Sparse sampling: theory, methods and an application in neuroscience.

The current methods used to convert analogue signals into discrete-time sequences have been deeply influenced by the classical Shannon-Whittaker-Kotelnikov sampling theorem. This approach restricts the class of signals that can be sampled and perfectly reconstructed to bandlimited signals. During the last few years, a new framework has emerged that overcomes these limitations and extends sampling theory to a broader class of signals named signals with finite rate of innovation (FRI). Instead of characterising a signal by its frequency content, FRI theory describes it in terms of the innovation parameters per unit of time. Bandlimited signals are thus a subset of this more general definition. In this paper, we provide an overview of this new framework and present the tools required to apply this theory in neuroscience. Specifically, we show how to monitor and infer the spiking activity of individual neurons from two-photon imaging of calcium signals. In this scenario, the problem is reduced to reconstructing a stream of decaying exponentials.