Analog circuit implementation of neuron model is an essential category of neuromorphic circuit since it can reproduce neuron firing patterns and assist in exploring neuron-based applications. The neuron models built by electrophysiological ion transport mechanism can effectively mimic the neuron firing patterns. However, they are rather difficult to implement on analog level because these neuron models involve complex exponential nonlinearities for characterizing the ion channels. Thanks to the superiorities of locally active memristor in constructing artificial neuron circuit, two locally active memristors are employed to characterize the sodium and potassium ion channels and then a memristive neuromorphic circuit is successfully built in this paper. Numerical simulations demonstrate that the memristive neuromorphic circuit can generate abundant firing patterns with respect to memristor-and stimulus-related parameters. Moreover, quasi-periodic bifurcation behavior and coexisting firing patterns are triggered by different memristor initial conditions. Employing off-the-shelf discrete circuit components, a PCB-based analog circuit is manually constructed and hardware experiments are executed. Experimental results captured from hardware measurements satisfactorily verify the numerically simulated ones and effectively show the feasibility of the memristive neuromorphic circuit in generating neuron firing patterns.