In: Symposium on Brain Theory and Modeling, II. World Congress 
of IBRO, Budapest, 1987 (Neuroscience)

Tensor Geometry as the Mathematical Language of Neuronal 
Networks: Brain Theory - Foundation for Neurobotics and 
Neurocomputers.

A. Pellionisz

Department of Physiology and Biophysics, New York 
University Medical Center, New York, N.Y. 10016, USA

Mathematical theory has a paramount importance in natural 
sciences: Philosophical attraction aside, a conceptually and formally 
homogeneous quantitative brain theory facilitates our 
understanding of the structuro-functional principles of distinct 
subsystems of the CNS (eg. those of the cerebellum and the role it 
plays in adaptive, coordinated sensorimotor  (neuronal network 
theories already started to yield new means, brain-like machines, 
both for production and control in neurobotics and neurocomputers). 
The basis of brain theory is the identification of the mathematical 
language that is inherent in the functioning of neuronal networks. A 
reductionalist view that nature's principal laws emerge from 
structure, (mathematically described "more geometrico"), 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. As for the 
formalism; tensor transformations connect such general coordinates, 
where one must distinguish between sensory- and motor-type 
vectorial representations (covariant and contravariant tensors) and 
needs to specify the features and the emergence of the metric tensor 
(comprising the multidimensional functional geometry) that 
transform these dual representations from one to another. 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). Once tensor network 
theory is elaborated and experimentally tested becomes a common 
language of neuroscience and robotics, and tensorial neuronal 
network algorithms lead to neurocomputer architectures and 
applications.