Serial dependence in time perception is attractive and non-linear

We begin this project by examining serial dependence and central tendency simultaneously in a widely employed temporal reproduction task, the “ready-set-go” task [6]. Participants observed a pair of visual events that defined a target interval (“Ready” and “Set”) and then made a single button press (“Go”), attempting to produce an interval between the Set and Go signals that reproduced the target interval (Fig 1A). The target durations were randomly sampled from a uniform distribution that ranged from 500 to 900ms.

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Fig 1. Trial and task structure.

(a) The stimulus sequence used in Experiment 1 (“ready-set-go” task). Two 100-ms stimuli flashed in sequence, signifying first a "Ready" signal and then a "Set" signal; the target duration was the interval between the onset times of the "Ready" and "Set" signals. Participants were instructed to press the space bar to reproduce the temporal interval after the “Set” signal. Performance feedback was conveyed for 50 ms via the color of the fixation cross (green = correct; red = incorrect). (b) The stimulus sequence used in Experiments 2–4. We used a duration reproduction task including both go and no-Go trials. A ripple-shaped stimulus was presented for a fixed duration denoting the temporal interval. After a 300 ms interval, the fixation point became either a "+" sign or “x” sign, signaling either a “go” or “no-go” trial, respectively. In Go trials, participants reproduced the temporal interval by holding down a key. When the key was released, the fixation point turned to a grey circle signaling the end of the trial. In the no-Go trial, participants were asked to withhold any movement and fixate until the "x" switched to a grey circle after 700ms.

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The reproduced durations exhibited robust regression towards the mean (Fig 2A), replicating the central tendency effect seen in previous studies [10, 29, 31]. Quantitatively, the slope of the reproduced duration to the target duration was significantly smaller than one (0.65 ± 0.20; t(11) = -6.08; p<0.001, S1A Fig). To examine serial dependence, we first calculated a “deviation” index, the difference between the reproduction of a target duration on a given trial and the individual’s mean reproduction to that target duration across all trials (see Fig 2A). The serial dependence effect is calculated by the change in the deviation index as a function of the difference between the previous and the current target durations. This method minimizes potential artifacts in the serial dependence function that may be induced by regression to the mean and reproduction biases [32, 33].

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Fig 2. Serial dependence in duration reproduction.

Experiment 1: (a) Reproduced durations of a representative participant. Grey dots represent the reproduced duration of each trial. The blue circle represents the participant’s mean reproduced duration for each target duration. (b) Serial dependence is evident in the non-linear relationship between the deviation index, the current reproduced duration minus the mean reproduced duration, as a function of the temporal difference between the previous target duration and the current target duration (positive values indicate the previous target was longer). Filled dots denote individual participants. Empty circles denote the mean of all 12 participants. The red solid line represents the best-fitted Derivative of Gaussian (DoG) model at the group level. (c) Half-amplitude of the best-fitted DoG in Experiment 1. Error bars represent an estimate of the standard error obtained from jackknife resampling. (d) Instantaneous slope obtained from the fitted DoG curve for trials n-1, n-2, n-3, and n+1 trials. Experiment 2: (e) Instantaneous slope obtained from the fitted DoG curve with respect to n-1 Go, n-1 no-Go trials, n-2 Go trial (when n-1 is no-Go trials), and n+1 (control condition). (f) Half-amplitude of the best-fitted DoG for trials n-1 and n-2.

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We found that the deviation is biased towards the previous stimulus (Fig 2B): When the target duration on trial n-1 was longer than the stimulus on trial n, the reproduced duration tended to be longer than average, and vice-versa. This indicates that the reproduction on the current trial is attracted towards the stimulus duration (or reproduced duration) of the previous trial. Notably, the shape of serial dependence is non-linear: The attraction effect peaks when the current stimulus differs from the previous stimulus by approximately 100ms and then falls off when the difference grows larger.

