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

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 σ and diffusive delay τ 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.