This article proposes a new method for interpreting computations performed by populations of spiking neurons. Neural firing is modeled as a rate-modulated random process for which the behavior of a neuron in response to external input can be completely described by its tuning function. I show that under certain conditions, cells with any desired tuning functions can be approximated using only spike coincidence detectors and linear operations on the spike output of existing cells. I show examples of adaptive algorithms based on only spike data that cause the underlying cell-tuning curves to converge according to standard supervised and unsupervised learning algorithms. Unsupervised learning based on principal components analysis leads to independent cell spike trains. These results suggest a duality relationship between the random discrete behavior of spiking cells and the deterministic smooth behavior of their tuning functions. Classical neural network approximation methods and learning algorithms based on continuous variables can thus be implemented within networks of spiking neurons without the need to make numerical estimates of the intermediate cell firing rates.