Discovering Low-Dimensional Descriptions of Multineuronal Dependencies

Abstract

Coordinated activity in neural populations is crucial for information processing. Shedding light on the multivariate dependencies that shape multineuronal responses is important to understand neural codes. However, existing approaches based on pairwise linear correlations are inadequate at capturing complicated interaction patterns and miss features that shape aspects of the population function. Copula-based approaches address these shortcomings by extracting the dependence structures in the joint probability distribution of population responses. In this study, we aimed to dissect neural dependencies with a C-Vine copula approach coupled with normalizing flows for estimating copula densities. While this approach allows for more flexibility compared to fitting parametric copulas, drawing insights on the significance of these dependencies from large sets of copula densities is challenging. To alleviate this challenge, we used a weighted non-negative matrix factorization procedure to leverage shared latent features in neural population dependencies. We validated the method on simulated data and applied it on copulas we extracted from recordings of neurons in the mouse visual cortex as well as in the macaque motor cortex. Our findings reveal that neural dependencies occupy low-dimensional subspaces, but distinct modules are synergistically combined to give rise to diverse interaction patterns that may serve the population function.

Keywords: copula; neural dependence structures; non-parametric vine copula; weighted NMF.

Figure 1

Mixed vine copula flows and NMF decomposition. (A) Spike train samples from two neurons can be decomposed into their margins and a copula. Empirical copulas are extracted by transforming the samples to uniform through the distributional transform. (B) Graphical illustration of a C-vine for 4 variables. Nodes and edges of the first tree denote the variables and bivariate dependencies, respectively. Edges of subsequent trees denote dependencies that condition on one or more variables. (C) Decomposition of pair copulas into non-negative coefficients for neuron pairs and copula factors.

Figure 2

Overlapping tail dependencies are challenging for standard NMF. (A) A matrix of 20 Frank copula flow densities and 20 Clayton copula flow densities (Top illustration) is reduced to 2 NMF factors with coefficients (blue bar plots) and copula modules. Bottom left tail dependence structures overlap, which does not affect the Clayton copula factor but leads to incorrect identification for the Frank copula left tail, which is highlighted with a red dashed circle. This is in contrast to (B), where the rotated (90°) Frank copula tail regions do not overlap with those of the Clayton copulas (top illustration), allowing for correct detection of both copula types.

Figure 3

WNMF outperforms standard NMF in identifying dependence structures with overlapping tails. (A) Train (blue) and validation (orange) MSE over 5 folds for 1 to 8 factors. Error is computed for the real data matrix against the reconstructed one for WNMF (left column of line plots) and for NMF (right column of line plots). Both methods were tested on data containing either 2, 4 or 6 different copulas. (B) Bar plots depict MSE of the real copula densities versus the copula modules identified by WNMF (left column of bar plots) and NMF (right column of bar plots). Y axes are in logarithmic scale. Bar colors correspond to different copula families, namely Frank (brown) and Clayton (beige), as well as their rotated versions used in the cases with 4 and 6 copulas, namely Frank 90° (navy blue), Clayton 90° (red), Clayton 180° (yellow) and Clayton 270° (teal). (C) Illustration of WNMF factorization for the case of 4 copulas. Bar plots depict WNMF coefficients and density plots depict the copula modules discovered by WNMF.

Figure 4

WNMF discovers structured and synergistic copula modules in mouse V1 responses. (A) Illustration of mouse navigating a virtual environment with grating stimuli until a designated reward zone (at 120–140 cm), where it is required to lick in order to receive a water reward. Neurons were recorded across a number of trials on each day of the experiment and their activity was binned with respect to the position of the mouse in the virtual corridor. (B) Average across 5 folds train (blue) and validation (orange) MSE for WNMF across different numbers of factors. (C) WNMF 4-dimensional representation of visual cortex copula dependence structures. Blue Bar plots depict WNMF coefficients specific to each neuron pair. Density plots depict copula modules discovered by WNMF.

Figure 5

WNMF factors are distinctly grouped across different trees. (A) WNMF coefficients from the same decomposition as Figure 4. The 4 factors have been plotted in a 2 by 2 arrangement where each block consists of rows of color-coded values of WNMF for a particular tree, starting from tree 1 at the top until tree 101 at the bottom. Warmer and colder colors illustrate the spatial divides across superficial and deeper trees with respect to copula modules per neuron pairs. Blank spaces denote independent neuron pairs. (B) Same WNMF copula modules as in Figure 4 depicted again here for illustration purposes.

Figure 6

WNMF discovers main and distributed copula modules in macaque motor cortex. (A) Illustration of a macaque monkey moving a lever as part of a delayed center-out reaching task [57]. Monitor has 8 different targets drawn as white squares, while the cued target is highlighted with yellow. (B) Average firing rate (Hz) across trials of different target presentations. Horizontal axis indicate the time interval we chose for analysis, i.e., −0.5 to 1.5 s with respect to the go cue for movement initiation. (C) Average across 5 folds train (blue) and validation (orange) MSE for WNMF across different numbers of factors. (D) WNMF 4-dimensional representation of motor cortex copula dependence structures. Blue bar plots depict WNMF coefficients specific to each neuron pair. Density plots depict copula modules discovered by WNMF.