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Tensor Geometry: A Language of Brains & Neurocomputers. Generalized Coordinates in Neuroscience & Robotics

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Neural Computers

Part of the book series: Springer Study Edition ((SSE,volume 41))

Abstract

Neurocomputers are implementations of mathematical paradigms performed by real neuronal networks. Thus, it is essential for their construction that the mathematical language of brain function be made explicit. Based on the philosophy that the brain, as a product of natural evolution, is a geometrical object (not a machine that is a product of engineering), tensor geometry is used to describe multidimensional general (tensor) transformations of natural coordinates that are intrinsic to the organism. Such an approach uses a formalism that not only generalizes existing Cartesian vector-matrix paradigms, but can unite neuroscience with robotics: general frames include both Natural coordinate systems (found by quantitative computerized anatomy) and those simple artificial ones that are selected in engineering for convenience. Utilizations of the tensor approach center on natural and artificial sensorimotor operations, promoting a co-evolution of coordinated (and intelligent) robots with Nature’s systems such as adaptive cerebellar compensatory reflexes. Such sensorimotor-based strategy enables also a cross-fertilization; eg. employing neurocomputers to implement a coordination-algorithm of cerebellar-networks, to be used for functional neuromuscular stimulation of paraplegics.

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References

  1. Anderson, J.A. and Mozer, M.C. (1981) Categorization and selective neurons. In: Parallel Models of Associateive Memory (Hinton, G.E. & Anderson, J.A. eds) pp. 213–236. Lawrence Erlbaum Assoc. N.J.

    Google Scholar 

  2. Babbage, C. (1889) Babbage’s Calculating Engines (London, Spon)

    Google Scholar 

  3. Bernstein, N.(1967) Co-ordination and regulation of Movements. Pergamon

    Google Scholar 

  4. Berthoz, A & Melvill-Jones G (1985) Adaptive Mechanisms in Gaze Control. Rev. in Oculomot. Res. Elsevier Amsterdam

    Google Scholar 

  5. Bloedel, J.R, Dichgans, J. and Precht, W. (1985). Cerebellar Functions. Berlin: Springer-Verlag.

    Book  Google Scholar 

  6. Caianiello, E.R. ed, (1968) Proceedings of the school on neuronal networks. In: Neuronal Networks. Springer-Verlag, N.Y

    Google Scholar 

  7. Cooper, L.N. (1974) A possible organization of animal memory and learning. In: Proceedings of Nobel Symposium on Collective Properties of Physiocal Systems. (Lundquist, B & Lundquist, S. eds), p. 252–264. Academic Press, N.Y.

    Google Scholar 

  8. Churchland, P.S. (1986). Neurophilosophy: Toward a Unified Science of the Mind-Brain. Cambridge, MA: MIT Press

    Google Scholar 

  9. Crick, F.H.C. (1979) Thinking about the brain. Sci. Amer., 241 (3) pp. 219–232.

    Article  Google Scholar 

  10. Daunicht, W. and Pellionisz, A. (1986) Coordinates Intrinsic to the Semicircular Canals and the Extraocular Muscles in the Rat. Soc. Neurosci. Absts. 12, p. 1089

    Google Scholar 

  11. Eckmiller, R. (1987) Neural Network Mechanism for Generation and Learning of Motor Programs. In: Proc. of IEEE 1st Internal Conf. on Neuronal Networks, San Diego, California

    Google Scholar 

  12. Edelman, G.M., V.B. Mountcastle, (1978) The Mindful Brain, MA: MIT Press

    Google Scholar 

  13. Fukushima, K., (1986) “Neocognitron: A Brain-Like System for Pattern Recognition”, Denshi Tokyo No. 25 NHK Science & Technical Research Laboratories 110-1, Kinuta, Setagaya, Tokyo 157, Japan

    Google Scholar 

  14. Farley BG & Clark WA (1954) Simulation of self-organizing system by digital computer. IRE Trans. Inf. Theor. 4, 76

    Article  MathSciNet  Google Scholar 

  15. Gielen, C.C.A.M. and van Zuylen, E.J. (1986). Coordination of arm muscles during flexion and supination: Application of the tensor analysis approach. Neuroscience, 17,527–539.

    Article  Google Scholar 

  16. Graf, H., Jackel, L, Howard, R., Straughn, B., Denker, J., Hubbard, W., Tennat, D., and Schwartz, D., (1987) “VLSI Implementation of a Neural Network Memory with Several Hundreds of Neurons”, AT&T Bell Laboratories, Holmdel, NJ

    Google Scholar 

  17. Grossberg S. (1982) Studies of Mind and Brain, Boston Studies in Phil. Sci. 70/Reidel

    Google Scholar 

  18. Gruner, J.A. (1986). Considerations in designing acceptable neuromuscular stimulation systems for restoring function in paralyzed limbs. Central Nervous System Trauma, 3(1), 37–47.

