The study aims to present the synchronization concerns of neural networks with stochastic packet dropouts utilizing a quantized memory sampled-data control (QMSDC) scheme and the problem of reachable set analysis (RSA) for NNs under the influence of external disturbances are discussed. To do this, a novel free-weighting matrix integral inequality (FWMII) is proposed with an augmented Lyapunov-functional. Furthermore, at this time, we show that the proposed inequality is less conservative due to the delay information. In the meantime, a novel enhanced looped function is created that utilizes not only sampling information but also incorporates both transmission delay information and a quantized sampling pattern. Next, using the proposed FWMII technique, several sufficient conditions in terms of linear matrix inequalities (LMIs) are derived to suggest how an augmented closed-loop system can achieves stochastic mean-square exponential synchronization under random packet loss situations. Finally, the proposed approach demonstrates its usefulness and superiority by comparing it with existing methods through numerical simulation of the secure communications.