Parametric Copula-GP model reveals tuning of neuronal and behavioral relationships to visual stimuli

Recent research in neuroscience shows that neural responses to stimuli are strongly modulated by behavioral context even as early as in sensory brain areas. Yet, the precise functional interactions underlying such modulation remain unknown. The analysis of this modulation proves challenging due to the different statistics and different timescales of the observed variables, e.g. discrete licks or neuronal spikes vs. continuous velocity or calcium signals. In order to overcome these hurdles, we propose parametric copula models with phase-dependent parameters, which separate the statistics of the individual variables from their dependence structure. For model fitting, we use a Bayesian framework with a Gaussian Process (GP) latent variable inference scheme, which is naturally suited to take into account the dependence of the copula parameters on the phase of the experiment (e.g. time or position) as well as the uncertainty of the parameter estimation given limited and variable amounts of data. We applied our model to calcium optical recordings from awake behaving mice performing a spatial navigation task in a virtual reality setup. We have found pairs of variables showing strong non-trivial changes in the dependence structure subject to the position in the virtual reality and related visual stimulus. This finding agrees with previous studies, showing that beyond independent responses, noise correlations carry additional visual information. Moreover, our results reveal that dependence structures are often non-Gaussian, both in noise correlations between neurons and in neurons vs. behavioural variables. By estimating information measures, we also show that modulations of these non-Gaussian tail dependencies can encode additional information about the stimuli. Therefore, contrary to other commonly applied methods, our Copula-GP approach makes stochastic (non-Gaussian) relationships explicit, representing them as the ‘tuning curves’ of the copula parameters, and generates interpretable models of dependencies between neural responses, visual stimuli, and behavior.