Task and behavioural results.

A. Experimental structure. Participants viewed a random dot stimulus that coherently moved either to the left or right during yellow evidence pulses, of which there was sometimes one, but usually two, separated by a variable gap in evidence indicated by blue dot colour. B. Grand-averaged (mean ± SEM) accuracy in two- pulse trials, sorted by pulse coherence (low, L/ high, H), order (LL, LH, HL, HH) and gap duration. Data points in the left-hand section indicate single-pulse accuracies. Overlaid horizontal dashed lines indicate the expected accuracy of a perfect accumulator, based on the grand-average performance on one-pulse trials. Note that a perfect accumulator predicts identical accuracies for the HL and LH conditions, and therefore the two lines overlap.

Neural correlates of evidence accumulation and motor preparation across temporal gaps.

A. Grand-averaged (±SEM) motor beta lateralisation (MBL) [Contra- Ipsilateral hemispheres with respect to correct response hand], sorted by accuracy (solid line correct, dashed line error). Participants started preparing a motor response in response to the first pulse, and MBL was sustained throughout the gap and until the time of behavioural response across all gap conditions in both experiments. B. Grand-averaged (±SEM) centroparietal positivity (CPP) across all temporal gap conditions. The CPP showed transient evoked responses following each pulse. Note that the evoked amplitudes by P1 (CPP-P1) were systematically higher than those evoked by P2 (CPP-P2) (Exp. 1: CPP-P2 = 52% of CPP-P1 peak amplitude; Exp. 2: CPP-P2 = 58% of CPP-P1 peak amplitude; see Fig. S4). C. Grand- averaged (±SEM) occipital alpha power for each of the gap conditions, baseline-corrected before the onset of the first pulse and collapsed across coherence. Occipital alpha power showed a steep decrease following pulse onset (which was clearly indicated by a change in the colour of the moving dots), and steep increase overshooting the baseline level once the dots changed colour again to indicate non-coherent. In all panels, shaded areas indicate the timing of evidence pulses for the various gap durations, which are colour coded (see top axis). Topographies illustrate MBL before response (right minus left responses, A) in correct trials, and CPP/alpha power 550 ms post stimulus onset (dashed line, B,C). Further, markers along the bottom indicate times where the signal significantly differed from zero (two-tailed cluster- based permutation test, p<0.005) in only correct trials for MBL for consistency of polarity (A), and for all trials pooled in the case of the CPP (B) and alpha (C). Note that evidence-dependent buildup dynamics are clearly delayed in MBL relative to the CPP, and this aligns with previous research supporting the contention that the CPP lies upstream from motor preparation (Steinemann et al. 2018); such time delays were not a focus of the present study because in such delayed-response conditions we do not have access to the time of decision commitment.

CPP amplitude following each pulse scales with its coherence, and the strength of evidence in pulse 1 additionally inversely modulates amplitudes following pulse 2.

A,C. Grand-averaged (±SEM) centroparietal activity after pulse 1 (CPP-P1), averaged across all gap conditions. B,D. Standardized regression coefficients illustrating the effect of P1 (solid line) and P2 (dashed line) coherence on evoked CPP-P1 amplitudes. Inset topographies illustrate the coherence effect [P1 high – P1 low] at [0.4-0.6s] after P1. E,G. Grand-averaged (±SEM) centroparietal activity after pulse 2 (CPP-P2), averaged over all non-zero gaps in the two experiments. For illustration purposes, we subtracted the average activity of long gap trials with the same P1 coherence from the epoched EEG traces, between [-0.2 to 1.4s] from P1 presentation, and baseline corrected traces [-100 to 0ms] before P2. This isolates the neural activity uniquely associated with the presentation of the second pulse in short gap trials (<1s), where potentials evoked by the first and second pulses overlapped, by removing the activity associated with the first pulse from the P2-evoked traces, including baseline trends (see Fig. S4). The qualitative patterns did not vary if this correction was not applied (see Fig S5). Note also the difference in the y axis scale compared to panels A,C. F,H. Standardized regression coefficients (mean ±SEM) illustrating the effect of P1 (solid line) and P2 (dashed line) coherence on evoked CPP-P2 amplitudes. Inset topographies illustrate the difference between [P1 high – P1 low], solid outline; [P2 high – P2 low], dashed outline) [0.4-0.6s] after P2. The negative effect of P1 illustrates higher amplitudes for P1-low trials. *Markers along the bottom of panels B,D,F and H indicate significant clusters where coherence effects differed from zero (two-tailed cluster-based permutation test, p<0.05). Shaded areas indicate the tested time window. See Fig. S3 for full-trial ERP traces, sorted by gap and coherence conditions.

