Neural Manifolds

Watched some vids and read some papers on this and was very interested, hoping to keep learning on this subject so maybe this page will just continue to be updated.

Comp. Neuroscience Intro Kinda

Neurons have generated an "action potential" when the electrical spike which transfers to other neurons occurs. It is thought that all sensory data, memory, and information is the result of these firings, based on different variables such as the strength of the spike, the temporal relation between it and other spikes, and so on. Using multielectrode arrays, we can read the signals of up to a few hundred neurons, and lay them all out together. Marking when spikes happen with a vertical line, we get what is called a "spike train". We then look at the firing rate of each neuron. We find the amount of times a spike occurs per unit time for each neuron. We then smooth the data to get a rising and falling curve for each neuron, like hundreds of little (smooth) richter scales. So if there are n neurons, the behavior of the system can be characterized by an amount of numbers called n, each one representing the instantaneous firing rate of a single neuron. The totality of these numbers forms an n-dimensional vector, which corresponds to a point in n-dimensional space. For instance: As an animal forages or does different activities, the instantaneous firing rate changes, if there was 3 neurons being recorded (wouldnt be enough info but go along) each would have a rate of activation that changes over time, so the instantaneous firing rate over time would be the value of the derivative of the function describing the firing rate. Take those 3 derivatives and put them together, each one describing how much in one direction the vector should go. So if you're following, you might realize different areas of the phase space graph exist on a gradient of increasing activation. The farther you are from the origin the more those neurons are going. So we can track a lot of information simply with a point moving through n-dimensional space. Now exciting part. Because the neurons being measured are part of the same system and their firing rates cannot be independent of each other, the trajectory of the point in n-dimensional space will be confined to a subspace. This subspace depends on the connectivity of the network, the strength of the connections between the neurons and the task being done. Current hypothesis which has been shown many times experimentally: Even though the ambient space is very high dimensional, the actual trajectory is confined to a very low dimensional structure.

Manifolds Yippeeee

Topology holds that shapes are homeomorphic if one can be bent twisted and stretched into the other. A manifold is (for our purposes) some shape in a euclidean space, which locally resembles a euclidean space of lower dimension; much like how on a sphere, at the local level, it is indistinguishable from a flat plane. The same is true of a torus. But if you pinch one side of the torus so that the two sides don't come together smoothly, instead collapsing into a single point, that is not a manifold. It stops being a manifold because at the local level around that point, it doesn't exactly look like a flat plane. It's funky. To analyze manifolds, we look at the properties that are invariant between homeomorphic shapes. Manifolds are defined by intrinsic and embedding dimension. The dimension the manifold is sitting in is its embedding dimension. The intrinsic dimension defines how many variables you need to describe your location if you lived on the manifold. A sphere floating in space is embedded in three-dimensional space but to describe location you only need two variables, latitude and longitude. There are neurons in your brain which are responsible for telling you your head direction. Each individual neuron in the system has a preferred direction, and increases firing rate when facing that way. We can assume then that similar neurons exist for all other activities and behave in much the same way, so we could theoretically get really accurate subconscious level info from people re: their feelings and such.

Putting it all Together

If the collective dimensions of a circuit encode a variable of a certain dimension and topology, then the activity of this network would be localized to a manifold of matching dimension and topology.

Tracking the neurons responsible for head direction then, we'd expect to see a manifold of 1 dimension, because it's one variable being encoded. We would also expect there to be a hole in the middle (neurons firing in the zero direction wouldnt make sense). So we'd expect to see something like a loop, which is homeomorphic to a circle. We see this happens perfectly, we can look at only experimental data, seeing no mouse at all, and clearly tell what way its head is facing. And all proccesses in the brain have such manifolds, if we could record the entire brain in this way, we would have access to a collection of manifolds which could characterize the functioning of the entire system. Additionally, by stimulating neurons consistent with the point on the manifold we can artificially make the subject (usually rats for these experiments) perceive conditions that aren't real. For example, have made a rat think they were at the beginning of the maze when they were at the end, but that's sort of mean so hopefully we don't do that too much. Computational neuroscience and topological data analysis are very exciting things. There's a little map of all your friends in your brain and every time something happens the little cursor moves to a different part on the map. Isn't that cute?
~Bone🦴