Synchronization of Hopfield Like Chaotic Neural Networks with Structure Based Learning

  • Nariman Mahdavi
  • Jürgen Kurths
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7664)


In this paper the issue of structure based learning of Hopfield like chaotic neural networks is investigated in such a way that all neurons behave in a synchronous manner. By utilizing the idea of structured inverse eigenvalue problem and the sufficient conditions on the coupling weights of a network which guarantee the synchronization of all neuron’s outputs, we propose a learning method for tuning the coupling weights of a network where not only synchronize all neuron’s outputs with each other but also brings about any desirable topology for the structure of the network. Specifically, this method is evaluated by performing simulations on the scale-free topology.


Synchronization Chaotic Neural Networks Structure Learning Scale-Free Networks 


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  1. 1.
    Hopfield, J.J.: Neural Networks and Physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. U.S.A. 81, 3088–3092 (1984)CrossRefGoogle Scholar
  2. 2.
    Roska, T., Chua, L.O.: Some novel capabilities of CNN: Game of life and examples of multipath algorithms. Int. J. Circuit Theory Appl. 20, 469–481 (1992)zbMATHCrossRefGoogle Scholar
  3. 3.
    Baldi, P., Atiya, A.F.: How delays affect neural deynamics and learning. IEEE Trans. Neural Networks. 5, 612–621 (1994)CrossRefGoogle Scholar
  4. 4.
    Cao, Y.J., Wu, Q.H.: A note on stability of analog neural networks with time delays. IEEE Trans. Neural Networks 7, 1533–1535 (1996)CrossRefGoogle Scholar
  5. 5.
    Wang, L., Pichler, E.E., Ross, J.: Oscillations and chaos in neural networks: an exactly solvable model. Proc. Natl. Acad. Sci. USA 87, 9467–9471 (1990)zbMATHCrossRefGoogle Scholar
  6. 6.
    Wang, L., Ross, J.: Interactions of neural networks: model for distraction and concentration. Proc. Natl. Acad. Sci. USA 87, 7110–7114 (1990)CrossRefGoogle Scholar
  7. 7.
    Churchland, P.S., Sejnowski, T.J.: The computational Brain. MIT Press, Cambridge (1989)Google Scholar
  8. 8.
    Babloyantz, A., Lourenco, C.: Computation with chaos: a paradigm for cortical activity. Proc. Natl. Acad. Sci. USA 91, 9027–9031 (1994)CrossRefGoogle Scholar
  9. 9.
    Wang, L.: Interactions between neural networks: a mechanism for tuning chaos and oscillations. Cognitive Neurodynamics 1, 185–188 (2007)zbMATHCrossRefGoogle Scholar
  10. 10.
    Aihara, K., Takada, T., Toyoda, M.: Chaotic Neural Networks. Physics Letters A 144, 333–340 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wang, L., Yu, W., Shi, H., Zurada, J.M.: Cellular Neural Networks with transient chaos. IEEE Trans. on Circuits and Systems-II: Express Briefs 54, 440–444 (2007)CrossRefGoogle Scholar
  12. 12.
    Ma, J., Wu, J.: Bifurcation and multistability in coupled neural networks with non-monotonic communication. Technical report, Math dept., York University (2006)Google Scholar
  13. 13.
    Nakano, H., Saito, T.: Grouping Synchronization in a Pulse-Coupled Network of Chaotic Spiking Oscillators. IEEE Trans. Neural Networks 15, 1018–1026 (2004)CrossRefGoogle Scholar
  14. 14.
    Lee, R.S.T.: A transient-chaotic auto associative Network based on Lee Oscilator. IEEE Trans. on Neural Networks 15, 1228–1243 (2004)CrossRefGoogle Scholar
  15. 15.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 440–442 (1998)CrossRefGoogle Scholar
  16. 16.
    Izhikevich, E.M.: Dynamical Systems in Neuroscience, ch. 10. MIT Press, Cambridge (2007)Google Scholar
  17. 17.
    Lü, J., Yu, X., Chen, G.: Chaos synchronization of general complex dynamical networks. Physica A 334, 281–302 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, Z., Chen, G.: Global synchronization and asymptotic stability of complex dynamical networks. IEEE Trans. on Circuits and Systems-II 53, 28–33 (2006)CrossRefGoogle Scholar
  19. 19.
    Nakano, H., Utani, A., Miyauchi, A., Yamamoto, H.: Synchronization-based data gathering scheme using chaotic pulse-coded neural networks in wireless sensor networks. In: Proceedings of IEEE Int. Joint Conf. Neural Network, Hong Kong, pp. 1116–1122 (2008)Google Scholar
  20. 20.
    Moody, T.C., Golub, G.H.: Inverse Eigenvalue Problems: Theory, Algorithms and Applications. Numerical Mathematics and Scientific Computation Series. Oxford Univ. Press (2005)Google Scholar
  21. 21.
    Mahdavi, N., Menhaj, M.B.: A New Set of Sufficient Conditions Based on Coupling Parameters for Synchronization of Hopfield like Chaotic Neural Networks. Int. J. Control Auto. and Syst. 9, 104–111 (2011)CrossRefGoogle Scholar
  22. 22.
    Barabasi, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Nariman Mahdavi
    • 1
  • Jürgen Kurths
    • 1
  1. 1.Potsdam Institute for Climate Impact ResearchPotsdamGermany

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