This non-linear function is well captured by a derivative of Gaussian (DoG) curve [17, 32]. To quantify how previous stimuli influence the current perception in experiment 1, we fitted a DoG to the function of the difference between the target duration in trial n and trial 1-back, 2-back, or 3-back, respectively. We used the instantaneous slope at zero to measure the sign of serial dependence (see Fig 2B). Positive instantaneous slopes indicate that the current perception is attracted towards the previous stimulus duration; negative slopes indicate that the current perception is repelled from the previous stimulus duration. We found a positive slope when calculating serial dependence based on trial n-1 (0.092 ± 0.094, t(11) = 3.39, p = 0.006; Fig 2D), but not when the calculation was based on trial n-2 (0.009 ± 0.079, t(11) = 0.39, p = 0.70) or n-3 (0.028 ± 0.095, t(11) = 1.00, p = 0.34). To better estimate the magnitude of the serial dependence effect in response to the previous stimuli, we fit a DoG at the group level. The DoG provided a good fit (Fig 2B), outperforming the null-model (ΔAIC_n = -46.3 ± 7.7) and linear-model (ΔAIC_l = -18.6 ± 4.6) for the trial n-1 function. The half-amplitude of this function is 7.6 ± 0.6ms, and the half-width is 95.8 ± 10.1ms (Fig 2C). When analyzed at the group level, we also get a significant serial dependence effect from trial n-2 (2.4 ± 0.8 ms, z = 3.00, p = 0.001), but not from trial n-3 (Fig 2C), indicating there might be a weak attractive effect that is not evident in the instantaneous slopes calculated at the individual level. We return to this issue below.

Serial dependence mainly originates from the temporal reproduction

In Experiment 2, we asked whether the serial dependence effect originates from temporal perception or temporal reproduction. To test this, we included No-Go trials in which participants were instructed not to respond. Temporal reproductions following Go trials could reflect biases arising from processes associated only with perception, only with motor production, or both. In contrast, temporal reproductions following No-Go trials should only be influenced by a perceptual bias. Importantly, the Go/No-Go signal was only presented after the target duration, ensuring that participants encoded the target duration on both Go and No-Go trials (Fig 1B).

In trials immediately following Go trials, we replicated the serial dependence effect observed in experiment 1 (instantaneous slope: 0.114 ± 0.042; Wilcoxon test: z = 2.03, p = 0.021; Fig 2E) with a half-amplitude of 7.4 ± 0.9ms (Fig 2F) and half-width of 111.0 ± 15.6ms (S3A Fig). In contrast, we did not find a serial dependence effect from the previous stimulus following a No-Go trial (instantaneous slope: 0.006 ± 0.049; Wilcoxon test: z = 0.01, p = 0.97; Figs 2E and S3B). This dissociation supports the idea that serial dependence in timing largely arises from factors associated with temporal reproduction. We recognize that this may reflect how reproduction is influenced by attention, decision making, memory, or the response itself. Consistent with this notion, when we examined trial triplets composed of Go-NoGo-Go trials, we found a serial dependence effect on the second Go trial towards the first Go trial (rather than the intervening NoGo trial; instantaneous slope, 0.080 ± 0.034, Wilcoxon test: z = 2.25, p = 0.012; Fig 2E–2F; S3C Fig).

Modeling central tendency and serial dependence

Having established robust signatures of the central tendency and serial dependence effects, we now ask whether these two effects are generated by a unitary mechanism or reflect separate mechanisms that are shaped by global and local statistics, respectively. To test this, we formalized our intuitions into a series of computational models.

To explain the central tendency effect, we begin with a Bayesian-least-square model [6] that assumes the observer’s global temporal expectancy is static (i.e., global prior), with bias constrained only by the global distribution of temporal stimuli (Global-Only model, Fig 3A and 3B). The observer’s temporal estimation on a given trial is the Bayesian integration of the global prior with the actual target duration (corresponding to the likelihood in Bayesian theory). As such, the estimated duration of each trial is biased towards the global prior, giving rise to the central tendency effect (Fig 4A). However, given that the Global-Only model uses a fixed prior, the observer’s estimate will not be influenced by the local context (e.g., recently experienced stimuli). That is, the basic Bayesian model cannot account for the serial dependence effect (Fig 4B).

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Fig 3. Schematics of the different models.

