The medial entorhinal cortex is part of a neural system for mapping the position of an individual within a physical environment1. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations2, and are organized in modules3 that collectively form a population code for the animal’s allocentric position1. The invariance of the correlation structure of this population code across environments4,5 and behavioural states6,7, independent of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern1,8–11. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained unclear owing to the small samples of cells obtained so far. Here, using simultaneous recordings from many hundreds of grid cells and subsequent topological data analysis, we show that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a two-dimensional CAN. Positions on the torus correspond to positions of the moving animal in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feedforward models12. This demonstration of network dynamics on a toroidal manifold provides a population-level visualization of CAN dynamics in grid cells.

The medial entorhinal cortex (MEC) is part of a neural system for mapping a subject's position within a physical environment. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations, and are organized in modules which collectively form a population code for the animal's allocentric position1. The invariance of the correlation structure of this population code across environments and behavioural states, independently of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained elusive due to the small samples of cells obtained to date. Here we show, with simultaneous recordings from many hundreds of grid cells, and subsequent topological data analysis, that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a two-dimensional CAN. Positions on the torus correspond to the moving animal's position in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained, with minimal distortion, between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feed-forward models where grid patterns are created from external inputs by Hebbian plasticity. This demonstration of network dynamics on a toroidal manifold provides the first population-level visualization of CAN dynamics in grid cells.

Neuroscience is pushing toward studying the brain during naturalistic behaviors with open-ended tasks. Grid cells are a classic example, where free behavior was key to observing their characteristic spatial representations in two-dimensional environments. In contrast, it has been difficult to identify grid cells and study their computations in more restrictive experiments, such as head-fixed wheel running. Here, we challenge this view by showing that shifting the focus from single neurons to the population level changes the minimal experimental complexity required to study grid cell representations. Specifically, we combine the manifold approximation in UMAP with persistent homology to study the topology of the population activity. With these methods, we show that the population activity of grid cells covers a similar two-dimensional toroidal state space during wheel running as in open field foraging, with and without a virtual reality setup. Trajectories on the torus correspond to single trial runs in virtual reality and changes in experimental conditions are reflected in the internal representation, while the toroidal representation undergoes occasional shifts in its alignment to the environment. These findings show that our method can uncover latent topologies that go beyond the complexity of the task, allowing us to investigate internal dynamics in simple experimental settings in which the analysis of grid cells has so far remained elusive.

This paper presents the computational challenge on differential geometry and topology that was hosted within the ICLR 2022 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide implementations of machine learning algorithms on manifolds that would respect the API of the open-source software Geomstats (manifold part) and Scikit-Learn (machine learning part) or PyTorch. The challenge attracted seven teams in its two month duration. This paper de-

We introduce Reeb complexes in order to capture how generators of homology flow along sections of a real valued continuous function. This intuition suggests a close relation of Reeb complexes to established methods in topological data analysis such as levelset zigzags and persistent homology. We make this relation precise and in particular explain how Reeb complexes and levelset zigzags can be extracted from the first pages of respective spectral sequences with the same termination.