Internal Models in Biological Control

Abstract

Rationality principles such as optimal feedback control and Bayesian inference underpin a probabilistic framework that has accounted for a range of empirical phenomena in biological sensorimotor control. To facilitate the optimization of flexible and robust behaviors consistent with these theories, the ability to construct internal models of the motor system and environmental dynamics can be crucial. In the context of this theoretic formalism, we review the computational roles played by such internal models and the neural and behavioral evidence for their implementation in the brain.

Figure 1. The roles of internal models in sensorimotor control.

Perception. Sensory input y is used to estimate the ball’s state z₀ which is uncertain due to noise along the sensory pathway and the inability to directly observe the full state of the ball (e.g. its spin and velocity). Bayes rule is used to calculate the posterior (an example of a posterior over one component of position and velocity shown in inset). Simulation. An internal dynamical model p_fw simulates the forward trajectory z of the ball. At short timescales, this internal modeling is necessary to overcome delays in sensory processing, while at longer timescales, the predictive distribution p_fw(z|z₀) of the ball’s trajectory can be used for planning. Motor planning. Internal simulation of the ball’s trajectory along with prospective movements are evaluated in order to generate an action plan. The player may have to decide between re-orienting their body in order to play a forehand or backhand. Optimal feedback control. Once a motor plan has been specified, motor commands u are generated by an optimal feedback controller which uses a state estimator to combine sensory feedback and forward sensory predictions (based on an efference copy of the motor command) in order to correct motor errors online in task-relevant dimensions (green arrows).

Figure 2. Minimum intervention principle and exploitation of redundancy.

A. Unperturbed movements (black traces show individual hand movement paths) to narrow or wide targets tend to be straight and to move to the closest point on the target. Hand paths during the application of mechanical loads (red traces in response to a force pulse that pushes the hand to the right) delivered immediately after movement onset, which disrupt the execution of the planned movement, obey the principle of minimum intervention. That is, for a narrow target (left), the hand paths correct to reach the target whereas, for a wide target (right), there is no correction and the hand just reaches to another point on the target. B. Participants make reaching movements to targets. In a two-cursor condition, each hand moves its own cursor (black dots) to a separate target. In a one-cursor condition, the cursor is displayed at the average location of the two hands and participants reach with both hands to move this common cursor to a single target. During the movement, the left hand could be perturbed with a leftward (red) or rightward (blue) force field or remain unperturbed (black). C. When each hand controls its own cursor there is only one combination of final hand positions for which there is no error (center of circle). Optimal feedback control predicts that there will be no correlation between the endpoint positions (black circle shows a schematic distribution of errors). When the two hands control the position of a single cursor, there are many combinations of final hand positions which give zero error (black diagonal line; task-irrelevant dimension). Optimal control predicts correction in one hand to deviations in the other leading to negative correlations between the final locations of the two hands, so that if one hand is too far to the left the other compensates by moving to the right (black ellipse). D. Movement trajectories shown for the left and right hand for perturbations shown in B (one-cursor condition). The response of the right hand to perturbations of the left hand shows compensation only for the one-cursor condition in accord with the predictions of optimal feedback control. In addition, negative correlations in final hand positions can be seen in unperturbed movements for the one-cursor but not two-cursor condition (not shown). Modified with permission from [52] (A) and [53] (B-D).

Figure 3. Physical reasoning.

Participants must decide if a complex scene of blocks will fall and if so the direction of the fall. A model of their performance combines perception, physical reasoning, and decision-making. Left. A Bayesian model of perception uses the sensory input y to estimate participant’s belief p(z₀|y) regarding environment states such as the position, geometry, and mass of the blocks. Middle. Stochastic simulations based on samples from the posterior are performed using a noisy and approximate model of the physical properties of the world. The simulations use a forward model to sample multiple (superscripts) state trajectories over time (subscripts) ${\widehat{\mathbf{\text{z}}}}^{(i)}=({\widehat{z}}_0^{(i)},\dots ,{\widehat{z}}_T^{(i)}).$ Right. The outputs of this “intuitive physics engine” can then be processed to make judgments, such as the probability that the tower block will fall (F_fall) and the direction of the fall (F_dir). Experiments have indicated that humans are adept at making rapid judgments regarding the dynamics of such complex scenes and these judgments are consistent with predictions generated using this model which includes approximate Bayesian methods combined with internal forward models. Modified with permission from [123].