The stochastic-decision model reproduces the distribution of waiting times as well as the characteristic shape and time course of the readiness potential. (A) Visual depiction of the model: After a stochastic exponential transition period, determined by the ratio of urgency and leak parameters, the leaky accumulator generates noisy trajectories whose threshold crossings determine movement times. When the threshold is crossed (t₀), we extract an “epoch” centered on t₀ and then reset the accumulator to zero and start the next trial. The “waiting time” is the time from trial onset to threshold crossing, and the “RP” is the average over all of the simulated epochs (sign reversed) time locked to threshold crossing. The shaded thick line in the foreground shows the mean trajectory over 1,000 simulated trials. (B) Mean waiting-time distribution (normalized to the mean for each subject) from the empirical data (shaded line) and the best fit of the output of the simulation (dashed solid line). Inset shows the distribution when the data from all subjects are pooled together (for comparison with ref. 16). (C) Mean empirical RP from the classic Libet task (minus the mean from −3.0 to −2.5 s; shaded line; n = 14) and the mean sign-reversed output of the simulation fit to the time range −3.0 to −0.15 s (r² = 0.96, P < 10⁻⁹). To avoid overfitting, the parameters of the model (threshold, drift, and leak) were chosen on the basis of the best fit to the empirical waiting-time distribution, and then those same parameters were used to fit the RP (C) and for all other analyses.

(A and B) Linear relationship between mean and SD of (A) threshold crossing times in the simulation and (B) empirical waiting times in the classic Libet task. This relationship is predicted by the drift-diffusion model and by data from reaction-time tasks (35). The values in A were obtained by varying the urgency parameter from 0.06 to 0.22 in steps of 0.02, while keeping the other parameters (leak and threshold) fixed at the values selected by fitting to the WT distribution (Fig. 1B). The presence of this relationship is evidence that the same mechanism (bounded integration) thought to be involved in perceptual decision-making tasks is also at work in a spontaneous movement task.

Readiness potential and waiting-time distribution from the classic Libet and interruptus paradigms. (A and B) The black traces show the mean RP from the classic Libet paradigm and the red traces show the RP from noninterrupted trials of the interruptus paradigm, for the simulated (A) and empirical (B) data, respectively. Insets show the distribution of waiting times for the classic (black) and interruptus (red) paradigms. The truncated distribution from the classic experiment is shown in B (gray) for comparison with that obtained from the interruptus experiment. There were no significant differences between the RPs from the two paradigms, in either the simulated or the empirical data. The empirical waiting-time distribution from the interruptus task (B, Inset, red line) differs from that predicted (B, Inset, gray line) for shorter waiting times (<8 s), because some subjects changed their behavior during this task, tending to make their movements earlier on average. No baseline correction was applied in B.

Libetus Interruptus experiment. (A) Premovement potential (from the same electrode, near the vertex, as the RP) for fast (orange) and slow (gray) responses to clicks (intermediate responses were similar to fast responses, being only slightly lower in amplitude during the preclick interval). All graded error boundaries extend out to 95% confidence. (C) Difference between fast and slow responses. The black asterisks at the top mark time points where the difference is significant (P < 0.01, two-sided signed-rank test). (B and D) Same as A and C, except that the data are time locked to the click rather than to the movement. We propose that the faster responses (top 33rd percentile) were faster because ongoing spontaneous activity was closer to threshold at the time of the interruption. When time locked to the click, an auditory evoked potential is evident (B), but this potential is canceled out in the difference (D). Because the variance in reaction times to the click was relatively small (subjects were asked to respond as quickly as possible), a diluted auditory evoked potential is also visible when the data are time locked to the movement (A). The weaker and inverted evoked potential in the difference between fast and slow responses time locked to the movement (C) is due to the difference in reaction time (i.e., delay between click and movement) for slow vs. fast responses (the auditory evoked potentials fail to completely cancel out as they do in D, where the data were time locked to the click). (E and F) Results of the simulation, time locked to threshold crossing (E) and time locked to the interruption (F). A speeded response was simulated by introducing a steep linear ramp at the time of the random interruption, which is visible at the end of each trace. Interruption times were chosen randomly from a uniform distribution extending from the minimum to the maximum of the WT distribution, just as was done in the experiment. In roughly half of simulated trials the output crossed the threshold before the interruption occurred, and these were treated as “spontaneous movement” trials. For details of the model, see Materials and Methods. For A and B the data for each subject were normalized to the overall mean and SD in the time range −2.5 to −0.3 s (A) or −2.5–0 s (B) to remove between-subject variance.

(Upper) The readiness potential is assumed to reflect an intentional process: (Left) The process begins with a “neural decision to initiate movement” 1 s or more before movement onset and builds up until the movement is initiated at t₀. The stochastic decision model (Right) offers a plausible alternative: In a spontaneous movement task, premotor activity fluctuates close to movement threshold due to an implicit imperative to produce a movement “sometime soon.” Random ongoing fluctuations determine the precise threshold-crossing time, and these are recovered in the time-locked average as an intentional-looking “buildup.” (Lower) The reasoning behind the Libetus interruptus experiment. A compulsory response cue may be delivered at any random time while the subject waits to produce a spontaneous movement. A cue delivered at t₂ will result in a faster reaction time than a cue delivered at t₁, because motor activity is closer to threshold (due to spontaneous fluctuations). Hence, fast responses to the cue should be preceded by a buildup. An interruption scheduled to be delivered at t₃ will never happen, because the threshold will already have been crossed before then, in which case a spontaneous movement will have been made. Temporal autocorrelation, a key assumption of the model, is a well-known property of spontaneous fluctuations in neural activity.