We enrolled 41 healthy human subjects, (22 females, 26.76 ± 4.30 years old, reported as mean ± sd). All participants were informed of the general purpose of the study but were naïve about the scientific questions and tasks, except for two who were the authors. Subjects had normal or corrected-to-normal vision and no history of neurological or psychiatric disorders. Written informed consent was obtained from all participants before the start of the study. Subjects performed three psychophysics tasks, each task with two different motor response types in separate sessions. Here we only report results from two of those tasks. We enrolled the same subjects in all tasks, and the order of tasks was counterbalanced between subjects. The Ethics Committee of the Institute for Research in Fundamental Sciences (IPM) approved this study.

The experiments were carried out on a computer running Linux operating system, on MATLAB (2016b), with Psychtoolbox 3 extension (Brainard, 1997; Pelli, 1997; Kleiner et al., 2007). Stimuli were presented on a monitor (17″) placed ~57 cm from the subject with a 60 Hz refresh rate. The subject sat comfortably on a chair in a dimly lit room to participate in this study, with the head stabilized by a head and chin rest. An EyeLink 1000 infrared eye tracking system (SR Research, Mississauga, Ontario) was used to record eye movements at 1 kHz.

Each trial began with the presentation of a central fixation point (diameter: 0.2°) and two peripheral targets (left target: 0.5° diameters, right target: 2° diameter, 10° eccentricity). After the subject acquired fixation within a ±1° of fixation point, a trial would start. After a random delay (500 ms plus a random sample from an exponential distribution with a mean of 250 ms), two similar wheel-like stimuli (2.5° diameter) were flashed (for 26.6 ms each) sequentially around the fixation point. The presented stimuli were a circle consisting of 6 sectors of equal size. The sectors were colored yellow (150 150 50 in RGB space) and purple (150 50 150), three each. The subject measured the time from the beginning of the first flash to the beginning of the second flash, sample interval or t_s, and produced a matching interval, t_r, by pressing the right arrow key or making a saccade to the right target, depending on the block. Across trials, t_s was sampled from one of 5 discrete values pseudorandomly (400, 500, 700, 1,100, 1900 ms, uniform distributions). At the end of each trial, we showed the response error (t_r − t_s) as feedback for 0.8 s to the subject. The inter-trial interval was 1.2 s, and each block contained 40 trials. Each subject participated in 6 blocks. In 3 blocks, they responded with their hand, and in 3 blocks, they responded with a saccadic eye movement. The order of response type was counterbalanced between subjects.

Each trial started with the presentation of a fixation point and two peripheral targets (the same as the reproduction task). After a random delay, the first interval (t_s1) started with the presentation of the first stimulus and ended with the presentation of the second stimulus (Figure 1B). The second interval (t_s2) immediately started with the second stimulus and ended with the presentation of the third stimulus. The subject measured the time between the beginning of successive flashes as described in the reproduction task, t_s1 and t_s2. The stimuli were the same as the ones we used in the reproduction task, but the order of yellow and purple sectors was different. During the t_s1 or t_s2 interval, only the fixation point was shown (i.e., empty interval). The subject had to compare t_s2 with t_s1; If t_s2 was longer than t_s1, then they had to press the right arrow key or make a saccade to the right target, if t_s2 was shorter than t_s1, then they had to press the left arrow key or made a saccade to the left target. Across trials, t_s1 was sampled from the same 5 discrete values as in the reproduction task. Duration of t_s2 was t_s1 duration ± [6, 12, 24, 48]% of t_s1 duration, selected pseudorandomly on each trial. Feedback was shown at the end of each trial for 0.8 s (a green circle with a 1.25° diameter for correct trials and a red circle of the same size for incorrect trials). The inter-trial interval was 1.2 s, and each block contained 40 trials. Each subject participated in 12 blocks, half with hand response and the other half with eye response. The order of response type was counterbalanced between subjects.

