Sparse sampling: theory, methods and an application in neuroscience
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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.
Keywords
Sampling theory FRI Spike train inference Calcium transient1 Introduction
The world is analogue, but computation is digital. The process that bridges this gap is known as the sampling process and has been instrumental to the digital revolution of the past 60 years. Without the sampling process, we could not convert real-life signals in digital form, and without digital samples, we could not use computers for digital computation. The sampling process is also ubiquitous in that it is present in any mobile phone or digital camera but also in sophisticated medical devices like MRI or ultrasound machines, in sensor networks and in digital microscopes just to name a few examples.
Over the last six decades, our understanding of the conversion of continuous-time signal in discrete form has been heavily influenced by the Shannon–Whittaker–Kotelnikov sampling theorem (Shannon 1949; Whittaker 1929; Kotelnikov 1933; Unser 2000) which showed that the sampling and perfect reconstruction of signals are possible when the Fourier bandwidth or spectrum of the signal is finite. In this case, the signal is said to be bandlimited and must be sampled at a rate (Nyquist rate) at least twice its maximum nonzero frequency in order to reconstruct it without error.
We are so used to this approach that we tend to forget that it comes with many strings attached. First of all, there are no natural phenomena that are exactly bandlimited (Slepian 1976). Moreover, we tend to forget that the Shannon sampling theorem provides sufficient but not necessary conditions for perfect reconstruction. In other words, this theorem does not claim that it is not possible to sample and reconstruct non-bandlimited signals. It is therefore incorrect to assume that the bandwidth of a signal is related to its information content. Consider for instance the function shown in Fig. 1a. This is a stream of short pulses and appears in many applications including bio-imaging, seismic signals and spread-spectrum communication. If the pulse shape is known a priori, the signal is completely determined by the amplitude and location of such pulses. If there are at most \(K\) pulses in a unit interval, then the signal is completely specified by the knowledge of these \(2K\) parameters per unit of time. Assume now that the duration of the pulses is reduced but that the average number of pulses per unit interval stays the same. Clearly, the information content of the signal is not changing (still \(2K\) parameters per unit of time); however, its bandwidth is increasing (bandwidth increases when the support of a function decreases).
Consider, as second example, the signal shown in Fig. 2c. This is given by the sum of a bandlimited signal with a step function. Clearly, the step function has only two degrees of freedom: the discontinuity location and its amplitude. So, its information content is finite. The bandlimited function has a finite number of degrees of freedom per unit of time since it is fully determined by its samples at points spaced by the sampling period (given by the inverse of the Nyquist rate). We thus say that they both have a finite rate of innovation. However, the combination of these two functions leads to a signal with infinite bandwidth (see Fig. 2d). Now, if we were to relate the information content of the signal to its bandwidth, we would conclude incorrectly that this signal has an infinite rate of information since it requires an infinite sampling rate for perfect reconstruction. Therefore, bandwidth and information content are not always synonyms.
The paper is organised as follows. In the next section, we define FRI signals and give some examples. Section 3 presents the framework for sampling and reconstructing some classes of FRI signals. Specifically, we show how to sample and perfectly reconstruct a stream of Diracs and what are the conditions that the acquisition device has to satisfy. We also extend this framework to the case of streams of decaying exponentials and present some denoising strategies. Section 4 presents an algorithm to reconstruct streaming signals where there is no clear separation between consecutive bursts of spikes. Section 5 describes an application of this theory to monitor neural activity from two-photon calcium images. Finally, conclusions are drawn in Sect. 6.
