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Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

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  • Anderson Hoff
  • Juliana Santos
  • Cesar Manchein
  • Holokx Albuquerque

Abstract

The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also presents multistability shown in the planes of the basin of attractions. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Anderson Hoff & Juliana Santos & Cesar Manchein & Holokx Albuquerque, 2014. "Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(7), pages 1-9, July.
    • Handle: RePEc:spr:eurphb:v:87:y:2014:i:7:p:1-9:10.1140/epjb/e2014-50170-9
    DOI: 10.1140/epjb/e2014-50170-9
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    Cited by:

    1. N. C. Pati, 2023. "Bifurcations and multistability in a physically extended Lorenz system for rotating convection," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(8), pages 1-15, August.
    2. Egorov, Nikita M. & Sysoev, Ilya V. & Ponomarenko, Vladimir I. & Sysoeva, Marina V., 2022. "Complex regimes in electronic neuron-like oscillators with sigmoid coupling," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    3. Achouri, Houssem & Aouiti, Chaouki & Hamed, Bassem Ben, 2022. "Codimension two bifurcation in a coupled FitzHugh–Nagumo system with multiple delays," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    4. Das, Saureesh, 2022. "Recurrence quantification and bifurcation analysis of electrical activity in resistive/memristive synapse coupled Fitzhugh–Nagumo type neurons," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    5. Wang, Guowei & Yu, Dong & Ding, Qianming & Li, Tianyu & Jia, Ya, 2021. "Effects of electric field on multiple vibrational resonances in Hindmarsh-Rose neuronal systems," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    6. Rao, Xiao-Bo & Zhao, Xu-Ping & Chu, Yan-Dong & Zhang, Jian-Gang & Gao, Jian-She, 2020. "The analysis of mode-locking topology in an SIR epidemic dynamics model with impulsive vaccination control: Infinite cascade of Stern-Brocot sum trees," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).

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    Keywords

    Statistical and Nonlinear Physics;

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