Sparse sampling: theory, methods and an application in neuroscience

Abstract

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.

Improved hyperacuity estimation of spike timing from calcium imaging

“…Hyperacuity spike-time estimation algorithm combining the generative model and supervised learning approaches. HA_time is an example of the so-called super-resolution techniques, which have proven successful in many fields of signal processing 37,38 . Such techniques use a forward model for signal distortion due to sensing and sampling and recover the lost information by estimation of the model parameters maximizing the likelihood of the estimates.…”

High frequency spike inference with particle Gibbs sampling

“…By combining the expressions in Eqs. (10), (11) and (12) with the gamma prior gamma(α, β) and by keeping only terms proportional to r 0,1 we obtain…”

“…Fluorescence indicators of calcium activity allow us to monitor the dynamics of neuronal populations both in vivo and in vitro. In the last decade there has been a proliferation of new methods to identify single spikes from fluorescence time series using template matching [1][2][3][4], linear deconvolution [5][6][7][8], finite rate of innovation [9,10], independent component analysis [11], non model-based signal processing [12], supervised learning [13][14][15][16][17], constrained non-negative matrix factorization [18][19][20], active set methods [21,22], convex and non-convex optimization methods [23][24][25][26][27], interior point method [28]. Model-based approaches allow to frame the problem of spike inference in a Bayesian context and use maximum-a-posteriori estimates [29][30][31][32].…”

A Hierarchical Approach for Audio Capture, Archive, and Distribution

“…In the several decades since both Shannon [129] and Nyquist [130], there has been considerable development in understanding of sampling theory [129][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144]. Shannon's theorem shows how appropriate band-limiting allows repeated resampling of a signal without build-up of alias products.…”

“…While these concepts might surprise some, the theory of sampling has evolved considerably since Shannon and Nyquist. Moreover, in several other disciplines, such as image processing or astronomy, it has been found that undersampling can increase resolution with careful applicationspecific thinking [131][132][133][134][135][136][137][138][139][140][141][142][143][144].…”