How higher-order interactions shape the dynamics of large-scale dual-star neural networks

Understanding large-scale neural network dynamics represents a fundamental challenge in both biological and artificial neural systems. Traditional pairwise interaction models inadequately capture the complex connectivity patterns observed in real neural networks, where groups of three or more neurons interact simultaneously. This paper presents a novel large-scale delayed neural network with dual-star architecture, incorporating both second-order and third-order interactions. Two analytical methods are employed to derive the characteristic polynomial for the proposed network: the Coates flow graph formula and the Schur complement method. Both approaches circumvent the computational intractability of conventional determinant-based methods for high-dimensional systems. Using the holistic element concept, rigorous conditions for local stability and Hopf bifurcation are established. Comprehensive numerical simulations validate theoretical findings and reveal that higher-order interactions exhibit non-monotonic effects on bifurcation thresholds, self-feedback mechanisms provide robust stabilization, and network size significantly influences bifurcation characteristics. Finally, the accurate function fitting achieved by our model provides new insights for the design and control of neural networks.