title: Bifurcations of synchronized responses in synaptically coupled Bonhöffer-van der Pol neurons.
abstract: The Bonhöffer-van der Pol (BvdP) equation is considered as an important model for studying dynamics in a single neuron. In this paper, we investigate bifurcations of periodic solutions in model equations of four and five BvdP neurons coupled through the characteristics of synaptic transmissions with a time delay. The model can be considered as a dynamical system whose solution includes jumps depending on a condition related to the behavior of the trajectory. Although the solution is discontinuous, we can define the Poincaré map as a synthesis of successive submaps, and give its derivatives for obtaining periodic points and their bifurcations. Using our proposed numerical method, we clarify mechanisms of bifurcations among synchronized oscillations with phase-locking patterns by analyzing periodic solutions observed in the coupling system and its subsystems. Moreover, we show that a global behavior of chaotic itinerancy or a phenomenon of chaotic transitions among several quasiattracting states can be observed in higher-dimensional systems of the synaptically four and five coupled neurons.
authors.affiliation: Department of Electrical and Electronic Engineering, Faculty of Engineering, The University of Tokushima, Tokushima, 770-8506 Japan. tsu@ee.tokushima-u.ac.jp
authors.forename: Kunichika
authors.initials: K
authors.lastname: Tsumoto
authors.forename: Tetsuya
authors.initials: T
authors.lastname: Yoshinaga
authors.forename: Hiroshi
authors.initials: H
authors.lastname: Kawakami