In this paper, we consider two different approaches of using deep neural networks for cough detection. The cough detection task is cast as a visual recognition problem and as a sequence-to-sequence labeling problem. A convolutional neural network and a recurrent neural network are implemented to address these problems, respectively. We evaluate the performance of the two networks and compare them to other conventional approaches for identifying cough sounds. In addition, we also explore the effect of the network size parameters and the impact of long-term signal dependencies in cough classifier performance. Experimental results show both network architectures outperform traditional methods. Between the two, our convolutional network yields a higher specificity 92.7% whereas the recurrent attains a higher sensitivity of 87.7%.