Each spatial footprint ${\mathit{𝒂}}_i$ should be spatially localized and sparse, since a given neuron will cover only a small fraction of the field of view, and therefore most elements of ${\mathit{𝒂}}_i$ will be zero. Thus, we need to incorporate spatial locality and sparsity constraints on $A$ (Pnevmatikakis et al., 2016). We discuss details further below.

Similarly, the temporal components ${\mathit{𝒄}}_i$ are highly structured, as they represent the cells’ fluorescence responses to sparse, nonnegative trains of action potentials. Following (Vogelstein et al., 2010; Pnevmatikakis et al., 2016), we model the calcium dynamics of each neuron ${\mathit{𝒄}}_i$ with a stable autoregressive (AR) process of order $p$, 

where $s_i(t)\ge 0$ is the number of spikes that neuron fired at the $t$-th frame. (Note that there is no further noise input into $c_i(t)$ beyond the spike signal $s_i(t)$.) The AR coefficients $\{{\gamma }_j^{(i)}\}$ are different for each neuron and they are estimated from the data. In practice, we usually pick $p=2$, thus incorporating both a nonzero rise and decay time of calcium transients in response to a spike; then Equation (3) can be expressed in matrix form as

The neural activity ${\mathit{𝒔}}_i$ is nonnegative and typically sparse; to enforce sparsity, we can penalize the ${\mathrm{\ell }}_0$ (Jewell and Witten, 2017) or ${\mathrm{\ell }}_1$ (Pnevmatikakis et al., 2016; Vogelstein et al., 2010) norm of ${\mathit{𝒔}}_i$, or limit the minimum size of nonzero spike counts (Friedrich et al., 2017b). When the rise time constant is small compared to the timebin width (low imaging frame rate), we typically use a simpler AR(1) model (with an instantaneous rise following a spike) (Pnevmatikakis et al., 2016).

In the above we have largely followed previously described CNMF approaches (Pnevmatikakis et al., 2016) for modeling calcium imaging signals. However, to accurately model the background effects in microendoscopic data, we need to depart significantly from these previous approaches. Constraints on the background term $B$ in Equation (1) are essential to the success of CNMF-E, since clearly, if $B$ is completely unconstrained we could just absorb the observed data $Y$ entirely into $B$, which would lead to recovery of no neural activity. At the same time, we need to prevent the residual of the background term (i.e. $B-\widehat{B}$, where $\widehat{B}$ denotes the estimated spatiotemporal background) from corrupting the estimated neural signals $AC$ in model (1), since subsequently, the extracted neuronal activity would be mixed with background fluctuations, leading to artificially high correlations between nearby cells. This problem is even worse in the microendoscopic context because the background fluctuation usually has significantly larger variance than the isolated cellular signals of interest (Figure 1D), and therefore any small errors in the estimation of $B$ can severely corrupt the estimated neural signal $AC$.

In (Pnevmatikakis et al., 2016), $B$ is modeled as a rank-$1$ nonnegative matrix $B=\mathit{𝒃}\cdot {\mathit{𝒇}}^T$, where $\mathit{𝒃}\in {ℝ}_{+}^d$ and $\mathit{𝒇}\in {ℝ}_{+}^T$. This model mainly captures the global fluctuations within the field of view (FOV). In applications to two-photon or light-sheet data, this rank-1 model has been shown to be sufficient for relatively small spatial regions; the simple low-rank model does not hold for larger fields of view, and so we can simply divide large FOVs into smaller patches for largely parallel processing (Pnevmatikakis et al., 2016; Giovannucci et al., 2017b). (See [Pachitariu et al., 2016] for an alternative approach.) However, as we will see below, the local rank-1 model fails in many microendoscopic datasets, where multiple large overlapping background sources exist even within modestly sized FOVs.

Thus, we propose a new model to constrain the background term $B$. We first decompose the background into two terms:

where $B^f$ represents fluctuating activity and $B^c={\mathit{𝒃}}_0\cdot {\mathrm{𝟏}}^T$ models constant baselines ($\mathrm{𝟏}\in {ℝ}^T$ denotes a vector of $T$ ones). To model $B^f$, we exploit the fact that background sources (largely due to blurred out-of-focus fluorescence) are empirically much coarser spatially than the average neuron soma size $l$. Thus, we model $B^f$ at one pixel as a linear combination of the background fluorescence in pixels which are chosen to be nearby but not nearest neighbors:

where ${\displaystyle {\mathrm{\Omega }}_i=\{j|\text{ dist}({\mathit{x}}_i,{\mathit{x}}_j)\in [l_n,l_n+1)\}}$, with $\text{dist}({\mathit{𝒙}}_i,{\mathit{𝒙}}_j)$ the Euclidean distance between pixel $i$ and $j$. Thus, ${\mathrm{\Omega }}_i$ only selects the neighboring pixels with a distance of $l_n$ from the $i$-th pixel (the green dot and black pixels in Figure 2B illustrate $i$ and ${\mathrm{\Omega }}_i$, respectively); here $l_n$ is a parameter that we choose to be greater than $l$ (the size of the typical soma in the FOV), e.g., $l_n=2l$. This choice of $l_n$ ensures that pixels $i$ and $j$ in Equation (6) share similar background fluctuations, but do not belong to the same soma.

