Signal processing and classification algorithms often have limited applicability resulting from an inaccurate model of the signal's un(cid:173) derlying structure. We present here an efficient, Bayesian algo(cid:173) rithm for modeling a signal composed of the superposition of brief, Poisson-distributed functions. This methodology is applied to the specific problem of modeling and classifying extracellular neural waveforms which are composed of a superposition of an unknown number of action potentials CAPs). Previous approaches have had limited success due largely to the problems of determining the spike shapes, deciding how many are shapes distinct, and decomposing overlapping APs. A Bayesian solution to each of these problems is obtained by inferring a probabilistic model of the waveform. This approach quantifies the uncertainty of the form and number of the inferred AP shapes and is used to obtain an efficient method for decomposing complex overlaps. This algorithm can extract many times more information than previous methods and facilitates the extracellular investigation of neuronal classes and of interactions within neuronal circuits.