Edge-of-Chaos Kernel and Dynamic Analysis of a Hopfield Neural Network with a Locally Active Memristor

Locally active memristors with an Edge-of-Chaos kernel (EOCK) represent a significant advancement in the simulation of neuromorphic dynamics. However, current research on memristors with an EOCK remains at the circuit level, without further analysis of their feasibility. In this context, we designed a memristor and installed it in a third-order circuit, where it showed local activity and stability under defined voltage and inductance parameters. This behavior ensured that by varying the input voltage and inductance, the memristor could effectively simulate various neural activities, including inhibitory postsynaptic potential and chaotic waveforms. By subsequently integrating the memristor with an EOCK into a Hopfield neural network (HNN) framework and substituting the self-coupling weight, we observed a rich spectrum of dynamic behaviors, including the rare phenomenon of antimonotonicity bubble bifurcation. Finally, we used hardware circuits to realize these generated dynamic phenomena, confirming the feasibility of the memristor. By introducing the HNN and studying its dynamic behavior and hardware circuit implementation, this study provides theoretical insights into and an empirical basis for developing circuits and systems that replicate the complexity of human brain functions. This study provides a reference for the development and application of EOCK in the future.