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Comment by frisco | Hacker News Reader

Yes definitely. Here the "space" doesn't refer to physical space, but an abstract vector space that neuron's tuning represents. For example, there is a famous paper[1] that showed neurons could be responsive to abstract concepts -- for example, one might fire for "Bill Clinton" regardless of whether the stimulus is a photo of him, his name written as letters, or even (with weaker activation) photos/text of other members of his family or other concepts adjacent to him. The neuron's activity gives a vector in this high dimensional concept space, and that's the "space" GP is referring to.

[1] https://www.nature.com/articles/nature03687

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Can I get an ELI5 on how physical neurons, stuck in a measly 3 dimensions, can possibly form higher-dimensional connections on a large scale?

I understand higher dimensional connections in theory (such as in an abstract representation of neurons within a computer), but I can’t imagine how more highly-connected neurons could all physically fit together in meat space.

Consider three neurons all connected together. Now consider that each of them may have some 'voltage' anywhere between 0 and 1. Using three neurons you could describe boxes of different shapes in three dimensions. Add more and you get whatever large dimension you want.

see related talk by the first author: "Dynamic representations reduce interference in short-term memory": https://www.youtube.com/watch?v=uy7BUzcAenw

> Can I get an ELI5 on how physical neurons, stuck in a measly 3 dimensions, can possibly form higher-dimensional connections on a large scale?

You can multiplex in frequency and time. I'm not sure if neurons do it, but it's certainly possible with computer networks.

Your stick of RAM is also stuck in 3 dimensions but it reifies a, say, 32-billion-dimensional vector over Z/2Z.

If you take a matrix of covariance or similarity between neurons based on firing pattern, and try to reduce it to the sum of a weighted set of vectors, the number of vectors you would need to accurately model the system gives you the dimensionality of the space.

This does not seem particularly like an "Explain Like I'm 5"-parsable comment that the posted asked for.

This isn’t about the 3 dimensional structure the neurons occupy, but about their operational degrees of freedom.

Think about how a CNC machine works, you can have CNC with more than 3 axis, for example a 4 axis CNC machine can move left/right up/down backwards/forwards and also have another axis which can rotate in a given plane.

From a more mathematical perspective just think about the number of parameters in a system (excluding reduction) each parameter would be a dimension.

Appreciate the attempt, but in this example the 4th axis is not independent since the motion along that axis can be achieved, with some complexity, by the motion along the other axes. Granted this is not very useful for a machinist because it will be very tedious to machine a part this way compared to the dedicated 4th rotating axis, but mathematically it is redundant.

I have found it easiest to think of a logical dimensions or configurations when thinking of higher dimensions. Physically it can be a row of bulbs (lighted or not) wherein N bulbs (dimensions) can represent 2^n states in total. The 2 here can be increased by having bulbs that can light up in many colours.

its not redundant. without rotation it could only ever drill downwards.

Smartphones eg. measure six dimensions of freedome, including rotation about every axis. 3 for location, 3 for orientation.

this has very little to do with synapses.

The vectors are in a configuration vector space, not a physical vector space.

I roughly understand what the article says about dimensional space (Reading higher mathematics books on the way to my meagre college course way back when, helps me a little, even if it is all half-remembered and a bit wrong -- this understanding is sufficient enough to satisfy me), however the poster above me doesn't, and clearly asked for a definition a 5 year old layman could understand.

The comment I am replying to, your comment in the tree, and the one next to you, does not seem to match that request in any sense.

Now, simplified definitions are an art, but Feynman managed it with Quantum Electrodynamics -- so it is not impossible to do it for complex subjects. And it seems to me the less you understand a subject, the less simple and more confusing your explanation will be, such as the explanations given by the other posters here. (fyi: I do not understand enough to properly convey my understanding clearly -- which is why I have not attempted to do so)

This isn't a matter of having an incomplete understanding, thanks for the offhanded aspersion though. The fundamental problem is that the concept of manifolds in state space isn't really something that has a non-tortured real world analogy, which is a prerequisite for a five year old to understand. It's probably possible to express more simply with a video demonstrating the covariance structure of a data set visually, then showing how that results from a small set of vectors, but I've read enough textbooks to be confident that a simple, concise explanation eludes words.

