Oscillatory networks with Hebbian matrix of connections

  • Kuzmina M. G. 
  • Manykin E. A. 
  • Surina I. I. 
Computational Models of Neurons and Neural Nets
Part of the Lecture Notes in Computer Science book series (LNCS, volume 930)


The systems of symmetrically coupled limit cycle oscillators admit the design of recurrent associative memory networks with Hebbian matrix of connections. Unlike the similar neural networks this matrix proved to be the complex-valued Hermitian one with nonzero diagonal. In the case of strong interaction in oscillatory system the memory vectors of the network are slightly perturbed properly normalized eigenvectors of matrix of connections. They can be calculated by perturbation method on the appropriate small parameter. The self-consistent analysis of dynamical system fixed points in the case of homogeneously all-to-all connected oscillators is presented. It is proved that for positive values of connection strength only a single memory vector can be stored. Some questions concerning the ”extraneous” memory of the networks are discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Y. Kuramoto, I. Nishakawa, Statistical macrodynamics of large dyamical systems. Case of phase transition in oscillator communities-J. Stat. Phys., Vol. 9, No. 3/4, pp. 569–605, 1987.Google Scholar
  2. 2.
    H. Sakaguchi, S. Shinomoto, Y. Kuramoto, Phase transitions and their bifurcation analysis in a large population of active rotators with mean field coupling-Progr. Teor. Phys., Vol. 79, No. 3, pp. 600–607, 1988.Google Scholar
  3. 3.
    S. H. Strogatz, R. E. Mirollo, Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies-Phys. D, Vol.31, No.2., pp. 143–168, 1988.Google Scholar
  4. 4.
    L. L. Bonilla, C. J. Perez Vicente, J. M. Rubi, Glassy synchronization in a population of coupled oscillators-J.Stat. Phys., Vol.70, No. 3/4, pp. 921–937, 1993.Google Scholar
  5. 5.
    Sompolinsky H.,Tsodyks M., Processing of sensory information by a network of oscillators with memory-Intern. J. Neur.Syst. Vol. 3, Supp., p.51–56, 1992.Google Scholar
  6. 6.
    A. Y. Plakhov, O. I. Fisun, An oscillatory model of neuron networks-Matematicheskoe modelirovanie, Vol. 3, No.3, pp. 48–54, 1991 (in Russian).Google Scholar
  7. 7.
    Kuzmina M.G., Manykin E.A., Surina I.I., Problem of Associative Memory in Systems of Coupled Oscillators-Second European Congress on Intelligent Techniques and Soft Computing. Aachen, Germany, September 20–23, 1994, Proceedings, Vol. 3, EUFIT'94, pp. 1398–1402, 1994.Google Scholar
  8. 8.
    Kuzmina M.G., Surina I.I., Macrodynamical approach for oscillatory networks-Proceedings of the SPIE. Optical Neural Networks, Vol. 2430 pp. 227–233, 1994.Google Scholar
  9. 9.
    Belov M.N., Manykin E.A. Photon-echo effect in optical implementation of neural network models.-In: Neurocomputers and Attention, eds. Holden A.V. and Kryukov V.I., Manchester University Press, Vol. 2, p.459–466, 1991.Google Scholar
  10. 10.
    J. Cook, The mean-field theory of a Q-state neural network model-J. Phys. Math. Gen., Vol.22, pp. 2057–2067, 1989.Google Scholar
  11. 11.
    A. J. Noest, Associative memory in sparse phasor neural networks-Europhys. Lett., Vol.6, No. 6 pp. 469–474, 1988.Google Scholar
  12. 12.
    Kuzmina M.G., Surina I.I., Oscillatory Networks of Associative Memory, submitted to IWANN'95.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Kuzmina M. G. 
    • 1
  • Manykin E. A. 
    • 2
  • Surina I. I. 
    • 2
  1. 1.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia
  2. 2.Superconductivity and Solid State Physics Institute of Russian Research CenterKurchatov InstituteMoscowRussia

Personalised recommendations