No evidence for the effect of entrainment's phase on duration reproduction and precision of regular intervals

Perception of time is not always veridical; rather, it is subjected to distortions. One such compelling distortion is that the duration of regularly spaced intervals is often overestimated. One account suggests that excitatory phases of neural entrainment concomitant with such stimuli play a major role. However, assessing the correlation between the power of entrained oscillations and time dilation has yielded inconclusive results. In this study, we evaluated whether phase characteristics of neural oscillations impact time dilation. For this purpose, we entrained 10‐Hz oscillations and experimentally manipulated the presentation of flickers so that they were presented either in‐phase or out‐of‐phase relative to the established rhythm. Simultaneous electroencephalography (EEG) recordings confirmed that in‐phase and out‐of‐phase flickers had landed on different inhibitory phases of high‐amplitude alpha oscillations. Moreover, to control for confounding factors of expectancy and masking, we created two additional conditions. Results, supplemented by the Bayesian analysis, indicated that the phase of entrained visual alpha oscillation does not differentially affect flicker‐induced time dilation. Repeating the same experiment with regularly spaced auditory stimuli replicated the null findings. Moreover, we found a robust enhancement of precision for the reproduction of flickers relative to static stimuli that were partially supported by entrainment models. We discussed our results within the framework of neural oscillations and time‐perception models, suggesting that inhibitory cycles of visual alpha may have little relevance to the overestimation of regularly spaced intervals. Moreover, based on our findings, we proposed that temporal oscillators, assumed in entrainment models, may act independently of excitatory phases in the brain's lower level sensory areas.


| INTRODUCTION
Time is a pivotal aspect of our everyday experience. It is so omnipresent and axiomatic that even when we close our eyes and try to abstract ourselves from the outer world, we cannot find a sense for the empty time (James, 1890, p. 619). Despite its gravity, the time has no dedicated sensory organ, and its estimation is subjected to distortions (Eagleman, 2008). One such well-replicated temporal illusion is an overestimation of stimuli that are presented at regular intervals. In the auditory modality, for example, it has been shown that the regularly spaced (isochronous) intervals are overestimated relative to intervals marked by two brief clicks (empty intervals; Nakajima, 1987), irregularly spaced (anisochronous) intervals (Horr & Di Luca, 2015a, 2015b and accelerating and decelerating intervals (Matthews, 2013). Similarly, in vision, where typically this phenomenon is termed flicker-induced time dilation (Herbst et al., 2015;Li et al., 2020;Matuschek et al., 2017;Okajima & Yotsumoto, 2016;Teghil et al., 2019), it has been shown that flickers compared to static stimuli of equal length are overestimated. In spite of its reproducibility in visual (Binetti et al., 2012;Droit-Volet & Wearden, 2002;Hashimoto & Yotsumoto, 2018;Kanai et al., 2006;Ortega & L opez, 2008;Treisman & Brogan, 1992;Yoshimatsu & Yotsumoto, 2021), auditory (Buffardi, 1971;Droit-Volet, 2017;Horr et al., 2016;Horr & Di Luca, 2015c;Nakajima, 1987) and tactile modalities (Buffardi, 1971;Khoshnoodi et al., 2008), the underlying mechanisms of this subjective time dilation are not well understood.
Pacemaker-accumulator models of interval timing (internal clock, Treisman, 1963Treisman, , 2013; scalar expectancy theory [SET], Church et al., 1994;Gibbon, 1977;Wearden, 2003) state that subjective estimation of time is based on a linear accumulation of pulses that are emitted by a pacemaker and are stored in memory for comparison. Assuming that the overall judgement of an interval is based on the weighted sum of segments (Nakajima, 1987), the linearity assumption of such interval models fails to fit the time dilation effect that is associated with regularly spaced intervals (Horr & Di Luca, 2015a; see Matthews, 2013, for a comparison of other models). Yet revising the original model by assuming a logarithmic function between physical and subjective duration has been successful in capturing the illusion (Horr & Di Luca, 2015a, 2015c. Nonetheless, an alternative account of time dilation favours entrainment models of time perception. According to this view, the estimation of time is based on the characteristics of an internal oscillator whose phase and period can be entrained by an external rhythm (McAuley & Jones, 2003;McAuley & Kidd, 1998). In fact, a multitude of electrophysiological studies have reported that rhythmic stimuli are capable of entraining brain oscillations to the frequency that matches the rate of stimuli presentation (e.g., Henry & Obleser, 2012;Herrmann, 2001;Thut et al., 2011; for a recent review, see Lakatos et al., 2019).
Considered as cyclical fluctuations of neural excitability (Buzs aki & Draguhn, 2004;Iemi et al., 2022), in the presence of rhythmic stimuli, neural oscillations align their ideal (highest excitability) phase with the incoming stimulus, which, in turn, amplifies the stimulus-related neural responses (Lakatos et al., 2005(Lakatos et al., , 2007Schroeder & Lakatos, 2009) and improves performance (Cravo et al., 2013;Henry et al., 2014;Stefanics et al., 2010). It has been suggested that the amount of energy used for the neural representation of a stimulus has a direct relationship with its subjective duration (efficient coding hypothesis, Eagleman & Pariyadath, 2009;Matthews et al., 2014). Thus, it can be assumed that increased stimulus-related neural responses due to the alignment of the ideal phase of oscillations with the entraining stimuli may play a role in time dilation. Such a hypothesis thus suggests that the overestimation of regularly spaced intervals can be correlated with amplitude or phase alignment of neural oscillations. In fact, some studies have supported this view by reporting correlations between phaselocking measures or amplitude indices of entrained oscillations and the amount of time dilation induced by rhythmic stimuli (Hashimoto & Yotsumoto, 2018;Horr et al., 2016). However, others have failed to report such a significant correlation. For example, Herbst et al. (2013), after removing between frequency variance, reported a non-significant correlation between the amplitude of steady-state visual-evoked potentials (SSVEPs) and perceived duration. Moreover, their findings indicated that flickers that evoked SSVEPs but were not consciously perceived as flickering did not induce time dilation. Yet, in a recent study, Li et al. (2020) by using beat (intermodulation) frequency reported a moderate time dilation effect that was induced by weak entrainment of flickers that were not consciously perceived as flickering. Therefore, despite its plausibility and consistency with characteristics of neural oscillations, the entrainment account of time dilation has received mixed empirical support.
One possible culprit might be the fact that studies reported here were merely assessing a correlational relationship between neural oscillation characteristics and the amount of induced time dilation. Therefore, in the current study, we attempted to approach a slightly more causal inference in determining the role of entrained oscillation in induced time dilation. To this end, we exploited the excitatory properties of neural oscillation's phase and induced a rhythmic context at the alpha frequency band. The choice of alpha as the entraining context was informed by both its selectivity in inducing flicker-induced time dilation (Hashimoto & Yotsumoto, 2015 and its general causal implications in temporal estimations (Mioni et al., 2020). While a wide range of cognitive functions have been ascribed to alpha (for a review, see Clayton et al., 2018), visual alpha oscillations recorded over occipital areas are generally regarded as rhythmic inhibitory oscillations whose cycles signify periods of higher and lower inhibition (pulsed inhibition theory; Jensen et al., 2012;Klimesch et al., 2007;Mathewson et al., 2011;Mazaheri & Jensen, 2010), which affect perceptual performances (Busch et al., 2009;Dugué et al., 2011;Fakche et al., 2022;Mathewson et al., 2009;Ronconi et al., 2018;Spaak et al., 2014). Designing our experiment based on this notion, after entraining alpha during the context interval, we manipulated the onset of subsequent to-be-timed flickers so that flickers would be presented either in-phase or out-of-phase relative to the entrained rhythm. We hypothesized that such manipulation would land flickers on opposite phases of entrained visual alpha oscillations and thus predicted that amount of time dilation will be different for flickers that land on lower inhibitory phases relative to ones that land on higher inhibitory phases. In addition to the relative amount of overestimation (i.e., (in)accuracy), the variability of reproduced durations (i.e., precision) was also analysed between conditions using the coefficient of variation (CV; Ferrara et al., 1997;Malapani et al., 1998;Meck, 2003). Thus, we additionally sought to explore how different inhibitory phases of visual alpha oscillations affect temporal precision. Based on efficient coding account (Eagleman & Pariyadath, 2009;Matthews et al., 2014), it can be hypothesized that because flickers relative to static stimuli of equal duration produce stronger neural responses (higher phase-locked power), their reproduction should be more precise. Moreover, it can be predicted that this enhancement in precision would be different for flickers landing on less (higher) inhibitory phases. Finally, in order to assess the generalizability of our findings to other modalities, in a separate behavioural experiment, we repeated the same design, yet with isochronous auditory stimuli.