1.1 Notations
For \(f(t) \in \varvec{L^2}(\mathbb {R})\), where \(\varvec{L^2}(\mathbb {R})\) is the Hilbert space of finite-energy functions, the Fourier transform of \(f(t)\) is denoted by \(\hat{f}(\omega )\) and is given by \(\hat{f}(\omega ) = \mathcal {F}\lbrace f(t)\rbrace = \int _{-\infty }^{+\infty } f(t) e^{-i \omega t} \hbox {d}t\). If \(f(t)\) is complex-valued, \(f^*(t)\) denotes its complex conjugate. The Hermitian inner product is \(\left\langle f\,,\,g\right\rangle = \int _{-\infty }^{+\infty } f(t) g^*(t) \hbox {d}t\). The indicator function is denoted by \(\varvec{1}_A(t)\) and is given by \(\varvec{1}_A(t)=1\) if \(t \in A\), and \(\varvec{1}_A(t)=0\) if \(t \notin A\). \(\delta _{i,j}\) denotes the Kronecker delta, which is defined as \(\delta _{i,j} = 1\) if \(i=j\) and 0 otherwise. \(\lfloor {\cdot }\rfloor \) and \(\lceil {\cdot }\rceil \) denote the floor and ceil functions.
2 Finite rate of innovation signals
- Stream of pulses: \(x(t)=\sum _k a_k \, p(t-t_k)\). For instance, stream of decaying exponentials:which are a good fit for calcium transient signals induced by neural activity in two-photon calcium imaging. Figure 1a, b are examples of such signals.$$\begin{aligned} x(t)=\sum _{k} a_k \, e^{-(t-t_k) / \tau } \, \varvec{1}_{t \ge t_k}, \end{aligned}$$(4)
- Piecewise sinusoidal signals (see Fig. 1c):$$\begin{aligned} x(t) = \sum _k \sum _r a_{k,r} \, e^{i (\omega _{k,r}t + \phi _{k,r})} \, \varvec{1}_{[t_k,t_{k+1})}(t). \end{aligned}$$(5)
- Stream of Diracs (see Fig. 1d):$$\begin{aligned} x(t) = \sum _k a_k \, \delta (t-t_k). \end{aligned}$$(6)
3 Sampling scheme
- Exponential reproducing property: Any function \(\varphi (t)\) that together with its shifted versions can reproduce exponential functions of the form \(e^{\alpha _m t}\) with \(\alpha _m \in \mathbb {C}\) and \(m = 0,1,\ldots ,P\):$$\begin{aligned} \sum _{n\in \mathbb {Z}} c_{m,n} \,\varphi (t-n) = e^{\alpha _m t}, \quad m = 0,1,\ldots ,P. \end{aligned}$$(9)
3.1 Exponential reproducing kernels
For the sake of clarity, in what follows, we restrict the analysis to the case where the parameter \(\alpha _m\) in (9) is purely imaginary, that is \(\alpha _m = i \omega _m\) for \(m = 0, 1, \ldots , P\), where \(\omega _m\in \mathbb {R}\). This analysis can easily be extended to the more general case where \(\alpha _m\) has nonzero real and imaginary parts, or is purely real.
3.1.1 Sampling with an exponential reproducing kernel
The choice of purely imaginary parameters \(\alpha _m = i \omega _m\) leads to an important family of sampling kernels. These design parameters directly determine the information of the input analogue signal \(x(t)\) that we acquire and allow us to perfectly reconstruct the input signal from the discrete samples \(y_n\) for some classes of signals. Specifically, the different \(\omega _m\) correspond to the frequencies of the Fourier transform of \(x(t)\) that we are able to retrieve from the only knowledge of samples \(y_n\). It can be shown that if parameters \(\alpha _m\) are real or appear in complex conjugate pairs, the corresponding E-spline is real. We thus impose that for all \(\alpha _m\) that are nonzero, their complex conjugates are also present in \(\varvec{\alpha }\). If parameters \(\alpha _m = i \omega _m\) in vector \(\varvec{\alpha }\) are sorted in increasing order of \(\omega _m\), we have that \(\alpha _m^* = \alpha _{P-m}\).
3.1.2 Computation of \(c_{m,n}\) coefficients
From (20) and (17), we can compute the \(c_{m,n}\) coefficients for our choice of \(\left( \alpha _m\right) _{m=0}^{P}\) and any value of \(n \in \mathbb {Z}\). By combining these coefficients with \(\lbrace \varphi (t-n)\rbrace _{n\in \mathbb {Z}}\), the exponentials \(\lbrace e^{\alpha _m t} \rbrace _{m=0}^P\) are perfectly reproduced as shown in Fig. 4.