In the reproduction task, we excluded outlier t_r which identified with Interquartile range method (for each subject; 166 from 10,080 trials in total). The mean, standard deviation and range of t_r was calculated for hand and eye blocks (Table 1). We plotted mean of reproduction time (t_r) as a function of interval duration (t_s) for each subject and fitted a linear regression function to them. The fitted regression line yielded the magnitude of the compressive bias [C = 1—the slope of the regression (Roach et al., 2017)] and the indifference point [IP, as the time value at which the fitted regression line intersects the diagonal unity line (Roach et al., 2017)]. We also evaluated the relationship between the interval duration (t_s1²) and timing variance (σ²) by fitting another linear regression, according to Weber’s law. The resulting slope and intercept correspond to the time-dependent (Weber’s fraction) and time-independent processes, respectively (Merchant et al., 2008). These analyses were done separately for blocks that subject responded by hand and saccadic eye movement.

In the discrimination task, psychometric curves were generated by plotting the proportion of long responses over conditions (t_s2) for each interval (t_s1). These points were then fitted by a Gaussian cumulative density function with the non-linear least squares method. The mean (μ) of the fitted curve provided the point of subjective equality (PSE; the time value at which subjects judged t_s2 was equal to t_s1). In order to evaluate how well the psychometric function fitted with data, we measured the goodness-of-fit. We excluded poorly fitted conditions (R ≤ 0.50, 57 from 410 conditions in total). Subjects with more than two excluded conditions were excluded entirely from all analyses (4 subjects). We characterized the relationship between interval duration (t_s1) and PSE by fitting a linear regression. The results of this analysis yielded C and IP, as described before. We also evaluated the relationship between the interval duration (t_s1²) and variance (σ²) of the fitted Gaussian cumulative density function by fitting another linear regression to calculate Weber’s parameters. These analyses were performed separately for blocks in which subjects responded by hand or saccadic eye movement.

We excluded subjects if the calculated indifference point was outside the range of presented intervals (i.e., shorter than 400 and longer than 1900 ms) in the reproduction task (4 subjects) or the discrimination task (4 subjects, one was shared with exclusion based on the goodness-of-fit). Based on these exclusion criteria, 11 subjects were excluded from all analyses, and 30 remained.

To examine whether these computed variables were different between sensorimotor domains and between effectors, for each variable, we performed a two-way repeated measure ANOVA. Wherever we found a significant interaction between sensorimotor domain and effector, we applied HSD Tukey statistical test to control for multiple comparisons.

We applied a Bayesian model with three stages (Jazayeri and Shadlen, 2010): measurement, estimation and motor response (Figure 2). In the first stage, an observer takes noisy measurements, t_m, from sample intervals, t_s. Measurement noise was modeled as a Gaussian function. We modeled the measurement stage as a Gaussian distribution with mean t_s and standard deviation w_mt_s, whereby the standard deviation grows as a constant fraction (w_m) of the mean. This stage is also known as the likelihood function, λ:

In the second stage, the Bayesian model combines the likelihood function and prior and use mean of the posterior to map the resulting posterior probability distribution onto an estimate, t_e. Bayes least-squares (BLS) was used as the mapping rule in our model.

In the third stage, the ideal observer uses t_e to respond, t_r. The relationship between t_r and t_e is characterized by motor noise, which was modeled by a Gaussian distribution with mean t_e and standard deviation w_rt_e.

In the reproduction task, our psychophysical data consisted of pairs of sample intervals and reproduction times (t_s and t_r). We derived a direct relationship between reproduction times and sample intervals in the Bayesian model (Jazayeri and Shadlen, 2010; Acerbi et al., 2012). This formulation was then used to describe subject’s responses in each effector, separately (Jazayeri and Shadlen, 2010; Acerbi et al., 2012). In the discrimination task, we transformed the psychometric function (Gaussian cumulative function) to response distribution (Gaussian probability function) with same parameters, as previous studies described (Getty, 1975; Keele et al., 1985; Merchant et al., 2008). Responses were generated from this distribution and we had pairs of sample intervals and responses. Then same Bayesian formulation was used to describe subjects’ responses for each effector:

We maximized the likelihood of model parameters w_m and w_r across all t_s and t_r values. Maximum likelihood estimation was performed with minimize function in SciPy library, using the Nelder–Mead downhill simplex optimization method. We evaluated the success of the fitting procedure by repeating the search with several different initial values.