| Participants
A total of 24 students participated in this study. Fourteen students (four female; age range, 22-30) volunteered to participate in the visual experiment (Experiment 1). Ten new students (five female; age range, 20-30) participated in the auditory experiment (Experiment 2). Data from one participant in the auditory experiment were excluded (see Method section in the supporting information). The number of participants per experiment was determined from a priori pilot studies to achieve 80% power with .05 alpha criterion (see Method section in the supporting information). Participants had normal or correctedto-normal vision. They reported no hearing issues and were naïve to the purpose of the study. All participants provided written informed consent prior to the experiment and were reimbursed for their participation. The study was approved by the institutional review boards of the University of Tokyo and was conducted in accordance with the Declaration of Helsinki.

| Apparatus
Participants sat on a partially reclining chair in a dark soundproof room while data were being collected. Stimuli were created using Psychophysics Toolbox V3 (Brainard, 1997;Kleiner et al., 2007;Pelli, 1997) in MATLAB environment and were presented on a 32-in. gamma-corrected LCD monitor with 1920 Â 1080 resolution at a refresh rate of 120 Hz (Display++, Cambridge Research Systems, UK). The subject's head was positioned at a 65-cm viewing distance. Electroencephalography (EEG) data were recorded using a 16-channel electrode DC amplifier (g.tec medical engineering GmbH, Schiedlberg, Austria) with active electrodes positioned on the scalp according to the international 10-20 system. Because visual entrainment results in strong SSVEPs over the occipital regions, data were recorded from Oz, POz and Pz channels, and the channel with the highest power at the entrained frequency was selected for further analysis (in accordance with previous studies conducted on visual entrainment; Li et al., 2020;Mathewson et al., 2012;Otero et al., 2020). The ground electrode was placed on the forehead at FPz, and the reference electrode was mounted on the left earlobe. The data were recorded at a sampling rate of 512 Hz. No online filter was applied. The timings of visual stimuli were recorded using a photocell attached to the monitor and a custom-built trigger box connected to the EEG acquisition system. In the auditory experiment, stimuli were delivered through headphones. The timings of auditory stimuli were recorded using a USB digital-to-analogue converter Focusrite audio interface Scarlett 2i4 1st Generation. The digital inputs from the Focusrite audio interface were received using UCA202 U-CONTROL (Behringer, Germany) audio interface, and the timings of the auditory stimuli were then verified using Audacity software. No EEG data were recorded during the auditory experiment.

| Stimuli and task
A time reproduction task was used in the current study (Grondin, 2010). In the visual experiment, participants had to reproduce the duration of a green circle (1 size, 30-cd/m 2 luminance; Figure 1) that was presented at the centre of the screen (labelled as 'to-be-timed interval').
The green circle was presented either continuously (static) or flickering at 10-Hz frequency (Figure 1). Each flicker consisted of 16.66-ms on-period (corresponding to two frames at the monitor refresh rate of 120 Hz) and 83.34-ms off-period (10 frames). The static stimuli were used as control stimuli so that the amount of flickers' time dilation could be compared against them. The to-be-timed F I G U R E 1 Stimuli dimensions and task design. The top panel displays the timeline of stimuli presentation. Each trial began with a fixation period followed by a context interval. Subsequently, either the 50-or 100-ms blank inter-onset interval (IOI) was presented. IOI was then followed by the to-be-timed interval, which consisted of either the flickering stimuli of 10 Hz (upper row) or the static stimuli (lower row). The standard duration of the to-be-timed interval was 216.66 ms ($217 ms). After a random interval of 500-1000 ms, a beep sound marked the beginning of the reproduction period. In the reproduction interval, for the duration that subjects held the response key, the static stimulus was presented. The bottom plot displays the last stimuli of each context and the subsequent to-be-timed flickers. In the alpha entraining condition, 100-and 50-ms IOI flickers (in-phase and out-of-phase flickers, respectively) landed on opposite phases of entrained alpha oscillation (a). In the unevenly spaced rhythmic condition, the last stimulus of the final sequence was omitted. Thus, in the absence of entrainment, the 100-and 50-ms IOI flickers had expected and unexpected onsets (b). In the empty-interval condition, to-be-timed flickers were presented after 100-or 50-ms IOIs, thus controlling for any potential masking effect (c). The task structure of the auditory experiment was identical to the visual experiment except that the flickering stimuli were replaced by auditory tones (see Section 2). interval was preceded by one of the three context conditions filled by presentation of white annuli (3 outer diameter, 1.05 inner diameter and 30-cd/m 2 luminance).
In the alpha entraining context (Figure 1), the white annuli were presented at a rate of 10 Hz for the duration of 1783.34 ms (16.66-ms on-period and 83.34-ms offperiod; a total of 17 annuli starting with the off-period). The period between the onset of the last entertainer and the onset of the to-be-timed interval was either 50 or 100 ms (termed as inter-onset interval [IOI]; Figure 1). When IOI was 50 ms, the to-be-timed flickers would be presented out-of-phase relative to the established entrainment. And, when IOI was 100 ms, the to-be-timed flickers would be presented in-phase relative to the entrainment. Thus, two conditions (in-phase and out-ofphase) were created (Figure 1a).
Alpha entraining context was the primary condition of interest regarding our hypothesis. However, in the current design, inference from this context is confounded by two factors. First, any time dilation difference between inphase and out-of-phase conditions can be attributed to the fact that the onset of the in-phase to-be-timed intervals was expected. Thus, it is conceivable that the same time dilation difference may be obtained from any condition that manipulates the expectancy of the to-be-timed intervals. Second, irrespective of the entrainment and expectancy, at the lowest sensory level, the time dilation difference between in-phase and out-of-phase conditions can be ascribed to the masking effect. Because in the current design, due to a limitation (maximum lasting duration of entrained oscillations; see Section 4), the outof-phase flickers are always presented at 50-ms IOI, their onset might have been masked relative to the 100-ms IOI flickers. Therefore, in order to control these confounding factors, two additional context conditions were added: unevenly spaced rhythmic (controlling for the onset expectancy) and empty-interval contexts (controlling for the masking effect).
The unevenly spaced rhythmic condition consisted of a set of annuli, which induced expectancy of the onset but were non-entraining at 10 Hz ( Figure 1). The presentation length of each annulus was 16.66 ms, and every three annuli formed a sequence. The three annuli were separated from each other by stimulus-onset asynchrony (SOA) of 50 and 100 ms. The inter-sequence interval was 241.66 ms, and a total of six sequences were presented. However, in the sixth sequence, the last annulus was omitted (therefore, similar to the alpha entrainment condition, a total of 17 annuli were presented), and thus, by presenting the to-be-timed interval at either 50-or 100-ms IOI, we created two conditions. When IOI was 50 ms, the onset of the to-be-timed interval was not expected. Yet when IOI was 100 ms, the onset was expected (Figure 1b).
The empty-interval condition consisted of 1783.34-ms empty interval marked by two annuli. The presentation length of each annulus was 16.66 ms. The IOI between the last annuli of the empty interval and the to-be-timed interval was either 50 or 100 ms to account for any possible masking effect (Figure 1c).
The standard duration of the to-be-timed interval was always 216.66 ms (hereafter, for the sake of simplicity, we refer to the duration of this interval as 217 ms). This choice was informed by the previous studies demonstrating that the entrained visual alpha oscillations last for approximately 3 cycles (200-300 ms) after stimulus offset (Mathewson et al., 2012;Otero et al., 2020;Spaak et al., 2014). Thus, 217 ms (added to the 50-or 100-ms IOI) was the maximum length we could employ for assessing the effects of entrained visual alpha oscillations on duration reproduction.
In the auditory experiment, the stimuli were 2500and 1500-Hz sinusoidal tones (1-ms rise/fall, 60-dB sound pressure level [SPL]), which were assigned to the context or the to-be-timed interval in a counterbalanced manner between subjects. The design in the auditory experiment was identical to the visual experiment with the timing and stimuli durations being the same as what is described for the visual experiment.

