Geometric model manifold of space, time, and belief in hippocampal cognitive maps

Cognitive maps are mental representations of spatial and conceptual relationships in an environment. The hippocampus, a key brain region for these maps, exhibits neural population activity that often evolves along nonlinear, low-dimensional manifolds. While cognitive maps have been characterized at the level of cell tunings to spatial location and decorrelations over time, a precise, unsupervised, and quantitative model of how this map forms and morphs at the population level remains elusive due to the large dimensionality and nonlinearity of neural activity. Current dimensionality reduction techniques have difficulty capturing this manifold in an interpretable way due to two shortcomings: they do not model regions of neural activity where there are no data, and the map from the embedding back to neural activity is too nonlinear. We solve these problems in a novel unsupervised technique using differential geometry to model the low-dimensional space in which the data reside as a gently curved manifold. We apply this method to activity from thousands of neurons across several days in the CA1 region of the hippocampus of mice learning to collect rewards under two task conditions on a linear maze. We show that the 3 relevant variables of position (space), trial (time), and task belief emerge as local directions on the population-level manifold that captures track topology, a consistent direction of representational drift within and between days, and task learning that evolves orthogonally to baseline drift (resulting in tuning decorrelation). Further, the manifold quantifies neuron-level features—such as changes in positional tuning across trials—at the population level, and we discover neurons tuned for generalized states including space, time and belief. We provide a novel technique and a unifying theory for how population-level neural manifolds encode generalized state-time information along gently curved manifolds, which we hypothesize is for efficient recall and co-localization in downstream regions.