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We can rewrite Equation (6) in matrix form:

where $W_{ij}=0$ if $\text{dist}({\mathit{𝒙}}_i,{\mathit{𝒙}}_j)\notin [l_n,l_n+1)$. In practice, this hard constraint is difficult to enforce computationally and is overly stringent given the noisy observed data. We relax the model by replacing the right-hand side $B^f$ with the more convenient closed-form expression

According to Equations (1) and (5), this change ignores the noise term $E$; since elements in $E$ are spatially uncorrelated, $W\cdot E$ contributes as a very small disturbance to ${\widehat{B}}^f$ in the left-hand side. We found this substitution for ${\widehat{B}}^f$ led to significantly faster and more robust model fitting.

We first use simulated data to illustrate the background model in CNMF-E and compare its performance against the low-rank NMF model used in the basic CNMF approach (Pnevmatikakis et al., 2016). We generated the observed fluorescence $Y$ by summing up simulated fluorescent signals of multiple sources as shown in Figure 1E plus additive Gaussian white noise (Figure 2A).

An example pixel (green dot, Figure 2A,B) was selected to illustrate the background model in CNMF-E (Equation (6)), which assumes that each pixel’s background activity can be reconstructed using its neighboring pixels’ activities. The selected neighbors form a ring and their distances to the center pixel are larger than a typical neuron size (Figure 2B). Figure 2C shows that the fluorescence traces of the center pixel and its neighbors are highly correlated due to the shared large background fluctuations. Here, for illustrative purposes, we fit the background by solving problem (P-B) directly while assuming $\widehat{A}\widehat{C}=0$. This mistaken assumption should make the background estimation more challenging (due to true neural components getting absorbed into the background), but nonetheless in Figure 2 we see that the background fluctuation was well recovered (Figure 2D). Subtracting this estimated background from the observed fluorescence in the center yields a good visualization of the cellular signal (Figure 2D). Thus, this example shows that we can reconstruct a complicated background trace while leaving the neural signal uncontaminated.

For the example frame in Figure 2A, the true cellular signals are sparse and weak (Figure 2E). When we subtract the estimated background using CNMF-E from the raw data, we obtain a good recovery of the true signal (Figure 2D,F). For comparison, we also estimate the background activity by applying a rank-$1$ NMF model as used in basic CNMF; the resulting background-subtracted image is still severely contaminated by the background (Figure 2G). This is easy to understand: the spatiotemporal background signal in microendoscopic data typically has a rank higher than one, due to the various signal sources indicated in Figure 1E), and therefore a rank-$1$ NMF background model is insufficient.

A naive approach would be to simply increase the rank of the NMF background model. Figure 2H demonstrates that this approach is ineffective: higher rank NMF does yield generally better reconstruction performance, but with high variability and low reliability (due to randomness in the initial conditions of NMF). Eventually as the NMF rank increases many single-neuronal signals of interest are swallowed up in the estimated background signal (data not shown). In contrast, CNMF-E recovers the background signal more accurately than any of the high-rank NMF models.

In real data analysis settings, the rank of NMF is an unknown and the selection of its value is a nontrivial problem. We simulated data sets with different numbers of local background sources and use a single parameter setting to run CNMF-E for reconstructing the background over multiple such simulations. Figure 2I shows that the performance of CNMF-E does not degrade quickly as we have more background sources, in contrast to rank-$1$ NMF. Therefore, CNMF-E can recover the background accurately across a diverse range of background sources, as desired.

Next, we used simulated data to validate our proposed initialization procedure (Figure 3A). In this example, we simulated 200 neurons with strong spatial overlaps (Figure 3B). One of the first steps in our initialization procedure is to apply a Gaussian spatial filter to the images to reduce the (spatially coarser) background and boost the power of neuron-sized objects in the images. In Figure 3C, we see that the local correlation image (Smith and Häusser, 2010) computed on the spatially filtered data provides a good initial visualization of neuron locations; compare to Figure 1B, where the correlation image computed on the raw data was highly corrupted by background signals.

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We choose two example ROIs to illustrate how CNMF-E removes the background contamination and demixes nearby neural signals for accurate initialization of neurons’ shapes and activity. In the first example, we choose a well-isolated neuron (green box, Figure 3A+B). We select three pixels located in the center, the periphery, and the outside of the neuron and show the corresponding fluorescence traces in both the raw data and the spatially filtered data (Figure 3D). The raw traces are noisy and highly correlated, but the filtered traces show relatively clean neural signals. This is because spatial filtering reduces the shared background activity and the remaining neural signals dominate the filtered data. Similarly, Figure 3E is an example showing how CNMF-E demixes two overlapping neurons. The filtered traces in the centers of the two neurons still preserve their own temporal activity.