(a) In all of the models, the target duration (green) is represented as a normal distribution (likelihood) centered at the target duration with perceptual noise. The global prior (blue) is based on the distribution of the target stimuli. The posterior (yellow) is obtained by multiplying the global prior by the likelihood. (b) The mean of the posterior is computed as a single-value estimate. The reproduced duration is sampled from a normal distribution centered on the estimate with motor noise. (c-d) In the BU and MGU models, the observer updates the prior based on the reproduced duration after the motor response. (c) For the MGU model, a scaled normal distribution centered at the reproduced duration is added to the old prior, and the new prior is normalized. (d) For the BU model, the prior is updated with a Kalman filter. (e) For the Local-Only model, the likelihood is integrated with the local prior in a non-linear manner, with the weight on the local prior increasing when the likelihood and local prior are similar and decreasing when they are dissimilar. In the DP model, the observer is assumed to hold two priors, one global and one local. After the posterior is computed based on the global prior and likelihood, the posterior is integrated with the local prior to generate a second posterior (posterior’). (f) In the second step of the DP and Local-Only models, the new local prior is fully determined by the motor response of the current trial. As with the other three models, the reproduced duration is drawn from a normal distribution centered at the mean of the posterior with motor noise.

https://doi.org/10.1371/journal.pcbi.1011116.g003

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Fig 4. Model simulations.

Predicted central tendency (a) and serial dependence (b) effects for the five models. (c) Predicted relationship between central tendency and serial dependence for the models. Each line is the prediction of one model, and each dot is the prediction with a specific learning rate parameter. The half-amplitude increases with the learning rate. The blue “+” represents the empirical data. The length of the bars indicates SE. (d) Illustration of how the MGU model generates a non-linear serial dependence. For illustration purposes, we depict the n-1 prior as a broad, uniform distribution ranging from 500-1900ms, the range used for the long condition in Exp 3. When the current and previous durations are close (left), the previous duration can influence the shape of the prior in the range around the current duration and make the current posterior shift more relative to the likelihood. In this region, the bias of the posterior increases with the distance between two successive stimuli. However, when the current and previous duration are distinct (Right), the previous duration cannot influence the shape of the prior around the likelihood, and thus, the attraction effect decreases.

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We next considered a unitary mechanism Bayesian Interference model that assumes that the observer integrates the likelihood and a local prior based solely on the local context (Local-Only model, Fig 3E and 3F). The weights of the local prior and likelihood are determined by the difference between distributions associated with the current sample and recently experienced samples. Specifically, the observer will rely less on the local prior when it is far away from the likelihood and more on the local prior when it is close to the likelihood. The Local-Only model can produce a non-linear serial dependence effect (Fig 4B). However, this model cannot simultaneously capture the observed serial dependence and central tendency effects (Figs 4A–4C and S4). Parameter values that generate the observed serial dependence effect produce a smaller than observed central tendency effect. Parameter values that generate the observed central tendency effect produce an unreasonably large serial dependence effect. Thus, a local prior by itself is not sufficient to explain adaptative behavior in the current experiments, indicating that the central tendency effect is not a by-product of serial dependence.

We then considered models capable of simultaneously generating serial dependence and central tendency effects. First, we considered a unitary mechanism model in which the global prior is updated across trials (Bayes-Updating model, BU). Given that serial dependence is driven by temporal reproduction (Exp 2), we assume that, following Bayes rule, the prior is updated by integrating a Gaussian centered at the reproduced duration. Since the priors and posteriors are both Gaussians, the Bayesian integration can be simplified into a linear weighted sum of the means (i.e., a Kalman filter, Fig 3D). This model is mathematically similar to what has been previously described as an internal reference model [10, 11, 26, 34, 35]. While this model predicts a central tendency effect and an attractive serial dependence effect, the predicted serial dependence function is near-linear (Fig 4B), inconsistent with the empirical results of Exps 1–2.

To get a non-linear serial dependence function, the prior can be updated in a non-Bayesian manner. We assume that the brain creates the prior distribution by summing multiple Gaussian distributions with different weights (Mixed Gaussian Updating model, MGU). When perceiving a new duration, the prior is updated by adding a Gaussian centered at the reproduced duration to the old prior, followed by normalization (Fig 3C). This kind of computation has previously been suggested to account for how Purkinje cells in the cerebellum adapt to the prior for representing temporal information [30]. Since the global prior is updated locally, the attraction effect from trial n-1 decreases when the duration of the current stimulus is far from the duration of the n-1 reproduction (Fig 4D). Thus, this model predicts a non-linear serial dependence function and serial dependence in accordance with the empirical results in experiments 1–2 (Fig 4A–4C).

