Paper:
Estimating Nonlinear Neural Response Functions using GP Priors and Kronecker Methods
Jointly characterizing neural responses in terms of several external variables promises novel insights into circuit function, but remains computationally prohibitive in practice. Here we use gaussian process (GP) priors and exploit recent advances in fast GP inference and learning based on Kronecker methods, to efficiently estimate multidimensional nonlinear tuning functions. Our estimator requires considerably less data than traditional methods and further provides principled uncertainty estimates. We apply these tools to hippocampal recordings during open field exploration and use them to characterize the joint dependence of CA1 responses on the position of the animal and several other variables, including the animal's speed, direction of motion, and network oscillations. Our results provide an unprecedentedly detailed quantification of the tuning of hippocampal neurons. The model's generality suggests that our approach can be used to estimate neural response properties in other brain regions.

Review:
The authors make use of a recently developed and fast variational algorithm to learn Gaussian processes (GP) from data with non-Gaussian likelihoods to learn the tuning properties of neurons in hippocampal cells. After quickly introducing the mathematical details on which the main algorithm is based, the authors show the robustness of their estimators on simulated data. Finally, they apply the technique on data from hippocampal area AC1 and discuss their findings.
In this work the authors make use of an already developed technique, so for this reason I don't find the paper innovative on the technical side. The authors emphasize that the focus is on the application of the method to infer multi-dimensional tuning functions of hippocampal neurons (line 40). Nevertheless they don't try to infer tuning functions in more than 3 dimensions (2 spatial + 1 non-spatial variables), and in my opinion this reduces the potential novelty that their results may have. In the next paragraphs I give a detailed opinion about the content and the structure of the paper.  Content  Subsection "Detailed parameter settings for experiments", page 4. The authors may consider to mainly reorganize this section, maybe also to remove it completely from here and merge it with the section in which they talk about the application of the method of neural data (see the following points in my review).   Section 3 "Results": In my opinion, there should be a clear separation between the estimator validation section (which may include all the experiments run on simulated data) and the section where the method is applied to data from neural recordings.  Subsection "Spectral mixture kernels for complex functional dependencies", page 5. I am not sure about the usefulness of the first experiment (extrapolating the tuning function outside rectangular region): the fact that using a spectral mixture kernel make it possible make extrapolations was already a well established fact in reference [9], and I would be surprise if this was not the case in this simple experiment. Rather, I would suggest to focus more on the second experiment, in which learning a correct representation of the tuning function is not trivial at all.  Subsection "Spatial and non-spatial modulation of CA1 responses": The authors may consider to reorganize the structure of this subsection, which represents the very core of the paper. A brief description of the data should be given at the very beginning of the paragraph (experimental techniques, number of recorder cells, few words about preprocessing the raw signals). I noticed that some of this information is already reported, but spread across the whole section, making it difficult for the reader to have a clear picture of what kind of data was used. An important analysis that in my opinion is missing in this sections is the estimation of the full joint posterior distribution over all the spatial and non-spatial variables at the same time. Only multiple 3D fields consisting of the spatial location plus an additional non-spatial variable are estimated. The authors claim that this choice was made for the sake of simplicity (Footnote #11), but the focus of the paper is "not on the methods per se but rather on their previously unacknowledged utility for estimating multidimensional nonlinear tuning functions", as stated by the authors (line 40). Therefore it is really important to see these methods applied to data which is really multidimensional (with dimension larger than 3). In this case, the 3D fields estimation would still be useful, using their predicted marginal posterior distributions to assess the quality of the the marginal posterior distributions predicted by the full multidimensional field.  General questions/remarks  I did not understand why the corners are missing in the plots in Figure 1d. It looks like this plots correspond to the walls of a box that was open and flatten on the plane, somehow corresponding to a 2D representation of a 3D plot. Is this the case? Please consider to specify it in the caption of the figure or to change the shape of the plot, because, as it is now, it is difficult to understand.  The authors should consider to reduce the number of footnotes. Overall, there are 14 footnotes, which sometimes contain also important information (#11, #12, #14).  Page 4, line 126: the authors refer to a parameter "k", stating that it takes discrete values. Nevertheless, the meaning of "k" is not defined until line 176.  Page 4, line 135: a reference would be important here.  On page 7: several correlations between pairs of variables are reported. In all cases, brackets are misaligned, and the last one is always missing (lines 207, 208, 211, 221, 222).