3.1.3 Approximate reproduction of exponentials
The generalised Strang-Fix conditions (10) impose restrictive constraints on the sampling kernel. This becomes a problem when we do not have control or flexibility over the design of the acquisition device. Recent publications (Urigüen et al. 2013; Dragotti et al. 2013) show that these conditions can be relaxed and still have a very accurate exponential reproduction, which is the property we require in order to reconstruct the analogue input signal. The first part of the Strang-Fix conditions, that is \(\hat{\varphi }(\omega _m) \ne 0\), is easy to achieve, but the second part is harder to guarantee when we do not have control over the sampling device.
In the case of the Gaussian filter, we can easily obtain the \(c_{m,n}\) coefficients of the exponentials to be reproduced since we have an analytical expression for its Fourier transform. When an analytic expression is unknown, we can still apply this approach since we only need knowledge of the transfer function of the acquisition device at frequencies \(\omega = \omega _m\). The \(c_{m,n}\) coefficients are then given by (22).
The approximate Strang-Fix framework is therefore very attractive since it allows us to use the theory discussed so far with any acquisition device.
3.2 Perfect reconstruction of FRI signals
In the previous section, we have seen some properties of exponential reproducing kernels. We have also seen that if the sampling kernel satisfies the exponential reproducing property, we can obtain some samples of the Fourier transform of the input analogue signal from the measurements \(\left( y_n\right) _{n=1}^{N}\) that result from the sampling process. We now show how this partial knowledge of the Fourier transform can be used to perfectly reconstruct some classes of band unlimited signals.
3.2.1 Perfect reconstruction of a stream of Diracs
We assume that the input signal is a stream of Diracs: \(x(t) = \sum _{k=1}^K a_k \, \delta (t-t_k)\), and that the sampling kernel \(\varphi (t)\) satisfies the exponential reproduction property for a choice of \(\varvec{\alpha } = \left( \alpha _m \right) _{m=0}^P\) such that \(\alpha _m = i \omega _m\), where \(\omega _m \in \mathbb {R}\) for \(m = 0, 1, \ldots , P\). We further impose the frequencies \(\omega _m\) to be equispaced, that is \(\omega _{m+1} - \omega _{m} = \lambda \), and to be symmetric, that is \(\omega _m = -\omega _{P-m}\). We thus have \(\omega _m = \omega _0 + \lambda m\) and \(\omega _P = -\omega _0\).
The system of equations (27) requires at least \(2K\) consecutive values \(s_m\). Recall that the sequence \(s_m\) is obtained as follows \(s_m = \sum _{n=1}^{N} c_{m,n}\,y_n\), with \(m=0,1,\ldots ,P\), where \(P+1\) is the number of exponentials reproduced by the sampling kernel. We thus have a lower bound on the number of exponentials that the sampling kernel has to reproduce: \(P+1 \ge 2K\). The perfect reconstruction of a stream of Diracs is summarised in the following theorem.
Theorem 1
Consider a stream \(x(t)\) of K Diracs: \(x(t) = \sum _{k=1}^{K} a_k \, \delta (t-t_k)\), and a sampling kernel \(\varphi (t)\) that can reproduce exponentials \(e^{\,i(\omega _0+\lambda \,m)t}\), with \(m=0,1,\ldots ,P\), and \(P+1\ge 2K\). Then, the samples defined by \(y_n = \left\langle x(t)\,,\,\varphi (t/T-n)\right\rangle \) are sufficient to characterise \(x(t)\) uniquely.
3.2.2 Perfect reconstruction of a stream of decaying exponentials
Streams of Diracs are an idealisation of streams of pulses. Although this example may seem limited, the framework presented so far can be applied to other classes of functions that model a variety of signals. For instance, calcium concentration measurements obtained from two-photon imaging to track the activity of individual neurons can be modelled with a stream of decaying exponentials. In this model, the time delays correspond to the activation time of the tracked neuron, that is, the action potentials (AP).