After initializing the neurons’ traces using the spatially filtered data, we initialize our estimate of their spatial footprints. Note that simply initializing these spatial footprints with the spatially filtered data does not work well (data not shown), since the resulting shapes are distorted by the spatial filtering process. We found that it was more effective to initialize each spatial footprint by regressing the initial neuron traces onto the raw movie data (see Materials and methods for details). The initial values already match the simulated ground truth with fairly high fidelity (Figure 3D+E). In this simulated data, CNMF-E successfully identified all 200 neurons and initialized their spatial and temporal components (Figure 3F). We then evaluate the quality of initialization using all neurons’ spatial and temporal similarities with their counterparts in the ground truth data. Figure 3G shows that all initialized neurons have high similarities with the truth, indicating a good recovery and demixing of all neuron sources.

Thresholds on the minimum local correlation and the minimum peak-to-noise ratio (PNR) for detecting seed pixels are necessary for defining the initial spatial components. To quantify the sensitivity of choosing these two thresholds, we plot the local correlations and the PNRs of all pixels chosen as the local maxima within an area of $\frac{l}{4}\times \frac{l}{4}$, where $l$ is the diameter of a typical neuron, in the correlation image or the PNR image (Figure 3H). Pixels are classified into two classes according to their locations relative to the closest neurons: neurons’ central areas and outside areas (see Materials and methods for full details). It is clear that the two classes are linearly well separated and the thresholds can be chosen within a broad range of values (Figure 3H), indicating that the algorithm is robust with respect to these threshold parameters here. In lower SNR settings, these boundaries may be less clear, and an incremental approach (in which we choose the highest-SNR neurons first, then estimate the background and examine the residual to select the lowest-SNR cells) may be preferred; this incremental approach is discussed in more depth in the Materials and methods section.

Using the same simulated dataset as in the previous section, we further refine the neuron shapes ($A$) and the temporal traces ($C$) by iteratively fitting the CNMF-E model. We compare the final results with PCA/ICA analysis (Mukamel et al., 2009) and the original CNMF method (Pnevmatikakis et al., 2016).

After choosing the thresholds for seed pixels (Figure 3H), we run CNMF-E in full automatic mode, without any manual interventions. Two open-source MATLAB packages, CellSort (https://github.com/mukamel-lab/CellSort; Mukamel, 2016) and ca_source_extraction (https://github.com/epnev/ca_source_extraction; Pnevmatikakis, 2016), were used to perform PCA/ICA (Mukamel et al., 2009) and basic CNMF (Pnevmatikakis et al., 2016), respectively. Since the initialization algorithm in CNMF fails due to the large contaminations from the background fluctuations in this setting (recall Figure 2), we use the ground truth as its initialization. As for the rank of the background model in CNMF, we tried all integer values between 1 and 16 and set it as 7 because it has the best performance in matching the ground truth. We emphasize that including the CNMF approach in this comparison is not fair for the other two approaches, because it uses the ground truth heavily, while PCA/ICA and CNMF-E are blind to the ground truth. The purpose here is to show the limitations of basic CNMF in modeling the background activity in microendoscopic data.

We first pick three closeby neurons from the ground truth (Figure 4A, top) and see how well these neurons’ activities are recovered. PCA/ICA fails to identify one neuron, and for the other two identified neurons, it recovers temporal traces that are sufficiently noisy that small calcium transients are submerged in the noise. As for CNMF, the neuron shapes remain more or less at the initial condition (i.e. the ground truth spatial footprints), but clear contaminations in the temporal traces are visible. This is because the pure NMF model in CNMF does not model the true background well and the residuals in the background are mistakenly captured by neural components. In contrast, on this example, CNMF-E recovers the true neural shapes and neural activity with high accuracy.

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We also compare the number of detected neurons: PCA/ICA detected 195 out of 200 neurons, while CNMF-E detected all 200 neurons. We also quantitatively evaluated the performance of source extraction by showing the spatial and temporal cosine similarities between detected neurons and ground truth (Figure 4C); we find that the neurons detected using PCA/ICA have much lower similarities with the ground truth (Figure 4C). We also note that the CNMF results are much worse than those of CNMF-E here, despite the fact that CNMF is initialized at the ground truth parameter values. This result clarifies an important point: the improvements from CNMF-E are not simply due to improvements in the initialization step. Furthermore, running the full iterative pipeline of CNMF-E leads to improvements in both spatial and temporal similarities, compared with the results in the initialization step.

