What's even a manifold? Let alone a neural one.
If you look into the current literature in computational and theoretical neuroscience, you’ll see neural manifolds mentioned a lot. But what are they? In short, they are the all the configurations a group of neurons can be active. Think of it like an orchestra. Sure, each person in the ensemble could play whatever they wanted on their instrument, but to make a song, everyone has to be coordinated relative to the group. And an orchestra can play many different songs, it just requires different arrangements of coordinated playing, but coordinated nonetheless.
Let’s take a step back and unpack what a manifold is before throwing neurons in the mix. The short textbook definition is a topological space that looks like Euclidean space at every point. So as humans, we live on the surface of the earth, a big three dimensional ball in space. But wherever we stand, we just see an expansive flat plane (if it’s sufficiently flat and grassy we call it a plain, haha English). This makes the surface of the earth a 2-manifold, because it looks like a flat 2 dimensional plane at every point. Most continuous functions you’ve probably seen in algebra are 1-manifolds. When you graph the equation it is a one manifold because if you zoom in really close it looks like a line. Same with or . BUT the equation is not a manifold because it’s pointy at so no matter how much you zoom in it will never look like a straight line.
All of these lines are 1-dimensional manifolds that are arranged differently on this 2-dimensional plane, just like how the surface of the earth is a 2-dimensional manifold sitting in 3-dimensional space. So we can say that a manifold is any smooth space of a certain dimension that lives in some space of a higher dimension.
Now, let’s look at how these relate to neuroscience.
If we have some population of let’s say, N neurons in our brain, then the “higher” dimensional space that our manifold would live in N-dimensional space. Think of this as each neuron, from 1 to N, as having its own axis that represents the activity of that neuron.
So now, if these neurons are from the motor cortex of a mouse, then we could let the mouse walk and see how this point moves around. It might look something like this
We can see that the trajectory of the combination of these three neurons’ activity sweeps out a circle in the three dimensional space, and does so about four times, probably because walking is a very cyclic activity so maybe the mouse took four steps while we were recording these three neurons. Although it’s a little messy, that’s probably because of the noisy nature of neurons, but it is very clearly a circle. This is a neural manifold. The important thing to note is that the activity of these neurons sweep out a manifold that is of a lower dimension than the ambient space (the number of neurons), here we have a 1-dimensional circle living in 3-dimensional space. (Remember, that even though a circle is 2-dimensional, we call this a 1-manifold because when we zoom in it looks like a line. If the neurons’ activity swept through the interior of the circle so it looked like a filled-in disk then that would be a 2-manifold).
These neural manifolds tell us that neural activity is not totally free and unhinged, these neurons’ activity is confined to this circle. And an interesting question in current research is if different manifolds are the created for different types of processing. Let’s say we recorded from the same three neurons but instead of having the mouse walk, we had it swim (mice are actually very skilled swimmers). Would these three neurons sweep out a different shape in the 3-dimensional space? Maybe it would still be a circle but at a different orientation? Maybe it would be a 2-manifold instead of a 1-manifold? These are very interesting questions being asked by researchers today.
Hopefully this was a good introduction of neural manifolds. There’s a lot of research being done in this area of neuroscience so now you can jump into some of the literature without being too afraid of the word manifold.