| Procedure
The visual and auditory experiments were conducted on different days with non-overlapping participants. The procedure in both experiments was identical. Each trial began with a 500-ms fixation cross followed by a randomly determined blank interval of the range 500-800 ms. Subsequently, either of the three context conditions was delivered. Next, after the IOI (50 or 100 ms), the to-be-timed interval was presented. After a random interval of 500-1000 ms, the reproduction interval was signalled by a beep sound (500 Hz, 300 ms, 60-dB SPL). Subjects were instructed to look at the centre of the screen but attend only to the to-be-timed interval and try to reproduce its duration as accurately as possible by holding the space key. In the visual experiment, for the duration subjects held the key, a static green circle (identical to the one used in the to-be-timed interval) was presented to guide the reproduction. In the auditory experiment, this was replaced with a continuous tone (800 Hz, 1-ms rise/fall, 60-dB SPL). The inter-trial interval (ITI) was set to a random duration of 1000-1500 ms.
The context conditions were presented in the random blocked design, and their order was counterbalanced between subjects. A total of six blocks (two blocks per context) were administrated. Each block contained 84 trials, which were equally allocated to static/flicker and 50/100-ms IOI combinations (21 trials per combination) and were randomly presented. Out of 21 trials per combination, 6 trials (29%) were catch trials whose duration was uniformly at random selected from the set of (100, 100, 100, 300, 300 and 400 ms, without replacement) intervals so that the mean duration of catch trials will roughly correspond to the length of standard duration (217 ms). Catch trials were intended to ensure that participants in fact were attending to the duration of tobe-timed intervals. Analysis of catch trials using repeated measures correlation (Bakdash & Marusich, 2017) revealed that participants were in fact attending to and reproducing to-be-timed intervals considerably well (visual experiment: r = .93, p < .01; auditory experiment: r = .98, p < .01). Catch trials were discarded from further analysis, and the following analyses were conducted on the to-be-timed intervals with the standard duration (217 ms). To avoid fatigue, per 42 trials, a mandatory 2-min break was inserted. Moreover, a self-paced break halfway through the experiment was also implemented.

| EEG
To compare power across contexts, the EEG signal was segmented into epochs of 3100-ms duration (600-ms prestimulus period) time-locked to the onset of the first stimuli in each context. The power of 10-Hz oscillation was then extracted from the span of context intervals and was compared across conditions and against the baseline (see Method section in the supporting information for detailed description of the hypothesis-driven and exploratory time frequency analysis). Moreover, to compare the 10-Hz power for flickers, EEG data were re-segmented time-locked to the onset of the first to-be-timed flicker. Subsequently, power was extracted from the span of tobe-timed intervals (217 ms) and was compared across context and IOI conditions. For the phase analysis, EEG data were first re-segmented time-locked to the onset of each three flickers presented during the to-be-timed interval. Next, corresponding single-trial phase angle values were extracted. Deviation from uniformity and difference in the mean direction of phase angles were then evaluated using the nonparametric Rao's spacing test (Rao, 1976) and Watson's large-sample nonparametric test (Watson, 1983, as cited in Pewsey et al., 2013), respectively. Moreover, group-level phase concentration was compared between conditions using the inter-trial phase clustering (ITPC) analysis (see Method section in the supporting information).

| Behavioural
Time reproduction data were analysed using a 3 (context) Â 2 (stimulus type) Â 2 (IOI) repeated-measure analysis of variance (rANOVA). To include single-trial data into the analysis, single-trial linear mixed-model (LMM) regression analyses were conducted. Moreover, in order to quantify the evidence in support of the null hypothesis, single-trial Bayesian LMM regression analyses were performed (see Method section in the supporting information). Reproduction precision was assessed using the CV and was analysed across conditions with the same rANOVA factors used for the reproduction data.

| RESULTS
To assess whether excitatory phases of oscillations, irrespective of the onset expectancy and masking effect, impact time dilation, three context intervals were created. In the alpha entraining context, rhythmic stimuli of 10 Hz were presented. In the unevenly spaced rhythmic context, a predictable (yet non-entraining at 10 Hz) sequence of stimuli was shown. And in the emptyinterval context, two stimuli marking the beginning and ending of the context interval were presented. Following the context interval with IOI of either 50 or 100 ms, the to-be-timed interval (composed of either 10-Hz regularly spaced or static stimuli) was introduced. By analysing EEG data in the visual experiment, we sought to make sure that in the alpha entraining condition, flickers have landed on opposite phases of high-amplitude entrained visual alpha oscillations. By analysing behavioural time reproduction data, we sought to investigate whether duration and/or precision of reproduced regularly spaced intervals were affected by the opposite phases of entrained visual alpha oscillation.

