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

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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.

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References

  1. Takens F (1981) Detecting strange attractors in turbulence. In: Dynamical systems and turbulence. Springer, Berlin, pp 366–381

    Google Scholar 

  2. Sauer T (1994) Reconstruction of dynamical systems from interspike intervals. Phys Rev Lett 72:3811–3814

    Article  Google Scholar 

  3. Saner T (1997) Reconstruction of integrate-and-fire dynamics. In: Nonlinear dynamics and time series. American Mathematical Society, Providence, p 63–75

    Google Scholar 

  4. Richardson KA, Imhoff TT, Grigg P, et al. (1988) Encoding chaos in neural spike trains. Phys Rev Lett 80:2485–2488

    Article  Google Scholar 

  5. Racicot DM, Longtin A (1977) Interspike interval attractors from chaotically driven neuron models. Physica D 104:184–204

    Article  Google Scholar 

  6. Suzuki H, Aihara K, Murakami J, et al. (2000) Analysis of neural spike trains with interspike interval reconstruction. Biol Cybern 82:305–311

    Article  MATH  Google Scholar 

  7. Keener JP, Hoppensteadt FC, Rinzel J (1981) Integrate-and-fire models of nerve membrane response to oscillatory input. SIAM J Appl Math 41:503–517

    Article  MATH  MathSciNet  Google Scholar 

  8. Coombes S (1999) Liapunov exponents and mode-locked solutions for integrate-and-fire dynamical systems. Phys Lett A 255:49–57

    Article  Google Scholar 

  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 63:059901

    Article  MathSciNet  Google Scholar 

  10. Coombes S, Owen MR, Smith GD (2001) Mode locking in a periodically forced integrate-and-fire-or-burst neuron model. Phys Rev E 64:041914

    Article  Google Scholar 

  11. Pakdaman K (2001) Periodically forced leaky integrate-and-fire model. Phys Rev E 63:041907.

    Article  Google Scholar 

  12. Rhodes F, Thompson CL (1986) Rotation numbers for monotone functions on the circle. J London Math Soc 34:360–368

    MATH  MathSciNet  Google Scholar 

  13. Rhodes F, Thompson CL (1991) Topologies and rotation numbers for families of monotone functions on the circle. J London Math Soc 43:156–170

    MATH  MathSciNet  Google Scholar 

  14. Budd CJ (1996) Non-smooth dynamical systems and the grazing bifurcation. In: Nonlinear mathematics and its applications. Cambridge University Press, Cambridge, pp 219–235

    Google Scholar 

  15. Hodgkin AL (1948) The local electric changes associated with repetitive action in a non-modulated axon. J Physiol 107:165–181

    Google Scholar 

  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. Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213

    Article  Google Scholar 

  18. Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase synchronization of chaotic oscillators. Phys Rev Lett 76:1804–1807

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, University of Tokyo, Tokyo, Japan

    Y. Ono & K. Aihara

  2. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Bunkyoku, 113-8656, Tokyo, Japan

    H. Suzuki

Authors
  1. Y. Ono
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  2. K. Aihara
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  3. H. Suzuki
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Corresponding author

Correspondence to H. Suzuki.

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Ono, Y., Aihara, K. & Suzuki, H. Grazing bifurcation and mode-locking in reconstructing chaotic dynamics with a leaky integrate-and-fire model. Artif Life Robotics 7, 55–62 (2003). https://doi.org/10.1007/BF02480886

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  • DOI: https://doi.org/10.1007/BF02480886

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