Think of it less as n-dimensional in meat space and more of n-dimensional in how it functions.

Do you mean due to the thickness of each connection, they would occupy too much space if the number of dimensions was too high? Not necessarily 4 or more, just very high because there are on the order of n^2 connections for n neurons?

In the visual cortex, neurons are arranged in layers of 2D sheets, so that perhaps gives an extra dimension to fit connections between layers.

If I’m recording from N neurons, I’m recording from an N-dimensional system. Each neuron’s firing rate is an axis in this space. If each neuron is maximally uncorrelated from all other neurons, the system will be maximally high dimensional. Its dimensionality will be N. Geometrically, you can think of the state vector of the system (where again, each element is the firing rate of one neuron) as eventually visiting every part of this N-dimensional space. Interestingly, however, neural activity actually tends to be fairly low dimensional (3, 4, 5 dimensional) across most experiments we’ve recorded from. This is because neurons tend to be highly correlated with each other. So the state vector of neural activity doesn’t actually visit every point in this high dimensional space. It tends to stay in a low dimensional space, or on a “manifold” within the N-dimensional space.

Would you have any further reading on this? Sounds fascinating.

Agreed, it's really cool :). A lot of this is very new -- it's only been in the past decade and a half or so that we've been able to record from large populations of neurons (on the order of hundreds and up, see [0]). But there are a lot of smart people working on figuring out how to make sense of this data, and why we see low-dimensional signals in these population recordings. Here are some good reviews on the subject: [1], [2], [3], [4], and [5].

[0]: https://stevenson.lab.uconn.edu/scaling/ [1]: https://www.nature.com/articles/nn.3776 [2]: https://doi.org/10.1016/j.conb.2015.04.003 [3]: https://doi.org/10.1016/j.conb.2019.02.002 [4]: https://arxiv.org/abs/2104.00145 [5]: https://doi.org/10.1016/j.neuron.2017.05.025

I'm curious about how much of this apparent low dimensionality is explained by (1) the physical proximity of the neurons being recorded, (2) poverty of the stimuli (just 4 sequences in this paper, if I'm not mistaken)

Both good questions. It could very well be that low dimensionality is simply a byproduct of the fact that neuroscientists train animals on such simple (i.e., low-dimensional) tasks. This paper argues that [0]. As for your first point, it is known that auditory cortex exhibits tonotopy, such that nearby neurons in auditory cortex respond to similar frequencies. But much of cortex doesn't really exhibit this kind of simple organization. Regardless, technological advancements are making it easier for us to record from large populations of neurons (as well as track behavior in 3D) while animals freely move in more naturalistic environments. I think these kinds of experiments will make it clearer whether low-dimensional dynamics are a byproduct of simple task designs.

[0]: https://www.biorxiv.org/content/10.1101/214262v1.abstract

This is basically just linear algebra.

For an abstract perspective, try Sheldon Axler's Linear Algebra Done Right.

For a more concrete perspective, Gilbert Strang's lectures: https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D

Look up state space, then neural population and neural coding.

This isn't really something about neurons per se, it's about systems.

Suppose I have a system that can be fully characterized (for my purposes) by two number: temperature and pressure. If I take every possible temperature and every possible pressure, these form a vector space. But notice that temperature and pressure are not positions in the real world. It's a "state space" or "configuration space". At any moment in time, I could measure my system's temperature and pressure, and plot a point at (temperature(t), pressure(t)). As the system changes through time according to whatever rules govern its behaviour, I could take snapshots and plot those points (temperature(t+1), pressure(t+1)), (temperature(t+2), pressure(t+2)). This would give a curve "trajectory" that represents the systems evolution over time.

Okay, that's a 2D state space. But imagine I had a simulation of 10 particles (maybe some planetary simulation for a game). For each point I have maybe a 3D position (x,y,z) and a 3D velocity (vx, vy, vz). So I need 6 numbers to fully describe the state of each particle, and I have 10 particles. Therefore to fully describe the state of the whole system, I need 60 numbers. I therefore have a 60-dimensional state space. But each of these dimensions does not represent a position measurement along some axis in the world. In fact, only 30 of them do (3 * 10), the other 30 represent velocities.

Same as a silicon chip stuck in 2 dimensions can.

