A reason for applying the direct method of Lyapunov to artificial neural networks (ANNs) is to design dynamical neural networks so that they exhibit global asymptotic stability. Lyapunov functions that frequently appear in the ANN literature include the quadratic function, the Persidskii function, and the Luré-Postnikov function. This contribution revisits the quadratic function and shows that via Krasovskii-like stability criteria, it is possible to have a very simple and systematic procedure to obtain not only new and generalized results but also well-known sufficient conditions for convergence established recently by non-Lyapunov methods, such as the matrix measure and nonlinear measure.