Simulated behavioural data

(N = 50000) of a simple bounded accumulator model with three parameters (B (bound), d1 (high drift rate), d2 (low drift rate)), fit to all data points for each experiment separately. A. Grand-averaged simulated (solid lines) accuracy in single- (left) and two-pulse (right) trials, sorted by pulse coherence and gap duration in experiments 1 (top) and 2 (bottom). Observed data are overlaid (dashed) for comparison. Simulated data recapitulated the observed order effects. For simplicity, the model was not endowed with any means to account for effects of gap duration. B. Proportion of simulated two-pulse trials that hit a bound during the first pulse, and therefore did not use the second pulse evidence to make a choice. C. Absolute value of the decision variable (DV) at the end of pulse 1 in the subset of trials where the bound was not hit during that pulse, sorted by condition.

Model-derived simulations of motor beta lateralisation (MBL) and centroparietal positivity (CPP) recapitulate empirical data.

A,D. Simulated MBL (top) and CPP (bottom) traces over the whole trial, sorted by gap duration. MBL traces are additionally sorted by accuracy (solid line = correct; dashed lines = error). The simulations recapitulate the bimodal shape of the CPP across the whole trial, with the second pulse reaching lower amplitudes than the 1st one in both experiments (see Fig. 2B). In turn, by assuming that MBL signals track the state of the DV throughout the trial and because no leak or decay is assumed, our simulations could also recapitulate the sustained nature of motor lateralisation signals throughout the gap. Note that DVs were simulated so that positive signs are associated with correct choices, and negative signs to errors. Dashed red lines indicate the bounds for error/correct trials, respectively. B,E. Simulated MBL-P1 (top) and CPP-P1 (bottom), sorted by P1 coherence, and averaged across all gap durations and accuracies. Note that the step-like changes in MBL traces are due to averaging the DV values over the various gap durations. No such effect is observed in simulated CPP potentials because traces are derived from P1 activity exclusively. C,F. Simulated MBL-P2 (top) and CPP-P2, sorted by P1 and P2 coherence, and averaged across all gap durations and accuracies. Our simulations recapitulate the observed coherence effects at both CPP-P1 (see Fig. 3A,C) and CPP-P2 (see Fig. 3E,G). Note that, for CPP simulations, we plot the estimated build-up at P2 assuming that activity started at 0 and not considering any potential overlap from earlier on in the trial, for consistency with the empirical traces where CPP-P1 activity is removed from the P2-evoked traces. In contrast, the MBL-P2 traces continue from their post-P1 state, accounting for the sustained nature of the signal (see Fig. S2,S3 for empirical MBL-P1/P2 traces). Note also that, for the sake of simplicity, our CPP simulations do not assume any delay between stimulus presentation and signal build up. Therefore, our simulated CPPs start building up immediately after pulse onset, while empirical ones start with approximately a 0.2s delay. Similarly, although motor preparation signals lag the motor independent buildup dynamics at the CPP level, we assumed instant transmission here for simplicity. Since only accuracy and not reaction time was to be captured in this delayed-response task, it was not necessary to estimate transmission delays.

Individual participants’ accuracy (grey bars) and average coherence (red line) for each block in Experiments 1 (A) and 2 (B).

On average, most participants performed close to 70% accuracy and clearly above chance (solid black line), and only required minimal coherence adjustments between blocks. The data of participants 1 and 21 in Experiment 2 were excluded from further analysis due to poor task performance in a majority of blocks.

