Mixed vine copulas as joint models of spike counts and local field potentials
Concurrent measurements of neural activity at multiple scales, sometimes performed with multimodal techniques, become increasingly important for studying brain function. However, statistical methods for their concurrent analysis are currently lacking. Here we introduce such techniques in a framework based on vine copulas with mixed margins to construct multivariate stochastic models. These models can describe detailed mixed interactions between discrete variables such as neural spike counts, and continuous variables such as local field potentials. We propose efficient methods for likelihood calculation, inference, sampling and mutual information estimation within this framework. We test our methods on simulated data and demonstrate applicability on mixed data generated by a biologically realistic neural network. Our methods hold the promise to considerably improve statistical analysis of neural data recorded simultaneously at different scales.
The authors develop tractable means of sampling from and calculating the likelihood of C-vine copulas which permit both discrete and continuous marginal distributions.  The proposed algorithms for sampling and likelihood calculation are extension of previous work (that of Aas et al. for likelihood calculation and  Panagiotelis et al. for sampling).  The benefit of the methods proposed is that while naive sampling strategies for calculating likelihoods and sampling would scale exponentially with the number of discrete random variables in a model, the proposed methods scaled quadratically.    The authors propose to apply such copulas to the study of neural data which consists of both continuous measurements (such as local field potentials) and discrete measurements (spike counts).  The authors demonstrate that data generated from such mixed copulas is better fit by these models than by models which assume independence between the marginal variables or use continuous distributions to model variables which are in fact discrete.  They also simulate LFP and spike data from a network driven by two different stimulus conditions and show that the mutual information between the stimulus and recorded neural signals differs substantially when it is calculated with the mixed copulas proposed in this work vs. a model which assume independence between marginal variables or a copula model which assume all marginal variables as continuous. 
The development of flexible methods to model the joint distribution between continuous and random variables is a important problem with many application areas, one of which, as the authors note, is neuroscience.  Copula which allow for both discrete and continuous random variables are one means of approaching this problem, and the development of general and computationally tractable methods for fitting and performing inference with such models is of broad interest.    The paper makes multiple methodological contributions, which I find valuable.  The proposed family of models seems flexible and likely useful in practice. While others have previously proposed pair copula constructions as well as efficient algorithms for sampling from discrete copulas, the development of pair copula constructions and associated efficient algorithms for sampling and inference for mixed discrete and continuous data is valuable.  However, I feel that other aspects of the paper could be improved.   I believe the paper would benefit from more clarity.  There were multiple parts of the paper that I found difficult to follow.  For example, equation 2 provides a general formula for the likelihood of data under a mixed copula model, but it was not until after reading some of the cited work that I was able to understand it was derived by taking differences of the discrete random variables.  While I was previously familiar with copula in general, this was my first exposure to C-vine copula.  The description of C-vines required additional reading for me to follow.  It is certainly understandable that it may be hard to fully explain all concepts in an 8 page paper, but I believe a general NIPS audience would benefit from the provision of a little more explanation and background.  There were also symbols which were used without introduction (the super script +/- signs which I believe indicate differences, "c" for the pdf of a copula, etc.).  While it was possible for me to follow the gist of paper, it was difficult to fully follow all the details.    It would be helpful to explain why a copula approach might be beneficial over existing approaches which have been applied in the neuroscience literature.  While it is not referenced in the present paper, there is established body of work in the neuroscience community which has examined the relationship between neural spiking and LFP (e.g., the first three references below and references therein).    Finally, I believe the results section would benefit from more depth.  If I understood correctly, in the first analysis data was generated from a particular mixed copula model.  The fit of mixed copula models, models which assumed independent random variable and copula models which modeled all marginal variables as continuous to this data was then compared.   Mixed copula models were shown to best fit the data.  While this is reassuring, this result also seems expected, as models which are subject to mismatch would not be expected to fit data as well as a model which is of the same family the data was generated from.  To show the utility of proposed methods, an application to more realistic data is beneficial, and the author's provide this by applying their methods to simulated spike and LFP data and show that mutual information (MI) between a stimulus and the simulated LFP and spike data is different under mixed copula models vs. models which assume independent random variables or require all random variables to be continuous.  I find the author's claim that a mixed copula model vs. the other two options should best fit the data believable, and seeing the difference in the values of MI between the calculated models is interesting.  However, if an actual application scenario, I believe experimentalists would not be primarily concerned with the value of MI returned - but by differences in MI returned by the same model in different experimental contexts.  For example, one might ask if MI between LFP and neurons in two areas changes when a subject is attending vs. not attending to a stimulus.  In such a scenario the primary question of interest is if a statistical method is sensitive enough to reveal differences between MI between the two conditions, and thus it would perhaps be more interesting to see an analysis showing the benefit of the proposed methods in such a setting.    References:   Lepage, K. Q., Gregoriou, G. G., Kramer, M. A., Aoi, M., Gotts, S. J., Eden, U. T., & Desimone, R. (2013). A procedure for testing across-condition rhythmic spike-field association change. Journal of neuroscience methods, 213(1), 43-62.  Kelly, R. C., Smith, M. A., Kass, R. E., & Lee, T. S. (2010). Local field potentials indicate network state and account for neuronal response variability. Journal of computational neuroscience, 29(3), 567-579.  Carlson, D. E., Borg, J. S., Dzirasa, K., & Carin, L. (2014). On the relations of LFPs & Neural Spike Trains. In Advances in Neural Information Processing Systems (pp. 2060-2068).  K. Aas, C. Czado, A. Frigessi, and H. Bakken. Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 44(2):182–198, 2009.  A. Panagiotelis, C. Czado, and H. Joe. Pair copula constructions for multivariate discrete data. Journal of the American Statistical Association, 107(499):1063–1072, 2012.