In order to respond appropriately to environmental stimuli, organisms must integrate over time spatiotemporal signals that reflect object motion and self-movement. One possible mechanism to achieve this spatiotemporal transformation is to delay or lag neural responses. This paper reviews our recent modeling work testing the sufficiency of delayed responses in the nervous system in two different behavioral tasks: (1) Categorizing spatiotemporal tactile cues with thalamic "lag" cells and downstream coincidence detectors, and (2) Predictive motor control was achieved by the cerebellum through a delayed eligibility trace rule at cerebellar synapses. Since the timing of these neural signals must closely match real-world dynamics, we tested these ideas using the brain based device (BBD) approach in which a simulated nervous system is embodied in a robotic device. In both tasks, biologically inspired neural simulations with delayed neural responses were critical for successful behavior by the device.