To examine the temporal dynamics of the twinkle-goes illusion, we estimated the perceived disappearance position, defined as the probe displacement yielding equal left/right judgments (i.e., PSE), as a function of probe timing.

In the dynamic background condition, the PSE depended on the asynchrony between the flashed probe and the vanishing of the moving bar (Fig. 4; main effect in ANOVA, F(7, 63) = 4.41, p < 0.001, ηp2 = 0.33). The PSE was indistinguishable from the null at the origin (t(9) = − 0.07, p = 0.94, Cohen’s d = − 0.02 for 0 ms), and gradually increased in the direction of motion with a delay in the probe of up to 120 ms (t(9) = 4.33, p = 0.002, Cohen’s d = 1.37 for 120 ms), after which the PSE became saturated at 0.16° on average (all t(9) > 2.02, p < 0.08, Cohen’s d > 0.64 for 160–400 ms; see Table S1 including Bayes factors). Multiple comparisons revealed that the PSE at 120 ms (t(9) = 3.89, pholm = 0.007, Cohen’s d = 1.08), 200 ms (t(9) = 3.45, pholm = 0.026, Cohen’s d = 0.96), 300 ms (t(9) = 3.64, pholm = 0.015, Cohen’s d = 1.01), and 400 ms (t(9) = 3.35, pholm = 0.035, Cohen’s d = 0.93) were significantly different from 0 ms. The difference between 40 ms and 120 ms (t(9) = 3.29, pholm = 0.039, Cohen’s d = 0.91) was also significant. The other pairs of time points did not exhibit any significant differences (all t(9) < 3.04, pholm >0.08, Cohen’s d < 0.85; Table S1). The slope within [0, 120] ms was consistently > 0 (1.33 ˚/s [SEM = 0.23 ˚/s], t(9) = 5.81, p < 0.001, Cohen’s d = 1.84), whereas the slope thereafter was not (− 0.01 ˚/s [SEM = 0.18 ˚/s], t(9) = − 0.06, p = 0.95, Cohen’s d = − 0.02).

In the static background condition, the PSE did not depend on the asynchrony (main effect in ANOVA, F(3, 27) = 1.71, p = 0.19, ηp2 = 0.16), with the slope within [0, 300] ms not differing from 0 (0.3 ˚/s [SEM = 0.20 ˚/s], t(9) = 1.55, p = 0.16, Cohen’s d = 0.49, BF10 = 0.77). Furthermore, the PSE did not significantly differ from zero at any asynchrony (all t(9) < 1.36, p > 0.20, Cohen’s d < 0.43 within [0, 300] ms; Table S1). The lack of position shifts in the static background condition was also consistent with a previous study that showed no position shifts, regardless of asynchrony, in a similar stimulus viewed with fixation (Kerzel 2000). This may be due to the inter-individual average collapsing the opposing effects of the representational momentum or some positional overshoot at motion termination (Freyd and Finke 1984; Kanai et al. 2004), which may also involve positional prediction (Hubbard 2005), and offset repulsion (Nakajima and Sakaguchi 2016). However, the data indicated no clear bimodality in the individual differences.

For the subset of asynchronies common between the two background conditions (0, 80, 160, and 300 ms), a two-way ANOVA on PSEs, with background condition and asynchrony as factors, revealed significant main effects of background (F(1, 9) = 10.83, p = 0.009, ηp2 = 0.55) and asynchrony (F(3, 27) = 4.92, p = 0.007, ηp2 = 0.35). While their interaction was not significant (F(3, 27) = 1.26, p = 0.31, ηp2 = 0.12), the slope within [0, 120] ms in the dynamic background condition was larger than the slope within [0, 300] ms in the static background condition (t(9) = 4.85, p < 0.001, Cohen’s d = 1.53, BF10 = 44.34). In addition, the pattern of statistical results was not affected even if the author’s data were excluded from the same analyses (for background, F(1, 8) = 8.32, p = 0.020, ηp2 = 0.51; for asynchrony, F(3, 24) = 6.91, p = 0.002, ηp2 = 0.46; for interaction, F(3, 24) = 0.99, p = 0.41, ηp2 = 0.11).

