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Reduction Theories Elucidate the Origins of Complex Biological Rhythms Generated by Interacting Delay-Induced Oscillations

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  • Ikuhiro Yamaguchi
  • Yutaro Ogawa
  • Yasuhiko Jimbo
  • Hiroya Nakao
  • Kiyoshi Kotani

Abstract

Time delay is known to induce sustained oscillations in many biological systems such as electroencephalogram (EEG) activities and gene regulations. Furthermore, interactions among delay-induced oscillations can generate complex collective rhythms, which play important functional roles. However, due to their intrinsic infinite dimensionality, theoretical analysis of interacting delay-induced oscillations has been limited. Here, we show that the two primary methods for finite-dimensional limit cycles, namely, the center manifold reduction in the vicinity of the Hopf bifurcation and the phase reduction for weak interactions, can successfully be applied to interacting infinite-dimensional delay-induced oscillations. We systematically derive the complex Ginzburg-Landau equation and the phase equation without delay for general interaction networks. Based on the reduced low-dimensional equations, we demonstrate that diffusive (linearly attractive) coupling between a pair of delay-induced oscillations can exhibit nontrivial amplitude death and multimodal phase locking. Our analysis provides unique insights into experimentally observed EEG activities such as sudden transitions among different phase-locked states and occurrence of epileptic seizures.

Suggested Citation

  • Ikuhiro Yamaguchi & Yutaro Ogawa & Yasuhiko Jimbo & Hiroya Nakao & Kiyoshi Kotani, 2011. "Reduction Theories Elucidate the Origins of Complex Biological Rhythms Generated by Interacting Delay-Induced Oscillations," PLOS ONE, Public Library of Science, vol. 6(11), pages 1-10, November.
  • Handle: RePEc:plo:pone00:0026497
    DOI: 10.1371/journal.pone.0026497
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