Benefits and Pitfalls in Analyzing Noise in Dynamical Systems - On Stochastic Differential Equations and System Identification

The search for a mathematical framework for describing motor behavior has a long but checkered history. Most studies have focused on recurrent, deterministic features of behavior. The use of dynamical systems to account for the qualitative features of end-effector trajectories of limb oscillations gained momentum in the last twenty-five years or so. There, salient characteristics of human movement served as guidelines for model developments. For instance, trajectories of limb cycling describing a bounded area in the position-velocity or phase plane may be interpreted as indicative of a limit cycle attractor, at least when modeling efforts are restricted to identifying deterministic forms, thereby disregarding variability. By using averaging methods, which are typically applied for first-order analyses of nonlinear oscillators, e.g., harmonic balance, Kay et al (1987, 1991) derived second-order nonlinear differential equations that mimic experimentally observed amplitude-frequency relations and phase response characteristics of rhythmic finger and wrist movements. The self-sustaining oscillators include weak dissipative nonlinearities that stabilize the underlying limit cycle, cause a drop of amplitude and an increase in peak velocity with increasing movement tempo or frequency.