3.3 FRI signals with noise
The denoising strategies that can be applied to improve the performance of the reconstruction process come from the spectral analysis community, where the problem of finding sinusoids in noise has been extensively studied. There are two main approaches. The first, named Cadzow denoising algorithm, is an iterative procedure applied to the Toeplitz matrix constructed from samples \(s_m\) as in (27). Let us denote by \(\varvec{S}\) this matrix. By construction, this matrix is Toeplitz, and in the noiseless case, it is of rank \(K\). The presence of noise makes this matrix be full rank. The Cadzow algorithm (Cadzow 1988) looks for the closest rank deficient matrix which is Toeplitz. At each step, we force matrix \(\varvec{S}\) to be of rank \(K\) by computing the singular value decomposition (SVD) and only keeping the \(K\) largest singular values and setting the rest to zero. This new matrix is not Toeplitz anymore, we thus compute a new Toeplitz matrix by averaging the diagonal elements. This last matrix might not be rank deficient, and we can thus iterate again. The next step is to solve equation (27). This is done computing the total least squares solution that minimises \(||\varvec{S} \varvec{h}||^2\) subject to \(||\varvec{h}||^2=1\), where \(\varvec{h}\) is an extended version of the vector in (27) and has length \(K+1\). If this vector is normalised with respect to the first element, we have that the following \(K\) elements correspond to the coefficients \(h_k\) in (26). This approach has successfully been applied in the FRI setup in (Blu et al. 2008).
The second approach is based on subspace techniques for estimating generalised eigenvalues of matrix pencils (Hua and Sarkar 1990, 1991). Such approach has also been applied in the FRI framework (Maravić and Vetterli 2005). This method is based on the particular structure of the matrix \(\varvec{S}\), which is Toeplitz and each element is given by a sum of exponentials. Let \(\varvec{S}_0\) be the matrix constructed from \(\varvec{S}\) by dropping the first row and \(\varvec{S}_1\) the matrix constructed from \(\varvec{S}\) by dropping the last row. It can be shown that in the matrix pencil \(\varvec{S}_0 - \mu \varvec{S}_1\) the parameters \(\left\{ u_k \right\} _{k=1}^K\) from (25) are rank reducing numbers, that is, the matrix \(\varvec{S}_0 - \mu \varvec{S}_1\) has rank \(K-1\) for \(\mu = u_k\) and rank \(K\) otherwise. The parameters \(\left\{ u_k \right\} _{k=1}^K\) are thus given by the eigenvalues of the generalised eigenvalue problem \((\varvec{S}_0 - \mu \varvec{S}_1)\varvec{v} = 0\).
Further variations of these two fundamental approaches have been proposed recently. See for example Tan and Goyal (2008), Erdozain and Crespo (2011), Hirabayashi et al. (2013).
4 Sampling streaming FRI signals
If the stream is made of clearly separable bursts, we can apply the previously described strategy by assuming that each burst has a maximum number of spikes. However, when this separation cannot be made because of the presence of noise, or due to the nature of the signal itself, this strategy is not valid. The infinite stream presents an obvious constraint due the number of parameters that have to be recovered. We have seen that the order of the sampling kernel, \(P\), and its support are directly related to the number of parameters to be estimated. However, we cannot increase \(P\) indefinitely. In order to handle this type of signals, we thus consider a sequential and local approach (Oñativia et al. 2013b).
