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What makes a reach movement effortful? Physical effort discounting supports common minimization principles in decision making and motor control

Fig 3

Effort depends on squared movement force (results of experiment 2).

The Bayesian fit of data from experiment 2 using Eq 1 and Eq 2 revealed a force exponent of 2 for linking effort and force. A and B. Subject-level fits of the choice behavior for subjects JP and MK. The fitted equivalent force curve (thick red line, Eq 3) represents the estimated test movement force FTeq at which the test movement felt as effortful as the corresponding double reference movement. The underlying psychometric functions in blue illustrate how this functional relationship is linked to the subject’s choice behavior. For each of the sampled reference movement forces FR (0, 3, 6, and 9 N, major x-axis), the estimated choice probability P(R|FT,FR) of choosing the reference movement over the test movement is plotted as function of test-movement force in the vertical plots (blue line). For a given FT (i.e., along the corresponding horizontal line), the width of the dark blue area represents the probability for the subject to pick the test movement, and the width of the light blue area represents the probability for the subject to pick the double reference movement. The darker region corresponds to the 95% credible interval extracted from the posterior samples. Note that the fitted equivalent force curve intersects the choice probability curves at their points of subjective equality (i.e., verifies P(R|FTeq,FR) = 0.5). The blue discs represent the subjects’ actual choice probabilities (disc areas are proportional to the number of trials. Equivalent forces computed as in experiment 1 are shown as red squares for comparison to the Bayesian fit. For reference, the unity line and FTeq = 2FR line are depicted with dotted red lines. C. Representation of the population-level compound posterior distribution for the force exponent α. This compound distribution represents the distribution of α’s posterior distributions, constructed from the sampled posterior distributions for α’s population mean and variance. The black line and the grey area represent the median of the compound distribution and its 95% credible interval. The dashed line and corresponding error bar indicate the median and 95% credible interval for the posterior of the population mean of the force exponent μα. Triangles represent the medians of the subject-level posteriors for the force exponent αi (red: subject shown in A; blue: subject in B). The value of 2 for the force exponent describes a squared force dependency of effort. D. Population-level compound posterior distribution for the effort offset. E. Population-level compound posterior distribution for the effort sensitivity γ. F. Distribution of residuals for the choice probability curves. The selected model (in grey, Eq 4) is compared to the alternative models (other colors). The residuals correspond to the signed distances between the blue curves and the blue discs on panels A and B. Residuals from all subjects are represented here. Data underlying this figure can be found at

Fig 3