BACKGROUND: Unsupervised identification of action potentials in multi-channel extracellular recordings, in particular from high-density microelectrode arrays with thousands of sensors, is an unresolved problem. While independent component analysis (ICA) achieves rapid unsupervised sorting, it ignores the convolutive structure of extracellular data, thus limiting the unmixing to a subset of neurons. NEW METHOD: Here we present a spike sorting algorithm based on convolutive ICA (cICA) to retrieve a larger number of accurately sorted neurons than with instantaneous ICA while accounting for signal overlaps. Spike sorting was applied to datasets with varying signal-to-noise ratios (SNR: 3-12) and 27% spike overlaps, sampled at either 11.5 or 23kHz on 4365 electrodes. RESULTS: We demonstrate how the instantaneity assumption in ICA-based algorithms has to be relaxed in order to improve the spike sorting performance for high-density microelectrode array recordings. Reformulating the convolutive mixture as an instantaneous mixture by modeling several delayed samples jointly is necessary to increase signal-to-noise ratio. Our results emphasize that different cICA algorithms are not equivalent. COMPARISON WITH EXISTING METHODS: Spike sorting performance was assessed with ground-truth data generated from experimentally derived templates. The presented spike sorter was able to extract approximately 90% of the true spike trains with an error rate below 2%. It was superior to two alternative (c)ICA methods ( approximately 80% accurately sorted neurons) and comparable to a supervised sorting. CONCLUSION: Our new algorithm represents a fast solution to overcome the current bottleneck in spike sorting of large datasets generated by simultaneous recording with thousands of electrodes.