    MathSciNet  Google Scholar 

  19. Gurfinkel VS (1987) Robotics and Biological Motor Control. Proc. of II. IBRO World Cong. Neurosci, Suppl. 22. S381.

    Google Scholar 

  20. Hebb, D.O. (1949) The Organization of Behaviour. John Wiley, New York

    Google Scholar 

  21. Hecht-Nielssen, R. (1987) Counterpropagation Networks. IEEE 1st Internal Conf. on Neuronal Networks, San Diego

    Google Scholar 

  22. Helmholtz (1896). Handbuch der Physiologischen Optik. Zweite Auflage. Lepizig: Voss.

    MATH  Google Scholar 

  23. Hopfield, J.J., D.I. Feinstein, R.G. Palmer, (1983) Nature 304:158–159

    Article  Google Scholar 

  24. Koenderink, J.J. (1984) Geometrical Structures Determined by the Functional Order in Nervous Nets. Biological Cybernetics. 50, 43–50

    Article  MathSciNet  MATH  Google Scholar 

  25. Kohonen, T. (1977) Associative Memory, Springer-Verlag Heidelb.-New York-Berlin

    MATH  Google Scholar 

  26. Kralj, A., and Grobelnik, S. (1973). Functional electrical stimulation — A new hope for paraplegic patients? Bulletin of Prosthesis Research. 10–20, 75–102.

    Google Scholar 

  27. Laczkó, J., Pellionisz, A.J. Peterson, B.W. and Buchanan, T.S. (1987) Multidimensional Sensorimotor “Patterns” Arising from a Graphics-Based Tensorial Model of the Neck-Motor System. Soc. Neurosci. Absts. 13.

    Google Scholar 

  28. Lestienne, A., Liverneaux, P., Laczkó, J., Pellionisz, A. (1987) Tensor Model of the Musculo-Skeletal Head-Neck System of the Monkey. Proc. of IBRO II. World Congress, Neurocience, Suppl. to Vol. 22, p.S658

    Google Scholar 

  29. Lestienne, F, Liverneaux, Ph., Pellionisz, A. (1987) Role of the Superficial and Deep Neck Muscles in the Control of Monkey Head Movement: Application of the Tensor Analysis Approach. Soc. Neurosci. Absts. 13.

    Google Scholar 

  30. Levi-Civita, T. (1926). The Absolute Differential Calculus (Calculus of Tensors). New York: Dover.

    Google Scholar 

  31. Liverneaux, Ph, Pellionisz, A.J., Lestienne, F.G. (1987) Morpho-Anatomy and Muscular Synergy of Sub-occipital Muscles in Macaca Mulatta: Study of Head-Trunk Coordination. Proc. IBRO II. World Congr, Neurosci, Suppl. to V.22, p. S847

    Google Scholar 

  32. Mauritz, K.H.(1986).Restoration of posture and gait by functional neuromuscular stimulation (FNS). In Disorders of Posture and Gait, ed. W. Bles and T. Brandt, Amsterdam: Elsevier, pp. 367–385.

    Google Scholar 

  33. McCulloch, W.S. and Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 5, 115–133.

    Article  MathSciNet  MATH  Google Scholar 

  34. Minsky, M and Papert, S (1969) Perceptrons, MIT Press, Cambridge, MA

    MATH  Google Scholar 

  35. Nashner, L.M. (1977). Fixed patterns of rapid postural responses among leg muscles during stance. Experimental Brain Research, 30,13–24.

    Article  Google Scholar 

  36. Palm, G. (1982) Neural Assemblies, Springer, Heidelberg-New York-Berlin

    Google Scholar 

  37. Palm, G. and Aertsen, A. (1986). Brain Theory: Proceedings of the First Trieste Meeting on Brain Theory, October 1–4, 1984. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag.

