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 study the effect of these distinctive features in a noiseless setup using recent results relating conic geometry to CS. We provide tight approximations to the number of measurements needed for perfect deconvolution for certain classes of spiking processes, and show that this number displays a "phase transition," similar to phenomena observed in more standard CS settings; however, in this case the required measurement rate depends not just on the mean sparsity level but also on other details of the underlying spiking process.