| Phase-locked 10-Hz oscillation power
To confirm that 10-Hz oscillation was only entrained in the alpha entraining context, phase-locked 10-Hz power was extracted from the context intervals. As expected, the extracted power was significantly different between context conditions (main effect of context, F(2, 26) = 15.86, p = .001) with the power of entrained 10-Hz oscillation being significantly higher in the alpha entraining context (M = 11.35 μV 2 , standard error [SE] = 2.61) relative to the unevenly spaced rhythmic (M = 1.75 μV 2 , SE = .85) and empty-interval (M = 1.07 μV 2 , SE = .30) conditions (p = .004 for both conditions; Figure 2, upper panels). Because the effect of IOI and stimulus type was not significant, the power in each context condition was averaged over these factors and was compared against the power extracted from the baseline. This comparison revealed that phase-locked 10-Hz oscillations were only entrained in the alpha entraining condition (t(13) = 4.18, p = .003; Figure 2a). The hypothesis-driven analysis focused on 10 Hz reported here was followed by an exploratory time frequency analysis (see Method section in the supporting information). The findings of the follow-up analysis closely matched the results reported here except that a significant phase-locked power and phase clustering were detected in the 7-Hz frequency band in the unevenly spaced rhythmic condition (see Results section in the supporting information).
Moreover, in order to make sure that the amplitude of entrained visual alpha was not different between inphase and out-of-phase flickers in the alpha entraining condition and that to-be-timed flickers did not entrain alpha in the control context conditions, the 10-Hz power was extracted from the to-be-timed intervals containing flickers (for the exploratory time frequency analysis, see Results section in the supporting information; Figure S2). The extracted power of to-be-timed flickers in each context condition interacted with the IOI (F(1.43, 18.59) = 5.77, p = .018). Post hoc analysis revealed that in the unevenly spaced rhythmic condition, the power of phaselocked 10-Hz oscillation was significantly higher for the F I G U R E 2 Phase-locked 10-Hz oscillation power. The upper panels show 10-Hz oscillation's power during the context intervals. The extracted 10-Hz power from the alpha entraining context was significantly different from the other contexts. Moreover, the power was significantly different from the baseline only in the alpha entraining context, confirming that 10-Hz oscillation was only entrained in this condition (a). No significant difference from baseline was detected in the unevenly spaced rhythmic (b) and empty-interval context conditions (c), confirming that these context intervals were non-entraining at 10 Hz. The bottom panels show 10-Hz oscillation power extracted from the to-be-timed intervals containing flickers. In the alpha entraining condition, the power of entrained oscillation was not significantly different for out-of-phase (50-ms inter-onset interval [IOI]) compared to in-phase flickers (100-ms IOI), confirming that the presentation of out-of-phase flickers did not significantly decrease the entrained 10-Hz oscillation power (d). In unevenly spaced rhythmic (e) and empty-interval contexts (f), the brief presentation of flickers led to entrained 10-Hz oscillation albeit in different IOI conditions (50-ms IOI for unevenly spaced rhythmic and 100-ms IOI condition for empty-interval context). Inset is the box plot of phase-locked 10-Hz power across all conditions. *p < .05; **p < .01. 50-ms IOI relative to 100-ms IOI flickers (t(13) = 2.72, p = .018; Figure 2e). Conversely, in the empty-interval context, the power of 100-ms IOI flickers was higher than that of 50-ms IOI flickers (t(13) = 2.95, p = .01; Figure 2f). More importantly, in the alpha entraining condition, the power of 100-and 50-ms IOI flickers was not significantly different (t(13) = 2.08, p = .057; Figure 2d). This insignificance thus confirms that outof-phase flickers have not suppressed the power of phaselocked 10-Hz oscillation, which was entrained by the preceding context. However, the fact that a significant increase in 10-Hz phase-locked power was also seen in the other contexts (albeit with different pattern of significance for IOI conditions) suggests that brief presentation of to-be-timed flickers in these contexts has in fact, to some extent, entrained 10-Hz oscillations (see Section 4).

| A 10-Hz phase analysis
To verify that the phases of to-be-timed flickers were non-uniformly structured only in the alpha entraining condition and that the in-phase and out-of-phase flickers were landed on opposite phases of alpha, 10-Hz phase F I G U R E 3 Entrained 10-Hz phase distribution of flickers presented during the to-be-time interval. Panels on the left (a, d and g) indicate the single-trial phase distribution of flickers presented during the to-be-timed interval pooled over all trials and subjects. Panels on the right (b, e, h, c, f and i) show the group-level inter-trial phase clustering values separately for each inter-onset interval (IOI) condition. In the alpha entraining condition (a-c), the phase distributions were non-uniform for both in-phase (100-ms IOI) and out-of-phase (50-ms IOI) flickers each landing on the opposite phases of entrained 10-Hz oscillation. In the unevenly spaced rhythmic (d-f) and empty-interval context (g-i), respectively, the 50-and 100-ms IOI flickers were non-uniformly distributed. This suggests that brief presentation of flickers during the to-be-timed interval in other contexts has partially entrained 10-Hz oscillation albeit in different IOI conditions. values were compared among conditions. Performing Rao's spacing test on phase values extracted from the three flickers that spanned the to-be-timed interval revealed a significant deviation from uniformity for 100-ms IOI (in-phase) and 50-ms IOI (out-of-phase) flickers in the alpha entraining condition (R = 164.03, p < .001 and R = 139.81, p < .01, respectively; Figure 3a). In the unevenly spaced rhythmic condition, a significant deviation from uniformity was detected for 50-ms IOI flickers (R = 158.17, p < .001; Figure 3d). However, the phase distribution for 100-ms IOI flickers was uniform (R = 132.75, p > .10; Figure 3d). In contrast, deviation from uniformity for 100-ms IOI flickers in the emptyinterval condition was significant (R = 143.41, p < .001; Figure 3g). Yet the phase distribution for 50-ms IOI flickers did not deviate from uniformity (R = 134.55, p > .10; Figure 3g). Because the in-phase and outof-phase flickers only in the alpha entraining condition showed significant organization (deviation from uniformity), the mean direction angle of these flickers was compared using Watson's large-sample nonparametric test (Watson, 1983, as cited in Pewsey et al., 2013). The comparison revealed that the mean direction of in-phase flickers (circular mean = 129.18 ) was significantly different from out-of-phase flickers (circular mean = 300.45 ; p < .0001; Figure 3a). This indicates that, in the alpha entraining condition, in-phase flickers were consistently presented at the peak of alpha (90 -270 , crossing 0 ) and that out-of-phase flickers were consistently presented at the trough of alpha (270 -90 , crossing 180 ). Lastly, by analysing ITPC, it was revealed that phase concentration in each context interacted with IOI (F(2, 26) = 14.61, p < .001). Phase concentration of 100-ms IOI flickers was significantly higher than that of 50-ms IOI flickers in the alpha entraining ( p < .006; Figure 3b vs. Figure 3c) and empty-interval contexts (p = .02; Figure 3h vs. Figure 3i). In contrast, in the unevenly spaced rhythmic condition, the phase concentration for the 50-ms flickers was significantly higher than that for the 100-ms flickers (Figure 3e vs. Figure 3f). Moreover, post hoc analysis following the main effect of context (F(2, 26) = 5.1, p = .014) indicated that the phase clustering of to-be-timed flickers in the alpha entraining condition was significantly higher than that in the empty-interval condition (t(13) = 2.93, p = .035).
Overall, analysis of phases pointed to the following results: Both 100-ms IOI (in-phase) and 50-ms IOI (outof-phase) flickers only in the alpha entraining condition were significantly non-uniformly distributed. The mean directions of these non-uniform distributions were opposite and were posited in the peak and trough of 10-Hz entrained oscillation, respectively. However, the amount of concentration of phase distribution for in-phase flickers was significantly higher than that for outof-phase flickers (ITPC results). Moreover, in the other two contexts (unevenly spaced rhythmic and empty-interval), either 50-or 100-ms IOI flickers also showed significant non-uniform distributions, which are indicative of partial 10-Hz entrainment (see Section 4).

