Grid cells are neurons typically described as being spatially selective, with increased activity in specific regions tessellating the environment with a hexagonal grid-like pattern. Grid cells can be grouped into distinct populations called modules, determined by their spatial scaling and orientation. We recorded neural activity from six modules across three rats during both free foraging and sleep, and, using persistent cohomology and circular coordinatization as our main tools, we depicted the toroidal structure of the population activity and decoded the time-varying toroidal positions encoded by the modules. We show that individual neurons are preferentially active at singular positions on the torus, and that these positions are maintained between environments and from wakefulness to sleep, as predicted by attractor network models of grid cells.