中国物理B ›› 2026, Vol. 35 ›› Issue (6): 60202-060202.doi: 10.1088/1674-1056/ae594c

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Hopf bifurcation and oscillatory dynamics in a delayed FitzHugh-Nagumo neuronal network on scale-free topologies

Zhan Shen(申瞻)1,2, Qianqian Zheng(郑前前)2,†, Jianwei Yang(杨建伟)1, and Jianwei Shen(申建伟)1,‡   

  1. 1 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China;
    2 School of Science, Xuchang University, Xuchang 461000, China
  • 收稿日期:2026-02-04 修回日期:2026-03-27 接受日期:2026-03-31 发布日期:2026-06-23
  • 通讯作者: Qianqian Zheng, Jianwei Shen E-mail:zqq@xcu.edu.cn;xcjwshen@gmail.com
  • 基金资助:
    This work was supported by the National Natural Science Foundation of China (Grant Nos. 12272135 and 11971416), the Natural Science Foundation of Henan Province (Grant No. 242300420661), the Training Program for Young Key Teachers in Colleges and Universities of Henan Province (Grant No. 2023GGJS144), the Funding Program of Henan Province for Merit-based Overseas Students (Grant No. 2023), and the Postgraduate Education Reform and Quality Improvement Project of Henan Province (Grant No. YJS2026AL002).

Hopf bifurcation and oscillatory dynamics in a delayed FitzHugh-Nagumo neuronal network on scale-free topologies

Zhan Shen(申瞻)1,2, Qianqian Zheng(郑前前)2,†, Jianwei Yang(杨建伟)1, and Jianwei Shen(申建伟)1,‡   

  1. 1 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China;
    2 School of Science, Xuchang University, Xuchang 461000, China
  • Received:2026-02-04 Revised:2026-03-27 Accepted:2026-03-31 Published:2026-06-23
  • Contact: Qianqian Zheng, Jianwei Shen E-mail:zqq@xcu.edu.cn;xcjwshen@gmail.com
  • Supported by:
    This work was supported by the National Natural Science Foundation of China (Grant Nos. 12272135 and 11971416), the Natural Science Foundation of Henan Province (Grant No. 242300420661), the Training Program for Young Key Teachers in Colleges and Universities of Henan Province (Grant No. 2023GGJS144), the Funding Program of Henan Province for Merit-based Overseas Students (Grant No. 2023), and the Postgraduate Education Reform and Quality Improvement Project of Henan Province (Grant No. YJS2026AL002).

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摘要： Neuronal oscillations arise from the interplay between intrinsic neuronal dynamics and network connectivity. In this work, we investigate the effects of network topology on oscillatory behavior in a FitzHugh-Nagumo (FHN) neuronal model with distributed delay, representing ion-channel memory effects, and diffusive delay, accounting for axonal transmission delays. The neurons are coupled through quasi-Laplacian interactions on Barabási-Albert (BA) scale-free networks. Using the multiple-time-scales (MTS) method, we derive amplitude equations near the Hopf bifurcation point and establish explicit relationships between oscillatory dynamics and network topology. The analysis shows that the smallest negative eigenvalue of the network governs the critical delay threshold for oscillation onset, while the distributed-delay parameter $\sigma$ and diffusive delay $\tau$ jointly regulate this threshold. The resulting oscillation frequencies are confined to the beta band (15-30 Hz), a frequency range often associated with pathological neural activity in Parkinson's disease. Extensive numerical simulations over 50 network realizations confirm the theoretical predictions. Hub nodes with higher degrees exhibit lower critical delays and larger oscillation amplitudes, whereas peripheral nodes display weaker and more heterogeneous responses. Statistical analysis further reveals a negative correlation between node degree and critical delay and a positive correlation between node degree and oscillation amplitude. These results demonstrate how delay effects and network topology jointly shape the emergence and spatial organization of collective oscillations, providing insights into synchronization phenomena in complex neuronal networks.

关键词: Hopf bifurcation, FitzHugh-Nagumo (FHN) model, delay, multiple-time-scales (MTS) method, quasi-Laplacian, Barabási-Albert (BA) scale-free network

Abstract: Neuronal oscillations arise from the interplay between intrinsic neuronal dynamics and network connectivity. In this work, we investigate the effects of network topology on oscillatory behavior in a FitzHugh-Nagumo (FHN) neuronal model with distributed delay, representing ion-channel memory effects, and diffusive delay, accounting for axonal transmission delays. The neurons are coupled through quasi-Laplacian interactions on Barabási-Albert (BA) scale-free networks. Using the multiple-time-scales (MTS) method, we derive amplitude equations near the Hopf bifurcation point and establish explicit relationships between oscillatory dynamics and network topology. The analysis shows that the smallest negative eigenvalue of the network governs the critical delay threshold for oscillation onset, while the distributed-delay parameter $\sigma$ and diffusive delay $\tau$ jointly regulate this threshold. The resulting oscillation frequencies are confined to the beta band (15-30 Hz), a frequency range often associated with pathological neural activity in Parkinson's disease. Extensive numerical simulations over 50 network realizations confirm the theoretical predictions. Hub nodes with higher degrees exhibit lower critical delays and larger oscillation amplitudes, whereas peripheral nodes display weaker and more heterogeneous responses. Statistical analysis further reveals a negative correlation between node degree and critical delay and a positive correlation between node degree and oscillation amplitude. These results demonstrate how delay effects and network topology jointly shape the emergence and spatial organization of collective oscillations, providing insights into synchronization phenomena in complex neuronal networks.

Key words: Hopf bifurcation, FitzHugh-Nagumo (FHN) model, delay, multiple-time-scales (MTS) method, quasi-Laplacian, Barabási-Albert (BA) scale-free network

中图分类号:  (Delay and functional equations)

  • 02.30.Ks
02.30.Oz (Bifurcation theory) 02.30.-f (Function theory, analysis)