Activation functions map data in artificial neural network computation. In an application, the activation function and selection of its gradient and translation factors are directly related to the convergence of the network. Usually, the activation function parameters are determined by trial and error. In this work, a Cauchy distribution (Cauchy), Laplace distribution (Laplace), and Gaussian error function (Erf) were used as new activation functions for the back-propagation (BP) algorithm. In addition, this study compares the effects of the Sigmoid type function (Logsig), hyperbolic tangent function (Tansig), and normal distribution function (Normal). The XOR problem was used in simulation experiments to evaluate the effects of these six kinds of activation functions on network convergence and determine their optimal gradient and translation factors. The results show that the gradient factor and initial weights significantly impact the convergence of activation functions. The optimal gradient factors for Laplace, Erf-Logsig, Tansig-Logsig, Logsig, and Normal were 0.5, 0.5, 4, 2, and 1, respectively, and the best intervals were [0.5, 1], [0.5, 2], [2, 6], [1, 4], and [1, 2], respectively. Using optimal gradient factors, the order of convergence speed was Laplace, Erf-Logsig, Tansig-Logsig, Logsig, and Normal. The functions Logsig (gradient factor = 2), Tansig-Logsig (gradient factor = 4), Normal (translation factor = 0, gradient factor = 1), Erf-Logsig (gradient factor = 0.5) and Laplace (translation factor = 0, gradient factor = 0.5) were less sensitive to initial weights, and as a result, their convergence performances were less influenced. As the gradient of the curve of the activation functions increased, the convergence speed of the networks showed an accelerating trend. The conclusions obtained from the simulation analysis can be used as a reference for the selection of activation functions for BP algorithm-based feedforward neural networks.