A Two-Dimensional Grid-Cell Code for Three-Dimensional Navigation in Freely Flying Bats

Heads up: only the abstract of this paper was available to me, so everything below builds intuition around the claims and mechanisms the abstract states. I flag where I'm extrapolating and I invent no numbers.

The one-sentence version: grid cells in flying bats keep the same two-dimensional toroidal code they use on flat ground, and bats can get away with that in 3D because their flight paths, moment to moment, lie in tilted 2D planes—so a 2D code aligned to the current plane of motion is enough to navigate a volume.

Why this is a real puzzle, not a curiosity

Grid cells are one of the cleanest examples we have of an abstract, structured neural code. As a rodent moves across a flat arena, a single grid cell fires whenever the animal is at any vertex of a regular triangular (hexagonal) lattice tiling the floor. Different cells share the same lattice spacing and orientation but are shifted in phase, so the population tiles all possible positions within one lattice period.

The deep result of the last decade is what that population does collectively. If you take an ensemble of co-modular grid cells (same spacing/orientation) and look at their joint activity as a point in high-dimensional firing-rate space, that point is constrained to move on a torus—a donut. This makes sense: position on a periodic hexagonal lattice is described by two phases (how far along each of two lattice axes, each wrapping around at the period). Two independent circular variables = a 2-torus. Gardner et al. (2022) confirmed this experimentally in rats using topological data analysis: the manifold really is a torus, and it persists even in sleep, which is strong evidence for a continuous attractor—the network's connectivity carves out a donut-shaped set of stable states regardless of input.

Now the problem. If grid cells are the brain's metric for space, what happens in 3D? The two obvious hypotheses both have trouble:

• A true 3D grid. A face-centered-cubic packing of firing fields would be the elegant 3D analog. But you can't build a 3D lattice from a 2-torus—you'd need a 3-torus and a fundamentally richer attractor. And experimentally, animals moving in 3D (rats climbing, bats crawling or flying) show loss of global periodicity: the beautiful hexagonal structure smears out. That looks like the code failing in 3D.

• A 2D grid that just doesn't work in 3D. If the code is stuck at 2D, how does the animal navigate a volume at all?

So the field has been stuck between "the 3D code is something we haven't characterized" and "the 2D code somehow degrades." What was missing is an account of how a genuinely 2D code could be sufficient for 3D behavior. That's the gap this paper claims to fill.

The key move: look at what the animal actually does with the third dimension

The insight is behavioral before it is neural. A bat foraging is not filling a volume like a gas molecule. Its trajectories, the authors report, are organized along transient two-dimensional planes of motion—arcs, swoops, and turns that at any moment lie approximately in a plane, even though that plane tilts and reorients over the course of a longer flight.

If that's true, then at each instant the behaviorally relevant space is 2D. And a 2D toroidal grid code carried along with the animal only needs to tile that plane. The third dimension isn't encoded by a third phase; it's absorbed by the fact that the plane itself reorients in 3D as the flight bends.

This reframes the "loss of periodicity" reports beautifully. If you record in a volume and analyze firing against fixed 3D coordinates, a grid that is locked to a reorienting plane will look scrambled—because you're projecting a clean 2D lattice through a constantly changing tilt onto your fixed reference frame. The periodicity was never lost; it was expressed in a moving frame. That's the kind of result that resolves a debate: the same data that looked like failure becomes evidence for a coherent, parsimonious code once you pick the right coordinates.

What they actually did and claimed to find

The experimental engine is the Yartsev lab's specialty: large-scale wireless recordings from medial entorhinal cortex in freely flying bats during spontaneous aerial foraging. Bats are the right animal—they are natural 3D navigators, unlike rodents that mostly live on surfaces, so any 3D grid story should show up here if anywhere.

The chain of claims, as stated in the abstract:

1. Robust periodic firing returns during structured flight trajectories. The grid is there when you look along the flight, not against arbitrary 3D coordinates.

