Internal representations of temporal statistics and feedback calibrate motor-sensory interval timing

Abstract

Humans have been shown to adapt to the temporal statistics of timing tasks so as to optimize the accuracy of their responses, in agreement with the predictions of Bayesian integration. This suggests that they build an internal representation of both the experimentally imposed distribution of time intervals (the prior) and of the error (the loss function). The responses of a Bayesian ideal observer depend crucially on these internal representations, which have only been previously studied for simple distributions. To study the nature of these representations we asked subjects to reproduce time intervals drawn from underlying temporal distributions of varying complexity, from uniform to highly skewed or bimodal while also varying the error mapping that determined the performance feedback. Interval reproduction times were affected by both the distribution and feedback, in good agreement with a performance-optimizing Bayesian observer and actor model. Bayesian model comparison highlighted that subjects were integrating the provided feedback and represented the experimental distribution with a smoothed approximation. A nonparametric reconstruction of the subjective priors from the data shows that they are generally in agreement with the true distributions up to third-order moments, but with systematically heavier tails. In particular, higher-order statistical features (kurtosis, multimodality) seem much harder to acquire. Our findings suggest that humans have only minor constraints on learning lower-order statistical properties of unimodal (including peaked and skewed) distributions of time intervals under the guidance of corrective feedback, and that their behavior is well explained by Bayesian decision theory.

Figure 1. Comparison of response profiles for different ideal observers in the timing task.

The responses of four different ideal observers (columns a–d) to a discrete set of possible stimuli durations are shown (top row); for visualization purpose, each stimulus duration in this plot is associated with a specific color. The behavior crucially depends on the combination of the modelled observer's temporal sensorimotor noise (likelihood), prior expectations and loss function (rows 2–4); see Figure 2 bottom for a description of the observer model. For instance, the observer's sensorimotor variability could be constant across all time intervals (column a) or grow linearly in the interval, according to the ‘scalar’ property of interval timing (column b–d). An observer could be approximating the true, discrete distribution of intervals as a Gaussian (columns a–b) or with a uniform distribution (columns c–d). Moreover, the observer could be minimizing a typical quadratic loss function (columns a–c) or a skewed cost imposed through an external source of feedback (column d). Yellow shading highlights the changes of each model (column) from model (a). All changes to the observer's model components considerably affect the statistics of the predicted responses, summarized by response bias, i.e. average difference between the response and true stimulus duration, and standard deviation (bottom two rows). For instance, all models predict a central tendency in the response (that is, a bias that shifts responses approximately towards the center of the interval range), but bias profiles show characteristic differences between models.

Figure 2. Time interval reproduction task and generative model.

Top: Outline of a trial. Participants clicked on a mouse button and a yellow dot was flashed ms later at the center of the screen, with drawn from a block-dependent distribution (estimation phase). The subject then pressed the mouse button for a matching duration of ms (reproduction phase). Performance feedback was then displayed according to an error map . Bottom: Generative model for the time interval reproduction task. The interval is drawn from the probability distribution (the objective distribution). The stimulus induces in the observer the noisy sensory measurement with conditional probability density (the sensory likelihood), with a sensory variability parameter. The action subsequently taken by the ideal observer is assumed to be the ‘optimal’ action that minimizes the subjectively expected loss (Eq. 1); is therefore a deterministic function of , . The subjectively expected loss depends on terms such as the prior and the loss function (squared subjective error map ), which do not necessarily match their objective counterparts. The chosen action is then corrupted by motor noise, producing the observed response with conditional probability density (the motor likelihood), where is a motor variability parameter.

Figure 3. Experiment 1: Short Uniform and Long Uniform blocks.

Very top: Experimental distributions for Short Uniform (red) and Long Uniform (green) blocks, repeated on top of both columns. Left column: Mean response bias (average difference between the response and true interval duration, top) and standard deviation of the response (bottom) for a representative subject in both blocks (red: Short Uniform; green: Long Uniform). Error bars denote s.e.m. Continuous lines represent the Bayesian model ‘fit’ obtained averaging the predictions of the most supported models (Bayesian model averaging). Right column: Mean response bias (top) and standard deviation of the response (bottom) across subjects in both blocks (mean s.e.m. across subjects). Continuous lines represent the Bayesian model ‘fit’ obtained averaging the predictions of the most supported models across subjects.

