Due to the limitation in speed and throughput of the traditional Von Neumann architecture, the interest in braininspired neuromorphic systems has been the focus of recent research activities. RRAM device has been extensively used as synapses in neuromorphic systems due to its many advantages including small size and compatibility with CMOS fabrication process. However, the RRAM device suffers from reliability soft-errors resulting from the stochastic nature of the oxygen vacancies of its conductive filaments. In this article, for the first time, using a combination of SPICE-based and BRIAN-based simulations, a novel framework is developed to model and assess the impact of RRAM reliability soft-errors on the performance of the neuromorphic systems. Simulation results show that the accuracy of a multi-perceptron RRAM-based neuromorphic system drops from 91.6% to 43% when the reliability softerrors are considered. To overcome this degradation in the system performance, a detailed analysis is conducted to modify the way the RRAM resistive state changes. In addition to this, a list of recommendations for the design of neuromorphic systems is also provided to overcome the RRAM reliability soft-errors.