Scaling the Poisson GLM to massive neural datasets through polynomial approximations
Recent advances in recording technologies have allowed neuroscientists to record simultaneous spiking activity from hundreds to thousands of neurons in multiple brain regions. Such large-scale recordings pose a major challenge to existing statistical methods for neural data analysis. Here we develop highly scalable approximate inference methods for Poisson generalized linear models (GLMs) that require only a single pass over the data. Our approach relies on a recently proposed method for obtaining approximate sufficient statistics for GLMs using polynomial approximations [7], which we adapt to the Poisson GLM setting. We focus on inference using quadratic approximations to nonlinear terms in the Poisson GLM log-likelihood with Gaussian priors, for which we derive closed-form solutions to the approximate maximum likelihood and MAP estimates, posterior distribution, and marginal likelihood. We introduce an adaptive procedure to select the polynomial approximation interval and show that the resulting method allows for efficient and accurate inference and regularization of high-dimensional parameters. We use the quadratic estimator to fit a fully-coupled Poisson GLM to spike train data recorded from 831 neurons across five regions of the mouse brain for a duration of 41 minutes, binned at 1 ms resolution. Across all neurons, this model is fit to over 2 billion spike count bins and identifies fine-timescale statistical dependencies between neurons within and across cortical and subcortical areas.
=== Update after author rebuttal === Thanks to the authors for their rebuttal. I have updated my score. One comment about the fully coupled GLM: since the validation data was taken from a specific portion of the experiment (minutes 45-46), then temporal non-stationarities in the data (e.g. changes in arousal state, adaptation, etc.) will make it harder to generalize. Instead, I would recommend that the authors hold out brief snippets of data throughout the experiment to use for validation and testing.  === Summary === This paper develops a method for scalable inference methods for Poisson generalized linear models (GLMs). These are particularly prevalent in the neuroscience literature. The authors use the polynomial approximate sufficient statistics (PASS-GLM) method to perform approximate Bayesian inference in these models. They apply this method to two neuroscience datasets: modeling primate retinal ganglion cell responses to white noise (here, the authors validate the approximate methods using exact inference) as well as a dataset of spontaneous activity recorded across brain regions in the mouse.  === Review ===  -- Clarity -- The paper is clear and well written. In particular, I appreciated the thorough background section, especially the comparison with the MELE.  -- Originality -- While the applications in neuroscience are new, much of the methods in this paper are directly taken from the PASS-GLM paper. At times, I found it hard to distinguish what (if any) of the method was new, versus deriving the PASS-GLM method for Poisson GLMs specifically.  Note that although the experiments in the PASS-GLM paper focus on logistic regression, that paper makes it clear that the method is appropriate for any model in the GLM family, including Poisson regression.  Therefore, it seems like much of the novelty of this paper is in the applications (specifically, the application to the large neuropixels dataset recorded across multiple mouse brain regions). Unfortunately, this occupies a small fraction of the overall paper.  -- Significance -- In this paper, the authors state (in the introduction) that their contributions are: (a) deriving the expressions for the MLE, MAP, and approximate posterior of a Poisson GLM using the PASS-GLM method, (b) validate the estimators on simulated data and retinal data, (c) introduce two novel extensions of the method, an analytic marginalization when using quadratic approximations and an adaptive procedure for selecting the interval of the polynomial approximation, and (d) apply these methods to study a large dataset recorded across multiple brain regions.  Given that most of the method is derived from the ideas in the PASS-GLM paper, I think (a) and (b) are less significant results.  Moreover, it is hard to judge the significance of the extensions to the method (c). For example, the authors did not report compare their results on using the adaptive intervals in section 4.2 to using fixed intervals, instead simply stating that they "find that this procedure provides large improvements" (line 185). Large improvements for which application? Simulated data? Retinal data? Mouse data? If the authors had a figure demonstrating this large improvement on real datasets with the adaptive interval, that would greatly increase the significance of this particular contribution.  Finally, while the application to mouse data (d) is certainly novel, it is equally hard to assess the significance of these findings. Namely, I think the authors should explicitly answer the following question: what scientific question would neuroscientists be able to answer with the approximate inference Poisson GLM that they could not answer otherwise? For example, am I unable to fit the GLM to the dataset using current techniques? Will it just take longer? How much longer? I would appreciate if there was a rough sense of how long it would take to do the exact GLM given N neurons and T time bins, compared to the approximate version.  This presupposed that I even want to fit a massive GLM. The application to the mouse dataset, to me, suggests that this might not be the best idea, as I was quite surprised at how much the GLM overfit to the dataset. The authors report that the model outperforms a homogenous GLM on only 65% of the neurons. Does this mean that the model is strongly overfitting? Shouldn't the hyperparameter search for the regularization parameters have dealt with this? Where was the 1 minute of held out data taken from (randomly throughout the recording, or at the beginning or end?). What if the authors did not fit coupling parameters between all neurons, instead fitting separate GLMs to each area? In that case, are there significant differences in predictive power? What if the full model was uncoupled (only spike history filters)? Also, I would have appreciated more interpretation of the model filters and coupling strengths presented in Fig 4d and 4c. Do these tell us something we did not already know? Do they look significantly different if I took subsets of neurons or data (such that I could do exact inference)?  I can appreciate if the authors' goal was to demonstrate that the approximate GLM works on large datasets, rather than analyze this particular GLM model fit, but it is hard to make sense of the significance of being able to fit a large GLM without more exploration of these details.  === General comments === This paper does a good job of taking a method recently published in the machine learning community (PASS-GLM) and shows how it may be relevant for neuroscience applications.  What I found most difficult about this work is that it is hard to tell which audience this is best suited for. I does not seem like the work is really about novel extensions to PASS-GLM that would be of interest to the machine learning community. Instead, it seems to target those in the neuroscience community who would be interested in building large scale GLMs of their data. However, for a neuroscientist, interested in applications, I think it is difficult to work through the derivations in sections 3 and 4 without a clear understanding of what the approximate GLM buys you, beyond being able to fit a GLM to a large recording (and it is unclear what that buys you in terms of new scientific understanding).  I think the paper could be improved by focusing more on applications to neural datasets, with more rigorous comparisons for the new ideas presented in this paper, as well as clear statements about what the approximate GLM buys you in terms of being able to ask new scientific questions.  The derivation of the approximate Poisson GLM (section 3), while valuable for those interested in applying the method, is harder to slog through when it is unclear what the method buys you (speaking as a neuroscientist). Perhaps focusing on applications and gains over previous methods first, and using that to motivate the derivation, would be more worthwhile for the community.