Geometrical approach to neural net control of movements and posture.

In one approach to modeling brain function, sensorimotor integration is described as geometrical mapping among coordinates of non-orthogonal frames that are intrinsic to the system; in such a case sensors represent (covariant) afferents and motor effectors represent (contravariant) motor efferents. The neuronal networks that perform such a function are viewed as general tensor transformations among different expressions and metric tensors determining the geometry of neural functional spaces. Although the non-orthogonality of a coordinate system does not impose a specific geometry on the space, this "Tensor Network Theory of brain function" allows for the possibility that the geometry is non-Euclidean. It is suggested that investigation of the non-Euclidean nature of the geometry is the key to understanding brain function and to interpreting neuronal network function. This paper outlines three contemporary applications of such a theoretical modeling approach. The first is the analysis and interpretation of multi-electrode recordings. The internal geometries of neural networks controlling external behavior of the skeletomuscle system is experimentally determinable using such multi-unit recordings. The second application of this geometrical approach to brain theory is modeling the control of posture and movement. A preliminary simulation study has been conducted with the aim of understanding the control of balance in a standing human. The model appears to unify postural control strategies that have previously been considered to be independent of each other. Third, this paper emphasizes the importance of the geometrical approach for the design and fabrication of neurocomputers that could be used in functional neuromuscular stimulation (FNS) for replacing lost motor control.

Spatial arrangement of the vestibular and the oculomotor system in the rat.

The treatment of the spatial aspects of vestibular sensation, ocular movement near primary position, and their neural processing requires numerical information about the directions of maximal sensitivity of the semicircular canals (SCC), the direction of gaze at primary position, and the directions of eye rotation generated by each individual extraocular muscle (EOM). A good approximation of this information can be gained from stereotaxic measurements of the geometrical arrangement of the canals' bony structure, from measurements of the pupil's orientation, and from measurements of the directions of muscle pull as well as of the center of eye rotation. The results of measurements in pigmented rats are given as unit sensitivity vectors and unit action vectors in head-fixed coordinate systems and compared with data from rabbit, cat, monkey, and human. The misalignment of 'coplanar' SCC with 2.5-15.6 degrees is second only to humans, while the misalignment of the vectors of 'antagonistic' EOM with 27.2-39 degrees is even more oblique than in humans and thereby even more so than in the other mammals. Misalignment of SCC and 'corresponding' EOM with 15.5-34.2 degrees again is largest, followed by that in humans and then the other mammals. The rat may therefore be useful in studying those mechanisms by which the central nervous system deals with the obliqueness of systems that play such an important role in humans, too.

Coordination: a vector-matrix description of transformations of overcomplete CNS coordinates and a tensorial solution using the Moore-Penrose generalized inverse.

Neuronal organisms express their function, such as a movement, by multicomponental actions. Thus, the problem of how the central nervous system (CNS) coordinates the elements of a single action is fundamental to our understanding of brain function. Coordinated activation of multijointed "limbs" has also become an acute problem in modern multivariable control theory and engineering, such as robotics. Thus, a coherent interdisciplinary approach is expected, one that arrives at concepts and formalisms applicable to this problem both in living and man-made organisms. By treating coordination with coordinates, tensor network theory of the CNS, which explains transformations through the neuronal networks of natural non-orthogonal coordinates that are intrinsic to living organisms, may successfully integrate the diverse approaches to this general problem. A link between tensor network theory of the CNS and multivariable control engineering can be established if the latter is formulated in generalized non-orthogonal coordinates, rather than in conventional Cartesian expressions. In general terms, the problem of coordinating an overcomplete (more than necessary) number of components of an action can be resolved by a three-step tensorial scheme. A key operation is a covariant-to-contravariant transformation executed by the Moore-Penrose generalized inverse when, in an overcomplete manifold, the covariant metric tensor is singular. In the neuronal organization of the CNS, it is assumed that the cerebellum plays this role of acting as a contravariant metric. A quantitative example is also provided, in order to demonstrate the viability of the numerical and network-implementations.