The vector here refers to the "feature vector" where the dimension is the number of elements in the vector. E.g. a feature vector of [size, length, width, height, color, shape, smell] has 7 dimensions. A feature vector for the space has 3 dimensions [x, y, z]. The term "higher dimension" just means the number of features encoded in the vector is higher than usual.

In the context of neurons, while the neurons are in the 3 spatial dimensions, the connections of each neuron can be encoded in a feature vector. Each connection can specialize on one feature, e.g. the hair color of the person. These connection features can be encoded in a vector. The number of connections becomes the dimension of the vector. Not to be confused with the physical 3D spatial dimensions of the neurons.

The nice thing about encoding things in vectors is that you can use generic math to manipulate them. E.g. rotation mentioned in this article, orthogonality of vectors implies they have no overlap, or dot product of vectors measures how "similar" they are. Apparently this article shows that different versions of the sensory data encoded in neurons can be rotated just like vector rotation so that they are orthogonal and won't interfere with each other.

Linear algebra usually deals with 2 or 3 dimensions. Geometric algebra works better on higher dimension vectors.

Don't conflate physical and logical, in this case we don't care about the physical dimensions, only how the logic is expressed. Even a 2D function can be expressed in N-dimensional parameters, such as

y = a1 * x + a2 * x^2 + a3 * x^3 + a4 * x^4

where you only have one input and one output, but 4 constants that can be adjusted. These 4 constants make up a 4D vector.

This is fun, I'm enjoying reading the replies :) I'm certainly no expert, but attempting an explanation helps me exercise my personal understanding, so here goes. Corrections welcome.

The "connections" you mention aren't the issue, in my understanding of the biology. Neurons are already very strongly interconnected by numerous synapses, so they already do physically fit together in their available 3D space, and appear capable of representing high-dimensional concepts. (See caveat below.)

The "higher dimensions" here are not where the neurons exist, only what they're capable of representing. If we think about a representation of the concept of a "dog" for example, there are many dimensions. Size, colour, breed, temperament, barking, growling, panting, etc etc. Those attributes are dimensions.

Take two dog attributes: size and breed. You can plot a graph of dogs, each dog being a mark on the graph of size vs breed. Add a third dimension and turn the graph into a cube: temperament. You can probably imagine plotting dogs inside this three dimensional space.

It's very difficult to imagine that graph extending into 4th, 5th or further dimensions. And yet, you can easily imagine, say, a dog that's a large, black, friendly Labrador with a deep bark who growls only rarely. We could say that dog can be represented as a point in 6-dimensional space (or perhaps a 6-dimensional slice through a space with even more dimensions, just a slice through 3D space could produce a 2D graph).

The number of connections between neurons may be related to the number of dimensions they can represent. In honesty, I don't know, and I guess that if there is a relationship it may not be linear. So neurons might be capable of representing 4 dimensions with fewer than 4 synapses, for example, I don't know. Seems possible to me, though.

Caveat: I think my reasoning here may be fallacious: "the fact that neurons are capable of representing high-dimension concepts demonstrates that they have adequate synapses to do so". It seems akin to anthropocentrism, I'm not sure. Perhaps it's just a circular argument. I think it provides an adequate basis for an ELI5 though.

I look forward to further comments!

the ELI 5 of higher dimensions explained mathematically in text is that a coordinate in R^3 is identified uniquely by a three tuple u = (x, y, z). A four touple simply adds one dimension. That might be a time coordinate, color, etc.

If I remember correctly, the integers Z form spaces, too. Z^2 can be illustrated as grid, where every node is uniquely identified again coordinates or by two of its neighbours, eitherway v = (a, b).

Adjency lists or index matrices are common ways to encode graphs. My modelnof a neuron network is then a graph.

I imagine that, since Neurons have many more Synapses, that's how you get a manifold with many more coordinates.

Each Neuron stores action potential much like color of a pixel and its state evolves over time, but that's when the model becomes limited.

How it actually represents complex information in this structure I don't know.

PS: Or very simply put, physics has more than three dimensions.

