This paper studies the construction and approximation of neural network operators with a centered bell-shaped Gaussian activation function. Using a univariate Gaussian function a class of Cardaliaguet-Euvrard type network operators is constructed to approximate the continuous function, and the Jackson type theorems of the approximation and some discussions about the convergence are given. Furthermore, to approximate the multivariate function, a class of bivariate Cardaliaguet-Euvrard type network operators isintroduced, and the corresponding estimates of the approximation rate are deduced.