Fitting T1-mGPLVM to the binned spike data, we found that the inferred latent state was highly correlated with the true head direction (Figure 5b). Here we make this connection more explicit. As described in the main text, the Lie algebrag of a groupG is a vector space tangent toG at its identity element. However,because the Lie algebra is isomorphic toRn, we have found it convenient in both our exposition and our implementation to work directly with the pair(Rn,ExpG), instead of(g,expG). We begin by noting thatSn is not a Lie group unlessn = 1 or n = 3, thus we can only apply the ReLie framework toS1 and S3.

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.