Results of Experiment 1. The abscissa shows asynchrony, defined as the relative time when the probe was turned off after the moving object vanished. The ordinate shows the PSE relative to the true vanishing position of the moving object, in degrees of visual angle, with the positive values indicating position shifts in the direction of motion. The solid and dashed curves indicate the dynamic and static background conditions, while the black and light gray curves indicate the mean and individual data, respectively. Error bars represent the Cousineau-Morey CI (Morey 2008)

The discrimination threshold did not significantly vary by asynchrony in either background condition (main effect in ANOVA, F(7, 63) = 0.67, p = 0.70, ηp2 = 0.07 in the dynamic background condition; F(3, 27) = 0.95, p = 0.43, ηp2 = 0.10 in the static background condition). For the asynchronies common to both background conditions (0, 80, 160, and 300 ms), a two-way ANOVA on discrimination thresholds with background condition and asynchrony as factors found no significant main effects of background (F(1, 9) = 2.59, p = 0.14, ηp2 = 0.22), asynchrony (F(3, 27) = 1.09, p = 0.37, ηp2 = 0.11), or their interaction (F(3, 27) = 0.21, p = 0.89, ηp2 = 0.02). This confirms that the present effects in PSE were not derived from any differences in discrimination threshold.

To examine whether neural oscillations contribute to the twinkle-goes illusion, we analyzed the relationship between trial-by-trial pre-vanishing EEG phase and perceived shift. This allowed us to test whether positional prediction is rhythmically updated in synchrony with neural activity. To allow the illusion to reach its maximum, the probe onset was delayed by 1.6 s after the moving objects’ vanishing. Consistent with the temporal dynamics observed in Experiment 1, the PSE was 0.08° (SEM = 0.02°) larger in the motion direction for the dynamic background condition than for the static background condition (t(19) = 3.84, p = 0.001, Cohen’s d = 0.86).

We computed the inter-trial circular–linear correlation between the EEG oscillation phase and the perceived position shift size at each of the time–frequency points, and analyzed the correlation difference between the dynamic and static background conditions. The scalp map, averaged across 3–5 Hz within [− 800, 0] ms, revealed a significant cluster over the right parietotemporal (TP10) and centroparietal (CP6) electrodes (cluster-corrected p = 0.016, maximum t(19) = 3.95, maximum Cohen’ d = 0.88 within the cluster; Fig. 5a). For visualization, we then plotted the time–frequency representation of the circular–linear correlation averaged across TP10 and CP6 (Fig. 5b–d). Importantly, although electrodes were selected based on the 3–5 Hz activity within [− 800, 0] ms, this selection alone does not specify whether the condition difference is confined to the 3–5 Hz band or extends to other frequencies, nor whether it persists across the entire [− 800, 0] ms interval or is limited to specific time points. To address this, we formally tested condition differences across the full time–frequency plane using cluster-based permutation statistics. This analysis revealed a significant cluster selectively in the theta band (3–5 Hz) and restricted to the − 800 to − 400 ms interval (cluster-corrected p = 0.0003, maximum t(19) = 5.15, maximum Cohen’s d = 1.15 within the cluster; Fig. 5d), indicating that the observed effect was restricted to a specific time–frequency window rather than uniformly present throughout the analysis range.

To provide a broader frequency context beyond the 3–5 Hz focus (Wutz et al. 2016), we repeated the scalp-map analysis separately for high-theta (5–8 Hz), alpha (8–13 Hz), and beta (13–30 Hz) activity within the same [− 800, 0] ms interval. This procedure allowed electrodes to be selected independently for each band, thereby testing whether condition differences might emerge at distinct scalp sites for other frequencies. However, no significant clusters were identified in these bands (all cluster-corrected p = 1), indicating that the illusion-related effect was specific to the 3–5 Hz range.

Results of Experiment 2 from the phase analyses of the EEG data. (a) Scalp map of the correlation difference between the dynamic and static background conditions at 3–5 Hz within [− 800, 0] ms, where the inter-trial circular–linear correlation was computed between the phase and the perceived shift. The squares indicate significant differences (cluster-corrected p < 0.05) identified in the permutation analysis. We selected the circular–linear correlation averaged between electrodes TP10 and CP6 to visualize the time–frequency data. The color scale is identical to that of panel (d), (b–d) time–frequency plot of the average inter-trial circular–linear correlation for the dynamic (b) and static (c) background conditions, as well as (d) their difference (b minus c; the static background condition served as a baseline for assessing the unsigned correlations observed in the dynamic background condition, helping to isolate the illusion-related component from spurious correlations due to the variability of internal noise). The saturated region of the color plot marks significant differences (cluster-corrected p < 0.01) identified in the permutation analysis. The dotted curve indicates the time points at which the wavelet’s time window starts to include the signals after the vanishing. In each of the time sequences illustrated at the bottom, the orange box depicts the analysis interval (within [−800, 0] ms) for the initial electrode selection