In many downstream analyses of calcium imaging data, pairwise correlations provide an important metric to study coordinated network activity (Warp et al., 2012; Barbera et al., 2016; Dombeck et al., 2009; Klaus et al., 2017). Since PCA/ICA seeks statistically independent components, which forces the temporal traces to have near-zero correlation, the correlation structure is badly corrupted in the raw PCA/ICA outputs (Figure 4D). We observed that a large proportion of the independence comes from the noisy baselines in the extracted traces (data not shown), so we postprocessed the PCA/ICA output by thresholding at the 3 standard deviation level. This recovers some nonzero correlations, but the true correlation structure is not recovered accurately (Figure 4D). By contrast, the CNMF-E results matched the ground truth very well due to accurate extraction of individual neurons’ temporal activity (Figure 4D). As for CNMF, the estimated correlations are slightly elevated relative to the true correlations. This is due to the shared (highly correlated) background fluctuations that corrupt the recovered activity of nearby neurons.

Next, we compared the performance of the different methods under different SNR regimes. Because of the above inferior results we skip comparisons to the basic CNMF here. Based on the same simulation parameters as above, we vary the noise level $\mathrm{\Sigma }$ by multiplying it with a SNR reduction factor. Figure 4E shows that CNMF-E detects all neurons over a wide SNR range, while PCA/ICA fails to detect the majority of neurons when the SNR drops to sufficiently low levels. Moreover, the detected neurons in CNMF-E preserve high spatial and temporal similarities with the ground truth (Figure 4F–G). This high accuracy of extracting neurons’ temporal activity benefits from the modeling of the calcium dynamics, which leads to significantly denoised neural activity. If we skip the temporal denoising step in the algorithm, CNMF-E is less robust to noise, but still outperforms PCA/ICA significantly (Figure 4G). When SNR is low, the improvements yielded by CNMF-E can be crucial for detecting weak neuron events, as shown in Figure 4H.

Finally, we examine the ability of CNMF-E to demix correlated and overlapping neurons. Using the two example neurons in Figure 3E, we ran multiple simulations at varying correlation levels and extracted neural components using the CNMF-E pipeline and PCA/ICA analysis. The spatial footprints in these simulations were fixed, but the temporal components were varied to have different correlation levels ($\gamma$) between calcium traces by tuning their shared component with the common background fluctuations. For high correlation levels ($\gamma >0.7$), the initialization procedure tends to first initialize a component that explains the common activity between two neurons and then initialize another component to account for the residual of one neuron. After iteratively refining the model variables, CNMF-E successfully extracted the two neurons’ spatiotemporal activity even at very high correlation levels ($\gamma =0.95$; Figure 5A,B). PCA/ICA was also often able to separate two neurons for large correlation levels ($\gamma =0.9$, Figure 5B), but the extracted traces have problematic negative spikes that serve to reduce their statistical dependences (Figure 4A).

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We now turn to the analysis of large-scale microendoscopic datasets recorded from freely behaving mice. We run both CNMF-E and PCA/ICA for all datasets and compare their performances in detail.

We begin by analyzing in vivo calcium imaging data of neurons expressing GCaMP6f in the mouse dorsal striatum. (Full experimental details and algorithm parameter settings for this and the following datasets appear in the Methods and Materials section.) CNMF-E extracted 692 putative neural components from this dataset; PCA/ICA extracted 547 components (starting from 700 initial components, and then automatically removing false positives using the same criterion as applied in CNMF-E). Figure 6A shows how CNMF-E decomposes an example frame into four components: the constant baselines that are invariant over time, the fluctuating background, the denoised neural signals, and the residuals. We highlight an example neuron by drawing its ROI to demonstrate the power of CNMF-E in isolating fluorescence signals of neurons from the background fluctuations. For the selected neuron, we plot the mean fluorescence trace of the raw data and the estimated background (Figure 6B). These two traces are very similar, indicating that the background fluctuation dominates the raw data. By subtracting this estimated background component from the raw data, we acquire a clean trace that represents the neural signal.

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To quantify the background effects further, we compute the contribution of each signal component in explaining the variance in the raw data. For each pixel, we compute the variance of the raw data first and then compute the variance of the background-subtracted data. Then the reduced variance is divided by the variance of the raw data, giving the proportion of variance explained by the background. Figure 6C (blue) shows the distribution of the background-explained variance over all pixels. The background accounts for around 90% of the variance on average. We further remove the denoised neural signals and compute the variance reduction; Figure 6C shows that neural signals account for less than 10% of the raw signal variance. This analysis is consistent with our observations that background dominates the fluorescence signal and extracting high-quality neural signals requires careful background signal removal.

The contours of the spatial footprints inferred by the two approaches (PCA/ICA and CNMF-E) are depicted in Figure 6D, superimposed on the correlation image of the filtered raw data. The indicated area was cropped from Figure 6A (left). In this case, most neurons inferred by PCA/ICA were inferred by CNMF-E as well, with the exception of a few components that seemed to be false positives (judging by their spatial shapes and temporal traces and visual inspection of the raw data movie; detailed data not shown). However, many realistic components were only detected by CNMF-E (shown as the green areas in Figure 6D). In these plots, we rank the inferred components according to their SNRs; the color indicates the relative rank (decaying from red to yellow). We see that the components missed by PCA/ICA have low SNRs (green shaded areas with yellow contours).