In contrast to the unitary mechanism models described above, an alternative way to produce both serial dependence and central tendency effect is to consider a hybrid model in which the two biases arise from two distinct processes (Dual Priors model, DP). Specifically, the DP model assumes that the current stimulus (likelihood) is integrated with a static global prior (Fig 3A), generating central tendency. The posterior is then integrated with a local prior (Fig 3F), inducing non-linear serial dependence. As such, the Dual Priors model can predict both the central tendency and non-linear serial dependence effects observed in the data (Fig 4A–4C). Note that these two effects are generated by two mechanisms that adapt separately to the global and local temporal context.

In sum, our computational analyses point to two candidate models that can account for the observed central tendency and serial dependence effects. These two models diverge in that the MGU model postulates that central tendency and serial dependence effects arise from a unitary mechanism (non-linear updates to a global prior). In contrast, the DP model postulates that these two effects arise from separate mechanisms (a fixed global prior paired with updates to a local prior). In the following sections, we sought to arbitrate between the MGU and the DP models.

Contextual effect on serial dependence

A key difference between the MGU and DP models is that the latter assumes the local prior is mainly determined by the duration of trial n-1; as such, a serial dependence effect from the n-2 trial will be very weak. In contrast, the MGU predicts a robust serial dependence effect from trial n-2, given that all (recent) observations are integrated into the prior. In experiment 1, the group level analysis indicated a positive serial dependence effect from the n-2 trial. However, this effect was not observed in the individual analyses of the instantaneous slope values.

To address this discrepancy, we sought to enhance the serial dependence effect and examine whether it will be manifest beyond trial n-1. The MGU and DP models both predict that extending the range of the durations in the test set will enhance the trial n-1 serial dependence effect (Fig 5). In the DP, sensory noise scales with duration, a form of Weber’s law. Because of this, the likelihood becomes relatively flat when the range is increased (Fig 5E). This will result in a greater influence of the previous production and, thus, a strengthening of the serial dependence effect (Fig 5F–5G). Sensory noise also influences the serial dependence effect in a similar way in the MGU. In addition, the MGU model postulates that the observer builds a concentrated prior when the range is limited (Fig 5D right). As such, the prior is resistant to updating, yielding a weak serial dependence effect. When the range is expanded, the prior becomes more distributed (Fig 5D left), resulting in a more pronounced local change after each update, and thus, a stronger serial dependence effect (Fig 5B and 5C). Importantly, as noted above, the MGU and DP models generate very different predictions regarding concerning a trial n-2 serial dependence effect (Fig 6A): The MGU predicts that the serial dependence effect should be observed from trial n-2, with a half-amplitude slightly attenuated relative to trial n-1, whereas the DP predicts almost no serial dependence effect from trial n-2.

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Fig 5. The range of the target duration distribution influences the serial dependence effect.

(a) Distributions of target durations in different contexts. (b-c) Simulations of the MGU model predict that the half-amplitude of the serial dependence effect will increase as the range of target duration is increased. (d) In the MGU model, changing the range of the target durations will impact the width of the prior and, therefore, the serial dependence effect. (e) In the PD model, since the scalar property in time perception such that the ratio between the SD and mean is a constant, the likelihood will become flatter as the target duration increases. (f-g) Simulations of the DP model also predict that the half-amplitude of serial dependence increases substantially as the range of target duration is increased. (h) Serial dependence effects for trial n-1 in the medium (left) and long condition (right). Filled dots represent individual participants. Blank circles represent the average of all participants. The turquoise and blue lines represent the best-fitted DoG in the medium and long conditions, respectively. The red and green dash lines represent the best-fitted DoG in Experiments 1 and 2, respectively. (i) Half-amplitude of the best-fitted DoG for serial dependence effect from trial n-1 in Experiments 1, 2, and 3 (medium and long conditions). Each filled dot represents an estimate from jackknife resampling. Error bars represent standard error.

https://doi.org/10.1371/journal.pcbi.1011116.g005

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Fig 6. The serial dependence effect becomes stronger when the range of the test distribution increases.

(a) Simulation of the half-amplitude with the MGU model (left) and the DP model(right). The upper two panels depict simulations for the long-range condition, and the lower two panels depict simulations for the medium-range condition. Simulations of the MGU model can produce robust serial dependence effects from trial n-2, whereas the DP model fails to predict this effect. (b) Instantaneous slope of the DoG-fitting curve for trials n-1, n-2, n-3, and n+1 (control condition) in the medium and long conditions. The p-values are with respect to the difference from zero. (c) Half-amplitude of the best-fitted DoG for the n-1 and n-2 trials in the medium (left) and long (right) conditions.