Sensitivity to evidence strength in the two pulses in motor beta lateralisation signals.

A. Grand-averaged (±SEM) motor beta lateralisation [Contra-Ipsilateral hemispheres with respect to correct response hand] after pulse 1 (MBL-P1), averaged across all gap conditions and across all trials regardless of accuracy. Motor preparation started shortly after the first pulse, and was sustained at a coherence-dependent level. B. Grand-averaged MBL after pulse 2 (MBL-P2), averaged over all non-zero gaps in the two experiments. C. Effect of P1 coherence on MBL after P2. Traces illustrate the grand-averaged (±SEM) aligned to pulse 2 (MBL-P2), averaged over all non-zero gaps and across P2 coherence. D. Effect of P2 coherence on MBL after P2. Traces illustrate the grand-averaged (±SEM) aligned to pulse 2 (MBL-P2), averaged over all non-zero gaps and across P1 coherence. In panels A-D, inset bar graphs illustrate the MBL excursion as the difference between activity at [0.6-0.8s, shaded area] after pulse onset, minus activity at baseline [-0.2-0s].

A,C. Motor beta lateralisation (MBL) for each gap condition, sorted by P1 coherence in experiments 1 (A) and 2 (C). Lateralisation is computed as a function of actual dot motion [Contra-Ipsilateral hemispheres with respect to correct response hand], sorted by gap duration and P1 coherence pooled across correct & error trials. B,D. Centroparietal activity for each gap condition, and sorted P1 coherence in experiments 1 (B) and 2 (D).

A,C. Centroparietal activity sorted by gap, P1 and P2 coherence in Experiments 1 and 2. B,D. Centroparietal activity after removing CPP-P1 activity. For each condition, the average of long-gap trials with the same P1 coherence was removed. That is, the average of the long gap HH & HL conditions was subtracted from HH and HL traces. Similarly, the average activity of the long gap LL & LH conditions was subtracted from LL and LH traces. This removes overlapping CPP-P1 potentials, and isolates centroparietal activity related to P2. Subtractions were performed between [-0.2 to 1.4s] from P1 presentation. Thus, CPP-P2 potentials in the long gap condition remain unaltered.

Replication of results in Fig. 3, now performing the analysis on CPP-P2 data (B- D) without removing CPP-P1 activity.

Note that baseline activity shows a drift consistent with the overlapping potentials from the preceding pulse, which is corrected for when removing CPP-P1 activity (cf. Fig. 3). Note also that the same patterns are qualitatively present in the uncorrected data. Panels C-D illustrate conditions pooled by P1 or P2 coherence, respectively, to highlight the different direction of the effect.

Model-predicted (A-B) and observed (C-D) CPP-P1 and -P2, in correct vs.

incorrect responses, for Experiments 1 (top) and 2 (bottom). Bar graph insets in A-B indicate the state of the simulated accumulator at the end of pulse 1 in those trials where the bound was not hit. Positive values indicate that the accumulator was leaning towards the correct answer, whereas negative values indicate that it was leaning towards an error. The model predicts stronger P1 coherence effects in correct compared to error trials (A-B, left), with this effect also being observed in empirical data across both experiments (C-D, left). For the sake of clarity, we do not plot LL and HH conditions in the empirical CPP-P2 data because no clear amplitude differences are predicted by the model in correct vs. error trials, and the trial count in the HH condition was insufficient in most participants. Further, the model predicted a strong interaction in CPP-P2 amplitudes in the HH vs. HL condition. While correct trials (which were the majority of trials in our task and the model) showed higher amplitudes at CPP-P2 for HL compared to the LH condition (A, right), consistent with the effect of P1 on CPP-P1 amplitudes reported in Fig 3, this effect reversed in incorrect trials (B, right). This interaction was also clearly visible in Experiment 2, although slightly less prominent in Experiment 1. Note however that Experiment 1 had fewer overall trials, and power to detect differences is likely to be insufficient after splitting conditions by accuracy.

Simulations (N=1000) illustrating the effect of various signal decay time (A-C) and decay delay (D-E) assumptions on the predicted CPP activity.