Figure 6E shows the spatial and temporal components of 14 example neurons detected only by CNMF-E. Here (and in the following figures), for illustrative purposes, we show the calcium traces before the temporal denoising step. For neurons that are inferred by both methods, CNMF-E shows significant improvements in the SNR of the extracted cellular signals (Figure 6F), even before the temporal denoising step is applied. In panel G we randomly select 10 examples and examine their spatial and temporal components. Compared with the CNMF-E results, PCA/ICA components have much smaller size, often with negative dips surrounding the neuron (remember that ICA avoids spatial overlaps in order to reduce nearby neurons’ statistical dependences, leading to some loss of signal strength; see (Pnevmatikakis et al., 2016) for further discussion). The activity traces extracted by CNMF-E are visually cleaner than the PCA/ICA traces; this is important for reliable event detection, particularly in low SNR examples. See Klaus et al., 2017) for additional examples of CNMF-E applied to striatal data.

We repeat a similar analysis on GCaMP6s data recorded from prefrontal cortex (PFC, Figure 7), to quantify the performance of the algorithm in a different brain area with a different calcium indicator. Again we find that CNMF-E successfully extracts neural signals from a strong fluctuating background (Figure 7A), which contributes a large proportion of the variance in the raw data (Figure 7B). Similarly as with the striatum data, PCA/ICA analysis missed many components that have very weak signals (33 missed components here). For the matched neurons, CNMF-E shows strong improvements in the SNRs of the extracted traces (Figure 7D). Consistent with our observation in striatum (Figure 6G), the spatial footprints of PCA/ICA components are shrunk to promote statistical independence between neurons, while the neurons inferred by CNMF-E have visually reasonable morphologies (Figure 6E). As for calcium traces with high SNRs (Figure 7E, cell 1-6), CNMF-E traces have smaller noise values, which is important for detecting small calcium transients (Figure 7E, cell 4). For traces with low SNRs (Figure 7, cell 7-10), it is challenging to detect any calcium events from the PCA/ICA traces due to the large noise variance; CNMF-E is able to visually recover many of these weaker signals. For those cells missed by PCA/ICA, their traces extracted by CNMF-E have reasonable morphologies and visible calcium events (Figure 7F).

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The demixing performance of PCA/ICA analysis can be relatively weak because it is inherently a linear demixing method (Pnevmatikakis et al., 2016). Since CNMF-E uses a more suitable nonlinear matrix factorization method, it has a better capability of demixing spatially overlapping neurons. As an example, Figure 7G shows three closeby neurons identified by both CNMF-E and PCA/ICA analysis. PCA/ICA forces its obtained filters to be spatially separated to reduce their dependence (thus reducing the effective signal strength), while CNMF-E allows inferred spatial components to have large overlaps (Figure 7G, left), retaining the full signal power. In the traces extracted by PCA/ICA, the component labeled in green contains many negative ‘spikes,’ which are highly correlated with the spiking activity of the blue neuron (Figure 7G, yellow). In addition, the green PCA/ICA neuron has significant crosstalk with the red neuron due to the failure of signal demixing (Figure 7G, cyan); the CNMF-E traces shows no comparable negative ‘spikes’ or crosstalk. See also Video 8 for further details.

In the previous two examples, we analyzed data with densely packed neurons, in which the neuron sizes are all similar. In the next example, we apply CNMF-E to a dataset with much sparser and more heterogeneous neural signals. The data used here were recorded from amygdala-projecting neurons expressing GCaMP6f in ventral hippocampus. In this dataset, some neurons that are slightly above or below the focal plane were visible with prominent signals, though their spatial shapes are larger than neurons in the focal plane.

This example is somewhat more challenging due to the large diversity of neuron sizes. It is possible to set multiple parameters to detect neurons of different sizes (or to e.g. differentially detect somas versus smaller segments of axons or dendrites passing through the focal plane), but for illustrative purposes here we use a single neural size parameter to initialize all of the components. This in turn splits some large neurons into multiple components. Following this crude initialization step, we updated the background component and then picked the missing neurons from the residual using a second greedy component initialization step. Next, we ran CNMF-E for three iterations of updating the model variables $A,C$, and $B$. The first two iterations were performed automatically; we included manual interventions (e.g. merging/deleting components) before the last iteration, leading to improved source extraction results (see Video 10 for details on the manual merge and delete interventions performed here). In this example, we detected 24 CNMF-E components and 24 PCA/ICA components. The contours of these inferred neurons are shown in Figure 8A. In total we have 20 components detected by both methods (shown in the first three rows of Figure 8B+C); each method detected extra components that are not detected by the other (the last rows of Figure 8B+C). Once again, the PCA/ICA filters contain many negative pixels in an effort to reduce spatial overlaps; see components 3 and 5 in Figure 8A–C, for example. All traces of the inferred neurons are shown in Figure 8D+E. We can see that the CNMF-E traces have much lower noise level and cleaner neural signals in both high and low SNR settings. Conversely, the calcium traces of the three extra neurons identified by PCA/ICA show noisy signals that are unlikely to be neural responses.

