Recurrent linear models of simultaneously-recorded neural populations
Population neural recordings with long-range temporal structure are often best understood in terms of a common underlying low-dimensional dynamical process. Advances in recording technology provide access to an ever-larger fraction of the population, but the standard computational approaches available to identify the collective dynamics scale poorly with the size of the dataset. We describe a new, scalable approach to discovering low-dimensional dynamics that underlie simultaneously recorded spike trains from a neural population. We formulate the Recurrent Linear Model (RLM) by generalising the Kalman-filter-based likelihood calculation for latent linear dynamical systems to incorporate a generalised-linear observation process. We show that RLMs describe motor-cortical population data better than either directly-coupled generalised-linear models or latent linear dynamical system models with generalised-linear observations. We also introduce the cascaded generalised-linear model (CGLM) to capture low-dimensional instantaneous correlations in neural populations. The CGLM describes the cortical recordings better than either Ising or Gaussian models and, like the RLM, can be fit exactly and quickly. The CGLM can also be seen as a generalisation of a lowrank Gaussian model, in this case factor analysis. The computational tractability of the RLM and CGLM allow both to scale to very high-dimensional neural data.
The authors present a new descriptive model of spiking 
      dynamics recorded in neural populations. The authors claim that the major 
      advantage of the new model is that it can be fitted easily and more 
      directly to experimental data than previous models and therefore it has a 
      better performance. The paper is generally well written, although 
      I found that the Results section is less clear than the presentation of 
      the models. Details about the training and testing procedure are missing. 
      The work is original and it makes an interesting progress in the models of 
      neural population data. Testing of the model against its alternatives is 
      not convincing in the present form and it needs to be clarified. 
      The authors use many acronyms for different models and it is not 
      always clear which one they are referring to. A tabular or a graphical 
      (tree-like) representation showing the similarities and differences 
      between these models could also be useful. (RLM, RNN, CLM, LDS, GLDS, 
      PLDS, gl-RLM, GLM) Specific comments: line 134: All GLDS have 
      an equivalent GRLM, but not all GRLM is equivalent to a GLDS. Putting this 
      in a different way, these parameters are not independent, W and S depends 
      on C. Similarly, is it true that a PLDS is a special case of RLMs with 
      Poisson observation? So later in the Results al Data generated by PLDS 
      could be, in principle, recovered by the appropriate class of RLMs? 
      line 172: The authors describe two ways to handle instantaneous 
      correlations in RLMs. It is not always clear which one is used later in 
      the Results section. line 193: W \hat{y}_t = W f(CA\hat{x}_t) that 
      could be (approximately) incorporated into the hidden dynamics. line 
      197: The authors use the acronym PLDS for the Poisson LDS model later. 
      line 235: vectors are usually typed in boldface in the paper, but I 
      got completely lost here. y_n is a scalar, y_{ < n} is a vector but the 
      typeset often suggests the opposite! line 238, Eq. 3: is \mu_n and 
      \mu_{ < n} = 0 ? If not, isn't it missing from (y_{ < n} - \mu_{ 
      < n})? line 245: I do not see why we have (\Sigma_{ < n, < 
      n})^{-1} instead of (\Sigma)^{-1} (same for line 255) line 257: I 
      could not find how this line follows from the previous one and from the 
      matrix inversion lemma, and I could not estimate how much we loose on the 
      approximation. line 270: Alternative models. Does this section refer 
      to alternatives of the CLM (it could be section 4.1. in this case) or in 
      general? Alternative models have already been mentioned in section 3.1. 
      line 281: Is B_i an NxN matrix? Or we have a single B matrix with NxN 
      parameters? line 289: The actual physical interpretation of the latent 
      variables in the RLM and CLM are also unclear and these models are also 
      more descriptive models than models describing the actual generative 
      process behind the data. line 307: see Tkacik et al., 2013 J. Stat 
      Mech for a simple and tractable way to describe low dimensional modulation 
      of spikes in Ising models. Section 6: Details of the fitting procedure 
      are missing here, and it is also sometimes unclear what kind of models are 
      fitted. e.g., what is the observation model for the RLM? Based on the 
      figures, I assume that dimensions of the latent, x is 3 and the 
      observations y is 5. Fig2ad: What is the observation model for the 
      RLM? Gaussian (GRLM) or Poisson? Does the model GLDS used on Fig 2D has 
      the same likelihood as the RLM? In this case is the data generated by the 
      RLM consistent with the GLDS - parametrised by A, C and Q? Or the 
      difference between PCA, GLDS and RLM are due to the different likelihood 
      used? Fig2be: Is RLM the same as on Fig2ad? Here PLDS is just a 
      special case of the RLM, so why the latter has a slightly lower 
      performance? Fig2cf: What is the performance of a trained PLDS model 
      on this dataset? It is clear that PCA and GLDS use a wrong likelihood 
      function. line 371: Does LDS refer to GLDS here? line 377: what is 
      the training/testing procedure? What is predicted here? Spike counts of a 
      single cell given the stimulus+observation of other neurons in the 
      population? It would be useful to show the actual data and the 
      predictions. Figure 3a: error bars are missing. Does adding PSTH 
      information to the other methods change their performance? Does the RLM 
      found during training correspond to a PLDS model? Is the difference 
      between PLDS and RLM because of the different training procedure or 
      because RLM has a greater flexibility in its parameters? Fig4 shows 
      only pooled data. It would be good to show cell-by-cell comparisons of 
      e.g., the decay or the amplitude of the CCG. line 412: this part 
      is a not really connected to the previous parts of the paper. I think that 
      the procedure of training/testing is slightly different here, but there is 
      no indication of this in the text. I appreciate the detailed 
      response of the authors and I think that including these 
      modifications/clarifications will certainly improve the paper. I modified 
      the scores accordingly. I still think that a table or graph showing the 
      similarities and differences between these models could also be 
      useful.