Simplified Discrete Two-Neuron Hopfield Neural Network and FPGA Implementation

Continuous Hopfield neural networks have been extensively studied in academic field and applied in various industrial fields, while discrete Hopfield neural networks have rarely been reported. In this study, we present a two-dimensional discrete map of the Hopfield neural network comprising of two neurons without self-connections. Through invariant point stability analysis, we qualitatively investigate the Neimmark-Sacker bifurcation behaviors along with the coexisting multiple attractors induced by the stability evolution. Using numerical methods, we explore the hyperchaotic bifurcation behaviors and reveal the polyhedral hyperchaotic attractors. Furthermore, we implement the discrete map on a field-programmable gate array (FPGA) hardware platform, validating our numerical findings with experimental results. Additionally, two hardware pseudorandom number generators are fabricated to provide random numbers. In summary, despite its simple algebraic structure, the discrete map exhibits hyperchaotic dynamics with polyhedral attractors, outstanding randomness, and ultra-wide parameter spaces, allowing it to be an ideal candidate for education, research, and practical applications.