Dissecting the multivariate structure of neural population activity with copula-based methods and tensor decompositions

Abstract

The advent of large high-dimensional datasets in neuroscience has been an important milestone for advancing our understanding of neural information processing and improving performance of brain computer interfaces. However, most existing methods of analysis fall short of capturing the complexity of multivariate neuronal interactions. Furthermore, collective neural dynamics are subject to adaptation in various time scales in correspondence with changing behavioural and environmental statistics. Novel techniques need to address these issues and be applicable in a wide range of neural data analysis scenarios.

Firstly, we tackle the challenge of modelling complex, heavy-tailed multivariate dependencies in recordings of neural population spiking activity. To that end, we introduced a non-parametric framework that combined copula-based methods with normalizing flows. This framework aimed to address limitations posed by parametric copula modeling such as the risk of misspecification of dependence structures. At a first stage, we demonstrate that our approach can accurately capture ground truth dependencies in simulated joint observations. We, then, applied the framework on neuronal responses recorded in the mouse primary visual cortex during a visual learning task and were able to capture low and higher order heavy tail dependencies in a subgroup of neurons. Given that such complex features of neuronal interactions are ignored with approaches based on linear correlations, our findings lay the groundwork for a more detailed investigation of coordinated neural activity.

Next, we sought to analyze single trial neural responses using a tensor decomposition approach. Population neural activity can exhibit intricate dynamics containing collective patterns unfolding both on slow and fast time scales. Therefore, averaging across trials can obscure how learning can slowly alter neural responses, and preclude identification of latent processes that shape and modulate the neural code. Tensor-based methods enable descriptions of neural activity with modules corresponding to neural assemblies, trial dynamics as well as temporal or position-related dynamics. We propose discovering these modules using a tensor-based method called non-negative Tucker-3 decomposition (NNTD). We show that NNTD improves upon existing tensor decomposition methods by introducing a core tensor which allows for the modules to interact, thus resulting in a more flexible parts-based representation. We applied this method on mouse primary visual cortex and macaque primary motor cortex. Our findings demonstrate that NNTD can yield a low dimensional, parts based and biologically interpretable representation of rich neural dynamics. Moreover, the core tensor allowed for less redundancy in NNTD representation compared to existing tensor-based methods.

Finally, we consider the challenge of compactly describing multivariate dependencies in neural populations with a non-parametric copula-based approach such as the one combining normalizing flows that we introduced earlier. The additional flexibility offered by this framework comes at the cost of a substantial number of copula densities to characterise. We alleviate this challenge by using a weighted non-negative matrix factorization (WNMF) procedure to leverage shared latent features in neural population dependencies. By applying this composite framework on responses from mouse primary visual cortex and macaque primary motor cortex we uncovered a rich picture of dependencies underpinned by a small number of copula modules that are synergistically combined to give rise to diverse interaction patterns that may serve the population function.