We study involuntary micro-movements of both eyes, in addition to saccadic macro-movements, as biometric characteristic. We develop a deep convolutional neural network that processes binocular eye-tracking signals and verifies the viewer's identity. In order to detect presentation attacks, we develop a model in which the movements are a response to a controlled stimulus. The model detects replay attacks by processing both the controlled but randomized stimulus and the ocular response to this stimulus. We acquire eye movement data from 150 participants, with 4 sessions per participant and conduct experiments on this new and legacy data sets with varying tracker precision and sampling rate. We observe that the model detects replay attacks reliably. For identification and identity verification, the model attains substantially lower error rates than prior work. We explore the relationships between training population size, training data volume, types of visual stimuli, number of training and enrollment sessions, interval between enrollment and probe sessions on one hand and the model performance on the other hand.