In this paper, we propose a deep neural network-based car-following model that has two distinctive properties. First, unlike most existing car-following models that take only the instantaneous velocity, velocity difference, and position difference as inputs, this new model takes the velocities, velocity differences, and position differences that were observed in the last few time intervals as inputs. That is, we assume that drivers' actions are temporally dependent in this model and try to embed prediction capability or memory effect of human drivers in a natural and efficient way. Second, this car-following model is built in a data-driven way, in which we reduce human interference to the minimum degree. Specially, we use recently developing deep neural networks rather than conventional neural networks to establish the model, since deep learning technique provides us more flexibility and accuracy to describe complicated human actions. Tests on empirical trajectory records show that this deep neural network-based car-following model yield significantly higher simulation accuracy than existing car-following models. All these findings provide a novel way to study traffic flow theory and traffic simulations.