This paper investigates the dynamics of a directed acyclic neural network by edge adding control. We find that the local stability and Hopf bifurcation of the controlled network only depend on the size and intersection of directed cycles, instead of the number and position of the added edges. More specifically, if there is no cycle in the controlled network, the local dynamics of the network will remain unchanged and Hopf bifurcation will not occur even if the number of added edges is sufficient. However, if there exist cycles, then the network may undergo Hopf bifurcation. Our results show that the cycle structure is a necessary condition for the generation of Hopf bifurcation, and the bifurcation threshold is determined by the number, size, and intersection of cycles. Numerical experiments are provided to support the validity of the theory.

中文翻译：

---

有向循环在有向神经网络中的作用

本文通过边缘添加控制研究了有向无环神经网络的动力学。我们发现受控网络的局部稳定性和Hopf分岔仅取决于有向环的大小和交集，而不取决于添加边的数量和位置。更具体地说，如果受控网络中不存在环路，则即使添加的边数足够多，网络的局部动态也将保持不变，并且不会发生Hopf分岔。然而，如果存在环路，则网络可能会发生Hopf分叉。结果表明，循环结构是Hopf分岔产生的必要条件，分岔阈值由循环的数量、大小和交集决定。提供数值实验来支持理论的有效性。