2. Toroidal topology at the ensemble level. Co-modular grid-cell ensembles show topological signatures consistent with a 2-torus—the same manifold rats show in 2D. This is presumably persistent-homology / TDA on the population activity, the Gardner-et-al. approach ported to bats. If it holds, it's the first demonstration that the toroidal attractor is conserved across species and, crucially, that it stays 2D in a 3D animal.

3. Behavior is planar. Flight paths in both the wild and the lab are organized along 2D planes of motion. This is the linchpin that makes a 2D code sufficient rather than merely present.

4. The phase is a plane wave on the plane of motion, and single-cell firing is a hexagonal lattice on that same plane. This ties it together: the toroidal phase advances linearly as the bat moves through the plane (a traveling plane wave), exactly as it would for a 2D grid, and the individual cell's fields form a hexagon in the plane of motion's coordinates. That's the concrete, falsifiable model—you could fit it yourself given position, plane-of-motion estimates, and spike times.

How to think about the mechanism

Here's the clean mental model. A continuous attractor network can only support the manifold its recurrent connectivity was wired for. If that manifold is a 2-torus, the network cannot represent a third independent spatial coordinate no matter what the animal does—there's no dimension available. The brain's apparent solution is not to grow a bigger attractor but to re-use the 2D one, yoking its two phase coordinates to whatever 2D subspace the animal is currently moving through. The third dimension is handled at the level of which plane is selected, presumably by head-direction-like or vestibular signals defining the plane's orientation, rather than inside the grid code itself.

This is elegant because it separates concerns: the grid provides a fine-grained metric within a plane; some other signal orients the plane in the volume. It's also biologically frugal—no need to evolve or wire a 3D attractor, which would be far more costly and, as the packing problem shows, geometrically awkward from a torus.

What would make this convincing, and where to be skeptical

The selection note names the right worries, and they're worth stating sharply.

Does planarity generalize beyond structured foraging? Foraging flights to and from feeders are exactly the kind of behavior that produces stereotyped, arcing, roughly-planar trajectories. The theory's force depends on planarity being a general property of bat flight, not an artifact of a task with fixed goals. A bat doing genuinely volumetric aerobatics—corkscrews, rapid pitch-and-roll pursuit of insects—is the adversarial test. If the plane-of-motion frame breaks down there and the grid smears with it, the account is a description of easy flights rather than a solution for 3D navigation. The abstract's phrase "structured flight trajectories" is doing quiet work and deserves scrutiny in the full paper.

How robust is the torus? Topological data analysis on neural populations is powerful but sensitive: sample size, the choice of co-modular cells, dimensionality reduction, and behavioral coverage all shape whether you recover a clean H_1 signature with the two independent loops of a torus versus something noisier. The strongest version of the claim would show the torus persisting independent of behavior (e.g., during rest or non-planar segments), as in the rodent work—that's what distinguishes an intrinsic attractor from a manifold inherited from planar behavior. Whether they show that is exactly what to check.

Correlation of frames. Because the plane of motion and the grid frame are both derived from the flight, there's a real risk of the analysis defining the plane such that the grid looks hexagonal by construction. The independence of the plane estimate from the neural fit matters.

What selects and reorients the plane? The abstract doesn't say what neural signal defines the current plane. That's the missing mechanistic piece; without it, "the grid tiles the plane of motion" is a strong phenomenological model but not yet a circuit.

What changes if it holds

If the toroidal code is confirmed in a second, phylogenetically distant, natively-3D species, the continuous attractor account of grid cells graduates from "robust in rats" to "conserved principle." And the plane-of-motion idea gives the field a concrete, testable resolution to the 3D grid debate that neither requires abandoning the 2D attractor nor invoking an uncharacterized 3D code: the code stays 2D and the behavior supplies the third dimension. For anyone building navigation models—biological or artificial—that's a clean design lesson: align a low-dimensional metric to the locally relevant subspace rather than paying to represent the full ambient space.

The section to read first, once the full text is out, is the behavioral analysis establishing planarity (claim 3) and the TDA establishing the torus (claim 2)—those two, and their independence from each other, are the whole argument.