Model-based targeted dimensionality reduction for neuronal population data
Summarizing high-dimensional data using a small number of parameters is a ubiquitous first step in the analysis of neuronal population activity. Recently developed methods use "targeted" approaches that work by identifying multiple, distinct low-dimensional subspaces of activity that capture the population response to individual experimental task variables, such as the value of a presented stimulus or the behavior of the animal. These methods have gained attention because they decompose total neural activity into what are ostensibly different parts of a neuronal computation. However, existing targeted methods have been developed outside of the confines of probabilistic modeling, making some aspects of the procedures ad hoc, or limited in flexibility or interpretability. Here we propose a new model-based method for targeted dimensionality reduction based on a probabilistic generative model of the population response data. The low-dimensional structure of our model is expressed as a low-rank factorization of a linear regression model. We perform efficient inference using a combination of expectation maximization and direct maximization of the marginal likelihood. We also develop an efficient method for estimating the dimensionality of each subspace. We show that our approach outperforms alternative methods in both mean squared error of the parameter estimates, and in identifying the correct dimensionality of encoding using simulated data. We also show that our method provides more accurate inference of lowdimensional subspaces of activity than a competing algorithm, demixed PCA.
The paper introduces an extension of an existing model, targeted dimensionality reduction (TDR), developed by Mante and colleagues in 2013. TDR is a regression based method to perform dimensionality reduction of neural data sets. Unlike PCA, which finds a basis with components sorted by the amount of total variance captured in the data, TDR identifies dimensions that specifically capture task-related variance, regardless of how much of the total variability they explain. This method assigns a single dimension per task variable, which as the authors explain, presents a clear limitation. There is no reason to believe that a given external covariate would elicit only one mode of activation in the neural population. Furthermore, this restriction implies that an important amount of variance in the data is potentially being missed. Therefore, by considering multi-dimensional task-related subspaces, and by identifying a low-rank factorization of the matrices defining such subspaces, the author's method MBTDR provides a means to discover additional structure in the population, which is of key importance in neuroscience.  The features that MBTDR incorporates and are worth being highlighted are  1. They model is a probabilistic extension of TDR that defines a generative model of the data. This allows to incorporate a wide variety of assumptions, such as specifying different noise models or pre-defining structure using different prior settings.  2. The model can be fit to single trial data, even when there are unequal amounts of trials or missing observations. This is often the case in many data sets, as neurons are typically not simultaneously recorded, so the models are fit to the PSTHs, which are averaged responses. This averaging can destroy valuable information about the correlational structure in the population.  3. The authors provide a means of estimating the rank of the task subspaces. This is not straightforward as it involves searching over many possible rank configurations. The greedy algorithm proposed seems to be a reasonable approach to obtain optimal estimates for dimensionality. Similar methods like dPCA do not provide such a feature, although this can be accounted for post-hoc by keeping the number of dimensions needed to explain a certain percentage of the variance. This procedure, however, does not assess optimality.  As the authors acknowledge, dPCA offers similar modelling advantages over TDR as MBTDR does. In particular dPCA is also able to identify multi-dimensional task subspaces. The authors go on by stating the limitations of dPCA with respect to their own method and compare the performance of the two models. They do so by testing how well the models recover the different parameter subspaces on generated data. However, given that the data was generated from the model itself -and in spite of the noise corruption-, I am not sure whether it is indeed surprising that their model can recover the subspaces better than dPCA. Similar arguments apply to figure 2A (although I guess this is partially addressed with the estimation under Poisson noise). Therefore, from the simulation studies, it was not clear to me whether MBTDR provides an advantage over dPCA at the task both dPCA and TDR were designed for: that of demixing variance from different task parameters. An example showing this would have been more illustrative, or at least a clearer reference to this in the text.   I also could not really interpret this couple of sentences: "It is notable that the principle angle is an extremely sensitive measure of errors between subspaces and that both methods provide reasonable results when checked by eye. It is also notable that any differences are observable at all, which give us confidence that these results are quite strong."  Finally, it is perhaps worth mentioning one limitation of these type of modelling approaches, which is that, no matter how much we increase the dimensionality, we are still dealing with a linear method, so non-linear relationships with respect to the external covariates cannot be captured.    Regarding:  -quality The model setup and fitting procedure seems rigorous. The authors used standard methods both for parameter estimation and rank estimation. However, I was not entirely familiar with some of the procedures and I did not check the math in detail, so I would not say I am confident about the methods in general. In any case, the authors seem to provide enough information to reproduce the study (provided that the code is released).  -clarity  The paper is well written, clear and easy to read. The figures are also good. I think most readers will appreciate figure 1A, nice visual illustration of the decomposition!  -originality and significance The model is an extension of an existing method, so the study is not meant to be particularly original... but all the additional features that the model incorporates makes it an important contribution.   minor clarification:  line 219, SNR (in log10 units) is negative. These are high levels of noise, right? Any criteria for how this noise was set? how did parameter recovery depend on this?  typos: line 169: TDA --> TDR line 192: decent --> descent line 300: a using (superfluous), dimension reduction —> dimensionality reduction figure 1B/2A: legend, EM --> ECME (if you wish...)   In conclusion, I believe that the study is an important contribution to both neuroscience and machine learning. I did not identify any methodological flaws and the analysis seemed accurate. My only objection is whether the paper makes a strong enough neuroscientific point. First, regarding its superiority to identify neural structure and second, demonstrating that it is able to trade-off variance from different parameters -this last point if one wish to compare it to dPCA and TDR-. Except from this issue, which hopefully the authors can address, I would accept the submission with confidence.  --------------------------------------------- Comments on authors' rebuttal   The authors have satisfactorily addressed all the issues we raised. If the authors include additional analysis on one of the publicly available datasets from the dPCA eLife paper, that would resolve our biggest concern. Finally, making this method available to the neuroscience community will definitely lead to new insights when applied to different data sets and can guide the development of additional methods to analyse neural data.