Spike sorting involves clustering spikes recorded by a micro-electrode according to the source neurons. It is a complicated task, which requires much human labor, in part due to the non-stationary nature of the data. We propose to automate the clustering process in a Bayesian framework, with the source neurons modeled as a non-stationary mixture-of-Gaussians. At a first search stage, the data are divided into short time frames, and candidate descriptions of the data as mixtures-of-Gaussians are computed for each frame separately. At a second stage, transition probabilities between candidate mixtures are computed, and a globally optimal clustering solution is found as the maximum-a-posteriori solution of the resulting probabilistic model. The transition probabilities are computed using local stationarity assumptions, and are based on a Gaussian version of the Jensen-Shannon divergence. We employ synthetically generated spike data to illustrate the method and show that it outperforms other spike sorting methods in a non-stationary scenario. We then use real spike data and find high agreement of the method with expert human sorters in two modes of operation: a fully unsupervised and a semi-supervised mode. Thus, this method differs from other methods in two aspects: its ability to account for non-stationary data, and its close to human performance.