Simulations are based on the fitted model in the main manuscript and for Exp. 1 only for illustration purposes. Given that it’s unclear whether the CPP fall-down to zero when evidence is paused follows the same dynamics as its fall-down when it hits a bound (where studies with immediate behavioural responses suggest that it reaches zero within approximately 200ms; Steinemann et al 2018), we simulated its CPP activity under three different assumptions about the duration of its decay upon evidence interruption (i.e. when blue dots appear onscreen after a pulse). We simulated the CPP assuming the signal takes 100 ms (A), 500 ms (B) or 1000 ms (C) to go back to baseline at pulse offset when a bound had not been reached, retaining a 200 ms fall-down upon bound-crossing. Longer signal decay times for trials where no bound was hit result in a protracted ramp down to baseline. In turn, it is unclear whether any post-decisional evidence accumulation is to be allowed to simulate the CPP in this context, given that previous studies have observed varying delays in signal decay after response (Steinemann et al. 2018; Afacan- Seref et al., 2011). In the manuscript, we assumed no delay between bound crossing and signal decay. Here, we simulated the CPP allowing it to continue building up 50ms (D) or 100ms (E) after bound crossing, while setting it to immediately decay upon blue dot offset. Allowing some post-decisional build-up resulted in higher average CPP amplitudes, and it also better recapitulated some features of the data, including a protracted build-up in the zero-gap condition compared to all others.

Simulated CPPs using the same computational model as in Fig. 5, but under the assumption that the signal was only reset to baseline if a bound was hit.

If the bound was not hit, the accumulator, like the downstream DV at the motor level, was set to stay stable at the final value reached by the end of pulse 1 with no decay and no added noise for the whole duration of the gap. Given that we do not model reaction times, we assumed no transmission delay between events on the screen and their impact on the accumulator. In this model, the partial fall down of the signal during the gap was driven by those trials that hit a bound following the first pulse. Conversely, the fact that it did not fall all the way to 0 on average during the gap is driven by those trials where the DV did not hit a bound during processing of pulse 1 and thus sustained through the gap. Accumulation resumed from that sustained level upon presentation of a second pulse, and the fact that on average CPP-P2 traces reach lower amplitudes than CPP-P1 evoked activities is explained by the fact that trials that were terminated during the first pulse did not engage in accumulation again, and therefore took value 0 in our model. Note that this model provides a qualitatively worse fit to the empirical CPP traces during the gap and CPP-P2 trials. In particular, the magnitude of the CPP-P2 potentials was predicted to be significantly smaller (Exp. 1: CPP-P2 = 19% of CPP-P1 peak amplitude; Exp. 2: CPP-P2 = 13% of CPP-P1 peak amplitude) than observed on the grand-averaged data (CPP-P2 = approximately 50% of CPP-P1 in both experiments; see Fig. 2).

Occipital alpha power tracks dynamics of attentional engagement and varies with temporal expectations.

