This work addresses the state estimation problem for recurrent neural networks over capacity-constrained communication channels. The intermittent transmission protocol is used to reduce the communication load, where a stochastic variable with a given distribution is used to describe the transmission interval. A corresponding transmission interval-dependent estimator is designed, and an estimation error system based on it is also derived, whose mean-square stability is proved by constructing an interval-dependent function. By analyzing the performance in each transmission interval, sufficient conditions of the mean-square stability and the strict $(Q,S,R)$ - $\gamma $ -dissipativity are established for the estimation error system. Finally, the correctness and the superiority of the developed result are illustrated by a numerical example.