The neuron as a Lie group germ and a Lie product
Author:
William C. Hoffman
Journal:
Quart. Appl. Math. 25 (1968), 423-440
DOI:
https://doi.org/10.1090/qam/99871
MathSciNet review:
QAM99871
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Abstract: A basic Lie group derived earlier that describes the “constancies” of visual perception is extended to one more dimension to take into account the nonlinear flow of an afferent volley of nerve impulses through the layers of the visual cortex. One is then led, via the usual determination of the solutions of a Lagrange partial differential equation in terms of an associated Pfaffian system of ordinary differential equations, to a correspondence between neuron cell body, Lie group germ, and critical point of the system of ordinary differential equations governing the orbits. The local phase portraits of the latter bear a marked resemblance to one or the other of the neuron types defined by Sholl. Since “brains are as different as faces", the concept of structural stability plays an important role in analyzing the connectivity of the neural network. Finally, Lukasiewicz’s theory of parentheses is used to obtain a graph-theoretic representation of the Jacobi identity, which then serves to explain the branching of neuronal processes (dendrites).
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C. Berge, The Theory of Graphs and Its Applications, Wiley and Sons, Inc., New York, 1962
G. H. Bishop, and J. L. O’Leary. Potential records from the optic cortex of the cat, J. Neurophysiol. 1, 391–404 (1938)
G. H. Bishop, Natural history of the nerve impulse, Physiol. Rev. 36, 376–399 (1956)
T. H. Bullock, Neuron doctrine and electrophysiology, Science 129, 997–1002 (1959)
E. Cassirer, The concept of group and the theory of perception, Philos. and Phenomenological Res. 5, 1–36 (1944)
H. T. Chang, Changes in excitability of cerebral cortex following single electrical shock applied to cortical surface, J. Neurophysiol. 14, 95–111 (1951)
P. M. Cohn, Lie groups, Cambridge Univ. Press, 1957
M. Colonnier, The tangential organization of the visual cortex, J. Anatomy (London) 98, 327–344 (1964)
J. L. Conel, The postnatal development of the human cerebral cortex. I; Cortex of the newborn. II; Cortex of the one-month infant. III; Cortex of the three-month infant. IV; Cortex of the six-month infant. V; Cortex of the fifteen-month infant. VI; Cortex of the 24-month infant. VII; Cortex of the four year old child, Harvard University Press, 1939, 1941, 1941, 1955, 1955, 1959, 1959
J. T. Culbertson, A neural analysis of behavior and consciousness, Wm. C. Brown Co., Dubuque, Iowa, 1950
J. T. Culbertson, Some uneconomical robots, in Automata Studies, C. E. Shannon and J. T. McCarthy, Eds., Princeton Univ. Press, Princeton, N. J., 1956, pp. 99–116
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J. F. Fulton, Physiology of the nervous system, 3rd ed., Oxford Univ. Press, New York, 1949
M. Gihr, Methode zur Rekonstruktion von Nervenzellen, J. f. Hirnforschung 5, 7–22 (1962)
H. W. Guggenheimer, Differential geometry, McGraw-Hill, New York, 1963
W. C. Hoffman, The Lie algebra of visual perception, J. Math. Psychol. 3, 65–98 (1966)
W. C. Hoffman, Comments on ’optical art’, Applied Optics 5, 873–874 (1966)
M. Kucharzewski, and M. Kuczma. Basic concepts of the theory of geometric objects, Rozprawy Matematyczne XLIII, Warsaw, 1964
I. Kupka, The singularities of integrable structurally stable Pfaffian forms, Proc. Nat. Acad. Sci. U.S.A. 52, 1431–1432 (1964)
C-L. Li, C. Cullen, and H. H. Jasper, Laminar microelectrode studies of specific somatosensory cortical potentials, J. Neurophysiol. 19, 111–130 (1956)
P. Mendelson, On phase portraits of critical points in n-space, Contributions to the Theory of Nonlinear Oscillations, Vol. IV, S. Lefschetz, Ed., Princeton Univ. Press, Princeton, N. J., 1958, pp. 167–199
K. Motokawa, Mechanisms for the transfer of information along the visual pathways, in Internat. Rev. Neurobiol. 5, C. C. Pfeiffer and J. R. Smythies, Eds., Academic Press, N. Y., 1963, pp. 121–181.
J. R. Munkres, Elementary differential topology, Princeton Univ. Press, Princeton, N. J., 1963
R. S. Palais, Global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc., No. 22, 1957
R. S. Palais, The classification of G-spaces, Mem. Amer. Math. Soc., No. 36, 1960
W. Pitts, and W. S. McCulloch. How we know universals. The perception of auditory and visual forms, Bull. Math. Biophys. 9, 127–147 (1947)
K. H. Pribram, A review of theory in physiological psychology, in Annual Revs. Psychol., Vol. 11, Palo Alto, Calif., 1960, pp. 1–40
K. H. Pribram, The new neurology: memory, novelty, thought and choice, in EEG and behavior, G. H. Glaser, Ed., Basic Books, N. Y., 1963, pp. 149–173
D. P. Purpura, Morphological basis of elementary evoked response patterns in the neocortex of the newborn cat, Annals N. Y. Acad. Sci., Vol. 92, Art. 3 (1961), pp. 840–859
S. Ramon y Cajal, Histologie du systeme nerveux de l’homme et des vertebres, Vol. II, Madrid, 1955
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N. Rashevsky, Mathematical biophysics: physico-mathematical foundations of biology, vol. II, 3rd rev. ed., Dover, N. Y., Chapter 11
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F. Rosenblatt, Perceptual generalization over transformation groups, in Self-Organizing Systems, M. Yovits and S. Cameron, Eds., Pergamon Press, 1960
D. A. Sholl, The organization of the cerebral cortex, Wiley & Sons, Inc., New York, 1956
J. R. Smythies, The stroboscopic patterns. II. The phenomenology of the bright phase and after-images, British J. Psychol. 50, 305–324 (1959)
M. Verzeano, and K. Negishi, Neuronal activity in cortical and thalamic networks, J. General Physiol. 43, Suppl., 177–195 (1960)
N. Wiener, Cybernetics, 2nd ed., MIT Press, Camb., Mass., 1961, Ch. VI: gestalt and universals
J. Z. Young, Two memory stores in one brain, Endeavour 24, 13–20 (1965)
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Article copyright:
© Copyright 1968
American Mathematical Society