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Identifying neurons and extracting their temporal activity is typically just the first step in the analysis of calcium imaging data; downstream analyses rely heavily on the quality of this initial source extraction. We showed above that, compared to PCA/ICA, CNMF-E is better at extracting activity dynamics, especially in regimes where neuronal activities are correlated (c.f. Figure 4D). Using in vivo electrophysiological recordings, we previously showed that neurons in the bed nucleus of the stria terminalis (BNST) show strong responses to unpredictable footshock stimuli (Jennings et al., 2013). We therefore measured calcium dynamics in CaMKII-expressing neurons that were transfected with the calcium indicator GCaMP6s in the BNST and analyzed the synchronous activity of multiple neurons in response to unpredictable footshock stimuli. We chose 12 example neurons that were detected by both CNMF-E and PCA/ICA methods and show their spatial and temporal components in Figure 9A–C. The activity around the onset of the repeated stimuli are aligned and shown as pseudo-colored images in panel D. The median responses of CNMF-E neurons display prominent responses to the footshock stimuli compared with the resting state before stimuli onset. In comparison, the activity dynamics extracted by PCA/ICA have relatively low SNR, making it more challenging to reliably extract footshock responses. Panel E summarizes the results of panel D; we see that CNMF-E outputs significantly more easily detectable responses than does PCA/ICA. This is an example in which downstream analyses of calcium imaging data can significantly benefit from the improvements in the accuracy of source extraction offered by CNMF-E. (sheintuch2017tracking recently presented another such example, showing that more neurons can be tracked across multiple days using CNMF-E outputs, compared to PCA/ICA.)

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Microendoscopic calcium imaging offers unique advantages and has quickly become a critical method for recording large neural populations during unrestrained behavior. However, previous methods fail to adequately remove background contaminations when demixing single neuron activity from the raw data. Since strong background signals are largely inescapable in the context of one-photon imaging, insufficient removal of the background could yield problematic conclusions in downstream analysis. This has presented a severe and well-known bottleneck in the field. We have delivered a solution for this critical problem, building on the constrained nonnegative matrix factorization framework introduced in Pnevmatikakis et al., 2016 but significantly extending it in order to more accurately and robustly remove these contaminating background components.

The proposed CNMF-E algorithm can be used in either automatic or semi-automatic mode, and leads to significant improvements in the accuracy of source extraction compared with previous methods. In addition, CNMF-E requires very few parameters to be specified, and these parameters are easily interpretable and can be selected within a broad range. We demonstrated the power of CNMF-E using data from a wide diversity of brain areas (subcortical, cortical, and deep brain areas), SNR regimes, calcium indicators, neuron sizes and densities, and hardware setups. Among all these examples (and many others not shown here), CNMF-E performs well and improves significantly on the standard PCA/ICA approach. Considering that source extraction is typically just the first step in calcium imaging data analysis pipelines (Mohammed et al., 2016), these improvements should in turn lead to more stable and interpretable results from downstream analyses. Further applications of the CNMF-E approach appear in (Cameron et al., 2016; Donahue and Kreitzer, 2017; Jimenez et al., 2016; Jimenez et al., 2018; Klaus et al., 2017; Lin et al., 2017; Murugan et al., 2016; Murugan et al., 2017; Rodriguez-Romaguera et al., 2017; Tombaz et al., 2016; Ung et al., 2017; Yu et al., 2017; Mackevicius et al., 2017; Madangopal et al., 2017; Roberts et al., 2017; Ryan et al., 2017; Roberts et al., 2017; Sheintuch et al., 2017).

We have released our MATLAB implementation of CNMF-E as open-source software (https://github.com/zhoupc/CNMF_E (Zhou, 2017a)). A Python implementation has also been incorporated into the CaImAn toolbox (Giovannucci et al., 2017b). We welcome additions or suggestions for modifications of the code, and hope that the large and growing microendoscopic imaging community finds CNMF-E to be a helpful tool in furthering neuroscience research.

