Appendix - Manifold GPLVMs for discovering non-Euclidean latent structure in neural data

To highlight the importance of unsupervised non-Euclidean learning methods in neuroscience and to illustrate the interpretability of the learned GP parameters, we consider a dataset from Peyrache et al. (2015b) recorded from the mouse anterodorsal thalamic nucleus (ADn; Figure 5a). This data has also been analyzed in Peyrache et al. (2015a), Chaudhuri et al. (2019) and Rubin et al. (2019). We consider the same example session shown in Figure 2 of Chaudhuri et al. (2019) (Mouse 28, session 140313) and bin spike counts in 500 ms time bins for analysis with mGPLVM. However, in contrast to the data considered in Section 3.1 and Section 3.2, this mouse dataset contains neurons with more heterogeneous baseline activities and tuning properties. This is reflected in the learned GP parameters which converge to small kernel length scales for neurons that contribute to the heading representation (Figure 5c, 'tuned') and large length scales for those that do not (Figure 5c, 'not tuned'). Finally, since mGPLVM does not require knowledge of behaviour, we also fitted mGPLVM to data recorded from the same neurons during a period of rapid eye movement (REM) sleep.