We performed Pearson correlation between subjects’ data and Bayesian model predicted data. To counteract the multiple comparisons problem, we performed Bonferroni correction. We showed the effect sizes using partial eta squared (η²) in the ANOVA and Bayes factor (BF) in the correlation analyses. Statistical analysis was performed with Python 3.8.5. A significant level was defined as p-value less than 0.05.

To further evaluate the effects of sensorimotor domain and effector on temporal processing and to understand the computations from which these effects might arise, we applied a Bayesian model with three stages (i.e., measurement, estimation and motor response) and two free parameters (i.e., measurement noise (w_m) and motor noise (w_r); Methods, Figure 2). We fitted the parameters of the Bayesian model, w_m and w_r, for each subject on the basis of t_r. Then we compared w_m and w_r between different tasks and different motor response types. While effector did not have any significant effect (F = 2.93, p = 0.10, η² = 0.01), tasks (F = 5.57, p < 0.03, η² = 0.04) had significant effects on w_m, and the interaction between effector and task reached significant level (F = 5.44, p < 0.03, η² = 0.02). Post-hoc analysis showed that w_m in the discrimination with hand (0.18 ± 0.03) was significantly smaller than in the reproduction with hand (0.21 ± 0.05, adjusted p = 0.03). Both effector (F = 10.31, p < 0.01, η² = 0.05) and tasks (F = 6.03, p = 0.02, η² = 0.05) had significant effects on w_r. Their interaction was significant as well (F = 19.66, p < 0.01, η² = 0.06). Additional analysis showed that w_r was significantly smaller in the reproduction with hand (0.16 ± 0.05) than in the discrimination with hand (0.24 ± 0.11, adjusted p < 0.01); than the discrimination with eye (0.24 ± 0.10, adjusted p < 0.01); and the reproduction with eye (0.24 ± 0.05, adjusted p < 0.01, Table 2).

To compare subjects’ responses to those predicted by the model, we simulated each subject’s responses using the fitted Bayesian model and compared model predictions to the actual responses using the bias and variability statistics. The correlations showed that our Bayesian model can predict subjects’ responses in each effector and each task (Figure 5). The model performed better in the standard deviation variable compared with bias (Figure 5A vs. Figure 5B). When we performed Bonferroni correction the Bayesian model did not describe subjects’ data in the discrimination task with hand (r = 0.378, p-unc = 0.039, p-corr = 0.157, BF = 1.7; Table 3).

To evaluate the effect of effector, we compared subjects’ responses in blocks using a key press to model predictions in blocks using a saccadic eye movement and vice versa in the same tasks. In reproduction, bias was significantly correlated between model from hand data to the data from the eye = 0.47, p < 0.01, BF = 6.1. The correlation between model from eye and data from hand was not significant (r = 0.31, p = 0.09, BF = 1.0). However, in the discrimination task, bias was not significantly correlated between subjects’ responses and model predictions with different effectors (Figure 5 and Table 3). In contrast, variability was significantly correlated between subjects’ responses and model predictions with different effectors in both tasks (r_s > 0.50, p_s < 0.01).

To evaluate the effect of tasks, we compared subjects’ responses in the reproduction task to model predictions in the discrimination task and vice versa with the same effector. Bias was not significantly correlated between subjects’ responses in each task and model fitted to the other task (Figure 5). Variability was significantly correlated between subjects’ responses and model predictions with different task in the eye blocks (model reproduction, data discrimination: r = 0.64, p < 0.01, BF = 218; model discrimination, data reproduction: r = 0.64, p < 0.01, BF = 230) but not in hand blocks (Figure 5 and Table 3). The scatter plot for each correlation analysis is shown in Figure 6.

