Artificial Life and Robotics

, Volume 7, Issue 1–2, pp 55–62 | Cite as

Grazing bifurcation and mode-locking in reconstructing chaotic dynamics with a leaky integrate-and-fire model

Original Article

Abstract

We examined the firing patterns of a chaotically forced leaky integrate-and-fire (LIF) model, and the validity of reconstructing input chaotic dynamics from an observed spike sequence. We generated inputs to the model from the Rössler system at various values of the bifurcation parameter, and carried out numerical simulations of the LIF model forced by each input. For both chaotic and periodic inputs, therotation numbers and the Lyapunov exponents were calculated to investigate the mode-locked behavior of the system. Similar behaviors as in the periodically forced LIF model were also observed in the chaotically forced LIF model. We observed (i)grazing bifurcation with the emergence of qualitatively distinct behaviors separated by a certain border in the parameter space, and (ii) modelocked regions where the output spike sequences are modelocked to the chaotic inputs. We found that thegrazing bifurcation is related to the reconstruction of chaotic dynamics with the LIF. Our results can explain why the shape of the partially reconstructed ISI attractor, which was observed in previous studies.

Key words

Leaky integrate-and-fire model Chaos Interspike interval Attractor reconstruction Grazing bifurcation Mode-locking 

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References

  1. 1.
    Takens F (1981) Detecting strange attractors in turbulence. In: Dynamical systems and turbulence. Springer, Berlin, pp 366–381Google Scholar
  2. 2.
    Sauer T (1994) Reconstruction of dynamical systems from interspike intervals. Phys Rev Lett 72:3811–3814CrossRefGoogle Scholar
  3. 3.
    Saner T (1997) Reconstruction of integrate-and-fire dynamics. In: Nonlinear dynamics and time series. American Mathematical Society, Providence, p 63–75Google Scholar
  4. 4.
    Richardson KA, Imhoff TT, Grigg P, et al. (1988) Encoding chaos in neural spike trains. Phys Rev Lett 80:2485–2488CrossRefGoogle Scholar
  5. 5.
    Racicot DM, Longtin A (1977) Interspike interval attractors from chaotically driven neuron models. Physica D 104:184–204CrossRefGoogle Scholar
  6. 6.
    Suzuki H, Aihara K, Murakami J, et al. (2000) Analysis of neural spike trains with interspike interval reconstruction. Biol Cybern 82:305–311MATHCrossRefGoogle Scholar
  7. 7.
    Keener JP, Hoppensteadt FC, Rinzel J (1981) Integrate-and-fire models of nerve membrane response to oscillatory input. SIAM J Appl Math 41:503–517MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Coombes S (1999) Liapunov exponents and mode-locked solutions for integrate-and-fire dynamical systems. Phys Lett A 255:49–57CrossRefGoogle Scholar
  9. 9.
    Coombes S, Bressloff PC (2001) Mode locking and Arnold tongues in integrate-and fire neural oscillators. Phys Rev E 60:2086–2096; erratum (2001) Phys Rev E 63:059901MathSciNetCrossRefGoogle Scholar
  10. 10.
    Coombes S, Owen MR, Smith GD (2001) Mode locking in a periodically forced integrate-and-fire-or-burst neuron model. Phys Rev E 64:041914CrossRefGoogle Scholar
  11. 11.
    Pakdaman K (2001) Periodically forced leaky integrate-and-fire model. Phys Rev E 63:041907.CrossRefGoogle Scholar
  12. 12.
    Rhodes F, Thompson CL (1986) Rotation numbers for monotone functions on the circle. J London Math Soc 34:360–368MATHMathSciNetGoogle Scholar
  13. 13.
    Rhodes F, Thompson CL (1991) Topologies and rotation numbers for families of monotone functions on the circle. J London Math Soc 43:156–170MATHMathSciNetGoogle Scholar
  14. 14.
    Budd CJ (1996) Non-smooth dynamical systems and the grazing bifurcation. In: Nonlinear mathematics and its applications. Cambridge University Press, Cambridge, pp 219–235Google Scholar
  15. 15.
    Hodgkin AL (1948) The local electric changes associated with repetitive action in a non-modulated axon. J Physiol 107:165–181Google Scholar
  16. 16.
    Rinzel J, Ermentrout GB (1989) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in neuronal modeling. MIT Press, Cambridge.Google Scholar
  17. 17.
    Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213CrossRefGoogle Scholar
  18. 18.
    Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase synchronization of chaotic oscillators. Phys Rev Lett 76:1804–1807CrossRefGoogle Scholar

Copyright information

© ISAROB 2003

Authors and Affiliations

  1. 1.Department of Complexity Science and Engineering, Graduate School of Frontier SciencesUniversity of TokyoTokyoJapan
  2. 2.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan

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