    MATH  Google Scholar 

  38. Pellionisz A (1983) Brain Theory: Connecting Neurobiology to Robotics.Tensor Analysis: Utilizing Intrinsic Coordinates to Describe, Understand and Engineer Functional Geometries of Intelligent Organisms. J. Theor. Neurob. 2, 185–211

    Google Scholar 

  39. Pellionisz, A. (1984) Coordination: A Vector-Matrix Description of Transformations of Overcomplete CNS Coordinates and a Tensorial Solution Using the Moore-Penrose Generalized Inverse. J. Theoret. Biol. 110, pp. 353–375

    Article  Google Scholar 

  40. Pellionisz, A. (1985) Tensorial Brain Theory in Cerebellar Modeling. In: Cerebellar Functions (ed. Bloedel, J., Dichgans, J. and Precht, W.), Springer, Heidelberg, pp.201–229

    Google Scholar 

  41. Pellionisz, A. (1985) Robotics Connected to Neurobiology by Tensor Theory of Brain Function. Proc. IEEE International Conf. on Systems, Man and Cybernetics. pp. 411–414

    Google Scholar 

  42. Pellionisz, A. (1986) Tensor Network Theory and its Application in Computer Modeling of the Metaorganization of Sensorimotor Hierarchies of Gaze. In: Neuronal Networks for Computing. AIP151, NY: Am. Inst. Physics. 339–344.

    Google Scholar 

  43. Pellionisz, A. (1986) Tensor Network Theory of the Central Nervous System and Sensorimotor Modeling. In: Brain Theory (eds. Palm G., and Aertsen, A.), Springer-Verlag, Berlin-Heidelberg-New York, pp. 121–145

    Chapter  Google Scholar 

  44. Pellionisz, A. (1987) Multidimensional Geometries Intrinsic to Cognitive CNS Hyperspaces, and their Metaorganization by a Sensorimotor Apparatus. Soc. Neurosci. Absts. 13.

    Google Scholar 

  45. Pellionisz, A. (1987) Sensorimotor Operations: a Ground for the Co-Evolution of Brain Theory with Neurobotics and Neurocomputers. In: Proc. IEEE 1st Ann. International. Conf. on Neural Networks, San Diego

    Google Scholar 

  46. Pellionisz, A. (1987) Tensor Geometry as the Mathematical Language of Neuronal Networks. Brain Theory — Foundation for Neurobotics and Neurocomputers. Proc. of IBRO ll.World Congress, Neuroscience Suppl.V.22, S101

    Google Scholar 

  47. Pellionisz, A. (1987) Tensor Geometry: Mathematical Brain Theory for Neurocomputers and Neurobots. In: Proc. of the Ninth Annual Conference of IEEE Engineering in Medicine and Biology Society. Boston

    Google Scholar 

  48. Pellionisz A (1987) Tensor Network Theory of the Central Nervous System. Enc. Neurosci. ed G. Adelman, Birkhauser

    Google Scholar 

  49. Pellionisz, A. (1987) Vistas from Tensor Network Theory: A Horizon from Reductionalistic Neurophilosophy to the Geometry of Multi-Unit Recordings. In: Computer Simulation in Brain Science (ed. R. Cotterill), Cambridge Univ. Press

    Google Scholar 

  50. Pellionisz, A. and Graf, W. (1987) Tensor Network Model of the “Three-Neuron Vestibulo-Ocular Reflex-Arc” in Cat. J. Theoretical Neurobiology, 5, 127–151.

    Google Scholar 

  51. Pellionisz, A. and Llinás, R. (1977) Computer Model of Cerebellar Purkinje Cells. Neuroscience, 2, pp. 37–48

    Article  Google Scholar 

  52. Pellionisz, A. and Llinás, R. (1979) Brain Modeling by Tensor Network Theory and Computer Simulation. The Cerebellum: Distributed Processor for Predictive Coordination. Neuroscience, 4, pp. 323–348

    Article  Google Scholar 

  53. Pellionisz, A. and Llinás, R. (1980) Tensorial Approach to the Geometry of Brain Function: Cerebellar Coordination via Metric Tensor. Neuroscience, 5, p. 1125–1136

    Article  Google Scholar 

  54. Pellionisz, A. and Llinás, R. (1982) Space-Time Representation in the Brain. The Cerebellum as a Predictive Space-Time Metric Tensor. Neuroscience, 7, pp. 2949–2970

    Article  Google Scholar 

  55. Pellionisz, A. and Llinás, R. (1985) Tensor Network Theory of the Metaorganization of Functional Geometries in the CNS. Neuroscience, 16, pp. 245–274

    Article  Google Scholar 

  56. Pellionisz, A., and Peterson, B.W. (1987) A Tensorial Model of Neck Motor Activation. In: Control of Head Movement (eds. Peterson, BW. and Richmond, F.), Oxford University Press

    Google Scholar 

  57. Pellionisz, AJ., Soechting, JF, Gielen, CCAM, Simpson, Jl, Peterson, BW, Georgopoulos, AP. (1986) Workshop: Multidimensional Analyses of Sensorimotor Systems. Soc. Neurosci. Absts. 12, p. 1

    Google Scholar 

  58. Piaget J. (1980) Structuralism. Basic Books.

    Google Scholar 

  59. Psaltis, D. and Abu-Mostafa, Y.S. (1987) Optical Neural Computers. Scientific American, March, 88–95

    Google Scholar 

  60. Purpura, D. P. (1975) Introduction and perspectives. In: Golgi centennial symposium: Perspectives in neurobiology, ed. M. Santini, XIII-XVII, New York: Raven

    Google Scholar 

  61. Reitboeck, H.J.P. (1983) A 19-channel matrix drive with individually controllable fiber microelectrodes for neurophysiological applications. IEEE Transactions on System, Man and Cybernetics, SMC-13,5, pp. 676–682.