There was a fun article in early March showing that the same is true for image recognition deep neural networks. They were able to identify nodes that corresponded with "Spider-Man", whether shown as a sketch, a cosplayer, or text involving the word "spider".

https://openai.com/blog/multimodal-neurons/

deep neural nets are an extension of sparse autoencoders which perform nonlinear principal component analysis [0,1]

There is evidence for sparse coding and PCA-like mechanisms in the brain, e.g. in visual and olfactory cortex [2,3,4,5]

There is no evidence though for backprop or similar global error-correction as in DNN, instead biologically plausible mechanisms might operate via local updates as in [6,7] or similar to locality-sensitive hashing [8]

[0] Sparse Autoencoder https://web.stanford.edu/class/cs294a/sparseAutoencoder.pdf

[1] Eigenfaces https://en.wikipedia.org/wiki/Eigenface

[2] Sparse Coding http://www.scholarpedia.org/article/Sparse_coding

[3] Sparse coding with an overcomplete basis set: A strategy employed by V1?https://www.sciencedirect.com/science/article/pii/S004269899...

[4] Researchers discover the mathematical system used by the brain to organize visual objects https://medicalxpress.com/news/2020-06-mathematical-brain-vi...

[5] Vision And Brain https://www.amazon.com/Vision-Brain-Perceive-World-Press/dp/...

[6] Oja's rule https://en.wikipedia.org/wiki/Oja%27s_rule

[7] Linear Hebbian learning and PCA http://www.rctn.org/bruno/psc128/PCA-hebb.pdf

[8] A neural algorithm for a fundamental computing problem https://science.sciencemag.org/content/358/6364/793

Wouldn't it be especially inelegant/inefficient to try and wire synapses for, say, a seven-dimensional cross-referencing system, when have to actually physically locate the synapses for this system in three-dimensional space?

(and when the neocortex that does most of the processing with this data is actually closer to a very thin, almost two-dimensional manifold wrapped around the sulci)

There has to be an information-theory connection between the physical form and the dimensionality of the memory lookup, even if they aren't referring to precisely the same thing, right?

Maybe, but my guess would be that there's a trade off made here. Either you can use higher dimensionality in the abstract, or you can have a much much bigger brain. A bigger brain processes slower merely because of volume and requires a lot more resources to support it.

Nature stumbled onto the path that it did because we don't have high enough nutrient food or fast enough neurons.

The issue with your question is that the dimensions of the configuration space and the physical form aren’t even approximately the same thing. Take, for example, a 100x100 grayscale image. It’s a flat image; the physical dimensions are 2. There are 10,000 different pixels though, and they are all allowed to vary independently of each other; the configuration-space dimensions are 10,000. Neurons are like the pixels in this analogy, not like the the physical dimensions.

Neurons are not random access. The analogy is otherwise pretty apt, except that an image doesn't store information about what it displays and I don't mean EXIF.

Neurons aren't allowed to vary independently of each other, and neither are pixels; A grayscale image with random pixels is static, not even recognizable as an image. The mind cannot decode those pixels in a seven-dimensional indexing scheme, it can't even decode them in the given two dimensions if you have an array size error and store the same data in an array 87 columns wide. In your analogy, if you put a stop sign into the upper right side of the image, that is always going to be recalled associativity with the green caterpillar you put in the lower left side of the image. These properties don't work so well for memories & imperfect/error-prone but statistically correct biological systems.

The average neuron has 1000 synapses, and for geometric reasons (Synaptic connections take up space) most of those are to other neurons that aren't very far away in 3D space.

You’re not contradicting my point, you’re just using the word “image” to refer to a different concept than I did. That “100x100 pictures of the real world” won’t reflect all the 10,000 dimensions available in “a 100x100 bitmap” is certainly the case—and that is what is meant by saying they lie on a lower-dimensional manifold within that space.

Similarly: yes, physics limits neuronal connectivity. The actual space of neuronal connections lies on a manifold inside the full “n squared, divided by 2” dimensions of connectivity of any old set of n points. That still doesn’t mean neurons can’t represent high-dimensional concepts, because your treatment of physical dimensions as the same thing as concept space is still mistaken. Taking your 1000 synapses number for granted, the input to a given neuron would be 1000-dimensional, not three. If you’re not arguing the concept space is 3d, and merely arguing against those who’d say neuronal connectivity isn’t limited by physical constraints, then I’d advise a reread of the ancestor comments; none of them are saying that.

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