Pseudo-Traveling Waves and Bumps in Quantum and Classical Hierarchical Cellular Neural Networks

Abstract

We study the existence of pseudo-traveling waves and bump solutions for two classes of hierarchical cellular neural networks (CNNs) defined over the ring of $p$-adic integers $\mathbb{Z}_{p}$. The first type is a $p$-adic CNN described by a reaction-diffusion equation, while the second type is its quantum analog obtained via Wick rotation. The $p$-adic CNNs are hierarchical versions of the classical Chua-Yang CNNs; these networks have a tree-like hierarchical architecture with infinitely many cells and hidden layers. The states are governed by integro-differential equations on $% \mathbb{Z}_{p}$. The $p$-adic traveling waves behave fundamentally differently from their Archimedean counterparts. A traveling wave restricted to a $p$-adic sphere yields a countably infinite collection of independent patterns. We introduce the notion of pseudo-traveling waves as finite truncations of this structure and prove their existence for both the classical and quantum networks. We further establish the existence of time-independent solutions (bumps) for both models. Our theoretical results are complemented by numerical simulations that approximate pseudo-traveling-wave solutions for quantum CNNs.