Recurrent neural network is a nonlinear dynamic system, which is widely used in the aspect of associative memory. The activation function of recurrent neural network is a typical piecewise linear function, and its hardware implementation is tedious. The memristor is nonlinear dynamic nanoscale device. If used for the neural network, it can achieve low power consumption and easy extension of the memristive activation function circuit. Using the relationship features between flux and charge of memristor, this paper puts forward a recurrent neural network based on the memristive activation function and applies it to the associative memory. First of all, the two value and the three value memristive activation function is designed, which provides an effective method for the realization of the recurrent neural network integrated circuit. Secondly, a recurrent neural network model based on memristive activation function is proposed by joining the matrix transfer function. Finally, we realize the simulation of auto-associative memory and hetero-associative memory of the binary and three values static image, and then the moving of the letters in the alphabet image, which can be considered as associative memory of the dynamic image. The study can simplify the recurrent neural network integrated circuits, and provide a new effective way for the realization of the video memory sports.