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Given the predictions of the two models, we extended the range of the target durations in experiment 3. We applied two test sets, one ranging from 520-1260ms (Medium range) and the other from 560-1860ms (Large range, Fig 5A). We compared the results with those obtained in Experiment 1, in which the target durations ranged from 520-880ms (Short range). Extending the range of target duration successfully enhanced the serial dependence effect from trial n-1. The best-fitted DoG of the medium condition (ΔAIC_n = -27.5 ± 4.2; ΔAIC_l = -10.8 ± 2.2) and long condition (ΔAIC_n = -20.7 ± 4.4; ΔAIC_l = -10.0 ± 2.1) had a higher peak and was broader than that for the short condition (Fig 5H). Correspondingly, the half-amplitude increased as the distribution became wider (Fig 5I; medium: 11.8 ± 0.9 ms; long: 22.6 ± 2.0 ms; Zs > 4.9, ps < 0.001). Similarly, the half-widths also increased as the distribution became wider (Zs > 10.5, ps < 0.001), indicating an influence from more distant productions on the previous trial when the test set range increased.

The key question in Experiment 3 centers on the trial n-2 data: Will the increase in the magnitude of the serial dependence effect from trial n-1 be accompanied by a stronger serial dependence effect from non-adjacent trials? In the analysis of the individual functions, there was a significant positive instantaneous slope for the trial n-2 data in both the medium and long conditions (Fig 6B). Trial n-3 also showed a tendency for a positive bias, although this did not reach significance. Moreover, fitting the DoG at the group level showed a significant positive half-amplitude for the serial dependence function from trial n-2 (Fig 6C). Thus, the results are consistent with the prediction of the MGU model and fail to support the DP model (Fig 6A). Moreover, the MGU is consistent with the central tendency effect in all conditions (short, medium, long), further supporting the idea that the process that produces the serial dependence effect uses local information to update the prior (S1F Fig).

We note that in the original version of the DP model, the local prior is decided by the posterior of the 1-back trial. Here we implemented a more general version of the DP model, the Dual-prior short-term memory (DPSM) model, in which the updating of the local prior after each trial incorporates information from previous trials with diminishing weight. The DPSM model generates a serial dependence from the 2-back trial (S5 Fig). However, the model predicts a near-linear serial dependence function, a prediction at odds with the data. Moreover, for the 1-back serial dependence function, the DPSM model predicts the direction of the serial dependence will flip when the current and previous stimuli are distinct from each other; this prediction is also not consistent with the data. In sum, adding a more graded working memory component to the dual prior model cannot account for the results of our experiments.

The variance of prior increases serial dependence

The MGU and DP models also make differential predictions when the variance of the target distribution is manipulated (Fig 7A). In Experiment 4, we set the mean of the target set distribution to 900 ms (the medium condition of Experiment 3). In one condition, we created a set with low variance (720-1080ms, short-900) compared to that used in experiment 3, where the variance was larger (540-1260ms). In the second condition, we used a bimodal distribution to increase variance. The GMU predicts that the serial dependence effect will be enhanced as the variance of the target distribution is increased (Fig 7B). The logic here is similar to that described above in terms of the range of the distribution: Because a low variance test set, by definition, is more concentrated, the effect of trial-by-trial updating of the prior will be smaller relative to when the variance is high. In contrast, the DP predicts that the variance of the test set will have little effect on serial dependence (Fig 7C).

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Fig 7. Serial dependence effect becomes stronger when the variance of the test distribution increases.

(a) Illustration of the distributions of target durations in experiment 4. The short-900 condition (purple) has the same mean as the medium condition (turquoise). The bimodal (orange) has the same range as the medium condition but has larger variability. (b-c) Predicted half-amplitudes of the DoGs for the different conditions by the MGU and the DP models. (d-e) Serial dependence effect for trial n-1 of the short-900 condition (d) and bimodal condition (e). Note the scale for the x-axis is different in d and e (and thus, the function for the medium condition looks different). Each filled dot represents a participant. The open circles indicate the average across participants. The purple, orange, and turquoise lines represent the best-fitted DoG for the short-900, the bimodal-medium, and the medium conditions, respectively. Each filled dot represents one participant. Error bars represent standard error. (f) Half-amplitude of the best-fitted DoG for the trial n-1 data in Experiments 1, 2, and 4. Each filled dot represents an estimate from jackknife resampling. Error bars represent standard error.

