Fig 1.
The presentation of a visual cue (a green bar) triggers the race of three independent units. The inhibitory unit can stop an early response. Importantly, both early and late responses can trigger pro- and antisaccades. Note that the PROSA model is a special case of the SERIA model in which πe = 1 and πl = 0, i.e. all early responses are prosaccades, whereas all late responses are antisaccades.
Fig 2.
The presentation of a visual cue (a green bar) triggers the race of four independent units. The inhibitory unit can stop an early response. The late decision process is triggered by the competition between two further units.
Fig 3.
Illustration of probability distributions used to model increase rates.
Left: Distribution of the rates based on different probability density functions: Normal (red), gamma (blue), inverse gamma (green), and log-normal (cyan). All distributions were matched to have equal mean and variance. Center: Probit plots of the same distributions. While the gamma and lognormal distributions are very close to the straight line induced by the normal distribution, the inverse gamma distribution diverges slightly more from linearity. Right: Arrival time distribution (scaled to ms).
Table 1.
Parametric density functions of the increase rates.
Fig 4.
After a variable fixation period of 500–1000ms (top) the cue (green rectangle) appeared on the screen for 500 ms. The orientation of the cue (horizontal or vertical) indicated the required response (prosaccade or antisaccade).
Table 2.
Model families with the respective increase-rate distributions.
Table 3.
Prior probability density functions.
Table 4.
Summary of trials per subject.
Fig 5.
Error rate and mean reaction time as a function of prosaccade trial probability (PP).
Left panel: Mean error rates for pro- and antisaccade trials. Right panel: Mean reaction time in ms. Error bars indicate standard errors of the mean. Only correct responses are displayed.
Fig 6.
Exemplary histogram of the reaction times of one subject in the PP50 condition.
Prosaccade trials are displayed in the upper half plane and antisaccade trials in the lower (negative) half plane. Prosaccade actions are depicted in red color, whereas antisaccade actions are shown in blue. Errors in prosaccade trials are antisaccades that for this subject occurred after the first peak of early prosaccades. Errors in antisaccade trials (lower half plane) occurred at a similar latency as early prosaccades in prosaccade trials. The histograms have been normalized to have unit probability mass, i.e., the sum of the area of all bars is one.
Table 5.
Summary of mean RTs and ERs.
Fig 7.
Top: Summed LME of all subjects for all 30 models. White bars show models with all parameters free, grey bars models with constrained parameters. LMEs are normalized by subtracting the lowest LME (m5). Model (constrained SERIAlr) exceeded all other models (ΔLME > 200). Bottom: Illustration of model probability for all subjects. The posterior model probabilities for all subjects are shown as black dots. In white shading are models with all parameters free, grey bars represent models with restricted parameters. Note that in nearly all subjects, the SERIAlr models with restricted parameters showed high model probabilities.
Fig 8.
Fits of best PROSA (m1), SERIA () and SERIAlr (
) models.
Columns display the normalized histogram of the RTs of pro- (red) and antisaccades (blue) in each of the conditions. Rows correspond to individual subjects named S1 to S4 for display purpose. As in Fig 6, prosaccade trials are displayed on the upper half plane, whereas antisaccade trials are displayed in the lower half plane. The predicted RT distributions based on the samples from the posterior distribution are displayed in solid (SERIAlr), broken (SERIA), and dash-dotted (PROSA) lines. Note that data from subject 3 in the PP50 condition is the same as shown in Fig 6. Early outliers are not displayed.
Fig 9.
Fits from the best models in each family ().
Model fits and RT histograms for each condition collapsed across subjects. For more details see Figs 6 and 8.
Fig 10.
Reciprobit plot of best models.
Predicted and empirical cumulative density function of the reciprocal RT in the probit scale for each condition and model collapsed across all subjects. The data shown are the same as in Fig 9, but split for trial types and illustrated as cumulative distributions. Note that the y-axis is in the probit scale and that nearly all differences between the model and the data occur at very small probability values of 5% or below.
Fig 11.
Empirical and predicted RT of corrective antisaccades.
Left: End time of erroneous prosaccades, RTs of corrective antisaccades, and time shifted predicted response time distribution of late antisaccades. The time shift was selected to be the difference between the empirical and predicted mean response time. Center: Quantile-quantile plot of the predicted and empirical distribution of corrective antisaccades, and a linear fit to the central 98% quantiles. There is a small deviation only at the tail of the distribution. Right: Reciprobit plot of the empirical and predicted cumulative density functions of the RT of corrective antisaccades. The scale of the horizontal axis is proportional to the reciprocal RT. The vertical axis is in the probit scale.
Fig 12.
Left: Mean arrival or response time and standard error of the early and inhibitory units and late pro- and antisaccades. Right: Probability of a late antisaccade p(Ua > Up) in prosaccade (red) and antisaccade (blue) trials in each condition in the probit scale.
Table 6.
Post hoc comparison of the effect of PP.
Fig 13.
Correlation between late arrival times and errors.
Left: Percentage of late errors against late antisaccades’ response times in antisaccade trials. Center: Percentage of late errors against late prosaccades’ response time in prosaccade trials. Left: Percentage of inhibitory failures against late antisaccades’ response time in antisaccade trials. The vertical axis is in the probit scale.
Fig 14.
Posterior distribution of late errors and inhibition failures.
The posterior distribution of the percentage of late and inhibition failures of two exemplary subjects (see Figs 8 and 13). Samples from the posterior distribution were obtained using MCMC. Histograms display the distributions of the samples in probit scale (horizontal axis). For these two subjects, the posterior distribution of late prosaccade and inhibitory failures clearly discriminates between the three PP conditions.
Fig 15.
Error sources and the correlation of response times.
Left: Error rate (black line) split into the two causes predicted by the model. Inhibition errors are early actions that always trigger prosaccades. Similarly as described by [23], late errors occur when a late response leads to a prosaccade. Right: Correlation between correct antisaccades' and late prosaccades' response times according to the best SERIAlc model. The best linear fit is depicted as a solid line. The mean ratio of pro- and antisaccade response times (s) is displayed on the right. Although late pro- and antisaccade response times are highly correlated, their ratio is different in each condition (interaction PP and late prosaccade response time F = 9.2, p < 0.001).
Fig 16.
Comparison between unconstrained SERIA and SERIAlr models.
Comparison between models m8 (broken lines; SERIA model) and m13 (solid lines; late race SERIAlr model.).