    Google Scholar 

  62. Robinson, D. (1982) The use of matrices in analyzing the three-dimensional behavior of the vestibulo-ocular reflex. Biol. Cybernetics, 46, 53–66

    Article  MATH  Google Scholar 

  63. Rosenblatt, F (1962) Principles of Neurodynamics, Spartan Books, NY

    MATH  Google Scholar 

  64. Rumelhart, D., (1986) “Parallel Distributed Processing”, (Vol. 1 & 2) MIT Press

    Google Scholar 

  65. Rumelhart, D., Hinton, G. and Williams, R.(1986) Learning Representations by Back-Propagating Errors, Nature 323 9

    Article  Google Scholar 

  66. Sage, J., Thompson, K. and Withers, R., (1987) Silicon Integrated Circuit Implementation of an Artificial Neural Network, MIT, Lincoln Lab Based on MNOS and CCD Technologies

    Google Scholar 

  67. Sejnowski TJ (1986) Higher-Order Boltzmann Machines: Neur. Netw. for Computing. NY: Am. Inst. Phys. 398–403

    Google Scholar 

  68. Shannon, C. (1948). A mathematical theory of communication. Bell System Technology J. 27,3–4.

    MathSciNet  Google Scholar 

  69. Sherrington, C. (1906). The Integrative Action of the Nervous System. New York: Scribner.

    Google Scholar 

  70. Simpson, J.I. and Graf, W. (1985). The selection of reference frames by nature and its investigators. In Adaptive Mechanisms in Gaze Control. Rev. of Oculomot. Res., V. 1, ed A. Berthoz and G. Melvill-Jones, Amsterdam Elsevier, 3–20

    Google Scholar 

  71. Steinbuch, K. (1961) Die Lernmatrix. Kybernetik, 1, 36–45.

    Article  MATH  Google Scholar 

  72. Voevodsky, J., (1987) “A Neural-Based Knowledge Processor”, Neuraltech, Mountain View, CA

    Google Scholar 

  73. von Braitenberg, V., and Onersto, N. (1961) The cerebellar cortex as a timing organ. Discussion of an hypothesis. Proc. 1st. Int. Conf. Med. Cybernet, pp. 1–19, Giannini, Naples

    Google Scholar 

  74. von Malsburg, C., (1986) “Disordered Systems and Biological Organization”, Springer

    Google Scholar 

  75. von Neumann, J. (1958). The Computer and the Brain. New Haven: Yale University Press

    MATH  Google Scholar 

  76. von Seelen, W., Mallot, H.A. Krone, G. and Dinse H.I (1984) On information processing in the cat’s visual cortex. In: Brain Theory, ed. by Palm, G. and Aertsen, A. Springer, Berlin, pp. 49–80

    Google Scholar 

  77. Wiener, N. (1948). Cybernetics, or Control and Commnication in the Animal and the Machine, Cambridge: MIT Press

    Google Scholar 

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Authors and Affiliations

  1. Department of Physiology and Biophysics, New York University Medical Center, 550 First Ave, New York City, N.Y., 10016, USA

    A. Pellionisz

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  1. A. Pellionisz
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Editors and Affiliations

  1. Abteilung Biokybernetik, Institut für Physikalische Biologie, Universität Düsseldorf, Universitätsstr. 1, D-4000, Düsseldorf 1, Germany

    Rolf Eckmiller

  2. Abteilung Neurobiologie, Max-Planck-Institut für Biophysikalische Chemie, D-3400, Nikolausberg, Göttingen, Germany

    Christoph v.d. Malsburg

  3. Dept. of Computer Science, University of Southern California, Los Angeles, CA, 90089-0782, USA

    Christoph v.d. Malsburg

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Pellionisz, A. (1989). Tensor Geometry: A Language of Brains & Neurocomputers. Generalized Coordinates in Neuroscience & Robotics. In: Eckmiller, R., v.d. Malsburg, C. (eds) Neural Computers. Springer Study Edition, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83740-1_39

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  • DOI: https://doi.org/10.1007/978-3-642-83740-1_39

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  • Print ISBN: 978-3-540-50892-2

  • Online ISBN: 978-3-642-83740-1

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