Ivancevic_Applied-Diff-Geom

1044 Applied Differential Geometry: A Modern IntroductionAll hierarchical subcortical neuro–muscular physiology, from the bottomlevel of a single muscle fiber, to the top level of cerebellar muscular synergy,acts as a temporal < out|in > reaction, in such a way that the higher levelacts as a command/control space for the lower level, itself representing anabstract image of the lower one:(1) At the muscular level, we have excitation–contraction dynamics [Hatze(1977a); Hatze (1978); Hatze (1977b)], in which < out|in > is given bythe following sequence of nonlinear diffusion processes: neural-actionpotential synaptic-potentialmuscular-action-potentialexcitationcontraction-coupling muscle-tension-generating [Ivancevic (1991);Ivancevic and Ivancevic (2006)]. Its purpose is the generation of muscularforces, to be transferred into driving torques within the jointanatomical geometry.(2) At the spinal level, < out|in > is given by autogenetic–reflex stimulus–response control [Houk (1979)]. Here we have a neural image of allindividual muscles. The main purpose of the spinal control level is togive both positive and negative feedbacks to stabilize generated muscularforces within the ‘homeostatic’ (or, more appropriately, ‘homeokinetic’)limits. The individual muscular actions are combined intoflexor–extensor (or agonist–antagonist) pairs, mutually controlling eachother. This is the mechanism of reciprocal innervation of agonists andinhibition of antagonists. It has a purely mechanical purpose to formthe so–called equivalent muscular actuators (EMAs), which would generatedriving torques T i (t) for all movable joints.(3) At the cerebellar level, < out|in > is given by sensory–motor integration[Houk et al. (1996)]. Here we have an abstracted image of allautogenetic reflexes. The main purpose of the cerebellar control levelis integration and fine tuning of the action of all active EMAs intoa synchronized movement, by supervising the individual autogeneticreflex circuits. At the same time, to be able to perform in new andunknown conditions, the cerebellum is continuously adapting its ownneural circuitry by unsupervised (self–organizing) learning. Its actionis subconscious and automatic, both in humans and in animals.Naturally, we can ask the question: Can we assign a single < out|in >measure to all these neuro–muscular stimulus–response reactions? We thinkthat we can do it; so in this Letter, we propose the concept of adaptivesensory–motor transition amplitude as a unique measure for this temporal