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Consistent with the prediction of the MGU, the serial dependence effect was modulated by the variance of the test set. For the trial n-1 data, the best-fitted DoG in the bimodal condition (ΔAIC_n = -78.6 ± 5.6; ΔAIC_l = -18.3 ± 3.6) yielded the highest and broadest serial dependence function of the three conditions (Fig 7E); the short-900 condition (short-900: ΔAIC_n = -48.3 ± 7.8) showed the lowest and narrowest function (Fig 7D). Statistically, the half-amplitudes of the functions increased as the variance of the prior increased (short-900: 8.7 ± 0.6 ms; medium: 11.8 ± 0.9 ms; bimodal: 18.7 ± 0.8ms, Zs>4.2, ps<0.001, Fig 7F). In the analysis of the individual functions, the serial dependence effect was found in both trial n-1 and trial n-2 in the bimodal condition, similar to what was found in the medium condition. In the short-900 condition, serial dependence effect was only significant from trial n-1 (S6 Fig). In sum, the results provide additional evidence that serial dependence is enhanced in higher variance test sets.

Taken together, the results from Experiments 3 and 4 are consistent with the predictions of the MGU model, and at odds with the predictions of the DP model. Manipulations of the distribution properties of the test set indicate that a unitary process gives rise to both the central tendency and serial dependence effects.

Ethics statement

All experimental protocols were approved by the institutional review board of the School of Psychological and Cognitive Sciences, Peking University, and carried out according to the approved guidelines. Written informed consent was obtained from all participants.

Participants

A total of sixty-four students at Peking University were recruited for the four experiments. All participants were right-handed with normal or corrected-to-normal vision. In Experiment 1, thirteen participants (8 females, mean age = 21.1, SD ± 1.0) were recruited for the two 1-hour sessions. One participant did not return for the second session. In Experiment 2, twelve participants (8 females, mean age = 24.1, SD ± 4.2) completed two 1.5-hour sessions. In Experiments 3 and 4, 52 participants (15 females, mean age = 21.1, SD ± 2.1, 13 for each of the four conditions) completed a 1-hour experiment. Participants received $10/h as compensation.

Testing was conducted in a dark room, and the stimuli were presented on a 27-inch LCD monitor (resolution of 1,024 × 768), viewed from a distance of 65 cm. The computer used a Windows 8 operating system with a refresh rate of 100 Hz. The experiment was written in MATLAB (Mathworks, Natick, MA; Psychophysics Toolbox Brainard, 1997;[44]).

Procedure and design

Data analysis

The logic of these experiments is predicated on the assumption that participants are familiar with the temporal context. Given this, the first block of each experiment (approx. 7–10 minutes of data collection) was treated as the familiarization phase and not included in the analysis. In addition, reproduced durations shorter than 0.3 s or longer than 1.5s (2.5s for Experiments 3–4) were considered outliers and excluded from the analyses (less than 0.1% of trials).

The central tendency bias or regression to the mean was quantified as the regression coefficient between the reproduced durations and the target durations. To analyze the serial dependence effect, we used a “deviation” index. For each individual, the average reproduced duration was calculated for each target duration. The deviation was defined as the reproduced duration for a given trial minus the mean reproduced duration of all trials with that target duration [32, 33]. Positive values indicate that the reproduced duration for the present trial was longer than the average reproduction for that target and negative values indicate that the reproduced duration was less than the average reproduction.

To quantify the magnitude of the serial dependence effect, a simplified DoG curve was fit to describe the deviation index as a function of the difference between the current target duration and reference duration, where the reference could be the target duration of the previous trial (n-1), two trials back (n-2), etc., as well as following trial (n+1, serving as a baseline): where y is the deviation, x is the relative target duration of the previous trial, a is half the peak-to-trough amplitude of the derivative-of-Gaussian, b scales the width of the Gaussian derivative, and c is a constant, , which scales the curve to make the a parameter equal to the peak amplitude. As a measure of serial dependence, we report half the peak-to-trough amplitude (half-amplitude) and half the width of the best-fitted derivative of a Gaussian. A positive value for the a parameter indicates a perceptual bias toward the target durations of the previous trials. A negative value for the a parameter indicates a perceptual bias away from the target durations of the previous trials.

