Exploring smoothness and discontinuities in human motor behaviour with Fourier analysis

The popular notion that human movements are smooth appears to be in contradiction to the fact that point-to-point movements must necessarily have discontinuities of some finite order at their onset and possibly offset. We explore discontinuities in the Fourier domain and show that the order and total strength of discontinuities in a trajectory can be measured from the slope and intercept of the envelope of the energy spectrum at high frequencies. In linear system models, the order of discontinuity is constrained by the motor command discontinuity and the order of the motor plant. We deduce that trajectories such as the minimum jerk are not smooth, and may even be the least smooth trajectories possible for biologically plausible motor plants. We further examine the role of discontinuities in optimal control and show that minimum square derivative profiles (such as minimum jerk) are time-optimal trajectories. This leads to the notion that point-to-point movements are a trade-off between duration and discontinuity strength, possibly reflecting neural command intensity or signal-dependent noise.