A. Grand-averaged (±SEM) occipital alpha power in the long gap conditions sorted by P1 coherence, again baseline-corrected before the onset of the first pulse. At evidence offset, alpha steeply increased, and, interestingly, attained and maintained a higher level during the longer gaps than the baseline before the first pulse (two-tailed cluster permutation test, p < 0.05 in both experiments; see Fig. 2A). One possible interpretation is that this reflects attentional disengagement from the decision occurring after early decision termination. However, if this was the case, we would expect this effect to be dependent on pulse 1 coherence, as it is more likely for a DV to have reached a bound after a high-coherence pulse (Fig. 4B), and we observed no such coherence-dependence in alpha power during the gap. Topographies illustrate the difference in alpha power between P1-high and P1-low coherence trials [-0.2 to 0s] before P2 onset. B. Occipital alpha power in the longest gap condition in the two experiments. In Experiment 2, where the event distribution was skewed towards short gap durations between the first and second pulses, participants’ occipital alpha was significantly higher (indicating lower attention) than in Experiment 1 (* p<0.05; two- sample t-test for the difference in alpha power between occipital activity [-0.1 0s] before P2 and the baseline [-0.2 0s] before P1. Alpha power has been linked to temporal expectation encoding (Rohenkohl & Nobre, 2011), as well as being a well-known index of attentional engagement (Thut et al., 2006). Thus, this difference is interesting in light of the difference in gap durations in our two experiments, and hence temporal expectations. In particular, in those trials where no second pulse had been presented after a 0.5s gap, the probability of the second pulse yet appearing was 0.67 in Experiment 1, compared to 0.6 in Experiment 2. Although another difference between experiments was that early terminations within pulse 1 occurred more often in Experiment 2, as predicted by our model (see Fig. 4B), the probability of early terminations is unlikely to be the key influence given the above finding that alpha power during the gap was not dependent on P1 coherence (A). Rather than reflecting switches between engagement and disengagement at the decision level, then, it may reflect a gating function at the sensory processing level that operates independent of whether early bound-crossings have happened, yet is modulated between contexts depending on the likelihood of a second pulse still to come. The dotted box indicates the test period. In all panels, shaded areas indicate the timing of evidence pulses for the relevant gap durations. *Markers along the bottom of panel A indicate significant clusters where alpha power differed from zero (two-tailed cluster-based permutation test, p<0.05). In panel B, they indicate significant clusters where alpha power differed in P1-high vs. P1-low coherence trials.

Model comparison investigating the effect of drift rate variability at the second pulse.

In our main analysis, we initially fit a model with two different drift rates for low (d1) and high (d2) coherence pulses: Model 1 (k = 3, b, d1, d2). Here, we explored whether increased temporal uncertainty about whether and when a second pulse might appear could have led to more variable attentional states before its appearance, and as a consequence more variable drift rates. To test this hypothesis, we extended the model with one additional free parameter introducing trial-by-trial variability to the drift rates applied to pulse 2: Model 2 (k = 4: b, d_low, d_high, dvar_p2). This model resulted in the lowest overall error as measured by G2, but the additional complexity was not warranted according to AIC/BIC metrics.

Weight on choice modulations by single-trial fluctuations in neural evidence accumulation and attention markers.

In this analysis, we sought to investigate whether single-trial fluctuations in the neural correlates of evidence accumulation and attention were linked to variability in choice behaviour. To do so, we exploited the trial-by-trial variability in the neural signals. A. Standardized regression coefficients (±SEM) resulting from Eq. 9, illustrating the effect of CPP-P1 (left) and CPP-P2 (right) variability on choice, averaged across all gap conditions. The magnitude of CPP-P1 modulated the impact of the first pulse on choices in both experiments while the magnitude of CPP-P2 only modulated the relative weight of P2 in Experiment 2. In our task, variability in the CPP may be related to trial-by-trial variability in motion energy, as well as random noise. Since the weight on choice of P1 and P2 may trade- off with one another due to a bound being set – i.e. on trials where P1 has a strong influence, P2 is likely to have a weaker one - we also computed the difference between the two modulatory terms ([P1coh*CPP - minus P2coh*CPP], for CPP-P1 and CPP-P2 respectively). This difference index effectively quantifies whether centroparietal responses to either pulse modulate the relative weight on choice of each of the pulses. A positive index indicates a relative increase of P1 weight on choice, whereas a negative index indicates a relative increase of P2 weight on choice. Black dashed lines correspond to the difference in regression coefficients for the [P1coh*CPP – P2coh*CPP modulations]. B. Standardized regression coefficients (±SEM) resulting from Eq. 10 for alpha aligned to pulse 1 (left) or pulse 2 (right), averaged across all gap conditions. The magnitude of occipital phasic alpha responses (baselined before P1) did not modulate the weight on choice of P1, and after P2 did modulate its weight on choice in Experiment 2 only. Black dashed lines correspond to the difference in regression coefficients for the P1*alpha – P2*alpha modulations, which effectively indicate a change in the relative weight for P1 and P2. *Asterisks along the bottom of the panels indicate significant (p<0.05) effects in the test period [0.2-0.8s post event].