Last time I told you about the reconstruction problem. Your brain receives spike trains — on/off electrical pulses — and from that, it builds a world. Color, depth, sound, the feeling of being you. And I told you about a mathematician named Floris Takens who proved something in 1981 that might explain how.
Let me show you what Takens figured out.
Imagine a car driving around a circular track. You’re standing by the side of the road, and you can’t see the track from above. All you have is a radar gun. So all you can measure is the car’s speed toward or away from you. That’s it. One number, changing over time.
What does that measurement look like? The car comes toward you, speeds up, passes you, slows down, goes around the far side, comes back. Speed toward you goes up, down, up, down. The short answer is that this is just a sine wave. If someone handed you just that sine wave, you might think: this is a thing that oscillates. That’s all I know.
But Takens showed you can do something clever. Take your measurement at time t. Now take the same measurement a little later — at time t plus some delay. Plot those two numbers against each other: speed-now on one axis, speed-a-moment-ago on the other.
If you do that over and over, for each pair of speeds (speed-now, speed-a-moment-ago), it turns out that the shape of that plot is… a circle.
The circular track — the actual shape of what the car is doing — pops out of a single measurement combined with a delayed copy of itself. You never saw the track. You never had a bird’s-eye view. You reconstructed the geometry of the system from one stream of data and time.
That’s Takens’ theorem.1 And it doesn’t just work for circles. Takens proved that this trick — combining a measurement with time-delayed copies of itself — can reconstruct the dynamical structure of arbitrarily complex systems. Strange attractors, chaotic systems, things with dozens of interacting variables. One measurement, enough delays, and the shape reveals itself.
The technical term for this operation is delay coordinate embedding — DCE. Delay because, well, obvious. Coordinate because that’s just another way to describe the value of the variable. Embedding is just the fancy math term for building something that looks like (x,y) from the time series data (x1, x2, x3 …). Remember DCE, because I’m going to use it a lot.
Now look at the brain again.
Last time I described how cortical neurons are massively recurrent — firing backward and sideways, not just forward. When a neuron receives input from other neurons that were influenced by the same neuron a few milliseconds ago, the current signal gets blended with echoes of the recent past.
That blending is delay coordinate embedding. The brain is doing Takens’ trick.
A cortical column receives a stream of spikes. Through its recurrent connections, the activity at any given moment reflects not just the current input but the input from 10 milliseconds ago, 50 milliseconds ago, 200 milliseconds ago — carried forward and then back by the loops. The column is automatically combining the signal with time-delayed copies of itself. And Takens’ theorem tells us that’s sufficient to reconstruct the dynamical structure of whatever generated the signal.
So when light from a red apple hits your retina and gets converted into a spike train — a single stream of electrical pulses — the recurrent circuits of your visual cortex perform delay coordinate embedding on that stream. And what pops out, mathematically, is a reconstruction of the thing that caused the signal. The geometry of the apple’s color. The structure of its motion. The shape of the world that produced what your photoreceptors are reporting.
The brain solves the inverse problem by doing exactly what Takens described: taking one-dimensional data, using time delays inherent in recurrent loops, and embedding it into a higher-dimensional space where the structure of reality is recoverable.
The brain didn’t read Takens. Evolution stumbled onto the same mathematics through the only design tool it has — variation and selection over deep time. Recurrent cortical architecture works because it implements delay coordinate embedding, and delay coordinate embedding works because Takens proved it must.
This idea, that DCE is what the brain does, is the core idea of my theory of consciousness. The idea first occurred to me in the summer of 2007, when I attended the Complex Systems Summer School in Santa Fe (which I affectionately refer to as nerd camp and was one of the most productive months of my life.)
Recently, evidence has emerged that this idea might not just be an abstract notion. Four independent research groups have recently shown that artificial neural networks, when trained to predict the next step in a time series, spontaneously learn the same trick.2 Two of those groups measured this embedding behavior in the neural networks. The other two groups mathematically proved that this embedding behavior “emerges” in certain kinds of neural network. The networks weren’t programmed to perform delay embedding. They “discovered” it. Apparently, if you’re trying to predict what comes next in a complex system, neural networks discover that reconstructing the system using this embedding is the best strategy there is. Prediction and embedding turn out to be the same thing.
Let me pause and tell you what we’ve got so far, because it’s already a lot.
Neurons receive spike trains. Cortical circuits are recurrent — they feed signals back into themselves with various time delays. This recurrence performs delay coordinate embedding, which Takens proved is sufficient to reconstruct the dynamical structure of the system being observed. The brain builds representations of reality — color, motion, depth, sound — by embedding sensory signals into state spaces that preserve the geometry of what’s out there.
I call these recurrent circuits “DCE engines.” They’re the building blocks of everything that follows.
A color DCE engine takes input from your three types of cone cells and embeds it into a three-dimensional space. Your experience of color is a position in that space. People with deuteranopia — red-green colorblindness — have two functioning cone types instead of three. Their color engine embeds into a two-dimensional space. Their palette is real but reduced. The quality of their color experience is literally, mathematically, a lower-dimensional embedding.
A motion DCE engine in area MT of your visual cortex takes sequences of position signals and embeds them into a space where trajectory is represented. You don’t just see where something is — you see it moving. That motion perception isn’t a logical inference (”it was there, now it’s here, so it must have moved”). It’s a direct reconstruction of temporal dynamics by a recurrent circuit doing exactly what Takens described.
This is what I mean by state space representation. Each DCE engine builds a space — a quality space, to borrow Nick Shea’s term — in which some aspect of reality is encoded as a position, a trajectory, a shape. Your experience of the world is built from these spaces. All of them constructed by the same fundamental operation: delay coordinate embedding in recurrent neural circuits.
Here’s the punchline, the key point, the thing that fundamentally changes the discussion when it comes to consciousness science:
This whole process takes time. Not time in the trivial sense that everything takes time. Time in the deep sense that the computation can’t happen without it. Delay coordinate embedding is defined on time series. The delays are the mechanism. Strip away the temporal unfolding and you don’t have a degraded version of the process — you have nothing. A snapshot of a recurrent circuit mid-computation is like a snapshot of a melody at a single instant. It’s not a frozen piece of music. It’s a pressure reading. The music exists only across time.
That’s going to turn out to be the most important sentence in this entire series. Everything that comes next falls out of this idea that what the brain is doing is fundamentally and irreducibly temporal.
Next time: why consciousness is a verb, not a noun.
This is Part 2 of a 10-part series on the State Space Theory of Consciousness. Part 1: “The Reconstruction Problem.” Part 3: “Consciousness Is a Verb.” All the papers are here.
Takens’ embedding theorem requires the embedding dimension m ≥ 2d + 1, where d is the dimension of the attractor. The brain likely does better than this bound suggests, because it uses multiple channels (many neurons, not one measurement) and because the relevant perceptual dynamics are often low-dimensional. A pure tone is a circle. A chord is a torus. The dimensional requirements for embedding these are modest and well within the capacity of even small cortical circuits.
The four research groups I mentioned: Uribarri & Mindlin (2022) showed trained LSTMs produce hidden states topologically equivalent to the original attractor. Ostrow, Eisen & Fiete (2024) showed that recurrent neural network state-space models trained on prediction learn viable delay embeddings and perform better than feedforward neural networks based on transformer technology. Duan (2023) and Hart (2025) developed the mathematical proofs related to recurrent neural network reservoir computing systems: that these systems predict on dynamical time series data via embeddings. Based on Hart’s work, we developed our own mathematical synthesis (O’Reilly-Shah & Selvitella, 2026) that proves that contracting recurrent networks driven by regular dynamics generically implement smooth (C¹) embeddings. This is a formal guarantee underlying a candidate mechanism for the sensory front-end of SST.