This article proposes a new compact model and presents a circuit-theoretical analysis for a third-order memristive neuromorphic element fabricated by Kumar et al. The proposed model mainly consists of a box-shaped resistor model and a simplified piecewise-linear memristor model. Since the dynamic behavior of the box-shaped hysteresis in the quasistatic current–voltage curve is mostly unexplored, we first extract the box-shaped resistor model and construct its oscillators. The coordinate system shifting method and dynamic route analysis method are used to reveal the operating mechanism of boxshaped resistor-based oscillators. Both the theoretical analysis and simulation verification indicate that the box-shaped hysteresis characteristic facilitates the generation of neuromorphic action potentials. The proposed new compact model not only captures quasi-static characteristics but also includes dynamic behaviors, such as action potential, periodic spiking, and periodic bursting. The influences of the model parameters are further investigated to reveal the mechanism of the neuromorphic behaviors in the box-shaped hysteresis and positive differential resistance regions. Simulation results manifest the feasibility of the proposed model and the correctness of the presented analysis methods, which pave the way to the optimized design of memristive devices and the research of neuromorphic dynamics.