In this paper, we introduce an alternative Memristor-based Oscillator (MBO) circuit design for implementing both artificial neurons and artificial synapses in the same MBO-based medium addressing also previous scalability issues. We utilize passive unipolar memristor devices fitted in Mott memristors by the JART memristor model to design MBO neurons, which found capable of reproducing bio-plausible spiking and bursting activities. The proposed MBO is also realized as artificial synapse incorporating synapse and axon mechanisms. Finally, the MBO neurons are adjusted to Parkinson-related neuron types and their direct bidirectional coupling presented that MBO neurons are capable of driving spiking activity to non-externally-stimulated MBO neurons. This enables them to be implemented in networks providing useful and low-cost neuromorphic emulators for brain-inspired applications including the design of novel Neurological-disease-related emulators.