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Biological Cybernetics

, Volume 71, Issue 2, pp 177–185 | Cite as

Synchronization in a neural network of phase oscillators with the central element

  • Yakov B. Kazanovich
  • Roman M. Borisyuk
Article

Abstract

A neural network model is considered which is designed as a system of phase oscillators and contains the central oscillator and peripheral oscillators which interact via the central oscillator. The regime of partial synchronization was studied when current frequencies of the central oscillator and one group of peripheral oscillators are near to each other while current frequencies of other peripheral oscillators are far from being synchronized with the central oscillator. Approximation formulas for the average frequency of the central oscillator in the regime of partial synchronization are derived, and results of computation experiments are presented which characterize the accuracy of the approximation.

Keywords

Neural Network Network Model Neural Network Model Average Frequency Central Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Borisyuk GN, Borisyuk RM, Kazanovich YB, Luzyanina TB, Turova TS, Cymbalyuk GS (1992) Oscillatory neural networks. Mathematics and applications (in Russian). Math Modeling 4:3–43Google Scholar
  2. Cowan N (1988) Evolving conceptions of memory storage, selective attention, and their mutual constraints within the human information-processing system. Psychol Bull 104:163–191CrossRefPubMedGoogle Scholar
  3. Daido H (1988) Lower critical dimension for populations of oscillators with randomly distributed frequencies: a renormalization-group analysis. Phys Rev Lett 61:231–234CrossRefPubMedGoogle Scholar
  4. Eckhorn R, Bauer R, Jordan W, Brosch M, Kruse W, Munk M, Reitboeck HJ (1988) Coherent oscillations: a mechanism of feature linking in the visual cortex? Biol Cybern 60:121–130CrossRefPubMedGoogle Scholar
  5. Gray CM, König P, Engel AK, Singer W (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338:334–337CrossRefPubMedGoogle Scholar
  6. Kammen DM, Holmes PJ, Koch C (1990) Origin of oscillations in visual cortex: feedback versus local coupling. In: Cotterill RMJ (eds) Models of brain function. Cambridge University Press, Cambridge, UK, pp 273–284Google Scholar
  7. Kazanovich YaB, Kryukov VI, Luzyanina TB (1991) Synchronization via phase-locking in oscillatory models of neural networks. In: Holden AV, Kryukov VI (eds) Neurocomputers and attention. I. Neurobiology, synchronization and chaos. Manchester University Press, Manchester, pp 269–284Google Scholar
  8. Kopell N, Ermentrout GB (1988) Coupled oscillators and the design of central pattern generators. Math. Biosci. 90:87–109CrossRefGoogle Scholar
  9. Kryukov VI (1991) An attention model based on the principle of dominanta. In: Holden AV, Kryukov VI (eds) Neurocomputers and attention. I. Neurobiology, synchronization and chaos. Manchester University Press, Manchester, pp 319–352Google Scholar
  10. Kuramoto Y, Nishikawa I (1987) Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities. J Stat Phys 49:569–605CrossRefGoogle Scholar
  11. Lumer ED, Huberman BA (1991) Hierarchical dynamics in large assemblies of interacting oscillators. Phys Lett A 160:227–232CrossRefGoogle Scholar
  12. Luzyanina TB (1992) Synchronization in an oscillator neural network model with time delayed coupling. Preprint, PushchinoGoogle Scholar
  13. Malsburg C von der (1981) The correlation theory of brain function. (Internal Report 81–2) Department of Neurobiology, Max-Planck-Institute for Biophysical Chemistry GöttingenGoogle Scholar
  14. Miller R (1991) Cortico-hippocampal interplay and the representation of contexts in the brain. Springer, Berlin Heidelberg New YorkGoogle Scholar
  15. Niebur E, Kammen DM, Koch C (1991) Phase-locking in 1-D and 2-D networks of oscillating neurons. In: Singer W, Schuster HG (eds) Non-linear dynamics and neuronal networks. VCH WeinheimGoogle Scholar
  16. Schmajuk N, DiCarlo J (1992) Stimulus configuration, classical conditioning, and hippocampal function. Psychol Rev 99:268–305CrossRefPubMedGoogle Scholar
  17. Sompolinsky H, Golomb D, Kleinfeld D (1990) Global processing of visual stimuli in a neural network of coupled oscillators. Proc Natl Acad Sci USA 87:7200–7204PubMedGoogle Scholar
  18. Strogatz SH, Mirollo RE (1988) Collective synchronization in lattices of non-linear oscillators with randomness, J Phys A Math Gen 21:L699-L705CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Yakov B. Kazanovich
    • 1
  • Roman M. Borisyuk
    • 1
  1. 1.Institute of Mathematical Problems of Biology, Russian Academy of SciencesPushchinoRussia

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