Bifurcation and Oscillations of a Multi-ring Coupling Neural Network with Discrete Delays

It is well known that brain neural networks are composed of hundreds of millions of neurons with complicated connections. It is noteworthy that there are a large number of neural circuits formed by the coupling relationship in neural networks. However, the vast majority of existing studies on ring-structured networks were restricted to models with fewer neurons or single-ring structure. Here, we design a multi-ring coupling network model, which includes a large number of neurons and multiple rings, to simulate the brain network with multiple neural circuits, and use bifurcation theory to investigate the dynamic behavior of bifurcations and oscillations of the proposed model. Instead of the traditional method, the Coates flow graph method is used to derive explicit expressions for the higher-order determinants of the model. Then, the stability and Hopf bifurcation of the model are studied with the sum of discrete delays of each ring as the bifurcation parameter, and the explicit formula for its critical value is deduced. Our results suggest that in some cases, the stability of the model gradually deteriorates with increasing time delay and eventually destabilizes the system at some critical value. More specifically, the development of the instability (increasing time delay) leads to limit cycle and periodic oscillations, and the amplitude and period of the system are also affected by it.