title: Reduction theories elucidate the origins of complex biological rhythms generated by interacting delay-induced oscillations.
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.
authors.affiliation: Graduate School of Frontier Science, The University of Tokyo, Chiba, Japan.
authors.forename: Ikuhiro
authors.initials: I
authors.lastname: Yamaguchi
authors.forename: Yutaro
authors.initials: Y
authors.lastname: Ogawa
authors.forename: Yasuhiko
authors.initials: Y
authors.lastname: Jimbo
authors.forename: Hiroya
authors.initials: H
authors.lastname: Nakao
authors.forename: Kiyoshi
authors.initials: K
authors.lastname: Kotani
mesh_terms: D000818:Animals; D001683:Biological Clocks; D004569:Electroencephalography; D004827:Epilepsy / Q000209:etiology; D006801:Humans; D008954:Models, Biological; D008962:Models, Theoretical; D017711:Nonlinear Dynamics
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