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Complex emergent properties in synchronized neuronal oscillations

  • Nathalie Corson
  • M. A. Aziz-Alaoui
Part of the Understanding Complex Systems book series (UCS)

Summary

This work adresses the dynamics and complexity of the Hindmarsh-Rose neuronal mathematical model. The general aim is the study of the asymptotic behaviour of neuron networks. In this paper, the analysis of these networks uses the synchronization theory via connections between neurons which give rise to emergent properties and self-organization. Our results lead to a classical law which describes many natural or artificial self-organized complex systems. This has been performed using numerical tools.

Keywords

Hindmarsh-Rose model synchronization complexity emergent properties 

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References

  1. 1.
    Aziz-Alaoui, M.A.: Synchronization of chaos. Encyclopedia of mathematical physics 5, 213–226 (2006)CrossRefGoogle Scholar
  2. 2.
    Aziz-Alaoui, M.A.: Complex emergent properties and chaos (De) synchronization. In: Emergent Properties in Natural and Artificial Dynamical Systems, Understanding complex systems, pp. 129–147. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Belykh, I., Lange, E., Hasler, M.: Synchronization of Bursting Neurons: What matters in the Network Topology. Phy. Rev. Lett. 94(18), 188101.1–188101.4 (2005)CrossRefGoogle Scholar
  4. 4.
    Boccaletti, S., Kurths, J., Osipov, G., Valladares, D., Zhou, C.: The synchronization of chaotic systems. Physics Reports 366, 1–101 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Corson, N., Aziz Alaoui, M.A.: Asymptotic dynamics of the slow-fast Hindmarsh-Rose neuronal model (to be submitted) (2008)Google Scholar
  6. 6.
    Gonzalez-Miranda, J.M.: Complex bifurcation structures in the Hindmarsh-Rose neuron model. International Journal of Bifurcation and Chaos 17(9), 3071–3083 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hindmarsh, J.L., Rose, R.M.: A model of the nerve impulse using two first-order differential equations. Nature 296, 162–164 (1982)CrossRefGoogle Scholar
  8. 8.
    Hindmarsh, J.L., Rose, R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Sc. Lond. B221, 87–102 (1984)CrossRefGoogle Scholar
  9. 9.
    Mosekilde, E., Lading, B., Yanchuk, S., Maistrenko, Y.: Bifurcation structure of a model of bursting pancreatic cells. BioSystems 63, 3–13 (2001)CrossRefGoogle Scholar
  10. 10.
    Innocenti, G., Morelli, A., Genesio, R., Torcini, A.: Dynamical phases of the Hindmarsh-Rose neuronal model: Studies of the transition from bursting to spiking chaos. Chaos 17, 043128, 1–11 (2007)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Izhikevich, E.M.: Dynamical systems in neuroscience - The geometry of excitability and bursting. The MIT Press, Cambridge (2007)Google Scholar
  12. 12.
    Pecora, L.M., Caroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109–2112 (1998)CrossRefGoogle Scholar
  13. 13.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization, A universal concept in nonlinear sciences. Cambridge Nonlinear Science Series 12 (2001)Google Scholar
  14. 14.
    Terman, D.: Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math. 51(5), 1418–1450 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Terman, D.: The transition from bursting to continuous spiking in excitable membrane models. J. Nonlinear Sc. 2, 135–182 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Wang, X.J.: Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle. Physica D 62(1-4), 263–274 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Yang, Z.Q., Lu, Q.S.: Characteristics of Period-Adding Bursting Bifurcation Without Chaos in the Chay Neuron Model Chin. Phy. Let. 21(11), 2124–2128 (2004)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Nathalie Corson
    • 1
  • M. A. Aziz-Alaoui
    • 1
  1. 1.Laboratoire de Mathématiques Appliquées du HavreLe HavreFrance

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