6 ms·

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 p

• PullJosh 5y ago
  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.
  • cochne 5y ago
    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.
  • deleted 5y ago
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  • andyxor 5y ago
    see related talk by the first author: "Dynamic representations reduce interference in short-term memory": https://www.youtube.com/watch?v=uy7BUzcAenw https://www.youtube.com/watch?v=uy7BUzcAenw
  • ajuc 5y ago
    > 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.
  • wyager 5y ago
    Your stick of RAM is also stuck in 3 dimensions but it reifies a, say, 32-billion-dimensional vector over Z/2Z.
  • CuriouslyC 5y ago
    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.
    • fao_ 5y ago
      This does not seem particularly like an "Explain Like I'm 5"-parsable comment that the posted asked for.
      • dogma1138 5y ago
        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.
        • anu7df 5y ago
          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.
          • posterboy 5y ago
            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.
  • fsociety 5y ago
    Think of it less as n-dimensional in meat space and more of n-dimensional in how it functions.
  • exporectomy 5y ago
    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.
  • dopu 5y ago
    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.
    • chadcmulligan 5y ago
      Would you have any further reading on this? Sounds fascinating.
      • dopu 5y ago
        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/ https://stevenson.lab.uconn.edu/scaling/ [1]: https://www.nature.com/articles/nn.3776 https://www.nature.com/articles/nn.3776 [2]: https://doi.org/10.1016/j.conb.2015.04.003 https://doi.org/10.1016/j.conb.2015.04.003 [3]: https://doi.org/10.1016/j.conb.2019.02.002 https://doi.org/10.1016/j.conb.2019.02.002 [4]: https://arxiv.org/abs/2104.00145 https://arxiv.org/abs/2104.00145 [5]: https://doi.org/10.1016/j.neuron.2017.05.025 https://doi.org/10.1016/j.neuron.2017.05.025
        • trott 5y ago
          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)
  • dboreham 5y ago
    Same as a silicon chip stuck in 2 dimensions can.
  • deleted 5y ago
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  • ww520 5y ago
    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.
  • Salgat 5y ago
    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.
  • austinjp 5y ago
    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!
  • posterboy 5y ago
    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.
• MereInterest 5y ago
  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/ https://openai.com/blog/multimodal-neurons/
  • andyxor 5y ago
    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 https://web.stanford.edu/class/cs294a/sparseAutoencoder.pdf [1] Eigenfaces https://en.wikipedia.org/wiki/Eigenface https://en.wikipedia.org/wiki/Eigenface [2] Sparse Coding http://www.scholarpedia.org/article/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/S0042698997001697 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-visual.html https://medicalxpress.com/news/2020-06-mathematical-brain-vi... [5] Vision And Brain https://www.amazon.com/Vision-Brain-Perceive-World-Press/dp/0262517736 https://www.amazon.com/Vision-Brain-Perceive-World-Press/dp/... [6] Oja's rule https://en.wikipedia.org/wiki/Oja%27s_rule https://en.wikipedia.org/wiki/Oja%27s_rule [7] Linear Hebbian learning and PCA http://www.rctn.org/bruno/psc128/PCA-hebb.pdf 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 https://science.sciencemag.org/content/358/6364/793
• mapt 5y ago
  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?
  • true_religion 5y ago
    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.
  • riwsky 5y ago
    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.
    • posterboy 5y ago
      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.
    • mapt 5y ago
      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.