Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Compensation type algorithms for neural nets: stability and convergence

  • 45 Accesses

  • 8 Citations

Abstract

Plasticity of synaptic connections plays an important role in the temporal development of neural networks which are the basis of memory and behavior. The conditions for successful functional performance of these nerve nets have to be either guaranteed genetically or developed during ontogenesis. In the latter case, a general law of this development may be the successive compensation of disturbances. A compensation type algorithm is analyzed here that changes the connectivity of a given network such that deviations from each neuron's equilibrium state are reduced. The existence of compensated networks is proven, the convergence and stability of simulations are investigated, and implications for cognitive systems are discussed.

This is a preview of subscription content, log in to check access.

References

  1. Allgower, E. L., Georg, K.: Simplicial and continuation methods for approximating fixed points and solutions to systems of equations. SIAM Rev. 22, 28–85 (1980).

  2. Changeux, J. P., Danchin, A.: Selective stabilization of developing synapses as a mechanism for the specification of neuronal networks. Nature 264, 705–712 (1976)

  3. Clark, J. W., Winston, J. V., Rafelski, J.: Self-organization of neural networks. Phys. Lett. 102A, no. 4 (1984)

  4. Cromme, L. J.: Neural nets with prescribed average firing activity. NAM-Bericht 57, Institut für Numerische und Angewandte Mathematik der Universität, Göttingen 1988

  5. Cromme, L. J., Diener, I.: Fixed point theorems for discontinuous mappings. Math. Programming, submitted

  6. Dammasch, I. E.: A mechanism for stabilizing intelligent properties of neural networks. In: Rose, J. (ed.) Proceedings of the Seventh International Congress of Cybernetics and Systems, pp. 864–871. Lytham St. Annes: Thales 1987

  7. Dammasch, I. E., Wagner, G. P.: On the properties of randomly connected McCulloch-Pitts networks: Differences between input-constant and input-variant networks. Cybern. Syst. 15, 91–117 (1984)

  8. Dammasch, I. E., Wagner, G. P., Wolff, J. R.: Self-stabilization of neuronal networks I. The compensation algorithm for synaptogenesis. Biol. Cybern. 54, 211–222 (1986)

  9. Dammasch, I. E., Wagner, G. P., Wolff, J. R.: Self-stabilization of neuronal networks II. Stability conditions for synaptogenesis. Biol. Cybern. 58, 149–158 (1988).

  10. Fukushima, K., Miyake, S.: Neocognitron. Pattern Recognition 15.6, 455–469 (1982)

  11. Griffith, J. S.: Mathematical neurobiology. London: Academic Press 1971

  12. Jacobson, M.: Development of specific neuronal connections. Science 163, 543–547 (1969)

  13. Jacobson, M.: Developmental neurobiology. New York: Plenum Press 1978

  14. Joó, F., Dames, W., Wolff, J. R.: Effect of prolonged sodium bromide administration on the fine structure of dendrites in the superior cervical ganglion of adult rat. Prog. Brain Res. 51, 109–115 (1979)

  15. Malsburg, C. von der: Nervous structures with dynamical links. Ber. Bunsenges. Phys. Chemie, 89, 703–710 (1985)

  16. McCulloch, W. S., Pitts, W. H.: A logical calculus of the ideas imminent in nervous activity. Bull. Math. Biophys. 5, 115–133 (1943)

  17. Minsky, M., Papert, S.: Perceptrons. Cambridge, Mass.: MIT Press 1969

  18. Reichardt, W.: Processing of optical information by the visual system of the fly. Vision Res. 26, 113–126 (1986)

  19. Todd, M.: The computation of fixed points and applications. Berlin Heidelberg New York: Springer 1976

  20. Torras i Genis, C.: Temporal-pattern learning in neural models. Berlin Heidelberg New York Tokyo: Springer 1985

  21. Wolff, J. R.: Some morphogenetic aspects of the development of the central nervous system. In: Immelmann, K., Barlow, G. W., Main, M., Petrinovich, L. (eds.) New York: Cambridge University Press 1981

  22. Wolff, J. R., Joó, F., Dames, W., Fehér, O.: Induction and maintenance of free postsynaptic membrane thickenings in the adult superior cervical ganglion. J. Neurocytol. 8, 549–563 (1979)

  23. Wolff, J. R., Wagner, G. P.: Self-organization in synaptogenesis: Interaction between the formation of excitatory and inhibitory synapses. In: Basar, E., Flohr, H., Haken, H., Mandell, A. J. (eds.) Synergetics of the brain, pp. 50–59. Berlin Heidelberg New York Tokyo: Springer 1983

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cromme, L.J., Dammasch, I.E. Compensation type algorithms for neural nets: stability and convergence. J. Math. Biology 27, 327–340 (1989). https://doi.org/10.1007/BF00275816

Download citation

Key words

  • Neural modeling
  • McCulloch-Pitts networks
  • Compensation algorithm
  • Cognitive systems
  • Fixed points (approximate)