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This paper studies the asymptotical stability for a class of neural networks (NNs) with time-varying delay. Under weaker assumptions on the activation functions, by defining a more general type of Lyapunov functionals and using a delay decomposition method and employing a new convex combination technique, a new less conservative stability criterion are established to guarantee the global asymptotical stability of the discussed NNs. The obtained conditions are dependent on the upper bound of the delay, and are expressed in terms of linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness and the less conservatism of the proposed conditions.