A class of differential equations for merging movements' kinematic optimality with geometric invariance
Year: 2014
Type: preprint
Abstract: Neuroscientific studies of execution of the drawing-like movements usually analyze neural representation of either geometric (eg. direction, shape) or temporal (eg. speed) features of trajectories rather than trajectory’s representation as a whole. This work is about mathematical ideas behind splitting and merging geometric and temporal features which characterize biological movements. Movement primitives supposedly facilitate the efficiency of movements’ representation in the brain and comply with different criteria for biological movements, among them kinematic smoothness and geometric constraint. Criterion for trajectories’ “maximal smoothness” of arbitrary order n is employed, n = 3 is the case of the minimum-jerk model. I derive a class of differential equations obeyed by movement paths for which nth order maximally smooth trajectories have constant speed. The speed is invariant under a class of geometric transformations. Equations’ solutions presumably serve as candidates for geometric movement primitives. The speed here is defined as the rate of accumulating geometric measurement along the drawn path. The geometric measurement may be chosen to be an arc in certain geometry. For example the two-thirds power-law model corresponds to piece-wise constant speed of accumulating equi-affine arc. The derived c of differential equations consists of two parts. The first part is identical for all geometric parameterizations of the path. The second part is parametrization specific and is needed to identify whether a solution of the first part indeed represents a curve. Corresponding counter-examples are provided. Equations in different geometries in plane and in space and their known solutions are presented. The derived class of differential equation is a novel tool for discovering candidates for geometric movement primitives.
Source: arXiv (Cornell University)
Author: Felix Polyakov
Institution: Bar-Ilan University
Language: English
Topic: Motor Control and Adaptation
Subfield: Cognitive Neuroscience
Field: Neuroscience
Domain: Life Sciences
Open Access status: green
OpenAlex ID: W2176597990