Sensori-motor transformations in the brain (with a critique of the tensor theory of cerebellum)

Section 1 lists 12 points which must be addressed by neural models of sensorimotor coordination. Section 2 addresses the problem of extrapolating motor output from noisy data or from sensory input. The Pellionisz-Llinas cerebellar lookahead module addresses this problem for the noise-free case, and we suggest theoretical and experimental tests of the model; we then suggest the investigation of neural analogs of the Kalman-Bucy filter. Section 3 offers a brief exposition of mechanics in a tensor framework to provide the irreducible minimum of mathematical machinery to evaluate the Pellionisz-Llinás tensor theory of brain function and to suggest fruitful new hypotheses. Our critique of this theory in section 4 leads us to conclude that what they offer is based on metaphorical use of terminology from Euclidean tensors, not on rigorous application of the mathematics of tensor analysis. The central claim of their theory--that the input is a covariant intention vector transformed by a metric tensor encoded in the cerebellum to a contravariant execution vector--has not been substantiated and probably cannot be substantiated. However, we do point the way to further use of tensor analysis in the study of neural control of movement. The concluding section then returns to the points raised in section 1 with a highly selective survey of models of cerebellum and tectum.