In problem (P-S), ${\mathit{𝒃}}_0$ is unconstrained and can be updated in closed form: ${\widehat{\mathit{𝒃}}}_0=\frac{1}{T}(\tilde{Y}-A\cdot \widehat{C}-{\widehat{B}}^f)\cdot \mathrm{𝟏}$. By plugging this update into problem (P-S), we get a reduced problem

where $\tilde{Y}=Y-{\widehat{B}}^f-\frac{1}{T}Y{\mathrm{𝟏𝟏}}^T$ and $\tilde{C}=\widehat{C}-\frac{1}{T}\widehat{C}{\mathrm{𝟏𝟏}}^T$. We approach this problem using a version of ”hierarchical alternating least squares’ (HALS; Cichocki et al., 2007), a standard algorithm for nonnegative matrix factorization. (Friedrich et al., 2017b) modified the fastHALS algorithm (Cichocki and Phan, 2009) to estimate the nonnegative spatial components $A,\mathit{𝒃}$ and the nonnegative temporal activity $C,\mathit{𝒇}$ in the CNMF model $Y=A\cdot C+\mathit{𝒃}{\mathit{𝒇}}^T+E$ by including sparsity and localization constraints. We solve a problem similar to the subproblem solved in Friedrich et al. (2017b):

where $P_k$ denotes the the spatial patch constraining the nonzero pixels of the $k$-th neuron and restricts the candidate spatial support of neuron $k$. This regularization reduces the number of free parameters in $A$, leading to speed and accuracy improvements. The spatial patches can be determined using a mildly dilated version of the support of the previous estimate of $A$ (Pnevmatikakis et al., 2016; Friedrich et al., 2017a).

In problem (P-T), the model variable $C\in {ℝ}^{K\times T}$ could be very large, making the direct solution of (P-T) computationally expensive. Unlike problem (P-S), the problem (P-T) cannot be readily parallelized because the constraints $G^{(i)}{\mathit{𝒄}}_i\ge 0$ couple the entries within each row of C, and the residual term couples entries across columns. Here, we follow the block coordinate-descent approach used in (Pnevmatikakis et al., 2016) and propose an algorithm that sequentially updates each ${\mathit{𝒄}}_i$ and ${\mathit{𝒃}}_0$. For each neuron, we start with a simple unconstrained estimate of ${\mathit{𝒄}}_i$, denoted as $\widehat{{\mathit{𝒚}}_i}$, that minimizes the residual of the spatiotemporal data matrix while fixing other neurons’ spatiotemporal activity and the baseline term ${\mathit{𝒃}}_0$,

where ${\displaystyle Y_{\text{res}}=Y-\hat{A}\hat{C}-{\hat{\mathit{b}}}_0{\mathbf{1}}^T-B^f}$ represents the residual given the current estimate of the model variables. Due to its unconstrained nature, ${\displaystyle {\hat{\mathit{y}}}_i}$ is a noisy estimate of ${\mathit{𝒄}}_i$, plus a constant baseline resulting from inaccurate estimation of ${\mathit{𝒃}}_0$. Given ${\widehat{\mathit{𝒚}}}_i$, various deconvolution algorithms can be applied to obtain the denoised trace ${\widehat{\mathit{𝒄}}}_i$ and deconvolved signal ${\widehat{\mathit{𝒔}}}_i$(Vogelstein et al., 2009; Pnevmatikakis et al., 2013; Deneux et al., 2016; Friedrich et al., 2017b; Jewell and Witten, 2017); in CNMF-E, we use the OASIS algorithm from (Friedrich et al., 2017b). (Note that the estimation of ${\mathit{𝒄}}_i$ is not dependent on accurate estimation of ${\mathit{𝒃}}_0$, because the algorithm for estimating ${\mathit{𝒄}}_i$ will also automatically estimate the baseline term in ${\widehat{\mathit{𝒚}}}_i$.) After the ${\mathit{𝒄}}_i$’s are updated, we update ${\mathit{𝒃}}_0$ using the closed-form expression ${\widehat{\mathit{𝒃}}}_0=\frac{1}{T}(\tilde{Y}-\widehat{A}\cdot \widehat{C}-{\widehat{B}}^f)\cdot \mathrm{𝟏}$.

Since problem (P-All) is not convex in all of its variables, a good initialization of model variables is crucial for fast convergence and accurate extraction of all neurons’ spatiotemporal activity. Previous methods assume the background component is relatively weak, allowing us to initialize $\widehat{A}$ and $\widehat{C}$ while ignoring the background or simply initializing it with a constant baseline over time. However, the noisy background in microendoscopic data fluctuates more strongly than the neural signals (c.f. Figure 6C and Figure 7B), which makes previous methods less valid for the initialization of CNMF-E.

Here, we design a new algorithm to initialize $\widehat{A}$ and $\widehat{C}$ without estimating $\widehat{B}$. The whole procedure is illustrated in Figure 10 and described in Algorithm 1. The key aim of our algorithm is to exploit the relative spatial smoothness in the background compared to the single neuronal signals visible in the focal plane. Thus, we can use spatial filtering to reduce the background in order to estimate single neurons’ temporal activity, and then initialize each neuron’s spatial footprint given these temporal traces. Once we have initialized $\widehat{A}$ and $\widehat{C}$, it is straightforward to initialize the constant baseline ${\mathit{𝒃}}_0$ and the fluctuating background $B^f$ by solving problem (P-B).

