<p>Memristors serve both as fundamental components of artificial neurons and as coupling elements that emulate synaptic connections in neuromorphic systems. The locally active memristor (LAM)-coupled neuron circuit, characterized by its unique nonlinear regulatory properties, exhibits rich dynamic behaviors, including complex oscillations. This circuit provides a novel approach to simulating synaptic coupling and developing efficient neuromorphic hardware. A network structure based on locally active memrisrtor-coupled third-order memristive neurons is proposed in this paper. Compared with lower-order models, third-order neuron models exhibit more complex dynamic behaviors. Moreover, the nonlinear coupling introduced by LAMs further enhances the system’s complexity. By integrating the locally active theory with the edge of chaos theory, the small-signal analysis method is employed to derive the edge of chaos domain of third-order memristive neuron, thereby revealing their distinctive dynamic properties. This work demonstrates that when two identical third-order memristive neurons are coupled via a locally active memristor, phenomena such as Smale paradox and transitions from period-doubling oscillations to chaos emerge. Moreover, tuning the coupling memristor’s parameters allows for dynamic control of the oscillation phase difference and response speed, offering theoretical support for asynchronous pulse coding. Finally, a locally active memristor-coupled third-order memristive neuron circuits are implemented in hardware. Experimental results align with theoretical analysis, validating both the feasibility of the proposed coupled neural network and its underlying dynamical mechanisms. This work advances understanding of neural nonlinear dynamics and enables physics-based platforms for neuroscience and neuromorphic computing.</p>