| Time reproduction
We analysed the time reproduction data (Figure 4) with a three-way rANOVA hypothesizing that presenting regularly spaced intervals (flickers and flutters, separately for visual and auditory experiments) during lower (higher) inhibitory phases of entrained visual alpha would result in an increased (decreased) time dilation relative to static stimuli (static visual stimuli and continuous tones) and that this effect would be dissociable from the patterns in control conditions (expectancy of the onset and masking). However, in the visual experiment, the three-way context Â stimulus type Â IOI interaction was insignificant (F(2, 26) = 2.60, p = .093). Thus, our hypothesis was not supported. Nonetheless, as expected, flickering relative to static stimuli was overestimated (main effect of stimulus type, F(1, 13) = 42.14, p < .001), indicating the flicker-induced time dilation. Moreover, in all contexts, stimuli (flickering and static) that were placed at 100-ms IOI were overestimated relative to stimuli at 50-ms IOI (main effect of IOI, F(1, 13) = 51.75, p < .001), revealing a general masking effect common to all contexts and stimulus types. Lastly, the amount of reproduction for stimuli (flicker and static) was different among conditions (context Â stimulus type interaction, F(2, 26) = 9.56, p = .001). Following the interaction, post hoc t tests with Bonferroni corrections indicated a trend (t(13) = 2.74, p = .051) for overestimation of static stimuli in the alpha entraining condition relative to the emptyinterval condition (speeding-up clock; see Section 4).
LMM regression analysis confirmed analysis of variance (ANOVA) results, pointing to a main effect of stimulus type (F(1, 13) = 42.21, p < .001), main effect of IOI (F(1, 4864.2) = 43.99, p < .001) and context Â stimulus type interaction (F(2, 4864.1) = 50.23, p < .001). Post hoc analysis following the interaction similarly revealed that static stimuli in the alpha entraining relative to the empty-interval context were overestimated (t(14.1) = 3.40, p = .012). Including the three-way interaction did not significantly improve the model performance relative to the model containing main effects and only context Â stimulus type interaction term (χ 2 (5) = 9.93, p = .07). Finally, our Bayesian analysis buttressed the previous results by showcasing extreme evidence (BF 10 > 100) in support of the main effect of stimulus type, the main effect of IOI and context Â stimulus type interaction over a wide range of priors ( Figure 5).
Moreover, the null result of the three-way interaction was favoured by BF 01 > 10 over a wide range of prior scales (.4-1.5), indicating strong to extreme evidence in support of the null effect. Moreover, restricting the F I G U R E 4 Single-trial distribution of reproduced standard duration (217 ms) per condition in the visual experiment. Black points represent the grand means. Error bars indicate 95% confidence intervals of within-subject standard error (Morey, 2008). IOI, inter-onset interval.
F I G U R E 5 Bayes factor robustness check for match-model effect analysis (visual experiment). The context Â Stimtype Â inter-onset interval (IOI) panel indicates that over a wide range of priors (.4-1.5), there was strong to extreme evidence supporting null effect of a threeway interaction (BF 01 > 10). The robustness of Jeffreys-Zellner-Siow's Bayes factor for each main effect and interaction was inspected by varying the scale of Cauchy prior from .2 to 1.5 in steps of .1. The scale values of √2/2, 1 and √2 are deemed to be 'medium', 'wide' and 'ultrawide' priors (see Method section in the supporting information). BF 01 represents evidence in support of the exclusion of an effect. BF 10 represents evidence in support of the inclusion of an effect. Plus and minus signs are added only for the sake of visibility. regression analysis only to the subjects whose individual alpha peak frequency (IAF) was 10 Hz did not change the significance of the effects reported here (see Results section in the supporting information). The only exception is that the context Â stimulus type interaction for 10-Hz IAF subjects was due to overestimation of flickers in the unevenly spaced rhythmic condition relative to the alpha entraining context (t(8.72) = 3.38, p = .02; Figure S10).
LMM regression analysis confirmed ANOVA results, pointing to a main effect of stimulus type (F(1, 8) = 7.61, p < .024), main effect of phase (F(1, 3119.12) = 16.36, p < .001), context Â IOI (F(2, 3119.12) = 7.95, p < .001) and stimulus type Â IOI interactions (F(1, 3119.15) = 8.67, p = .003). In addition, the context Â stimulus type interaction was significant (F(2, 3119.13) = 7.31, p < .001) but was not supported by any of the post hoc t tests with Bonferroni correction. Finally, adding the three-way interaction did not improve model performance relative to the best fitting model (χ 2 (2) = 3.17, p = .204). The results of the Bayesian analysis of auditory experiment data ( Figure S7), however, were not as decisive as visual data, showing moderate to anecdotal evidence (1 < BF 10 < 10) for the main effect of stimulus type, stimulus type Â IOI and context Â stimulus type interactions. Nevertheless, regarding our hypothesis, there was strong to extreme evidence (BF 01 > 10) in support of a null three-way interaction effect over a wide range of priors (.3-1.5). Furthermore, there was strong to extreme evidence (BF 10 > 10) supporting the main effect of phase and context Â IOI interaction effect.
Overall, the analysis of time reproduction data, contrary to our hypothesis, did not indicate any three-way interactions. Instead, in the visual experiment, a main effect of IOI (in absence of higher level interaction) was observed indicating that irrespective of context and stimulus type, intervals presented at 100-ms IOI were more overestimated relative to intervals in 50-ms IOI condition. This finding suggests that a potential masking effect, common to all stimulus types and contexts, led to less overestimation of 50-ms IOI intervals. In the auditory experiment, in addition to the common masking effect (main effect of IOI), IOI interacted with the context revealing that stimuli in the 50-ms IOI (compared to the 100-ms IOI) were less overestimated in the emptyinterval context. This shows that the masking effect of auditory tones was stronger in the empty-interval condition relative to other contexts.
One issue with study designs where the effect of interest lies in interactions is that such studies often do need large-sample sizes for detecting the true effects (Brysbaert, 2019;Rouder & Haaf, 2018). However, with the frequentist approach, it is impossible to quantify the amount of evidence in support of the null hypothesis (Lee & Wagenmakers, 2014). Thus, a null effect in a frequentist approach may arise either due to the study being underpowered or due to the fact that a null effect is genuinely present. However, in the Bayesian framework, these two are separable as a lack of enough observations in underpowered studies results in uncertain Bayes factors (BFs ≤ 10), which are highly dependent on prior specifications (Brysbaert, 2019;Kruschke, 2015). Thus, in this study, we supplemented our frequentist approach with Bayesian analyses to assess whether the null effect found in our study receives certain support. Adhering to this approach, we restricted our interpretation of the results only to the effects that received strong evidence (BF ≥ 10) over a wide range of plausible priors. By looking at Figures 5 and S7, it is evident that in both auditory and visual experiments, there is strong evidence against the three-way interaction of context, stimulus type and IOI (BF 01 ≥ 10) over a wide range of priors.
Moreover, due to the limited window of entrainment persistence (see Section 4) and the need to keep the temporal structure of the task similar across the subjects, we could not entrain the alpha rhythm tailored to each individual's alpha peak frequency (IAF). Because IAF has been linked to temporal mechanisms (Mioni et al., 2020;Samaha & Postle, 2015), we reanalysed our data taking IAF into account (see Results section in the supporting information). Based on studies suggesting that IAF can vary depending on the task demands (Klimesch et al., 1993;Wutz et al., 2018), we extracted each subject's IAF only considering the alpha entraining condition's epochs. This was necessary because our primary interest was to examine if subjects with 10-Hz IAF during the alpha entraining condition were in fact different from the full sample with regard to the temporal performance. Yet restricting our analysis to the subset of subjects (N = 8) whose IAF during alpha entraining context was 10 Hz did not change the pattern of results obtained from the full sample (N = 14).

