How Do Higher-Order Interactions Affect the Dynamic Evolution of Layered Neural Networks

In recent years, with the wide application of artificial neural networks (ANNs) in the field of artificial intelligence, the study of neural network bifurcation dynamics has received much attention and has achieved a large number of results, but the current research only focuses on the binary interactions between neurons and does not take into account the higher-order interactions (HOIs) that exist between neurons. Furthermore, most neural network models focus on ring, star and chain structures, but it is more practical to study multi-layer neural networks. For this reason, this paper develops a three-layer neural network with multiple time delays under HOIs. Firstly, the characteristic equation of network is obtained, and the time delay is selected as the bifurcation parameter. Subsequently, the stability of the neural network and the sufficient condition for the occurrence of Hopf bifurcation are established. Then, the correctness of the theoretical results is verified through numerical simulations. The simulation results demonstrate that the increase of the time delay leads to Hopf bifurcation, which consequently leads to the system oscillation and instability. In addition, the stability of the network is closely related to the higher-order coupling coefficient, self-feedback coefficient and unified connection weight. An increase in the higher-order coupling coefficient expands the stability domain of the network, an increase in the self-feedback coefficient enlarges the stability domain of the network, and an increase in the unified connection weight narrows the stability domain of the network. Finally, our model performs well in terms of function fitting, which also provides some insights into the subsequent modelling and analysis of higher-order neural networks.