Synchronization is a key topic of research in neuroscience, medicine, and artificial neural networks; however, understanding its principle is difficult, both scientifically and mathematically. Specifically, the synchronization of the FitzHugh-Nagumo network with a hierarchical architecture has previously been studied; however, a mathematical analysis has not been conducted, owing to the network complexity. Therefore, in this paper, we saught to understand synchronization through mathematical analyses. In particular, we consider the most common types of hierarchical architecture and present a condition of the hierarchical architecture to induce synchronization. First, we provide mathematical analyses of a Lyapunov function for each layer, from which we obtain sufficient conditions guaranteeing synchronization and show that the Lyapunov function decreases exponentially. Moreover, we show that the internal connectivity critically affects synchronization in the first layer; however, in the second and subsequent layers, the internal connectivity is not important for synchronization, and the connectivity up to the first layer critically affects synchronization. We expect that the results and mathematical methodology can be applied to study other similar neural models with hierarchical architectures