We fit the Gaussian derivative at the group and individual level using constrained nonlinear minimization of the residual sum of squares. Jackknife resampling was applied to estimate the variation of the parameters for the group-level fit, where each participant was systematically left out from the pooled sample. The standard deviation of those estimates represented the standard error of the parameter at the group level. The half-amplitude and half-width of the best-fitted DoG were compared between groups with a t-test, where the t-value was computed with the mean and the variance of the parameters estimated from the jackknife resampling procedure. Bonferroni correction was applied for multiple comparisons. To test the goodness-of-fit of our model, we computed the ΔAIC for the DoG model compared with either a non-model (y = 0, ΔAIC_n) or a linear model (y = kx, ΔAIC_l). A negative ΔAIC indicates DoG performed better than the alternative models.

To determine the extent of serial dependency, we fitted individual serial dependence functions in which the reference could be the target duration of the previous trial (n-1), two trials back (n-2), etc. We also tested the serial dependence function for the n+1 trial. Given that this reference stimulus has not been experienced, there should be no serial dependence effect here, providing a test of whether the deviation measure is a valid index to analyze serial dependence. Since individual serial dependence functions do not always show a strong nonlinearity, a parameter (half-amplitude) can become unreasonably large. Thus, we opted to use the instantaneous slope of the DoG at the inflection point (abc) when estimating the presence of a serial dependence effect at the individual level. Note that this index measures the sign of the serial dependence independent of whether or not the function is linear.

In Experiment 2, the Go trials were sorted into two groups based on whether the reproduced interval on that trial was preceded by a Go trial or a No-Go trial. We calculated the average reproduced durations and the deviation using the same protocol as Experiment 1 and then fit DoG functions separately for the Go/Go trial sequence and the No-Go/Go trial sequence to measure the serial dependence effect from trial n-1. For each function, we calculated the instantaneous slope of the DoG at the inflection point. Note that if the serial dependence effect is dependent on a motor response, there should be no serial dependence effect when the preceding trial was a No-Go trial. For Go trials preceded by a No-Go trial, we also analyzed the serial dependence effect from n-2 Go trials. Group-level DoG fitting was performed on n-1 and n-2 Go trials separately to quantify the amplitude of serial dependence.

One-sample t-tests and paired t-tests were applied at the group level for comparisons. Normality and equal variance assumptions were assessed prior to the t-tests. The Wilcoxon Sign-rank test was applied when the normality assumption was violated. Two-tailed P values are reported for all statistic tests, and the significance level was set as p < 0.05. All analyses were performed with MATLAB 2018b (The MathWorks, Natick, MA).

Models

To account for central tendency and serial dependence effects, we implemented five Bayesian models: Two single process models (Global-only, Local-Only), a model with both a static global prior and a local prior (Dual-prior), and two unitary models that capture how a global prior is dynamically updated by recent experience (Bayes-Updating & Mix-Gaussian-Updating).

Global-Only model

This model is based on a Bayesian observer model that uses a Bayes-Least-square as the mapping rule [6]. We assume that the observer builds up an internal prior, π(t_s) based on the target durations (t_s) observed during an initial exposure phase, and subsequent judgments are made by reference to this static prior. The likelihood function describes the probability of the perceived duration (t_p) given t_s. (1) where N(x|m, s) represents a normal distribution with mean m and standard deviation s, and v_p scales the sensory noise. The posterior, π(t_s|t_p), is the product of the prior multiplied by the likelihood function and appropriately normalized.

(2)

The loss function, l(t_e, t_s), was used to convert the posterior into a single estimate, t_e, the mean of the posterior in the present situation.

(3)

(4)

The Bayesian observer makes a response based on t_e: (5) where t_r is the reproduced duration, and v_m scales the motor noise. Previous studies [6, 9] have shown that scalar forms of perceptual and motor noise provide a better fit of the behavior compared to when v_p and v_m are treated as constants. Thus, we set v_p as n_p*t_s, and v_m = n_m*t_e, where n_p and n_m are constants.

For this baseline, Global-only model, we simulated the results under the assumption that the prior was established based on a data set that would be experienced during the first 7–10 minutes of the experiments, and then remained fixed for the duration of the experiment. We assumed that participants learn the true distribution of target durations as the prior: (6) where U(x, [y, z]) represents a uniform distribution ranging from y to z.

