The intentional spring: a strategy for modeling systems that learn to perform intentional acts.

In motor task learning by instruction, the instructor's skill and intention, which, initially, are extrinsic constraints on the learner's perceiving and acting, eventually become internalized as intrinsic constraints by the learner. How is this process to be described formally? This process takes place via a forcing function that acts both as an anticipatory (informing) influence and a hereditary (controlling) influence. A mathematical strategy is suggested by which such intentions and skills might be dynamically learned. A hypothetical task is discussed in which a blindfolded learner is motorically instructed to pull a spring to a specific target in a specific manner. The modeling strategy involves generalizing Hooke's law to the coupled instructor-spring-Learner system. Specifically, dual Volterra functions express the anticipatory and hereditary influences passed via an instructor-controlled forcing function on the shared spring. Boundary conditions (task goals) on the instructor-spring system, construed as a mathematical (self-adjoint) operator, are passed to the learner-spring system. Psychological interpretation is given to the involved mathematical operations that are passed, and mathematical (Hilbert-Schmidt's and Green's function) techniques are used to account for the release of the boundary conditions by the instructor and their absorption by the learner, and an appropriate change of their power spectra.