Previous studies also reported that temporal precision is correlated across different sensorimotor domains (Spencer and Zelaznik, 2003) and different effectors (Keele et al., 1985). We also found similar results. Compression bias, indifference point, and Weber’s parameters were not different between these behavioral contexts. Although these observations may qualitatively suggest that subjects who are good timers in one behavioral context are also good timers in another one, converging data from psychophysical studies remain controversial (for review see Merchant et al., 2013). We used a mechanistic model to address this controversy. Our results demonstrate that the Bayesian model for each effector could not describe bias in other effector in either reproduction (Figure 5a₁) or discrimination (Figure 5a₂) tasks. Similarly, in each task the model predicted data could not describe bias in other task in either hand (Figure 5a₃) or eye (Figure 5a₄) blocks. This absence of correlation suggests that temporal accuracy is context-specific in the sensorimotor and effector domains. As we mentioned earlier, the first two stages of Bayesian model contribute to response bias. These findings and the presence of a significantly different measurement noise (w_m) between different tasks suggest that timing computation in the measurement stage is different between behavioral contexts. In other words, the measurement stage is context-specific in the sensorimotor and effector domains which suggests that the brain might use different mechanisms or computations for timing when it knows that it has to engage in sensory vs. motor timing tasks with different effectors. We also found that the Bayesian model for each effector could describe standard deviation in other effector in either reproduction (Figure 5b₁) or discrimination (Figure 5b₂) tasks, which suggest temporal precision is context-invariant in the effector domain, unlike temporal accuracy. We found similar results in each task in eye blocks (Figure 5b₄), but not in hand blocks (Figure 5b₃). It seems that the temporal precision between reproduction and discrimination tasks in hand blocks (Figure 5b₃) showed different pattern when compared with other behavioral contexts (Figures 5b_1,2,4). Different temporal and non-temporal mechanisms might contribute to this different pattern of temporal precision between the reproduction and the discrimination tasks. Sensory and motor timing tasks have different instructions and different temporal and non-temporal computations which might be recruited in these tasks, for example in the discrimination task subject should memorize and sort two different intervals. Different working memory components between sensory and motor timing tasks may contribute to these different computations in sensorimotor domain. Another contributing factor may be inter-stimulus interval. Previous studies showed that different inter-stimulus intervals change temporal computations in the sensory timing tasks (Karmarkar and Buonomano, 2007; Sadibolova et al., 2021). Both sensory and motor timing tasks in this study did not have inter-stimulus interval (between intervals), however we cannot rule out that the lack of inter-stimulus interval has the same effect on sensory and motor timing tasks.

In sensory and motor timing, the observer’s task can be explained as accumulating evidence in the time domain and comparing it to a bound, similar to what has been shown in decision making (Simen et al., 2011). This evolving decision variable would then lead to a motor action when reaches a bound. In our sensory timing task, temporal decisions and motor responses are made by choosing between two alternatives. However, in the motor timing task, decisions and motor responses are made by choosing when to act. de Lafuente et al. (2015) used a two-alternative forced choice (2-AFC) random dots motion discrimination task showed that when decisions are communicated by different effectors neurons in the medial intraparietal (MIP) area exhibit different firing activity. Instead, decision-related activity was observed in the lateral intraparietal (LIP) area in both hand and eye blocks. We argue that at the computational level, when a temporal decision (or maybe any other decision) has to be made in a 2-AFC task, temporal measurement (the first stage in our model) raises from different computations in eye vs. hand blocks. But when an observer wants to decide when to act, a 1-AFC task, temporal measurement had the same computations between different effectors.