Figure 4. Experiment 2: Medium Uniform and Medium Peaked blocks.

Very top: Experimental distributions for Medium Uniform (light brown) and Medium Peaked (light blue) blocks, repeated on top of both columns. Left column: Mean response bias (average difference between the response and true interval duration, top) and standard deviation of the response (bottom) for a representative subject in both blocks (light blue: Medium Uniform; light brown: Medium Peaked). Error bars denote s.e.m. Continuous lines represent the Bayesian model ‘fit’ obtained averaging the predictions of the most supported models (Bayesian model averaging). Right column: Mean response bias (top) and standard deviation of the response (bottom) across subjects in both blocks (mean s.e.m. across subjects). Continuous lines represent the Bayesian model ‘fit’ obtained averaging the predictions of the most supported models across subjects.

Figure 5. Experiment 2: Difference in response between Medium Peaked and Medium Uniform blocks.

Difference in response between the Medium Peaked and the Medium Uniform conditions as a function of the actual interval, averaged across subjects ( s.e.m.). The experimental distributions (light brown: Medium Uniform; light blue: Medium Peaked) are plotted for reference at bottom of the figure. The dashed black line represents the average response shift (difference in response between blocks, averaged across all subjects and stimuli), with the shaded area denoting s.e.m. The average response shift is significantly different from zero ( ms; two-sample t-test ), meaning that the two conditions elicited consistently different performance. Additionally, the responses were subject to a ‘local’ (i.e. interval-dependent) modulation superimposed to the average shift, that is, intervals close to the peak of the distribution (675 ms) were attracted towards it, in addition to the average shift, while intervals far away from the peak were less affected. (*) The response shift at 600 ms and 825 ms is significantly different from the average response shift; .

Figure 6. Bayesian observer and actor model components.

Candidate (i) sensory and (ii) motor likelihoods, independently chosen for the sensory and motor noise components of the model. The likelihoods are Gaussians with either constant or ‘scalar’ (i.e. homogeneous linear) variability. The amount of variability for the sensory (resp. motor) component is scaled by parameter (resp. ). iii) Candidate priors for the Medium Uniform (top) and Medium Peaked (bottom) blocks. The candidate priors for the Short Uniform (resp. Long Uniform) blocks are identical to those of the Medium Uniform block, shifted by 150 ms in the negative (resp. positive) direction. See Methods for a description of the priors. iv) Candidate subjective error maps. The graphs show the error as a function of the response duration, for different discrete stimuli (drawn in different colors). From top to bottom: Skewed error ; Standard error ; and Fractional error . The scale is irrelevant, as the model is invariant to rescaling of the error map. The squared subjective error map defines the loss function (as per Eq. 1).

Figure 7. Nonparametrically inferred priors (Experiment 1 and 2).

Top row: Short Uniform (red) and Long Uniform (green) blocks. Bottom row: Medium Uniform (light brown) and Medium Peaked (light blue) blocks. Left column: Nonparametrically inferred priors for representative participants. Right column: Average inferred priors. Shaded regions are s.d. For comparison, the discrete experimental distributions are plotted under the inferred priors.

Figure 8. Nonparametrically inferred priors (Experiment 3 and 4).

Top row: Medium Uniform (light brown) block. Bottom row: Medium High-Peaked (dark blue) block. Left column: Nonparametrically inferred priors for representative participants. Right column: Average inferred priors. Shaded regions are s.d. For comparison, the discrete experimental distributions are plotted under the inferred priors.

Figure 9. Experiment 5: Medium Bimodal and Wide Bimodal blocks, mean bias and nonparametrically inferred priors.

Very top: Experimental distributions for Medium Bimodal (dark purple, left) and Wide Bimodal (light purple, right) blocks. Top: Mean response bias across subjects (mean s.e.m. across subjects) for the Medium Bimodal (left) and Wide Bimodal (right) blocks. Continuous lines represent the Bayesian model ‘fit’ obtained averaging the predictions of the most supported models across subjects. Bottom: Average inferred priors for the Medium Bimodal (left) and Wide Bimodal (right) blocks. Shaded regions are s.d. For comparison, the experimental distributions are plotted again under the inferred priors.