We demonstrate that single-variable integrate-and-fire models can quantitatively capture the dynamics of a physiologically-detailed model for fast-spiking cortical neurons. Through a systematic set of approximations, we reduce the conductance based model to two variants of integrate-and-fire models. In the first variant (non-linear integrate-and-fire model), parameters depend on the instantaneous membrane potential whereas in the second variant, they depend on the time elapsed since the last spike (Spike Response Model). The direct reduction links features of the simple models to biophysical features of the full conductance based model. To quantitatively test the predictive power of the Spike Response Model and of the non-linear integrate-and-fire model, we compare spike trains in the simple models to those in the full conductance-based model when the models are subjected to identical randomly fluctuating input. For random current input, the simple models reproduce 70-80 percent of the spikes in the full model (with temporal precision of +/- 2ms over a wide range of firing frequencies. For random conductance injection, up to 73 percent of spikes are coincident. We also present a technique for numerically optimizing parameters in the Spike Response Model and the non-linear integrate-and-fire model based on spike trains in the full conductance-based model. This technique can be used to tune simple models to reproduce spike trains of real neurons.