This article discusses the issue of delay-dependent stability in generalized neural networks (GNNs). First, a suitable Lyapunov-Krasovskii functional (LKF), including more state information on time delays, was constructed, effectively reducing the system's conservatism. Second, a novel stability condition was established by utilizing the suitable LKF and the matrix-valued cubic polynomials to determine the negative definite conditions. Finally, the experimental simulation results were validated using three typical numerical examples. The experimental results demonstrated the efficiency and merits of our proposed approach compared with the existing methods.