The assessment of retinal and choroidal thickness derived from spectral domain optical coherence tomography (SD-OCT) images is an important clinical and research task. Current OCT instruments allow the capture of densely sampled, high-resolution cross-sectional images of ocular tissues. The extensive nature of such datasets makes the manual delineation of tissue boundaries time-consuming and impractical, especially for large datasets of images. Therefore, the development of reliable and accurate methods to automatically segment tissue boundaries in OCT images is fundamental. In this work, two different deep learning methods; convolutional neural networks (CNN) and recurrent neural networks (RNN) are evaluated to calculate the probability of the retinal and choroidal boundaries of interest to be located in a specific position within the SD-OCT images. The method is initially trained using small image patches centred around the three boundaries of interest. After that, the method can be used to provide a per-layer probability map that marks the most likely location of the boundaries. To convert each layer-probability map into a boundary position, the map is subsequently traced using a graph-search method. The effect of the network architecture (CNN vs RNN), patch size, and image intensity compensation on the performance and subsequent boundary segmentation is presented. The results are compared with manual boundary segmentation as well as a previously proposed method based on standard image analysis techniques.