  We theoretically study an artificial neuron circuit containing a quantum
memristor in the presence of relaxation and dephasing. The charge transport in
the quantum element is realized via tunneling of a charge through a quantum
particle which shuttles between two terminals -- a functionality reminiscent of
classical diffusive memristors. We demonstrate that this physical principle
enables hysteretic behavior of the current-voltage characteristics of the
quantum device. In addition, being used in artificial neural circuit, the
quantum switcher is able to generate self-sustained current oscillations. Our
analysis reveals that these self-oscillations are triggered only in quantum
regime with a moderate rate of relaxation, and cannot exist either in a purely
coherent regime or at a very high decoherence. We investigate the hysteresis
and instability leading to the onset of current self-oscillations and analyze
their properties depending on the circuit parameters. Our results provide a
generic approach to the use of quantum regimes for controlling hysteresis and
generating self-oscillations.
