Entorhinal grid cells, so-called because of their hexagonally tiled spatial receptive fields, are organized in modules which, collectively, are believed to form a population code for the animal’s position. Here, we apply topological data analysis to simultaneous recordings of hundreds of grid cells and show that joint activity of grid cells within a module lies on a toroidal manifold. Each position of the animal in its physical environment corresponds to a single location on the torus, and each grid cell is preferentially active within a single “field” on the torus. Toroidal firing positions persist between environments, and between wakefulness and sleep, in agreement with continuous attractor models of grid cells.

Natural systems provide both puzzles to unravel and demonstrations of what's possible. The natural world is full of complex systems of dynamically interchangeable, individually unreliable components that produce effective and reliable outcomes at the group level. A complementary goal to understanding the operation of such systems is that of being able to engineer artifacts that work in a similar way. One notable type of collective behavior is collective construction, epitomized by mound-building termites, which build towering, intricate mounds through the joint activity of millions of independent and limited insects. The artificial counterpart would be swarms of robots designed to build human-relevant structures. I will discuss work on both aspects of the problem, including studies of cues that individual termite workers use to help direct their actions and coordinate colony activity, and development of robot systems that build user-specified structures despite limited information and unpredictable variability in the process. These examples illustrate principles used by the insects and show how they can be applied in systems we create.

In the past decade evidence has accumulated in favor of the hypothesis that multiple sub-networks in the medial entorhinal cortex (MEC) are characterized by low-dimensional, continuous attractor dynamics. Much has been learned about the joint activity of grid cells within a module (a module consists of grid cells that share a common grid spacing), but little is known about the interactions between them. Under typical conditions of spatial exploration in which sensory cues are abundant, all grid-cells in the MEC represent the animal’s position in space and their joint activity lies on a two-dimensional manifold. However, if the grid cells in a single module mechanistically constitute independent attractor networks, then under conditions in which salient sensory cues are absent, errors could accumulate in the different modules in an uncoordinated manner. Such uncoordinated errors would give rise to catastrophic readout errors when attempting to decode position from the joint grid-cell activity. I will discuss recent theoretical works from our group, in which we explored different mechanisms that could impose coordination in the different modules. One of these mechanisms involves coordination with the hippocampus and must be set up such that it operates across multiple spatial maps that represent different environments. The other mechanism is internal to the entorhinal cortex and independent of the hippocampus.