In this section we are going to expand our mechanistic insight into explicit timing using the results of our study. As we discussed earlier our results showed combination of context-specific and context-invariant computations across sensorimotor and effector domains. These observations suggest overlapping and distributed computations as the underlying mechanism of timing in different contexts. Previous neuroimaging and electrophysiological studies also support this idea at the implementational level of analysis (for review see Coull and Nobre, 2008; Paton and Buonomano, 2018; Tsao et al., 2022). Several studies showed that depending on the task and timescale, many areas had been implicated in different temporal contexts (Coull et al., 2004; Jantzen et al., 2004; Bengtsson et al., 2005; Garraux et al., 2005; Jantzen et al., 2005; Pouthas et al., 2005; Jantzen et al., 2007). The results of these previous studies suggest that time is encoded in both context invariant and context specific areas. In this framework, context specific time representation can be encoded through changes in neural population activity over time in distributed neural circuits (Paton and Buonomano, 2018; Tsao et al., 2022). However, a more detailed understanding of the neural substrates of context invariant computations is lacking. These computations might be implemented in overlapping neural circuits. The effects of neuromodulatory systems on distributed neural circuits can also play a critical role in context invariant computations of time. Previous studies reported that the basal ganglia are activated in timing tasks with different effectors (Bengtsson et al., 2005), sensorimotor domains (Schubotz and von Cramon, 2001; Bueti et al., 2008), and duration scales (Jahanshahi et al., 2006). These studies suggested that context invariant computations might be implemented in the basal ganglia. Another study also reported that optogenetic manipulation of substantia nigra pars compacta (SNc) dopamine neurons can modify timing behavior (Soares et al., 2016). The results of this study suggested that dopaminergic projections from the SNc to the striatum could modify striatal population dynamics. Considering that the basal ganglia is interconnected with widespread regions of the cerebral cortex and subcortical areas, it is possible that basal ganglia can modify distributed neural circuits in timing tasks. Future studies are needed to address this question and to understand how these different timing computations are implemented in the brain.

In this study we showed different context-specific and context-invariant temporal computations. Previous studies investigated interval timing in neurological and psychiatric disorders (for review see Allman and Meck, 2012). Malapani et al. (1998) studied a temporal reproduction task in patients with Parkinson’s disease and tested patients while they were ON or OFF their levodopa medication. They showed in the OFF state, temporal reproduction was impaired in both accuracy (bias) and precision (variance). Singh et al. (2021) also showed that impaired cue-evoked midfrontal ~4 Hz activity predicts increased timing variability and both timing variability and midfrontal ~4 Hz rhythms were correlated with overall cognitive impairments in patients with Parkinson’s disease. Timing tasks can be used as a proxy to reveal pathophysiological changes in different neurological and psychiatric disorders and also these studies can reveal mechanistic insights into temporal computations. Future studies can investigate different context-specific and context-invariant temporal computations in different pathophysiological conditions and expand the mechanistic understanding of these different temporal computations.

This study has some limitations. We studied explicit timing tasks and our results cannot be generalized to implicit timing tasks. We also used sensorimotor and effector (i.e., hand and eye response) domains as behavioral contexts however future studies are needed to investigate the effects of other behavioral contexts including other factors on interval timing. The effects of temporal and spatiotemporal learning (Emmons et al., 2020; Gür et al., 2023) on the contextual temporal computations are also interesting for future experiments. In this study we did not compare different computational models of interval timing. We showed that the Bayesian model can accurately describe behavioral data in different sensorimotor domains and different effectors. Our results also showed that the model capture standard deviation better than bias (Figure 5A vs. Figure 5B). We did not investigate the underlying reason of this observation, one reason might be that one parameter, w_m, is responsible for catching the bias and two parameters, both w_m and w_r, are responsible for standard deviation. Previous studies also reported that Bayesian models with the Bayes least-squares strategy in the estimation stage described behavioral data in temporal reproduction task with hand response (Jazayeri and Shadlen, 2010; Acerbi et al., 2012). Our results supported these studies and extended these findings to sensory timing and saccadic eye movement. However, as we discussed earlier the Bayesian models are not the only way to investigate temporal computations. Future studies can compare different computational models of interval timing across different behavioral contexts.