Amplification schemes and gain prescriptions for hearing aids have thus far been developed and evaluated based on perceptual criteria such as speech intelligibility, sound comfort, and loudness equalization. Finding amplification strategies that simultaneously optimize all of these perceptual metrics has proven difficult, despite decades of research. Furthermore, novel amplification schemes based on rough conceptual models of the normal and impaired auditory physiology have often proven to be unsuccessful. In this talk, I will describe studies directly employing physiologically accurate computational models to evaluate more rigorously what hearing aids are likely doing to the neural representation of speech. The results of these investigations indicate that (i) a physiologically accurate auditory-nerve (AN) model can predict optimal linear and nonlinear amplification gains, (ii) optimal gains are dependent on both the spike-timing and mean-rate representations of speech in the AN, (iii) the proportion of outer hair cell and inner hair cell dysfunction in an impaired ear can affect optimal gains, and (iv) slow wide dynamic range compression (WDRC) or automatic gain control better restores the neural representation of speech than does fast WDRC.