Regular and Chaotic Dynamics

, Volume 20, Issue 6, pp 701–715 | Cite as

Sequential dynamics in the motif of excitatory coupled elements

  • Alexander G. Korotkov
  • Alexey O. Kazakov
  • Grigory V. Osipov


In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of the generalized Lotka–Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of the generalized three-dimensional Lotka–Volterra model.


Neuronal motifs Lotka–Volterra model heteroclinic cycle period-doubling bifurcation Feigenbaum scenario strange attractor Lyapunov exponents 

MSC2010 numbers

37G35 70K05 70K50 70K55 


  1. 1.
    Rabinovich, M. I., Varona, P., Selverston, A. I., and Abarbanel, H.D. I., Dynamical Principles in Neuroscience, Rev. Mod. Phys., 2006, vol. 78, no. 4, pp. 1213–1266.CrossRefGoogle Scholar
  2. 2.
    Hahnloser, R.H.R., Kozhevnikov, A. A., and Fee, M. S., An Ultra-Sparse Code Underlies the Generation of Neural Sequences in a Songbird, Nature, 2002, vol. 419, pp. 65–70.CrossRefGoogle Scholar
  3. 3.
    Galan, R., Sasche, S., Galicia, C.G., and Herz, A. V., Odor-Driven Attractor Dynamics in the Antennal Lobe Allow for Simple and Rapid Olfactory Pattern Classification, Neural Comput., 2004, vol. 16, no. 5, pp. 999–1012.CrossRefMATHGoogle Scholar
  4. 4.
    Levi, R., Varona, P., Arshavsky, Y. I., Rabinovich, M. I., and Selverston, A. I., Dual Sensory-Motor Function for a Molluskan Statocyst Network, J. Neurophysiol., 2004, vol. 91, no. 1, pp. 336–345.CrossRefGoogle Scholar
  5. 5.
    Guckenheimer, J. and Holmes, P., Structurally Stable Heteroclinic Cycles, Math. Proc. Cambridge Philos. Soc., 1988, vol. 103, no. 1, pp. 189–192.CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Stone, E. and Holmes, P., Random Perturbations of Heteroclinic Attractors, SIAM J. Appl. Math., 1990, vol. 50, no. 3, pp. 726–743.CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Postlethwaite, C. M. and Dawes, J.H.P., Resonance Bifurcations from Robust Homoclinic Cycles, Nonlinearity, 2010, vol. 23, no. 3, pp. 621–642.CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Driesse, R. and Homburg, A. J., Resonance Bifurcation from Homoclinic Cycles, J. Differential Equations, 2009, vol. 246, no. 7, pp. 2681–2705.CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Seliger, P., Tsimring, L. S., and Rabinovich, M. I., Dynamics-Based Sequential Memory: Winnerless Competition of Patterns, Phys. Rev. E(3), 2003, vol. 67, no. 1, 011905, 4 pp.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Afraimovich, V. S., Rabinovich, M. I., and Varona, P., Heteroclinic Contours in Neural Ensembles and the Winnerless Competition Principle, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2004, vol. 14, no. 4, pp. 1195–1208.CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Afraimovich, V. S., Zhigulin, V.P., and Rabinovich, M. I., On the Origin of Reproducible Sequential Activity in Neural Circuits, Chaos, 2004, vol. 14, no. 4, pp. 1123–1129.CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Arneodo, A., Coullet, P., and Tresser, C., Occurence of Strange Attractors in Three-Dimensional Volterra Equations, Phys. Lett. A, 1980, vol. 79, no. 4, pp. 259–263.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Arneodo, A., Coullet, P., Peyraud, J., and Tresser, C., Strange Attractors in Volterra Equations for Species in Competition, J. Math. Biol., 1982, vol. 14, no. 2, pp. 153–157.CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Vano, J.A., Wildenberg, J.C., Anderson, M.B., Noel, J.K., and Sprott, J. C., Chaos in Low-Dimensional Lotka–Volterra Models of Competition, Nonlinearity, 2006, vol. 19, no. 10, pp. 2391–2404.CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Wang, R. and Xiao, D., Bifurcations and Chaotic Dynamics in a 4-Dimensional Competitive Lotka — Volterra System, Nonlinear Dynam., 2010, vol. 59, no. 3, pp. 411–422.CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Feigenbaum, M. J., Universal Behavior in Nonlinear Systems, Phys. D, 1983, vol. 7, nos. 1–3, pp. 16–39.CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • Alexander G. Korotkov
    • 1
  • Alexey O. Kazakov
    • 2
  • Grigory V. Osipov
    • 1
  1. 1.Lobachevsky State University of Nizhny NovgorodNizhny NovgorodRussia
  2. 2.National Research University Higher School of EconomicsNizhny NovgorodRussia

Personalised recommendations