Fig 1.
Smooth pursuit eye movements and explicit direction predictions in a volatile switching environment.
We test the capacity of human participants to adapt to a volatile environment. (A) We use a 3-layered generative model of fluctuations in target directions (TD) that we call the Binary Switching model. This TD binary variable is chosen using a Bernoulli trial of a given probability bias. This probability bias is constant for as many trials until a switch is generated. At a switch, the bias is chosen at random from a given prior. Switches are generated in the third layer as binary events drawn from a Bernoulli trial with a given hazard rate (defined here as 1/40 per trial). We show one realization of a block of 200 trials. (B) The eye-movements task was an adapted version of a task developed by [48]. Each trial consisted of sequentially: a fixation dot (FIX, of random duration between 400 and 800 ms), a blank screen (GAP, of fixed duration of 300 ms) and a moving ring-shaped target (TARGET, with 15°/s velocity) which the observers were instructed to follow. The direction of the target (right or left) was drawn pseudo-randomly according to the generative model defined above. (C) In order to titrate the adaptation to the environmental volatility of target direction at the explicit and conscious level, we invited each observer to perform on a different day a new variant of the direction-biased experiment, where we asked participants to predict, before each trial, their estimate of the forthcoming direction of the target. As shown in this sample screenshot, this was performed by moving a mouse cursor (black triangle) on a continuous rating scale between “Left”, to “Unsure” and finally “Right”.
Fig 2.
Binary Bayesian Change-Point (BBCP) detection model.
(A) This plot shows a synthesized sequence of 13 events, either a leftward or rightward movement of the target (TD). Run-length estimates are expressed as hypotheses about the length of an epoch over which the probability bias was constant, that is, the number of trials since the last switch. Here, the true probability bias switched from a value of .5 to .9 at trial 7, as can be seen by the trajectory of the true run-length (blue line). The BBCP model tries to capture the occurrences of a switch by inferring the probability of different possible run-lengths. At any new datum (trial), this defines a Hidden Markov Model as a graph (trellis), where edges indicate that a message is being passed to update each node’s probability (as represented by arrows from trial 13 to 14). Black arrows denote a progression of the run-length at the next step (no switch), while gray lines stand for the possibility that a switch happened: In this case the run-length would fall back to zero. The probability for each node is represented by the grey scale (darker grey colors denote higher probability) and the distribution is shown in the inset for two representative trials: 5 and 11. Overall, this graph shows how the model integrates information to accurately identify a switch and produce a prediction for the next trial (e.g. for t = 14). (B) On a longer sequence of 200 trials, representative of a trial block of our experimental sequence (see Fig 1A), we show the actual events which are observed by the agent (TD), along with the (hidden) dynamics of the true probability bias Ptrue (blue line), the value inferred by a leaky integrator (Pleaky, orange line) and the results of the BBCP model in estimating the probability bias PBBCP (green line), along with .05 and .95 quantiles (shaded area). This shows that for the BBCP model, the accuracy of the predicted value of the probability bias is higher than for the leaky integrator. Below, we show the belief (as grayscales) for the different possible run-lengths. The green and orange line correspond to the mean run-length which is inferred, respectively, by the BBCP and leaky models: Note that in the BBCP, while it takes some trials to detect switches, they are in general correctly identified (transitions between diagonal lines) and that integration is thus faster than for the leaky integrator, as illustrated by the inferred value of the probability bias.
Fig 3.
Behavioral results, qualitative overview.
For one trial block of 200 trials, we compare the different model-estimated probabilities with respect to the behavioral results. The top row represents the sequence of target directions (TD) that were presented to observers and agents, as generated by the binary switching model (see Fig 1A). We show the evolution of the value of the (true) probability bias Ptrue (blue line) which is hidden to observers and that is used to generate the TD sequence above. We have overlaid the results of the probability bias predicted with a leaky integrator (Pleaky, orange line) and with the BBCP model (PBBCP, see Fig 2B, green line). Bottom two rows display the raw behavioral results for the n = 12 observers, by showing their median (lines) and the .25 and .75 quantiles (shaded areas): First, we show the anticipatory pursuit eye velocity, as estimated right before the onset of the visually-driven pursuit. Below, we show the explicit ratings about the expected target direction (or bet scores). These plots show a good qualitative match between the experimental evidence and the BBCP model, in particular after the switches. Note that short pauses occurred every 50 trials (as denoted by vertical black lines, see main text), and we added the assumption in the model that there was a switch at each pause. This is reflected by the reset of the green curve close to the 0.5 level and the increase of the uncertainty after each pause.
Fig 4.
Behavioral results, quantitative analysis across participants (n = 12).
To analyze the relation between these behavioral data with the predictions made by models, we first looked at the variability of all these measures conditioned on the predicted probability and gathered over 5 equal partitions of the [0, 1] probability segment. For the 12 participants, we collected an estimate of (A) the amplitude of anticipatory pursuit (aSPEM) and (B) the bet score value. As a regressor, we have used the true probability (, blue color), and the probability bias estimates obtained with a leaky integrator (Pleaky, orange color) and by the BBCP model (PBBCP, green color). We display these functional relations using an error-bar plot showing the median with .25 and .75 quantiles over the 5 partitions. This shows a monotonous dependency for both behavioral measures with respect to the probability, close to a linear regression, but with different strengths. Second, we summarize in insets quantitative measures of the strength of this dependence for each participant individually, by computing the squared Pearson correlation coefficient r2 and the mutual information (MI). Dots correspond to these measures for each individual observer, while the bar gives the median value over the population. This confirms quantitatively that for both experimental measures, there is a strong statistical dependency between the behavioral results and the prediction of the BBCP model, but also that this dependency is significantly stronger than that obtained with the true probability and with the estimates obtained with the leaky integrator (stars denote significative differences, see text for details).
Fig 5.
Analysis of inter-individual differences.
(A) We analyzed the behavior of the n = 12 participants individually, by searching for each participant the best value of the model’s single free parameter, the hazard rate h. Estimates were performed independently on both experiments, such that we extracted different estimates of haSPEM and hbet respectively for the anticipatory pursuit velocity and the rating value. The dots correspond to independent estimates of the hazard rate for each individual participant are shown as dots, while the radius is proportional to the squared Person’s correlation coefficient. This plot shows that best fit hazard rates have a median value of and
. The values are in general higher than the ground truth (blue line), and in general higher for eye movements (below the diagonal). Note that the dispersion of hazard-rate best-fit estimates is narrower for the eye movement session than for the bet experiment. Such an analysis may suggest that participants ultimately have different mechanisms at the implicit (anticipatory pursuit) and explicit (ratings) levels for guiding their tendency of exploitation versus exploration. (B) To illustrate the models corresponding to these best-fitted values of the hazard rates, we show the predicted probability to the same sequence of TDs, with the lowest (
), optimal (
) and highest (
) hazard rates (respectively from top to bottom).