Figure 10

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We use iterative matrix updates to estimate model variables in CNMF-E. This strategy gives us the flexibility of integrating prior information on neuron morphology and temporal activity during the model fitting. The resulting interventions (which can in principle be performed either automatically or under manual control) can in turn lead to faster convergence and more accurate source extraction. We integrate 5 interventions in our CNMF-E implementation. Following these interventions, we usually run one more iteration of matrix updates.

The simplest pipeline for running CNMF-E includes the following steps:

1. Initialize $\widehat{A},\widehat{C}$ using the proposed initialization procedure.

2. Solve problem (P-B) for updates of ${\widehat{\mathit{𝒃}}}_0$ and ${\widehat{B}}^f$.

3. Iteratively solve problem (P-S) and (P-T) to update $\widehat{A},\widehat{C}$ and ${\mathit{𝒃}}_0$.

4. If desired, apply interventions to intermediate results.

5. Repeat steps 2, 3, and 4 until the inferred components are stable.

In practice, the estimation of the background $B$ (step 2) often does not vary greatly from iteration to iteration and so this step usually can be run with fewer iterations to save time. In practice, we also use spatial and temporal decimation for improved speed, following (Friedrich et al., 2017a). We first run the pipeline on decimated data to get good initializations, then we up-sample the results $\widehat{A},\widehat{C}$ to the original resolution and run one iteration of steps (2-3) on the raw data. This strategy improves on processing the raw data directly because downsampling increases the signal-to-noise ratio and eliminates many false positives.

Step 4 provides a fast method for correcting abnormal components without redoing the whole analysis. (This is an important improvement over the PCA/ICA pipeline, where if users encounter poor estimated components it is necessary to repeat the whole analysis with new parameter values, which may not necessarily yield improved cell segmentations.) The interventions described here themselves can be independent tasks in calcium imaging analysis; with further work we expect many of these steps can be automated. In our interface for performing manual interventions, the most frequently used function is to remove false positives. Again, components can be rejected following visual inspection in PCA/ICA analysis, but the performance of CNMF-E can be improved with further iterations after removing false positives, while this is not currently an option for PCA/ICA.

We have also found a two-step initialization procedure useful for detecting neurons: we first start from relatively high thresholds of $P_{\min }$ and $L_{\min }$ to initialize neurons with large activity from the raw video data; then we estimate the background components by solving problem (P-B); finally we can pick undetected neurons from the residual using smaller thresholds. We can terminate the model iterations when the residual sum of squares (RSS) stabilizes (see Figure 4B), but this is seldom used in practice because computing the RSS is time-consuming. Instead we usually automatically stop the iterations after the number of detected neurons stabilizes. If manual interventions are performed, we typically run one last iteration of updating $B,A$ and $C$ sequentially to further refine the results.

For all experimental data used in this work, we ran both CNMF-E and PCA/ICA. For CNMF-E, we chose parameters so that we initialized about 10–20% extra components, which were then merged or deleted (some automatically, some under manual supervision) to obtain the final estimates. Exact parameter settings are given for each dataset below. For PCA/ICA, the number of ICs were selected to be slightly larger than our extracted components in CNMF-E (as we found this led to the best results for this algorithm), and the number of PCs was selected to capture over 90% of the signal variance. The weight of temporal information in spatiotemporal ICA was set as 0.1. After obtaining PCA/ICA filters, we again manually removed components that were clearly not neurons based on neuron morphology.

We computed the SNR of extracted cellular traces to quantitatively compare the performances of two approaches. For each cellular trace $\mathit{𝒚}$, we first computed its denoised trace $\mathit{𝒄}$ using the selected deconvolution algorithm (here, it is thresholded OASIS); then the SNR of $\mathit{𝒚}$ is

For PCA/ICA results, the calcium signal $\mathit{𝒚}$ of each IC is the output of its corresponding spatial filter, while for CNMF-E results, it is the trace before applying temporal deconvolution, that is, ${\widehat{\mathit{𝒚}}}_i$ in Equation (9). All the data can be freely accessed online (Zhou et al., 2017).

All analyses were performed with custom-written MATLAB code. MATLAB implementations of the CNMF-E algorithm can be freely downloaded from https://github.com/zhoupc/CNMF_E (Zhou, 2017a). We also implemented CNMF-E as part of the Python package CaImAn (Giovannucci et al., 2017b), a computational analysis toolbox for large-scale calcium imaging and behavioral data (https://github.com/simonsfoundation/CaImAn [Giovannucci et al., 2017a]).

The scripts for generating all figures and the experimental data in this paper can be accessed from https://github.com/zhoupc/eLife_submission (Zhou, 2017b).