Neural responses are variable: even under identical experimental conditions, single neuron and population responses typically differ from trial to trial and across time. Recent work has demonstrated that this variability has predictable structure, can be modulated by sensory input and behaviour, and bears critical signatures of the underlying network dynamics and computations. However, current methods for characterising neural variability are primarily geared towards sensory coding in the laboratory: they require trials with repeatable experimental stimuli and behavioural covariates. In addition, they make strong assumptions about the parametric form of variability, rely on assumption-free but data-inefficient histogram-based approaches, or are altogether ill-suited for capturing variability modulation by covariates. Here we present a universal probabilistic spike count model that eliminates these shortcomings. Our method builds on sparse Gaussian processes and can model arbitrary spike count distributions (SCDs) with flexible dependence on observed as well as latent covariates, using scalable variational inference to jointly infer the covariate-to-SCD mappings and latent trajectories in a data efficient way. Without requiring repeatable trials, it can flexibly capture covariate-dependent joint SCDs, and provide interpretable latent causes underlying the statistical dependencies between neurons. We apply the model to recordings from a canonical non-sensory neural population: head direction cells in the mouse. We find that variability in these cells defies a simple parametric relationship with mean spike count as assumed in standard models, its modulation by external covariates can be comparably strong to that of the mean firing rate, and slow low-dimensional latent factors explain away neural correlations. Our approach paves the way to understanding the mechanisms and computations underlying neural variability under naturalistic conditions, beyond the realm of sensory coding with repeatable stimuli.
Neuroscience faces a growing need for scalable data analysis methods that reduce the dimensionality of population recordings yet retain key aspects of the computation or behaviour. To extract interpretable latent trajectories from neural data, it is critical to embrace the inherent topology of the features of interest: head direction evolves on a ring or torus, 3D body rotations on the special orthogonal group, and navigation is best described in the intrinsic coordinates of the environment. Accordingly, we recently proposed the manifold Gaussian process latent variable model (mGPLVM) to simultaneously infer latent representations on non-Euclidean manifolds and how neurons are tuned to these representations. This probabilistic method generalizes previous Euclidean models and allows principled selection between candidate latent topologies. While powerful, mGPLVM makes two unjustified approximations that limit its practical applicability to neural datasets. First, consecutive latent states are assumed independent a priori, whereas behaviour is continuous in time. Second, its Gaussian noise model is inappropriate for positive integer spike counts. Previous work in Euclidean LVMs such as GPFA has shown significant improvements in performance when modeling such features appropriately (Jensen et al., 2021). Here, we extend mGPLVM by incorporating temporally continuous priors over latent states and flexible count-based noise models. This improves inference on synthetic data, avoiding negative spike count predictions and discontinuous jumps in latent trajectories. On real data, we also mitigate these pathologies while improving model fit compared to the original mGPLVM formulation. In summary, our extended mGPLVM provides a widely applicable tool for inferring (non-)Euclidean neural representations from large-scale, heterogeneous population recordings. We provide an efficient implementation in python, relying on recent advances in approximate inference to e.g. fit 10,000 time bins of recording for 100 neurons in five minutes on a single GPU.
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