Neuromuscular invariants in action execution and perception: Comment on “Motor invariants in action execution and perception” by Torricelli et al

Kinematic invariants, such as those comprehensively reviewed by Torricelli and colleagues [1], represent regu-larities observed in biological movements. They do not identify specific movements, but rather they describe a set of constraints within the large space of kinematic solutions of a given task, or some functional relationship between kinematic and behavioral variables. Consider straight paths and bell-shaped velocity profiles characterizing reaching movements. The specific path depends on the position of the target and, for a given target, the velocity profile depends on the movement time. Then, the specific kinematic solutions observed in a reaching task are not invariant. However, they represent a small subset of all the possible solutions. Mathematically, a spatial trajectory of the hand moving from point A to point B in time T can be described by a vector-valued function of time, i.e., a point in the infinite-dimensional space of such functions. Trajectories along a straight path are a subset of that space, and those with bell-shaped velocity profiles are contained in an even smaller subset that includes smooth scalar function of times of a specific form, which can often be specified by just a few parameters. For example, given A, B, and T, a minimum-jerk trajectory is uniquely determined. Thus, the observation of kinematic invariants may be understood as the existence of a low-dimensional representation of the solutions adopted by a biological controller to perform a given task. They may simplify sensorimotor control while preserving its flexibility because they allow mapping a goal onto a small set of parameters which identify an adequate solution.