A growing algorithm of neural network based on axon growth is developed in this article. At the beginning, the Newtonian gravitation and the Brownian motion are employed to establish an axon growth model, and the constraint function of the neural axon growth is derived according to the gravitational coefficient and the Brown coefficient. And then, the growing algorithm of neural network is obtained on the basis of the output function of neural network. Simulation experiments show that the developed algorithm can simulate the process that different neurons connect each other by their axon growth to establish neural network. And at the end, the developed algorithm is applied to the rhythmic motion control of a quadruped robot to verify the effectiveness in walk gait and in trot gait, respectively.