4.1 Sliding window approach
5 Application to neuroscience
To understand how neurons process information, neuroscientists need accurate information about the firing of action potentials (APs of spikes) by individual neurons. We thus need techniques that allow to monitor large areas of the brain with a spatial resolution that distinguishes single neurons and with a temporal resolution that resolves APs. Of the currently available techniques, only multiphoton calcium imaging (Denk et al. 1990, 1994; Svoboda et al. 1999; Stosiek et al. 2003) and multielectrode array electrophysiology (Csicsvari et al. 2003; Blanche et al. 2005; Du et al. 2009) offer this capability. Of these, only multiphoton calcium imaging currently allows precise three-dimensional localisation of each individual monitored neuron within the region of tissue studied, in the intact brain. Populations of neurons are simultaneously labelled with a fluorescent indicator, acetoxy-methyl (AM) ester calcium dyes (Stosiek et al. 2003). This allows simultaneous monitoring of action potential-induced calcium signals in a plane (Ohki et al. 2005) or volume (Göbel and Helmchen 2007) of tissue. The calcium concentration is measured with a laser-scanning two-photon imaging system.
A number of methods have previously been used to detect spike trains from calcium imaging data, including thresholding the first derivative of the calcium signal (Smetters et al. 1999), and the application of template-matching algorithms based on either fixed exponential (Kerr et al. 2005, 2007; Greenberg et al. 2008) or data-derived (Schultz et al. 2009; Ozden et al. 2008) templates. Machine learning techniques (Sasaki et al. 2008) and probabilistic methods based on sequential Monte Carlo framework (Vogelstein et al. 2009) or fast deconvolution (Vogelstein et al. 2010) have also been proposed. Some broadly used methods such as template matching or derivative-thresholding have the disadvantage that they do not deal well with multiple events occurring within a time period comparable to the sampling interval. Our spike detection algorithm is based on connecting the calcium transient estimation problem to the theory of FRI signals. The calcium concentration model in (38) is clearly a FRI signal, we can thus apply the techniques presented in the previous sections.
5.1 Spike inference algorithm
The spike inference algorithm is based on applying the sliding window approach presented in Sect. 4.1 combined with the reconstruction of streams of decaying exponentials presented in Sect. 3.2.2. One major issue of the framework presented so far is that we have assumed the number \(K\) of spikes within a time window to be known a priori. In practice, this is a value that has to be estimated.
To overcome these inaccuracies, we make the algorithm more robust by applying a double consistency approach. We run the sliding window approach presented in Sect. 4.1 twice. First, with a sufficiently big window where we estimate \(K\) from the singular values of \(\varvec{S}\). Second, with a smaller window where we assume that we only capture one spike and therefore we always set \(K=1\). We then build a joint histogram out of all the locations retrieved from both approaches and estimate the spikes from the peaks of the histogram. This approach is illustrated in Figs. 12 and 13 with real data.
This technique is fast and robust in high noise and low temporal resolution scenarios. It is able to achieve a detection rate of 84 % of electrically confirmed AP with real data (Oñativia et al. 2013a), outperforming other state of the art real-time approaches. Due to its low complexity, tens of streams can be processed in parallel with a commercial off-the-shelf computer.
6 Conclusions
We have presented a framework to sample and reconstruct signals with finite rate of innovation. We have shown that it is possible to sample and perfectly reconstruct streams of Diracs, and more importantly, streams of decaying exponentials. The latter offer a perfect fit for calcium transients induced by the spiking activity of neurons. The presented approach is sequential, and the reconstruction is local. These two features make the overall algorithm resilient to noise and have low complexity offering real-time capabilities.
The theoretical framework, where perfect reconstruction can be achieved, is also extended to the more realistic case where we do not have full control over the sampling kernel. In this case, perfect reconstruction cannot be guaranteed, but we can still reconstruct the underlying analogue signal with high precision if the sampling kernel can reproduce exponentials approximately.
Footnotes
- 1.
For appropriate functions \(f\), the Poisson summation formula is given by: \(\sum _{n=-\infty }^{+\infty } f (t - nT) = \frac{1}{T} \sum _{k=-\infty }^{+\infty } \hat{f} \left( \tfrac{2 \pi k}{T}\right) \, e^{i 2\pi k t / T}\).
Notes
Acknowledgments
This work was supported by European Research Council (ERC) starting investigator award Nr. 277800 (RecoSamp).
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