Electroencephalography (EEG) is a noninvasive way to record brain activity by means of measuring electrical fields arising from neural activation. Being relatively inexpensive, safe, and readily available, EEG-based techniques have been studied as potential methods for controlling brain-machine interfaces. Previous attempts to analyze EEG signals have focused on well-characterized sensorimotor data features. However, the brain-machine interface field seems to have stagnated in improving motor decoding using this method. One way to overcome this hurdle is to use neural-network-based classification methods to analyze brain-activity data. In this paper, we describe the novel neural networks we created for analyzing existing EEG data. Although these neural networks were programmed, trained, and tested in a conventional central processing unit or graphics processing unit environment, their novelty lies in their full compatibility with IBM's recently introduced ultralow power, neuromorphic TrueNorth chip infrastructure, thus, constituting the analytical units in the next generation of neurobionic mobile devices. We report on the development of a new EEG signal classifier built on a spiking neural network that runs on the TrueNorth platform. Using a modified back-propagation training method that employs trinary weights, we demonstrate state-of-the-art classification accuracy.