A kinematic theory of rapid human movement. Part IV: a formal mathematical proof and new insights

A few years ago a kinematic theory was proposed to study and analyze rapid human movements. The theory relies on a model of a synergy made up of two neuromuscular systems, one agonist and the other antagonist to the movement. Representing these systems with lognormal impulse responses, it is predicted that the velocity profile of a fast movement will be described by a delta-lognormal equation. So far, many studies have been conducted to test and empirically validate the theory. This paper presents an extended mathematical proof of the model. The proof is based on the Central Limit Theorem under the assumption that a law of proportionate effect governs the cumulative time delays of a sequence of dependent subprocesses constituting a neuromuscular system. Furthermore, a detailed interpretation of the parameters of the delta-lognormal equation, in terms of movement time and amplitude, response time and time delays, is discussed, providing new insights into the properties of the model with respect to neuromuscular system activity and movement generation.