Abstract
The role of symmetry in simplifying the theory of complex neural systems is argued. When the structural symmetries of a network are expressed as an ismorphism group, implications emerge for the dynamics. Various qualitative possibilities concerning stability of uniform motion in homogeneous nets are discussed and an approach to neural hierarchies is outlined.
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Literature
Amari, S. 1974. “A Method of Statistical Neurodynamics.”Kybernetik 14, 201–215.
Arbib, M. A. 1969.Theories of Abstract Automata. New York: Prentice-Hall.
Cowan, J. D. 1982. “Spontaneous Symmetry-breaking in Large Scale Nervous Activity.”Int. J. Quantum Chem. XXII, 1059–1082.
Ermentrout, G. B. and J. D. Cowan. 1979. “A Mathematical Theory of Visual Hallucination Patterns.”Biol. Cybernet. 34, 137–150.
— and —. 1980. “Secondary Bifurcation in Neuronal Nets.”SIAM J. appl. Math. 39, 323–340.
Griffith, J. S. 1971.Mathematical Neurobiology. London: Academic Press.
Hartline, H. K. and F. Ratcliff. 1972. “Inhibitory Interaction in the Retina ofLimulus.” InHandbook of Sensory Physiology, Ed. M. G. F. Fuortes, Vol. VII/2, pp. 381–447. Berlin: Springer.
Harth, E. M., T. J. Csermely, B. Beck and R. D. Lindsay. 1970. “Brain Functions and Neural Dynamics.”J. theor. Biol. 26, 93–120.
Hebb, D. O. 1949.The Organization of Behaviour. New York: Wiley.
Kobuchi, Y. 1976. “Signal Propogation in 2-dimensional Threshold Cellular Space.”J. math. Biol. 3, 297–312.
Minsky, M. 1969. “The Society Theory.” InA.I. An MIT Perspective, Ed. P. H. Winston and Brown, Vol. 1. Cambridge, MA: MIT Press.
Muir, A. 1979. “A Simple Case of the Wilson-Cowan Equations.”Biophys. J. 27, 267–276.
— 1981. “The Construction of Homogeneous Nets.”Bull. math. Biol. 43, 415–426.
— and M. W. Warner. 1981. “The Decomposition of Tolerance Automata.”Kybernetes 9, 265–273.
Palm, G. 1981. “Towards a Theory of Cell Assemblies.”Biol. Cybernet. 39, 181–194.
— and V. Braitenberg. 1979. “Tentative Contributions of Neuroanatomy to Nerve Net Theories.” InProgress in Cybernetics and Systems Research, Ed. R. Trappl, G. J. Klir and L. Ricciardi, Vol. 3, p. 369. London: Advance Publ.
Pribram, K. H. 1971.Languages of the Brain. Englewood Cliffs, NJ: Prentice-Hall.
Sattinger, D. H. 1978. “Group Representation Theory, Bifurcation Theory and Pattern Formation.”J. funct. Anal. 28, 58–101.
Szenthagothai, J. and M. A. Arbib. 1974. “Conceptual Models of Neural Organisation.”N.R.P. Bull. 12, 305–510.
van Rotterdam, A., F. H. Lopes da Silva, J. van den Ende, M. A. Viergever and A. J. Hermans 1982. “A Model of the Spatial-Temporal Characteristics of the Alpha Rhythm.”Bull. math. Biol. 44, 283–305.
Warner, M. W. 1981. “Tolerance Nets.”Bull. math. Biol. 43, 641–650.
Wilson, H. R. and J. D. Cowan. 1972. “Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons.”Biophys. J. 12, 1–24.
— and —. 1973. “A Mathematical Theory of the Functional Dynamics of Cortical and Thalmic Nervous Tissue.”Kybernetik 13, 55–80.
Zeeman, E. C. 1961. “The Topology of the Brain and Visual Perception.” InThe Topology of 3-manifolds, Ed. M. K. Fort, pp. 240–256. New York: Prentice-Hall.
Zvonkin, A. K. and L. A. Levin. 1970. “The Complexity of Finite Objects and the Development of the Concepts of Information and Randomness by Means of the Theory of Algorithms.”Russ. Math. Surveys 25, 83–124.
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Muir, A., Warner, M.W. The dynamics of symmetric nets. Bltn Mathcal Biology 45, 781–792 (1983). https://doi.org/10.1007/BF02460049
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DOI: https://doi.org/10.1007/BF02460049