The local active memristor with an Edge of Chaos Kernel represents a significant advancement in the simulation of neuromorphic dynamics. Initially, we designed and analyzed a memristor circuit that demonstrates local activity and stability within defined voltage and inductance parameters. By varying the input voltage and inductance, this memristor effectively emulates diverse neural activities, including inhibitory postsynaptic potentials and chaotic waveforms. Subsequently, by integrating the EOCK memristor into the Hopfield neural network (HNN) framework, we substituted the self-coupling weight and observed a rich spectrum of dynamic behaviors. Finally, we implemented hardware circuits to realize these generated dynamic phenomena. This research introduces a novel hardware approach to brain-like computing, providing both theoretical insights and empirical foundations for developing circuits and systems that replicate the complexity of human brain function. The integration of EOCK is anticipated to inspire innovative methodologies for simulating the varied neural activities inherent in human cognition.