| CV
We hypothesized that because neural entrainments are concomitant with regularly spaced stimuli (flicker and flutters, separately for visual and auditory experiments), the signal amplification characteristics of entrainment would boost reproduction precision of these stimuli relative to non-entraining static stimuli (static visual stimuli and continuous tones). Moreover, if this hypothesis holds, the precision should be higher for the alpha entraining condition relative to other contexts and that, in this condition, in-phase flicker and flutters should have higher precision. In visual experiment (Figure 6), the 3 (context) Â 2 (stimulus type) Â 2 (IOI) rANOVA only indicated a main effect of stimulus type (F(1, 13) = 31.87, p < .001). Pairwise post hoc comparisons revealed that flickers in general had higher precision (i.e., lower CV; t(13) = 5.65, p < .001). The results obtained with 10-Hz IAF subjects were identical (see Results section in the supporting information). In the auditory experiment ( Figure S8), the rANOVA only pointed to a significant context Â stimulus type interaction (F(3, 16) = 3.96, p = .04). Post hoc t tests revealed that for the static stimuli, the reproduction precision was significantly lower (as indicated by higher CV) in the empty-interval condition relative to the unevenly spaced rhythmic context (t(9) = 3.05, p = .047).

| DISCUSSION
In this study, we aimed to assess whether the neural entrainment concomitant with regularly spaced stimuli has a major impact on the time dilation effect that is associated with these types of stimuli. To do so, we first provided a visual context in which we entrained the neural oscillations at 10 Hz. Subsequently, we experimentally manipulated the presentation of 10-Hz flickers so that such flickers would be presented either in-phase or outof-phase relative to the established rhythm. In doing so, we took advantage of the previously known phenomenon that entrained oscillations in the alpha band persist for durations of at least 3 cycles (200-300 ms;de Graaf et al., 2013;Mathewson et al., 2012;Otero et al., 2020;Spaak et al., 2014). This fact provided an opportunity for us to directly shed light on the role of excitatory phases of oscillations in flicker-induced time dilation. Because flickering stimuli, due to their rhythmic identity, naturally entrain brain oscillations as they unfold over time F I G U R E 6 Average coefficient of variation (CV) per condition in the visual experiment. Across all contexts and all inter-onset interval (IOI) conditions, flickers were reproduced with higher precision (as indicated by smaller CV values) relative to static stimuli. Circles represent subject averages. Stars represent grand averages. Error bars represent 95% confidence interval of within-subject standard error (Morey, 2008). (Herrmann, 2001), it is almost impossible to dissociate the entraining nature of such stimuli and observe how much time dilation would be impacted in the presence or absence of entrained oscillations. However, building our experiment based on entrainment residuals, we exploited the opportunity to place upcoming flickers in phases of higher/lower excitability of entrained oscillation to assess whether time dilation would be affected by the cyclical fluctuations of excitability that accompany flickering stimuli. Yet, because entrainment residuals only last for a brief time, the window for manipulation of flicker presentations is limited. Thus, inadvertently, the out-of-phase flickers should be positioned at either a proportionately far distance relative to the offset of the entrainment (150-ms IOI for 10 Hz) or relatively close to the offset of the entrainment (50-ms IOI). Because the entrainment residuals would have been lost with the far IOIs, we chose to set the IOI to the closer value (50 ms). However, we added a control condition that would explain away the effect of masking the last entertainer might impose on timing of the onset of the out-of-phase flickers. Moreover, regarding the onset, in addition to the sensory-level masking effect, another higher level confounding factor existed in the current design. Namely, the onset of in-phase flickers is predicted by the context; yet the onset of out-of-phase flickers is rather unexpected. Such an expectancy factor can complicate inference about any differences observed between in-phase and out-of-phase flickers. Therefore, a predictable context (labelled as unevenly spaced rhythmic) was created to regress out the effect of expectancy from the results.
Our EEG analysis confirmed that, only in the alpha entraining condition, the power of phase-locked 10-Hz oscillation was significantly different from the baseline and thus was significantly higher than the other two context conditions (Figure 2, upper panels; Figure S1). Moreover, this increase in power at the entrained 10-Hz oscillation was not significantly different during the time interval when in-phase and out-of-phase flickers were presented (Figures 2d and S2a,d). Thus, the brief presentation of out-of-phase flickers did not suppress the power of the residual entrainment. Moreover, the phase analysis also corroborated that the phase distribution of both inphase (50-ms IOI) and out-of-phase (100-ms IOI) flickers was non-uniformly distributed only in the alpha entraining context and had opposite directions (Figure 3a). It has been suggested that the peak and trough of alpha oscillations are associated with minimal and maximal inhibition, respectively (Mathewson et al., 2011). In line with this hypothesis, studies have shown that detection of near-threshold stimuli is facilitated (impeded) during peak (trough) of visual alpha oscillations (Mathewson et al., 2009(Mathewson et al., , 2012. Similarly, in our study, flickers that were in-phase relative to the entraining context landed on peaks of 10-Hz oscillation, while out-of-phase flickers landed on the troughs, therefore coinciding lower and higher inhibitory phases of alpha, respectively. If maximal (minimal) inhibitory phases of entrained visual alpha oscillation had an impact on the time dilation, a three-way interaction would be expected. However, our Bayesian analysis pointed to strong evidence against such a three-way interaction in both the visual and auditory experiments.
Furthermore, it is worthy of mentioning that contrary to our expectation, the brief presentation of flickers during the to-be-timed intervals in the unevenly spaced rhythmic and empty-interval contexts partially entrained 10-Hz phase-locked oscillations. In the unevenly spaced rhythmic condition, the power and phase concentration of 50-ms IOI flickers yielded significant results, whereas in the empty-interval condition, the power and phase concentration of 100-ms IOI flickers were significant. Nonetheless, this incidental entrainment does not necessarily jeopardize the expected three-way interaction. That is, even though 10-Hz phase-locked oscillations were entrained by different IOI flickers in each of these contexts, the mean phase direction of the significant nonuniform distribution for both contexts was identical (aligning to the peak of alpha; Figure 3f,h). Hence, if phases of lower inhibition indexed by the peak of alpha affected time dilation, 100-versus 50-ms IOI flickers in these conditions should have been differentially overestimated.
Our null finding has implications both for the role of entrained visual alpha oscillations in perceptual performances and for the time-perception models. In recent years, in addition to studies that have reported positive findings (e.g., Busch et al., 2009;de Graaf et al., 2013;Dugué et al., 2011;Fakche et al., 2022;Mathewson et al., 2009;Ronconi et al., 2018), there has been a trend in reporting null results regarding the role of inhibitory phases of entrained alpha oscillations in perception (de Graaf et al., 2020;Vigué-Guix et al., 2020; also for null findings regarding the phase of spontaneous alpha oscillations, see Benwell et al., 2017;Ruzzoli et al., 2019). Such null findings essentially adjust our understandings of what neural oscillations are and what roles they play in perception. It has been suggested that the type of task used in studies can have an impact on perceptual consequences of alpha oscillation's phase (Lin et al., 2021). For example, the majority of studies that reported a significant role for alpha oscillation's phase in perception have used a range of tasks that target lower level sensory detection (Busch et al., 2009;Dugué et al., 2011;Fakche et al., 2022;Mathewson et al., 2009Mathewson et al., , 2012Ronconi et al., 2018;Sherman et al., 2016), and thus, such phase-dependent perceptual facilitation might not be expected in tasks that involve higher level perceptual mechanisms such as time perception. Our results thus indicate that entrained inhibitory phases of visual alpha oscillations may not play a role in tasks where time must be estimated consciously (contrary to Lin et al., 2021, suggestion). Moreover, it has been hypothesized that alpha oscillations are not regular sinusoidal oscillations; rather, they are governed by non-sinusoidal pulses of inhibition (Mazaheri & Jensen, 2010). Pulsating oscillations are characterized by modulation of only peaks and not troughs over time (Jensen et al., 2012). In this regard, it can be speculated that the amplitude asymmetry of entrained 10-Hz oscillations (Mazaheri & Jensen, 2008, 2010 did not provide the maximal inhibitory contrast that otherwise would be expected across cycles of sinusoidal oscillations. This submaximal inhibitory contrast thus might not have been strong enough to cause reliable differences in the timing of in-phase versus out-of-phase flickers. Thus, future research by entraining oscillations in other bands (such as theta, delta and beta frequencies) and varying the task properties (such as targeting lower level vs. higher level perceptual performances such as the time estimation mechanism in this study) should shed more light on qualification of neural oscillation's phase impact on perception.
