Dynamics of a Three Neurons Network with Excitatory-Inhibitory Interactions

  • Carlos Aguirre
  • Juan I. Cano
  • Eloy Anguiano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6686)


Sets of coupled neurons can generate many different patterns in response to modulatory or sensory inputs. The study of how these patterns have been generated from the inputs has been object of great interest in the literature. These studies have been mainly performed by means of computer simulations, based on differential models or phenomenological models. However complete descriptions of the behaviour of sets of coupled neurons are hard to obtain due to the complex behaviour of the dynamics generated even by the simplest neuron models and for the high number of parameters involved. Here we present a study of a three neuron network that appears in models of Central Pattern Generators. The use of a lineal model allows a complete dynamical description of the system, identifying the relevant situations and drawing some conclusions concerning the dynamics of the network.


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  1. 1.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and application to conduction and excitation in nerve. Journal of Physiology 117, 165–181 (1954)Google Scholar
  2. 2.
    Rose, R.M., Hindmarsh, J.L.: The assembly of ionic currents in a thalamic neuron, I The three dimensional model. Proceedings of The Royal Society of London B 237, 267–288 (1989)CrossRefGoogle Scholar
  3. 3.
    Rulkov, N.F.: Modeling of spiking-bursting neural behavior using two–dimensional map. Physical Review E 65, 041922 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Izhikevich, E.M.: Simple Model of Spiking Neurons. IEEE Trans. on Neural Networks 68, 052901 (2003)Google Scholar
  5. 5.
    Shilnikov, A.L., Rulkov, N.F.: Subthreshold oscillations in a map–based neuron model. Physics Letters A 327, 177–184 (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Aguirre, C., Campos, D., Pascual, P., Serrano, E.: Neuronal Behavior with Sub-threshold Oscillations and Spiking/Bursting Activity Using a Piecewise Linear Two-Dimensional Map. In: Duch, W., Kacprzyk, J., Oja, E., Zadrożny, S. (eds.) ICANN 2005. LNCS, vol. 3696, pp. 103–108. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Kuramoto, Y.: International Symposium on Mathematical Problems in Theoretical Physics. In: Araki, H. (ed.) Lecture Notes in Physics, vol. 39 (1975)Google Scholar
  8. 8.
    Latorre, R., Rodríguez, F.B., Varona, P.: Characterization of Triphasic Rhythms in Central Pattern Generators (I): Interspike Interval Analysis. In: Dorronsoro, J.R. (ed.) ICANN 2002. LNCS, vol. 2415, pp. 160–166. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Rodríguez, F.B., Latorre, R., Varona, P.: Characterization of Triphasic Rhythms in Central Pattern Generators (II): Burst Information Analysis. In: Dorronsoro, J.R. (ed.) ICANN 2002. LNCS, vol. 2415, pp. 167–173. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Gerstner, W., Kistler, W.M.: Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge (August 2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    Izhikevich, E.M.: Neural Excitability, Spiking and Bursting. International Journal of Bifurcation and Chaos 10, 1171–1266 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Aguirre, C., Campos, D., Pascual, P., Serrano, E.: Effects of different connectivity patterns in a model of cortical circuits. In: Mira, J., Álvarez, J.R. (eds.) IWANN 2003. LNCS, vol. 2686, pp. 78–85. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Bernarder, O., Koch, C., Usher, M.: Synaptic background activity determines spatio–temporal integration in single pyramidal cells. Proceedings of the National Academy of Sciences USA 88, 11569–11573 (1991)CrossRefGoogle Scholar
  14. 14.
    van Vreeswijk, C., Sompolinsky, H.: Chaotic Balanced State in a Model of Cortical Circuits. Neural Computation 10, 1321–1372 (1998)CrossRefGoogle Scholar
  15. 15.
    Elson, R.C., et al.: Synchronous Behavior of Two Coupled biological Neurons. Physical Review Letters 81(25), 5692 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Carlos Aguirre
    • 1
  • Juan I. Cano
    • 2
  • Eloy Anguiano
    • 2
  1. 1.GNB, Escuela Politécnica SuperiorUniversidad Autónoma de MadridMadridSpain
  2. 2.CRL, Escuela Politécnica SuperiorUniversidad Autonoma de MadridMadridSpain

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