Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships
Fig 5
Validation of Copula-GP method on neuronal population activity and behavioral variables from awake mice.
Copula-GP accurately models the neuronal and behavioral heavy tailed dependencies in the data from the visual cortex of awake mice, and quantifies more mutual information between various combinations of variables than the alternative methods. A Schematic of the navigational experimental task [5, 16] in virtual reality; B Example traces from ten example trials: x is a position in virtual reality, y is a vector of neuronal (blue) and behavioral (red) variables; these traces show that variables have different timescales and different signal-to-noise ratios, which result in different distributions of single variables yi (i.e. different marginal statistics). C-D Copula probability density plots for: the noise correlation between two neurons (number 3 and 63) (C) and for the correlation between one neuronal activity (60) vs. one behavioral variable (licks) (D); Black outlines show empirical copula, shades of blue—the best fitting Copula-GP model: a mixture of Gaussian + 90°-rotated Clayton copula in (C) and a mixture of Frank + 0°-rotated Clayton + 270°-rotated Gumbel copula in (D) (see S4 Text for model parameters). Similarly to the example with two dynamically coupled neurons (Fig 3G), these copulas are heavy-tailed. The goodness-of-fit for these models is measured with the proportion of the variance explained , which is indicated in the upper-right corner of each plot corresponding to a range of positions in virtual reality; E-G Conditional entropy for the bivariate examples (E-F) and the population-wide statistics (G) all peak in the reward zone; this entropy is equivalent to the mutual information between variables, given the position x, which means that the variables carry the most information about each other when the animal is in the reward zone. H Comparison of Copula-GP method (“integrated”) vs. non-parametric MI estimators (MINE [55] and KSG [53]) on estimating the amount of information about the location x from the subsets of variables ux. While the true I(x, ux) is unknown, the validation on synthetic data (Fig 4) suggest that Copula-GP “integrated” does not overestimate the amount of mutual information. Yet, Copula-GP “integrated” quantifies more information about the position x from the large subsets of data ux than MINE and KSG methods.