Regarding time-perception models, our findings failed to provide support for entrainment models of time perception. However, a distinction between types of timeperception models that posit a role for entrainment is needed. The classical entrainment model (McAuley & Jones, 2003) assumes that temporal information is encoded within the phase and period of a self-sustained oscillator. In the presence of rhythmic structure, this oscillator adjusts its phase and period to the context rhythm so that the peaks of the attentional rhythm would coincide with the rhythmic stimuli (Large, 2008;McAuley & Fromboluti, 2014;McAuley & Jones, 2003). Therefore, the accuracy of time estimation is a function of how close the rhythmic stimuli occur relative to the peaks of this oscillator (McAuley & Jones, 2003). While this model makes predictions about empty-interval timing, its implication for regularly spaced intervals is not clear. Notwithstanding, there is accumulated evidence showing that regularly spaced intervals dilate the subjective time (e.g., Hashimoto & Yotsumoto, 2018;Herbst et al., 2013;Horr et al., 2016;Kanai et al., 2006;Khoshnoodi et al., 2008;Matthews, 2013). Thus, the entrainment of the oscillator to the rhythmic stimuli, rather than providing accurate estimations, results in overestimations. In short, our results are not in agreement with the attentional rhythm hypothesis and entrainment model that is proposed by the dynamic attending theory (DAT; Large, 2008), indicating that in-phase versus out-of-phase flickers (where presumably the peak and troughs of the attentional rhythm should be located) are not differentially estimated. One contemporary entrainment model (Horr et al., 2016;Horr & Di Luca, 2015a, 2015b, 2015c stems from implications of neural oscillation studies (Lakatos et al., 2005(Lakatos et al., , 2007. According to this model, regularly spaced stimuli result in the entrainment of neural oscillations. These oscillations, in turn, align their ideal phase with the rhythmic stimuli and thus will result in amplified neural responses. Based on the efficient coding hypothesis (Eagleman & Pariyadath, 2009;Matthews et al., 2014), these amplified signals then are assumed to result in time dilation. However, as the authors (Eagleman & Pariyadath, 2009) have conceded, it is not well understood which neural signals in what part of the brain need to be amplified for time dilation to occur. Thus, our results indicate that the presumable suppression of flicker's neural representations by maximal (minimal) inhibitory phases of visual alpha oscillations has little relevance to time dilation. Alternatively, one interpretation of the results is that the neural representation of flickers intended by the efficient coding hypothesis is in fact the power of entrained oscillation. In this sense, because the power of entrained 10-Hz oscillations was not significantly different between in-phase and outof-phase flickers, expectedly, the amount of time dilation difference between these two conditions was not different either. This latter interpretation of our results thus suggests that while entrainment may play a role in time dilation, its working mechanism is phase independent; that is, excitatory cycles of the entrainment have no impact on time dilation. This interpretation is potentially in agreement with one variety of entrainment models originally proposed by Matell and Meck (2004) known as striatal beat frequency (SBF). SBF proposes that there is an array of oscillators in the brain (mostly in the prefrontal cortex) that possess various frequency responses. At the time of an interval's onset, these oscillators are phase-reset and at the offset, their activation pattern is read out by striatal medium spiny neurons serving as a detector (Kononowicz & van Wassenhove, 2016). Simulation studies have shown that changing the intrinsic frequency of the oscillators by external entrainment in fact can predict flicker-induced time dilation (Hashimoto & Yotsumoto, 2015). This, together with the fact that in SBF, the main locus of the oscillators is presumed to be in the prefrontal cortex (Buhusi & Meck, 2005), suggests that excitatory phasic characteristics of lower level visual areas do not play a prominent role in SBF. Thus, because the time estimation in this entrainment model is dependent neither on excitatory properties of oscillations (as was the case with the model implied by Horr et al., 2016) nor on attentional rhythm (DAT), we suggest that our results are not in disagreement with SBF model. Moreover, one secondary finding of the visual experiment was that static stimuli in the alpha entraining condition relative to the empty-interval condition were overestimated (a trend was seen, p = .051; see Section 3). This phenomenon that is known as 'speed up the clock' in interval timing literature (e.g., Jones & Ogden, 2016;Wearden et al., 2017) can also be explained by a phaseindependent entrainment model. Such a model would predict that the residual entrainment (which lasts for 3 cycles after the offset of entraining context) can make static stimuli appear longer. In fact, a supplementary analysis paralleled the behavioural results revealing that the residual power of phase-locked 10-Hz oscillation (extracted from the duration of to-be-timed static stimuli) was significantly higher for the alpha entraining condition relative to the empty-interval condition (t(13) = 3.04, p = .028). Moreover, for the subset of subjects with IAF of 10 Hz, where this effect was absent, the power comparison showed a trend (t(7) = 3.14, p = .05). Nevertheless, this suggestion requires further controlled experiments (see Section 4.1).
In addition to reproduced durations, another timeperception feature that may be modulated by entrainment is the precision of duration reproduction, which is measured by CV. CV is a close approximation to the Weber fraction (Burr et al., 2013) and thus is a measure of relative sensitivity (i.e., the smallest detectable duration difference scaled by the mean duration; Ortega & L opez, 2008). One indirect implication of the efficient coding hypothesis (Eagleman & Pariyadath, 2009) is that the amount of neural energy used for representing a stimulus in the brain may also affect the precision (sensitivity) of duration estimations (Horr & Di Luca, 2015b). This is justified given that stimuli with higher evoked neural responses are often perceived more accurately than stimuli that elicit weaker neural responses (Fakche et al., 2022;Mathewson et al., 2012). Thus, because regularly spaced intervals, due to their rhythmic nature, are characterized by an increase in power of entrained oscillation (thus higher energy spent), they may in fact be reproduced with higher precision. Moreover, from a perspective that is in line with the model of entrainment implied by Horr et al. (2016), it can be inferred that the precision should also be a function of entrained oscillation's phase because for flickers that land on higher excitatory phases of entrained oscillations, the neural representation of flickers must have been amplified (Horr et al., 2016;Horr & Di Luca, 2015a, 2015b, 2015c. Our results provide partial support for the former hypothesis and failed to provide any support for the latter. Based on our data, flickers' duration, irrespective of context and IOI, was consistently reproduced with the highest precision (as indicated by lower CV). However, in a supplementary analysis assessing the relation between CV and the power of phase-locked 10-Hz oscillations for flickers (in the to-be-timed interval), we find no significant correlation. Moreover, this stimulus-specific enhancement of precision was not observed in the auditory experiment, suggesting intra-modal differences in reproduction precision of regularly spaced intervals exists. Previous studies assessing flicker's influence on the sensitivity of duration estimation have yielded inconclusive results, with some reporting a decrease in sensitivity (e.g., Plomp et al., 2012) and others reporting no effect (e.g., Droit-Volet & Wearden, 2002). In this experiment, however, we found a robust enhancement of reproduction's precision for flickers, which were presented for a brief duration (217 ms). Understanding the underlying mechanisms that yield enhanced precision for flickers is yet to be explored. By comparing various durations and task types that capitalize on different perceptual and motor features, future research should elucidate what underlying perceptual and motor components give rise to the more (less) precise reproduction of flickers.

