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Human representation of visuo-motor uncertainty as mixtures of orthogonal basis distributions

Nature Neuroscience volume 18, pages 1152–1158 (2015)Cite this article

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Abstract

In many laboratory visuo-motor decision tasks, subjects compensate for their own visuo-motor error, earning close to the maximum reward possible. To do so, they must combine information about the distribution of possible error with values associated with different movement outcomes. The optimal solution is a potentially difficult computation that presupposes knowledge of the probability density function (pdf) of visuo-motor error associated with each possible planned movement. It is unclear how the brain represents such pdfs or computes with them. In three experiments, we used a forced-choice method to reveal subjects' internal representations of their spatial visuo-motor error in a speeded reaching movement. Although subjects' objective distributions were unimodal, close to Gaussian, their estimated internal pdfs were typically multimodal and were better described as mixtures of a small number of distributions differing only in location and scale. Mixtures of a small number of uniform distributions outperformed other mixture distributions, including mixtures of Gaussians.

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Figure 1: Computation of expected gain.
Figure 2: Task and design of experiment 1.
Figure 3: Internal pdfs in the choice task of experiment 1.
Figure 4: Model fits of all subjects' internal pdfs in experiment 1.
Figure 5: Experiment 2.
Figure 6: Experiment 3.
Figure 7: U-mix simulation for ref. 31.

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Acknowledgements

The authors would like to thank J. Tee for inspiring discussions. H.Z. and L.T.M. were supported by grant EY019889 from the US National Institutes of Health and L.T.M. by an award from the Alexander v. Humboldt Foundation. N.D.D. was supported by a Scholar Award from the McKnight Foundation and a James S. McDonnell Foundation Award in Understanding Human Cognition.

Author information

Authors and Affiliations

  1. Department of Psychology and Beijing Key Laboratory of Behavior and Mental Health, Peking University, Beijing, China

    Hang Zhang

  2. PKU-IDG/McGovern Institute for Brain Research, Peking University, Beijing, China

    Hang Zhang

  3. Peking-Tsinghua Center for Life Sciences, Peking University, Beijing, China.,

    Hang Zhang

  4. Department of Psychology, New York University, New York, New York, USA

    Hang Zhang, Nathaniel D Daw & Laurence T Maloney

  5. Center for Neural Science, New York University, New York, New York, USA

    Hang Zhang, Nathaniel D Daw & Laurence T Maloney

  6. Institute for the Interdisciplinary Study of Decision Making, New York University, New York, New York, USA

    Hang Zhang, Nathaniel D Daw & Laurence T Maloney

Authors
  1. Hang Zhang
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  3. Laurence T Maloney
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Contributions

H.Z. designed and performed the experiments, analyzed the data and wrote the manuscript. N.D.D. and L.T.M. supervised the project and improved the manuscript.

Corresponding author

Correspondence to Hang Zhang.

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Competing interests

The authors declare no competing financial interests.

Integrated supplementary information

Supplementary Figure 1 Visualization of subjects’ internal pdfs in Experiment 1.

Each panel is for one subject. Shaded regions denote ±SEM. is in the unit of the subject’s horizontal standard deviation estimated from the reaching task.

Supplementary Figure 2 Equivalent ratio as a function of Triple Number in Experiment 1.

Each panel is for one subject. In the choice task, there were 12 different Triples, whose paired Singles were adjusted in width by adaptive procedures (Fig. 2e). For each subject and each Triple, we defined the equivalent width of a Triple as the width of the Single that the subject judged to be as “hittable” as the Triple. We defined the equivalent ratio as the ratio of the equivalent width of the Triple to the total width of its three rectangles. The equivalent ratio provides a measure for the probability density function in subjects’ internal pdf, roughly indicating the mean probability density over the side rectangles of the Triple relative to the probability density around the center. The 12 Triples in the plot are sorted by the horizontal position of the right rectangle in the Triple. Note the measured equivalent ratios (black dots and line) had abrupt changes at adjacent Triples. The U-mix prediction (green line) captured the abrupt changes better than the predictions of the Gaussian prediction (blue line). The predictions of other mixture models (especially mG-mix) were close to that of U-mix and were omitted in the plot for simplicity.

Supplementary Figure 3 Visualization of subjects’ internal pdfs in Experiment 2.

Each panel is for one subject. Shaded regions denote ±SEM. is in the unit of the subject’s vertical standard deviation estimated from the reaching task.

Supplementary Figure 4 Model fits of subjects’ internal pdfs in Experiment 2.

(a) mG-mix model. (b) U-mix model. Each panel is for the probability density function of one subject. is in the unit of the subject’s vertical standard deviation estimated from the reaching task. Subjects are in the same order as in Supplementary Figure 3.

Supplementary Figure 5 Model fits of subjects’ internal pdfs in Experiment 3 for the mG-mix model.

Each panel is for one subject. The pair of numbers inside each panel denotes the numbers of components for the horizontal and vertical directions. Green and gray curves respectively denote probability density functions in the horizontal and vertical directions. is in the unit of the subject’s standard deviation (averaged across the horizontal and vertical directions) estimated from the reaching task.

Supplementary Figure 6 Model fits of subjects’ internal pdfs in Experiment 3 for the U-mix model.

Each panel is for one subject. The pair of numbers inside each panel denotes the numbers of components for the horizontal and vertical directions. Green and gray curves respectively denote probability density functions in the horizontal and vertical directions. is in the unit of the subject’s standard deviation (averaged across the horizontal and vertical directions) estimated from the reaching task. Subjects are in the same order as in Supplementary Figure 5.

Supplementary Figure 7 mG-mix simulation for Körding and Wolpert31.

Similar to Figure 7, except that the U-mix representation was replaced by an mG-mix representation (Methods online). (a) Illustration of the mG-mix representation. (b) Simulated aim points vs. data.

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Zhang, H., Daw, N. & Maloney, L. Human representation of visuo-motor uncertainty as mixtures of orthogonal basis distributions. Nat Neurosci 18, 1152–1158 (2015). https://doi.org/10.1038/nn.4055

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