  A common problem in neuroscience is to elucidate the collective neural
representations of behaviorally important variables such as head direction,
spatial location, upcoming movements, or mental spatial transformations. Often,
these latent variables are internal constructs not directly accessible to the
experimenter. Here, we propose a new probabilistic latent variable model to
simultaneously identify the latent state and the way each neuron contributes to
its representation in an unsupervised way. In contrast to previous models which
assume Euclidean latent spaces, we embrace the fact that latent states often
belong to symmetric manifolds such as spheres, tori, or rotation groups of
various dimensions. We therefore propose the manifold Gaussian process latent
variable model (mGPLVM), where neural responses arise from (i) a shared latent
variable living on a specific manifold, and (ii) a set of non-parametric tuning
curves determining how each neuron contributes to the representation.
Cross-validated comparisons of models with different topologies can be used to
distinguish between candidate manifolds, and variational inference enables
quantification of uncertainty. We demonstrate the validity of the approach on
several synthetic datasets, as well as on calcium recordings from the ellipsoid
body of Drosophila melanogaster and extracellular recordings from the mouse
anterodorsal thalamic nucleus. These circuits are both known to encode head
direction, and mGPLVM correctly recovers the ring topology expected from neural
populations representing a single angular variable.