| Limitations and future direction
We should mention that this study suffers from important limitations. First, due to the limited duration of entrainment persistence, the to-be-timed interval was restricted only to 217-ms length. Thus, the generalizability of our findings to durations longer than this length remains speculative. Moreover, due to this time limit, we had no other choice but to use the time reproduction task so that we can minimize the number of trials and make the task less susceptible to fatigue. This is because, in other time estimation tasks, normally, a standard duration is compared against a range of comparison durations. Considering that we had a 3 (context) Â 2 (stimulus type) Â 2 (phase) design, we could not use time estimation paradigms that contain varying comparison intervals because, otherwise, the number of trials would impractically be multiplied. One potential confounding factor with reproduction task results is that they combine the perceptual and motor noises (Shi et al., 2013). Thus, it remains crucial for the findings of the current study to be replicated with types of temporal tasks that only target the perceptual components (van Wassenhove et al., 2019). Furthermore, in this study, we utilized the entrainment persistence, building our experiment based on the assumption that the underlying mechanisms of entrainment persistence are identical to the actual entrainment. Although this concern might not be fully justified (Otero et al., 2020), it still can cast doubt on the generalizability of our null finding. Lastly, we strongly suggest that a study design by using transcranial alternating current stimulation (tACS) can address all the abovementioned concerns. This is because tACS directly entrains neural oscillations without the need of resorting to entrainment persistence. Therefore, a tACS design by using temporal tasks that solely contain perceptual components can target a wide range of durations in an efficient way.
Moreover, a current debate regarding the origins of brain responses to regularly spaced stimuli is whether such responses are a mere linear superposition of evoked responses (Capilla et al., 2011;Keitel et al., 2014) or they truly reflect the entrainment of endogenous neural oscillations (Thut et al., 2011). Accordingly, in our task design, two possibilities arise: either our manipulation entrained endogenous oscillations that persisted during the to-be-timed interval (the assumption our task design is built upon) or our manipulation failed to establish persistent entrainment (if any), and thus, our results are inconclusive regarding the effect of entrainment phase on flicker-induced time dilation. While we cannot utterly rule out the possibility of the latter interpretation, we present evidence in support of persistent neural entrainment in response to regularly spaced rhythmic stimuli.
In vision and with a focus on alpha band frequency,  used two physics concepts regarding oscillators to investigate the underlying mechanisms of SSVEPs. They found out that the degree of synchronization of neural oscillations with the external stimulation of visual flickers followed a pattern that is consistent with the Arnold tongue concept (i.e., the intensity of a driving force and its distance to the intrinsic frequency predict the degree of synchronization of an oscillator). Moreover, at around the borders of the Arnold tongue, the phase locking showed a nonlinear intermittency pattern that is only predicted by an entrainment model and not by the evoked model. Citing evidence that rhythmic visual stimulations (flickers) at alpha frequency reduce blood oxygen level-dependent (BOLD) responses relative to arrhythmic stimulations (Parkes et al., 2004),  directly tested whether neural entrainment underlies phase-dependent inhibitory behaviour of alpha oscillations. Their approach being informed by the Arnold tongue concept provided evidence in support of the entrainment hypothesis. In the auditory modality, a seminal study directly tested the oscillatory model against the evoked model (Doelling et al., 2019). Focusing on the phase lag between the neural responses and the rhythmic stimuli across different stimulation rates, this study showed that an oscillatory (and not the evoked) model better matches patterns of magnetoencephalography (MEG) data (Doelling et al., 2019). Lastly, a recent systematic review has provided theoretical and complementary evidence in support of involvement of endogenous neural oscillations in brain responses to rhythmic stimuli (Zoefel et al., 2018).
Considering this nontrivial evidence, we hypothesized and consequently designed the experiment based on the notion that brain responses to rhythmic stimuli entail the entrainment of endogenous neural oscillations. In fact, our task design was strongly influenced by Mathewson et al. (2012) and . Moreover, we would like to stress that this study by no means was designed to address the oscillatory versus evoked debate.
Nonetheless, in a separate post hoc analysis focused on persistence as an indication of neural entrainment (Capilla et al., 2011;Hanslmayr et al., 2019;Henao et al., 2020;Obleser & Kayser, 2019;Thut et al., 2011), in the alpha entraining condition, we examined the persistence of entrained 10-Hz neural oscillations during the static to-be-timed interval (where no 10-Hz flicker was presented). We found evidence suggesting that the entrained 10-Hz oscillations in the alpha entraining condition persisted during the static to-be-timed interval ( Figure S3). Yet this analysis can be criticized based on temporal smearing of filters applied and/or by the fact that the analysis of persistence was conducted during additional stimulation (onset of static to-be-timed stimuli). Therefore, future studies should address this issue by devising task designs that can maximally disentangle endogenous entrainment from mere evoked potentials.

| CONCLUSIONS
The current study did not provide evidence for the effect of inhibitory phases of entrained visual alpha oscillations on the overestimation of regularly spaced intervals, implying that phase characteristics of visual alpha waves may not play a role in the reproduction of such intervals. This lack of evidence thus adds up to the extant literature of null findings regarding the effect of phase of alpha on perception (Benwell et al., 2017;de Graaf et al., 2020;Ruzzoli et al., 2019;Vigué-Guix et al., 2020), eventually clarifying the nature and scope of alpha oscillations impact on perception. Moreover, by revisiting the oscillator-based models of time perception, we conclude that our null finding is not necessarily against the role of entrainment in temporal performance. Rather, it suggests that temporal oscillators may act in a manner that is independent of excitatory cycles of oscillations in lower level sensory brain areas. We propose that such interpretation favours SBF over other entrainment models.
Finally, we observed a robust enhancement of precision in reproducing flickering intervals that did not fit well with predictions of phase-dependent entrainment models. Thus, future research should elucidate the conditions and the underlying mechanisms that yield such enhancement in temporal precision.