Hundreds of cells firing every which way, and yet the whole thing draws a single line on a thin sheet: a brain has far fewer moves than it has parts.
Close your eyes, turn halfway around, and stop. You still know which way you're facing. Nobody told you, and you didn't count degrees in your head. This isn't stored in some "heading" slot in the brain. It lives in a ring of cells whose activity forms a bump parked somewhere on that ring; when you turn, the bump slides around the ring by the same angle. This issue changes the question we ask of the brain: stop asking what a single cell represents, and ask where the whole population is heading right now. Once you make that switch, why a memory can hold, why movement needs preparing, and why some things simply cannot be learned today all fall out of the same picture.
Start with a change of notation. Say you record 100 cells at once, and right now each one has a number: how fast it's firing. Line those 100 numbers up and you have the population's state at this instant. Think of it as a point in a 100-dimensional space — too many dimensions to draw, but the idea is exactly the same as locating a point with an x and a y, only here it takes 100 numbers. Let time run and the point moves, tracing a trajectory.
The notation by itself says nothing new. The finding is this: the point does not roam. A hundred cells could in principle give a hundred independent degrees of freedom; what you actually measure is that the state varies along only a handful of directions — a dozen or so, often fewer. The rest are essentially empty. That low-dimensional sheet the point is pinned to is called a neural manifold.
An analogy: a hundred people on a stage waving looks like a hundred freedoms, but they're dancing the same dance, and only a few things actually vary — the beat, the facing, the size of the gesture. You don't need to watch each person; watch those few things. The brain gives the same impression: enormously many parts, far fewer postures, and those postures joined into one continuous surface.
"Looks impossibly high-dimensional, actually lives on a thin low-dimensional sheet" is the manifold hypothesis in machine learning, one of the field's foundations. A one-megapixel photo of a face has a million degrees of freedom on paper; real faces occupy a thin slice of that, varying in angle, lighting, expression — a countable handful of things. The activity of a deep network's hidden layer behaves the same way: thousands of numbers, effective variation along very few directions. Neuroscience and AI reach for the same dimensionality-reduction tools here (PCA, which pulls out the directions of greatest variation, for a start), and the resulting plots look alike.
Once you have states and trajectories, you can describe a brain in a very economical language: give the current state, plus a rule for where it moves next, and you have a landscape with the state rolling across it like a ball. Wherever it settles is an attractor.
The simplest kind of basin is a single point. Someone reads you a phone number and you have to hold it for a few seconds — the input is long gone, yet a group of prefrontal cells is still firing (prefrontal cortex). They hold each other up by mutual excitation: they've been pushed into a basin. The value of a basin is noise tolerance: random fluctuation in the surrounding cells shoves the ball off-centre and it rolls back by itself — exactly the kind of stability "remembering" requires. In 1982 Hopfield turned this into a network that stores things: each memory is a basin in the landscape, and recalling is starting from a partial cue and letting it roll downhill. That line of work earned him a share of the 2024 Nobel Prize in Physics.
A basin need not be a point. What if it's an entire ring? In the 1990s, to explain head-direction cells, theorists proposed arranging them in a ring where neighbours excite each other and distant ones inhibit each other. Activity then collapses into a bump that can sit anywhere on the ring, and a turning signal pushes the bump around it. That was pure paper. Two decades later the ring was seen directly, in a fly: a structure at the centre of the fly brain called the ellipsoid body, imaged while the fly turned, shows a bump that rotates by exactly the angle the fly turned (Seelig & Jayaraman 2015; Kim et al. 2017 confirmed the ring-attractor dynamics). The same ring was later found in mice — and it keeps turning while the animal is asleep. There is a heading in the dream too.
Prettier still: run the same analysis on a population of grid cells and the manifold you get is the surface of a doughnut (Gardner et al. 2022) — precisely the shape demanded by "walk across a plane, and one grid-spacing later you're back where you started" (hippocampus & entorhinal cortex). The shape is forced by what the task requires; it isn't a coincidence.
