Proceedings of the Ninth Annual Conference of IEEE/Engineering in Medicine and Biology Society. Boston, 13-16. November, 1987.
Tensor Geometry: Mathematical Brain Theory for Neurocomputers and Neurobots.
Andras J. Pellionisz*, Department of Physiology and Biophysics, New York University Medical Center, New York, N.Y. 10016, USA
Conceptually and formally homogeneous quantitative brain theory serves as a mathematical basis for our understanding of the structuro-functional principles of distinct subsystems of the CNS, e.g. those of the cerebellum and gaze systems and the role they play in adaptive, coordinated sensorimotor operations; cf. Pellionisz, 1986 in "Brain Theory", Springer). Further, however, theory also leads from knowledge to its utilization. Mathematical neuronal network theories already started to yield new means, brain-like machines, both for production and control in the fields of neurobotics (robots equipped with brain-like controllers; cf. Pellionisz, 1984, in: IEEE SMC Int. Congress) and neurocomputers (new bread of non von Neumann-type "computers" implementing brain-like functions; cf. Pellionisz, 1987, IBRO II World Congress, Pergamon). A main role for brain theory is to identify the mathematical language that is inherent in the functioning of neuronal networks. The reductionalist view that nature's principal laws emerge from structure compelled the adoption of the axiom that CNS function should be understood in brain's own terms, the coordinates intrinsic to the physical geometry of the organism. Mathematically, these coordinates are different from those used in engineering: a variety of intrinsic frames are utilized, typically with non-orthogonally arranged overcomplete number of axes. This realization leads to Tensor Network Theory, based on the formalism of tensor transformations (cf. Pellionisz, 1987 in: Encyclopedia of Neurosci., Birkhauser Press). Tensors connect such general coordinates, where one must distinguish between sensory- and motor-type vectorial representations (covariant and contravariant tensors) and one also needs to specify the features and the emergence of tlie metric tensor that comprises a multidimensional functional geometry that transform these dual representations from one to another. The theory has implications in several subfields of neuroscience (cf. Pellionisz 1987, in: Computers in Brain Science, Cambridge U.P). As for the structural geometry, the intrinsic coordinate systems are to be established by the emerging field of quantitative computerized anatomy. As for specific neuronal networks, such as cerebellar, head-, gaze- and postural control systems, experimental research has to identify the functional coordinate systems intrinsic to neuronal expression. In turn, theoretical analysis must reveal how external geometries organize internal functional representations, expressed by networks (cf. Metaorganization-principle: cf. Pellionisz & Llinas 1985 Neurosci.). Once tensor network theory is elaborated and experimentally tested (Gielen & Zuylen, 1986 Neurosci, Pellionisz & Peterson 1987 in: Head Control, Oxford U.P) it becomes a common language of neuroscierice and robotics, and tensorial neuronal network algorithms lead to applications including neurocomputer architectures and neuronal-type controls used in robotics.