Local-Only model

To capture the serial dependence effect, we adapted a Bayesian integration model from a previous study that examined serial dependence in magnitude estimation [26]. In the current context, this model assumes that the observer integrates the previous response (t_r) with the current stimulus (t_s) when estimating the duration of the current stimulus: (7) where t_e,i and t_s,i indicate the t_e and t_s of trial i, and t_r,i−1 is the reproduced duration of trial i-1. W_i−1 is the weight the observer assigns to the trial n-1 response when estimating the current duration. Following Bayes rule to integrate the two samples dictating the current percept, the observer specifies W_i−1 as (8) where σ is the variance of the estimate. The weight is influenced by the distance between the two stimuli, (9)

The variance is assumed to follow a power law.

(10)

Assuming that time perception follows a scalar rule, α = 1. K is a free parameter regulating the behavior of the model.

Dual Prior model (DP)

We combined the Local-Only and Global-Only models to create a Dual Prior model. It contained two Bayesian processes. For the first integration, the observer integrates the current stimulus (t_s) with the prior π(t_s) to get an estimation t_e′, following Global-Only model (Eqs [4]–[7]). The second integration estimates t_e,i based on t_r,i−1, t_e,i′ and t_e,i−1′ following the Local-Only model. The Formulas [7] and [9] are rewritten as (11) (12)

Duel Prior Short-term Memory model (DPSM)

In the DP model, we assume the local prior is determined by the 1-back reproduction. Here, we introduce a more general version of this model that assumes the local prior is decided by short-term memory. Specifically, the local prior (t_prior) is updated after each reproduction following a Kalman-filter: (13) where l is the learning rate that decides how fast the prior is updated. Eq [7] can be rewrite as (14)

Bayes Updating model (BU)

We also considered two unitary process models in which a global prior is updated in a dynamic manner. For the Bayes Updating model, the prior is updated following Bayes rule after each observation. Since the prior and likelihood are Gaussian, this model can be expressed as a Kalman filter. On each trial, the estimated duration (t_e) is a weighted sum of t_o and the duration of the stimulus (t_s).

(15) where the weight w is determined by the sensory noise and variance of the prior. As such, as w increases, the weight given to the prior will increase (i.e., attraction to central tendency). Participants make a motor response based on t_e with Gaussian motor noise (see Eq [5]). Given that Experiment 2 showed that serial dependence is primarily induced by the reproduction component rather than the perceptual component, we assumed the prior is updated according to t_r. After the motor response, t₀ is updated based on t_r following another Kalman filter: (16) where is the new reference point for the next trial, and k represents the learning rate. To address the scalar property of timing noise (Weber law), t_r, t_s, t_e, and t_o are taken to be the log value of the respective durations. Note, that an alternative version of this model could have the observer directly use the posterior as a new prior (what is known as a fully iterative model). However, as with the Local-Only model, a fully iterative model will largely overestimate serial dependence given a reasonable central tendency effect.

Mixed-Gaussian Updating model (MGU)

In a second unitary model, the prior is represented as a Gaussian mixture model. This model is identical to the Global-Only model (Eqs [1–5]) except that the prior is updated after each trial to generate a better estimate of the temporal context. We applied a simple updating rule here, in which the observer adds a normal distribution centered at t_r with a standard deviation of v_p to the old test set. The posterior is then appropriately normalized: (17) where π′(t_s) is the new test set, and r is the learning rate.

Simulation procedure

Simulations of each model were conducted to evaluate the results of Experiment 1. The data from 100 pseudo-participants were generated for each simulation. To determine the values of the free parameters for the simulations, we referred to a previous study that used a similar design to that employed here and evaluated the results with the BLS model, or what we refer to as the Global-Only model [6]. Based on their results, we set n_p = 0.10 and n_m = 0.06 for the Global-Only, MGU, and DP models. Other parameters were determined to make the amplitude of the serial dependence function roughly similar to the behavioral results. Specifically, we set w = 0.7 and k = 0.3 for the BU model, the learning rate r = 0.3 for the MGU model, and K = 0.06 for the Local-Only and DP models. We designed Experiments 3 and 4 to focus on the MGU and DP models, the two models that produce both central tendency and non-linear serial dependence effects. For simulations of the short conditions, the step between every two adjacent stimuli was the same as what was used in the other experiments (40 ms). For simulations of the medium and the long conditions, the step size was set to a smaller value (70 ms) than used in the experiments to improve resolution. The parameters n_p and n_m were fixed at the values used in the previous simulations, while a series of r and K values were tested.