Hopfield's network is the prototype of associative memory: you don't fetch by address, you hand it a corrupted cue and let it roll into the nearest basin, which completes it — the same family as today's vector retrieval. Better still is what came later: when the network is pushed to its "modern" high-capacity version, its retrieval formula turns out to be all but identical to the equations of attention in a large language model (each step compares the current query against a pile of stored keys and pulls back a similarity-weighted mixture). "Rolling into a basin" and "retrieving by similarity" are two ways of writing the same move.
You reach for the cup on the table. How does motor cortex get that done? The textbook version: cells encode direction, downstream circuits read the direction out and send it to the muscles (motor system & cerebellum). That story has always had an embarrassment attached: for two or three hundred milliseconds before the movement, motor cortex is already busy, and the hand hasn't moved. What is all that activity doing?
In 2012 Churchland's group changed the question: never mind what each cell represents, look at the trajectory of the whole population. The answer came out clean — on every reach, the population state rotates on the manifold, like a wheel that's been given a spin. Which makes the point of the preparatory period obvious: it sets the state to a particular starting position. Let go, and the circuit's own dynamics unwind the trajectory; the muscle command is that trajectory read out. Reaching in a different direction doesn't change "what command to issue" — it changes where the starting point sits.
Then why doesn't such strong preparatory activity move the hand? Because it lives in an output-null subspace — those cells' changes are arranged so that they cancel exactly when the downstream sum is taken (Kaufman et al. 2014). The brain picks the directions the muscles can't hear, and says everything it needs to say in advance.
This one barely needs translating. Artificial networks include a family called recurrent networks — they feed their own output back into their input, so they have a state and evolve step by step, which makes them dynamical systems too. In 2013 Sussillo and Barak trained such networks on memory and decision tasks and then pried them open by finding their fixed points — fixed points being the basin floors from earlier — and mapping the slopes around them, which exposes the algorithm the network learned. What they found is that the trained networks had grown the same repertoire: point attractors to hold an answer, a slow line along a shallow valley to accumulate evidence, rotation to emit an action. Analysing monkey motor cortex and analysing a recurrent network now use one toolbox — a rare thing between two fields.
Cold water first. In 2019 Stringer's group recorded more than ten thousand cells in mouse visual cortex while showing thousands of natural images — and the dimensionality was not low at all: effective directions kept coming, the variance falling off as a slow power law. Which is a reminder of something easy to forget: how many dimensions you measure depends heavily on how many cells you recorded and how hard a thing you asked the animal to do. A monkey reaching back and forth between a few fixed targets obviously needs only a few degrees of freedom. So the honest statement is that a manifold is the effective low dimensionality under a given task, not proof that the brain itself has a dozen degrees of freedom.
But manifolds do have causal bite, and the evidence is hard. In 2014 Sadtler's group had monkeys drive a cursor through a brain–computer interface: read motor cortex live, convert activity to cursor motion through a mapping (motor system & cerebellum). Once the animal was fluent, the experimenters quietly swapped the mapping. Swap in a new mapping that still lives inside the existing manifold and the monkey relearns within hours. Swap in one that requires activity outside the manifold and it cannot be learned that day, no matter how much practice. Follow-up work (Oby et al. 2019) showed such mappings aren't unlearnable forever — they take days, while new activity patterns slowly grow. What you can learn today is fenced in by the manifold you already have; getting past that fence takes time.
The other end of it is good news. Bring the same monkey back years later to do the same movement and individual cells will have drifted — a cell that once preferred leftward may no longer. Yet project the population onto the manifold and the latent dynamics are strikingly stable (Gallego et al. 2020). That has a concrete payoff for brain–computer interfaces: rather than clinging to a particular set of cells, align the manifold, and a decoder keeps working across months or years (the BCI issue covered the engineering of the read-out side; this is why it can stay stable over the long run).
"Don't push every step — build a slope and let it roll" is an idea that has been arrived at independently, far from neuroscience: