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Bifurcation between Superstable Periodic Orbits and Chaos in a Simple Spiking Circuit

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Advances in Neuro-Information Processing (ICONIP 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5506))

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Abstract

This paper studies typical nonlinear dynamics of spiking circuit including two capacitors. Applying impulsive switching depending on both state and time, the circuit can exhibit rich chaotic/periodic phenomena. We pay special attention to superstable periodic orbits and related bifurcation phenomena. The circuit dynamics can be simplified into a piecewise linear one-dimensional return map that enables us to analyze basic bifurcation phenomena precisely.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Mirollo, R.E., Strogatz, S.H.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50, 1645–1662 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Izhikevich, E.M.: Resonate-and-fire neurons. Neural Networks 14, 883–894 (2001)

    Article  Google Scholar 

  4. Toyoizumi, T., Aihara, K., Amari, S.: Fisher information for spike-based population coding. Phys. Rev. Lett. 97, 098102 (2006)

    Article  Google Scholar 

  5. Lindner, B., Chacron, M.J., Longtin, A.: Integrate-and-fire neurons with threshold noise: A tractable model of how interspike interval correlations affect neuronal signal transmission. Physical Review E 72, 021911 (2005)

    Article  MathSciNet  Google Scholar 

  6. Nakano, H., Saito, T.: Grouping synchronization in a pulse-coupled network of chaotic spiking oscillators. IEEE Trans. Neural Networks 15(5), 1018–1026 (2004)

    Article  Google Scholar 

  7. Izhikevich, E.M.: Weakly pulse-coupled oscillators, FM interactions, synchronization, and oscillatory associative memory. IEEE Trans. Neural Networks 10(3), 508–526 (1999)

    Article  Google Scholar 

  8. Campbell, S.R., Wang, D., Jayaprakash, C.: Synchrony and desynchrony in integrate-and-fire oscillators. Neural Comput 11, 1595–1619 (1999)

    Article  Google Scholar 

  9. Sushchik, M., Rulkov, N., Larson, L., Tsimring, L., Abarbanel, H., Yao, K., Volkovskii, A.: Chaotic pulse position modulation: a robust method of communicating with chaos. IEEE Comm. Lett. 4, 128–130 (2000)

    Article  Google Scholar 

  10. Miyachi, K., Nakano, H., Saito, T.: Response of a simple dependent switched capacitor circuit to a pulse-train input. IEEE Trans. Circuits Systs. I 50(9), 1180–1187 (2003)

    Article  MathSciNet  Google Scholar 

  11. Kobayashi, Y., Nakano, H., Saito, T.: A Simple Chaotic Circuit with Impulsive Switch Depending on Time and State. Nonlinear Dynamics 44, 73–79 (2006)

    Article  MATH  Google Scholar 

  12. Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise, 2nd edn. Springer, Heidelberg (1994)

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. EE Dept., Hosei University, Koganei, Tokyo, 184-0002, Japan

    Yuji Kawai & Toshimichi Saito

Authors
  1. Yuji Kawai
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  2. Toshimichi Saito
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Editor information

Editors and Affiliations

  1. Kyushu Institute of Technology, Network Design and Research Center, 680-4 Fukuoka, 820-8502, Kawazu, Iizuka, Japan

    Mario Köppen

  2. Knowledge Engineering and Discovery Research Institute (KEDRI), School of Computing and Mathematical Sciences, Auckland University of Technology, 350 Queen Street, 10110, Auckland, New Zealand

    Nikola Kasabov

  3. Department of Electrical and Computer Engineering, Robotics Laboratory, Auckland University of Technology, 38 Princes Street, 1142, Auckland, New Zealand

    George Coghill

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© 2009 Springer-Verlag Berlin Heidelberg

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Kawai, Y., Saito, T. (2009). Bifurcation between Superstable Periodic Orbits and Chaos in a Simple Spiking Circuit. In: Köppen, M., Kasabov, N., Coghill, G. (eds) Advances in Neuro-Information Processing. ICONIP 2008. Lecture Notes in Computer Science, vol 5506. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02490-0_103

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  • DOI: https://doi.org/10.1007/978-3-642-02490-0_103

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02489-4

  • Online ISBN: 978-3-642-02490-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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