This paper is concerned with the synchronization problem for a class of delayed chaotic neural networks with discontinuous activations. First a lemma which concerns stability in general decay rate is constructed. Based on this lemma, the general decay synchronization stability criteria of discontinuous neural networks are derived via a designed controller. The general decay synchronization is obtained by introducing a decay function and it contains exponential synchronization and polynomial synchronization as its two special cases. Finally, two examples are given to verify the effectiveness of the obtained results.