Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships
Fig 3
Copula-GP describes the noise correlations between dynamically coupled neurons with a Clayton copula.
A i. GLM model of two coupled neurons (excitatory and inhibitory) that receive the same time-dependent input x(t); ii. the spike history coupling filters h12 and h21; B-C simulated calcium transients (fluorescence across time) showing dynamic responses to the stimulus x(t) for excitatory and inhibitory neurons, respectively; D calcium transients of two neurons (y1(t), y2(t)) projected onto a unit cube by the probability integral transform based on unconditional marginals; colored points show transformed samples (u1, u2) corresponding to times t (color-coded). The clusters of similarly colored points (e.g. green) illustrate that the copula c(u) depends on time t; the particular shape and the location of the clusters depends on the function x(t). E same as D, but based on conditional marginals Fi(yi|t). The resulting copula describes ‘noise correlations’ between two neurons. The colored data-points (,
) are not uniformly distributed on the unit square, which suggests that the noise correlation between these neurons and the copula c(ut) itself depends on time t. F Clayton copula parameter (θ) that characterizes the strength of the non-linear noise correlation between neurons (see Methods for details); G probability density plots illustrating the stimulus-dependent shape of noise correlations. The empirical dependence estimated from data samples is shown with black outlines, while the predictions of the Clayton copula model are shown in shades of blue. The proportion of the variance explained
is indicated in the upper-right corner for each time interval. Orange circles indicate the heavy tail of the distribution, which can be best seen in the range t ∈ [0.6, 1.0] where the variables are stronger correlated.