Conventional CMOS technology is slowly approaching its physical limitations and researchers are increasingly utilizing nanotechnology to both extend CMOS capabilities and to explore potential replacements. Novel memristive systems continue to attract growing attention since their reported physical realization by HP in 2008. Unique characteristics like non-volatility, re-configurability, and analog storage properties make memristors a very promising candidate for the realization of artificial neural systems. In this work, we propose a memristor-based design of bidirectional transmission excitation/inhibition synapses and implement a neuromorphic computing system based on our proposed synapse designs. The robustness of our system is also evaluated by considering the actual manufacturing variability with emphasis on process variation.