Neural conduction delay is a serious issue for organisms that need to act in real time. Perceptual phenomena such as the flash-lag effect (FLE, where the position of a moving object is perceived to be ahead of a brief flash when they are actually colocalized) suggest that the nervous system may perform extrapolation to compensate for delay. However, the precise neural mechanism for extrapolation has not been fully investigated. The main hypothesis of this dissertation is that facilitating synapses, with their dynamic sensitivity to the rate of change in the input, can serve as a neural basis for extrapolation. To test this hypothesis, computational and biologically inspired models are proposed in this dissertation. (1) The facilitatory activation model (FAM) was derived and tested in the motion FLE domain, showing that FAM with smoothing can account for human data. (2) FAM was given a neurophysiological ground by incorporating a spike-based model of facilitating synapses. The spike-based FAM was tested in the luminance FLE domain, successfully explaining extrapolation in both increasing and decreasing luminance conditions. Also, inhibitory backward masking was suggested as a potential cellular mechanism accounting for the smoothing effect. (3) The spike-based FAM was extended by combining it with spike-timing-dependent plasticity (STDP), which allows facilitation to go across multiple neurons. Through STDP, facilitation can selectively propagate to a specific direction, which enables the multi-neuron FAM to express behavior consistent with orientation FLE. (4) FAM was applied to a modified 2D pole-balancing problem to test whether the biologically inspired delay compensation model can be utilized in engineering domains. Experimental results suggest that facilitating activity greatly enhances real time control performance under various forms of input delay as well as under increasing delay and input blank-out conditions. The main contribution of this dissertation is that it shows an intimate link between the organism-level problem of delay compensation, perceptual phenomenon of FLE, computational function of extrapolation, and neurophysiological mechanisms of facilitating synapses (and STDP). The results are expected to shed new light on real-time and predictive processing in the brain, and help understand specific neural processes such as facilitating synapses.