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

Ikuhiro Yamaguchi, Yutaro Ogawa, Yasuhiko Jimbo, Hiroya Nakao and Kiyoshi Kotani

PLOS ONE, 2011, vol. 6, issue 11, 1-10

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.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:plo:pone00:0026497

DOI: 10.1371/journal.pone.0026497

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