KPK, IF, ISH, JNI, and DMW conceived and designed the experiments. KPK and IF performed the experiments. KPK and IF analyzed the data. ISH and JNI contributed reagents/materials/analysis tools. KPK and DMW wrote the paper.
The authors have declared that no conflicts of interest exist.
Making choices is a fundamental aspect of human life. For over a century experimental economists have characterized the decisions people make based on the concept of a utility function. This function increases with increasing desirability of the outcome, and people are assumed to make decisions so as to maximize utility. When utility depends on several variables, indifference curves arise that represent outcomes with identical utility that are therefore equally desirable. Whereas in economics utility is studied in terms of goods and services, the sensorimotor system may also have utility functions defining the desirability of various outcomes. Here, we investigate the indifference curves when subjects experience forces of varying magnitude and duration. Using a two-alternative forced-choice paradigm, in which subjects chose between different magnitude–duration profiles, we inferred the indifference curves and the utility function. Such a utility function defines, for example, whether subjects prefer to lift a 4-kg weight for 30 s or a 1-kg weight for a minute. The measured utility function depends nonlinearly on the force magnitude and duration and was remarkably conserved across subjects. This suggests that the utility function, a central concept in economics, may be applicable to the study of sensorimotor control.
Economists use the concept of a utility function, which increases with increasing desirability of the outcome, to characterize human decision making. This concept is shown here to apply to the control of movement.
In real world situations we often have to choose between possible actions that lead to different outcomes. To provide a computational framework for such a decision process, the notion of a utility function is often used (
In sensorimotor control, utility functions that depend on several variables occur frequently. Consider, for example, unpacking a car after a snowboarding vacation. We could carry all the suitcases at the same time, reducing the time to unpack but maximizing the weight we have to lift concurrently. At the other extreme we could transport each item individually, which would minimize the magnitude of the force required at the expense of a long unpacking duration. The chosen solution is likely to lie somewhere between these two extremes and may reflect an optimal decision based on a utility function that depends on duration and magnitude of the forces. Once a utility function is specified, the decision problem becomes one of solving an optimal control problem, finding the actions that maximize the utility.
A number of studies in the field of optimal sensorimotor control have proposed loss functions (the negative of utility) and derived the optimal actions given these proposed loss functions. For example, the minimum jerk model (
Different movements may be associated with different costs or utility. For example, a utility function could assign a numerical value to each possible movement, characterizing how costly it is to the organism. Here, we have examined how the utility associated with producing a force depends on two parameters, the duration and the magnitude of a force profile (see
In a two-alternative forced-choice paradigm, subjects chose which of two experienced force profiles they preferred to experience a second time (see
The subject's hand position (pink circle) was visible on the screen. The hand movement was restricted to stay within a small area (blue box). The direction of the force is represented by the blue arrow, and the temporal profile of the force is shown by the blue curve.
Various hypothesized utility functions predict different choices and thus different indifference lines. The first model we hypothesized was that subjects would minimize the integrated force they are using
(A–C) The predicted indifference lines are shown that minimize (A) the integrated force
(D) Experimental data from a single subject. The open circles are the reference forces. The blue full circles connected by the black lines represent indifference points. Error bars denote the 95% confidence intervals. Force profiles are illustrated (blue curves for single forces, pink curve for doubling points).
(E) Inferred color plot of the loss function (warmer colors represent greater cost).
A single subject's results are shown in
We furthermore measured how much smaller a force profile needed to be (scaled uniformly in duration and force) so that experiencing it twice had the same utility as experiencing the unscaled profile once. For the four open-circle reference points in
The black curves are the iso-loss curves. Error bars denote the standard error of the mean over the population. The color plot represents the inferred loss function (warmer colors represent greater loss) obtained by interpolating the data from the double-hump forces.
By applying the methodology developed by economists, we have shown that fundamental properties of the nervous system, such as loss functions, can be inferred by the choices humans make in a sensorimotor task. In general, these loss functions will depend on a large number of factors that were not measured in our experiment. For example, there are subjective emotional components to human decision making (
After providing written informed consent, five right-handed subjects (aged 20–40 y) participated in this study. The experiments were carried out in accordance with institutional guidelines. A local ethics committee approved the experimental protocols.
While seated, subjects held the handle of robotic manipulandum with two degrees of planar freedom. This was a custom-built device (vBot) consisting of a parallelogram constructed mainly from carbon fiber tubes that were driven by rare earth motors via low-friction timing belts. High-resolution incremental encoders were attached to the drive motors to permit accurate computation of the robot's position. Care was taken with the design to ensure it was capable of exerting large end-point forces while still exhibiting high stiffness, low friction, and also low inertia.
The robot's motors were run from a pair of switching torque control amplifiers that were interfaced, along with the encoders, to a multifunctional I/O card on a PC using some simple logic to implement safety features. Software control of the robot was achieved by means of a control loop running at 1,000 Hz, in which position and force were measured and desired output force was set.
A virtual reality system was used that prevented subjects seeing their hand, and allowed us to present visual images into the plane of the movement (for full details of the setup see
The experiment consisted of trials in which the robot generated force profiles on the subjects' hands. The force profiles experienced were parameterized by their duration
On each trial the subjects experienced two different force profiles and then could choose which of the two profiles to experience for a second time. Using such a forced-choice procedure allowed us to determine the indifference curves.
Subjects saw a starting sphere and two selection spheres (see
To obtain four indifference curves, we chose four reference profiles that had durations
On each trial, one of the two force profiles,
The above procedure allowed us to obtain indifference lines—where the utility has equal value. However, to obtain a full utility function we need to join up these lines and determine the relative utility of one indifference line to another. To achieve this we performed a two-alternative forced-choice paradigm in which the reference force was as before, but the test force was selected from the
We would like to thank Josh Tenenbaum, Peter Dayan, and Zoubin Ghahramani for helpful discussions. This work was supported by the Wellcome Trust, McDonnell Foundation, and Human Frontiers Science Programme.