number PM task.

Although it is an essential property, little research has explored the role of time in prior PM research. The number PM task provided an opportunity to explore and isolate the role of time. Previous researchers have usually applied position PM tasks to explore people’s CTEs estimation (Baurès et al., 2010, 2021; Bennett & Benguigui, 2013; Brenner & Smeets, 2015; DeLucia et al., 2016). However, many researchers have pointed out that participants could track visible stimuli with pursuit eye movements and continue to track as well as possible across an occlusion in the position PM tasks (Barnes, 2008; Makin & Poliakoff, 2011). In other fields of research, when eye movements were inhibited, researchers found that the oculomotor system might drive covert tracking with visuospatial attention (cf. Rizzolatti et al., 1987). Therefore, to avoid the interference of tracking, we used the number PM task to explore the role of time in the number PM task. The results showed that when the number sequence was not presented in order, people could use the time structure heuristics to estimate CTEs. People could sub-vocally count through mental number space when the sequence was presented in order.

Footnotes

Acknowledgements

The authors thank Saifang Liu, Quan Xu, Xinyi Zhang, and Jiayu Cui for the valuable suggestions. They express their appreciation to Dr Makin and Dr Baurès for manuscript review and for many helpful comments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (32000753), Young Talent Fund of University Association for Science and Technology in Shaanxi (202008), Fundamental Research Funds for the Central Universities (GK202003098), and Key Research and Development Program of Shaanxi (2021SF-481).

ORCID iDs

Kuiyuan Qin

Yuan Li

Data accessibility statement

The data and materials from the present experiment are publicly available at the Open Science Framework website: .

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