Spike Sorting by Convolutional Dictionary Learning

Spike sorting refers to the problem of assigning action potentials observed in extra-cellular recordings of neural activity to the neuron(s) from which they originate. We cast this problem as one of learning a convolutional dictionary from raw multi-electrode waveform data, subject to sparsity constraints. In this context, sparsity refers to the number of neurons that are allowed to spike simultaneously. The convolutional dictionary setting, along with its assumptions (e.g. refractoriness) that are motivated by the spike-sorting problem, let us give theoretical bounds on the sample complexity of spike sorting as a function of the number of underlying neurons, the rate of occurrence of simultaneous spiking, and the firing rate of the neurons. We derive memory/computation-efficient convolutional versions of OMP (cOMP) and KSVD (cKSVD), popular algorithms for sparse coding and dictionary learning respectively. We demonstrate via simulations that an algorithm that alternates between cOMP and cKSVD can recover the underlying spike waveforms successfully, assuming few neurons spike simultaneously, and is stable in the presence of noise. We also apply the algorithm to extra-cellular recordings from a tetrode in the rat Hippocampus.