Uncovering 2-D toroidal representations in grid cell ensemble activity during 1-D behavior.

Minimal experiments, such as head-fixed wheel-running and sleep, offer experimental advantages but restrict the amount of observable behavior, making it difficult to classify functional cell types. Arguably, the grid cell, and its striking periodicity, would not have been discovered without the perspective provided by free behavior in an open environment. Here, we show that by shifting the focus from single neurons to populations, we change the minimal experimental complexity required. We identify grid cell modules and show that the activity covers a similar, stable toroidal state space during wheel running as in open field foraging. Trajectories on grid cell tori correspond to single trial runs in virtual reality and path integration in the dark, and the alignment of the representation rapidly shifts with changes in experimental conditions. Thus, we provide a methodology to discover and study complex internal representations in even the simplest of experiments.

Toroidal topology of population activity in grid cells.

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.