Sparse nonnegative deconvolution for compressive calcium imaging: algorithms and phase transitions
We propose a compressed sensing (CS) calcium imaging framework for monitoring large neuronal populations, where we image randomized projections of the spatial calcium concentration at each timestep, instead of measuring the concentration at individual locations. We develop scalable nonnegative deconvolution methods for extracting the neuronal spike time series from such observations. We also address the problem of demixing the spatial locations of the neurons using rank-penalized matrix factorization methods. By exploiting the sparsity of neural spiking we demonstrate that the number of measurements needed per timestep is significantly smaller than the total number of neurons, a result that can potentially enable imaging of larger populations at considerably faster rates compared to traditional raster-scanning techniques. Unlike traditional CS setups, our problem involves a block-diagonal sensing matrix and a non-orthogonal sparse basis that spans multiple timesteps. We provide tight approximations to the number of measurements needed for perfect deconvolution for certain classes of spiking processes, and show that this number undergoes a "phase transition," which we characterize using modern tools relating conic geometry to compressed sensing.
[EDIT] I read the rebuttal and removed point 3 from my 
      review since it has been sufficiently addressed. Although I'm still not 
      100% happy with the responses to points 1 and 2 I upped the score to 8. 
      The authors propose a framework for two-photon imaging based 
      on compressed sensing. In the first half of the paper they use simulations 
      to demonstrate the feasibility of estimating calcium traces, non-negative 
      firing rates (closely corresponding to spike times in a high-SNR 
      situation) and spatial intensity profiles of the imaged neurons based on 
      random projections of the data. In the second part they show that in the 
      noiseless case the quality of reconstruction undergoes a phase transition. 
      They derive an approximation to the phase transition curve, which they 
      show to be quite accurate under some conditions. I am generally 
      enthusiastic about the approach since it could have a lot of potential to 
      substantially improve the efficiency of two-photon imaging experiments, 
      which generally suffer from the inherent trade-off between data quality 
      and number of neurons that can be imaged. Although I am unable to verify 
      the math, in particular in section 3, judging from the successful 
      verification through simulations I believe it's correct. Besides 
      the stylistic complaint that the paper is written in a rather technical 
      way with a lot of jargon, my major concern is that the authors did not do 
      a good job at convincing the reader that the approach will actually be 
      feasible or even useful in practice. There are three main issues related 
      to this concern: 1. Since the method is sensitive to noise (Fig. 
      1) it is important to test it with realistic noise levels. In practice, 
      noise levels are very high with calcium imaging due to shot noise, in 
      particular so if one images fast as the authors propose. Although I don't 
      know for sure what a typical SNR (as defined in line 189) would be for a 
      real experiment, my guess would be that it's closer to 0 dB than to 20 dB 
      as used in the Fig. 1 and I wouldn't be surprised if it's even 
      substantially lower than 0 dB. Since this is a very fundamental problem, 
      the authors should present some convincing evidence that the SNRs they use 
      are realistic. 2. For estimating the locations, it looks like 
      although the neurons were overlapping there was no background (neuropil) 
      activity between the neurons and the neurons were actually covering the 
      major fraction of the imaged space. However, in practice the situation is 
      dramatically different. In a 3d volume cell bodies make up at most 10% 
      (probably less) of the voxels while the remainder consists of neuropil 
      (axons and dendrites), which display calcium activity as well (although 
      usually highly correlated and with faster dynamics). Although I would 
      guess that the result would be a few (large) singular values that could be 
      excluded based on their spatial profile, I cannot judge whether the 
      algorithm (P-NN) is really not affected in a major way by this. In 
      addition, using SVD to identify neurons assumes they fire uncorrelated, 
      which is unlikely to be true for most experiments. Since one of the main 
      goals of imaging many neurons at the same time is to characterize their 
      joint activity, this could be a major caveat and should at least be 
      discussed. [3. removed since clarified by rebuttal] The 
      above concerns would be somewhat less of a problem if the goal of the 
      paper was to just lay out the basic mathematical framework, but then the 
      authors should be more upfront about the various practical problems that 
      may arise. Currently, the manuscripts reads somewhat like the method could 
      be used to build an actual imaging setup, but I think we are still 
      relatively far away from that goal.