Take a binary array of length N, where N is in the hundreds to thousands range. Choose 2% of the bits to set to 1. Now you have a "sparse array".
Now, you want to use this sparse array to represent a note in a song. So you need every note to consistently map to a distinct* sparse array.
However, you also want to be able distinguish a note as being in one song or another. The representation should tell you not only that this is note A but note A in song X.
How might you do that? Well some portion of the ON bits could be held consistent for every A note and some could be used to represent specific contexts.
Stable and variable bits of you will.
Now if you look at two representations of the note A from two songs you'll see they're different. How different are they? Well you could just count the bits they have in common or not, or you can treat them as vectors. (Lines in high dimensional space) Then you can calculate the angle between those two lines. As that angle increases its easier to distinguish the two lines. They won't ever get to full "right angles" between them because of the shared stable bits, but they can be more or less orthogonal.
That's what's happening here. The brain is encoding notes in a way that it can both recognize A, but also recall it in different contexts.
*But not perfectly consistent, we use sparse representations because the brain is noisy and it's more energy efficient. Pretty close is good enough in the brain and you can encode a lot of values in 1000 choose 20 options.
It would make an interesting essay or monograph to descrbibe the various similes and metaphors that people have variously used to describe the working of the human mind. Descartes, I believe, thought the brain worked via a series of hydraulics, and the Victorians thought that it worked by some variant of electricity, both of these being "new" and possibly exciting technologies for their times. Notwithstanding the scientific versimillitude of these previous theories, it seems only fitting that computers occupy that role today.
This really would have been harder for me to understand had I not taken linear and abstract algebra courses a few years ago. That area of maths reused common words like "rotation" but with more generalized definitions, which made it was jarring and confusing to hear and take in at the time. When someone said the word "rotate" my mind as if by reflex was already trying visualize a 3d or 2d rotation even when it made no sense for the problem at hand. Being an English speaker my whole life I thought I understood what a rotation was or could be but I didn't.
Same goes for what's being alleged here: Is there even a way to visualize this that makes mathematical sense?
What will be the corollaries to this discovery simply as a result of what the mathematics of rotations will dictate?
Reminds me of how orthogonally polarized waves can inhabit the same bit of space without interfering with each other (and can be cleanly separated later using 2 polarized filters at 90 degrees).
"After my one course in linear algebra, I knew eigenvectors and eigenvalues like the back of my head. If your instructor was anything like mine, you recall solving problems involving eigendoohickeys, but you never really understood them."
-- Jonathan Richard Shewchuk, from An Introduction to
the Conjugate Gradient Method Without the Agonizing Pain
I would say that one can still interpret it geometrically, but the difficulty is that it's not in 2D or 3D, but in a higher dimensional space.
Imagine you have a population of 100 neurons. To each one you associate a real number representing its activity. So a particular snapshot of firing would have 100 real values. It's a vector in a 100 dimensional space. Each individual neuron is one dimension. The high dimensionality of the space is what makes it unintuitive. But, after some familiarization you learn heuristics for reasoning about high-dimensional spaces and combine that with your 3D spatial intuition.
The 45 degree rotation is just some nice art, but you could think of it as representing a projection down to 2D.
Sure, but the neural activity is actually low-dimensional (see Extended Fig 5e). By day 4, the first two principal components of the neural activity explains 75% of the variance in response. ~3-4 dimensions is not particularly high dimensional.
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.
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.
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.
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].
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.
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.
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.
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.
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 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.
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.
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.
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.
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".
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]
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?
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 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.
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.
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.
If you think of groups of neurons in arbitrary dimensions, where some groups fire together for some things, and a different group with some overlap fire for other things, then it's like two dimensions where a line is a sense or thought and the lines are crossing where they fire for both memories. So two thoughts along two dimensions can cross and light up that subset of neurons. If the two thoughts, or lines, are orthogonal, then not many neurons are both firing for thoughts. If you have many many neurons, and many many memories, then the dimensionality, or possible subsets of firing neurons, is huge. Like our two lines but now in three dimensions, there are a lot of ways for them not to overlap. So the possibility that many things in that space are orthogonal is likely. In a highly dimensional space, a whole lot of things don't overlap.
Yes, grid cells in the hippocampus [0] form a coordinate system that is used for 4D spatiotemporal navigation [1], as well as navigation in abstract high-dimensional "concept space" [2]
Yes but only in aggregate, like how adding a column to a database table is also adding a "dimension" to said data.
I'm not convinced the author's analogy of cross-writing to fit more information on a page is actually going to be helpful to most people's understanding. It led me at least to try to imagine visually what's going on, to picture the input being physically rotated. This is more akin to the more abstract but inclusive concept of rotation from linear algebra, where more dimensions (of information, not space or time) makes sense.
No? If the samples are randomly chosen then you'd expect the cosign similarity to be low, but there's no such assumption here, in fact it's the exact opposite.
"Independence" is defined for random variables, so "randomness" is implied. However, this approximate orthogonality property depends on the distribution. It holds for the normal distribution and many others, although not in general.
Not sure about the neuroscience context, but if you have two large ("high-dimensional") vectors of variables that have a population correlation of zero ("independent"), then the dot product of a sample is likely to be close to zero ("orthogonal") due to the law of large numbers.
Curious. I cannot understand it clearly. Lets take for example "my wife and my mother-in-law" illusion[1]. It is known for it's property that one cannot see both women at once. If we assume that it has something to do with such a coding in neurons, would it mean that those women are orthogonal, or it would mean that they refuse to go orthogonal?
Sorry, I'm pretty tired, but I fail to see the relation to this article, how does that example apply?
I thought that was more of a case of a human's facial recognition being a special function, and we're not able to process two or more people's faces at the same time. Like, see the details in them, recognize that it's their face.
You're either looking at one person, or the other, but if you try to look at both of them at the same time, they become "blurry", unrecognizable, even though you remember all the other information about them both.
But that's not related to memory integrity and new emotions/sensations?
It is a work of human visual perception at work. Somehow you mind chooses how to interpret sensations from a retina, and shows you one of women. Then you mind chooses to switch interpretations and you see the other one. Both interpretation are somewhere in memory. So it may be connected with this research.
Like with those chords in a research. Mice hear one chord, and by association from memory it expects other chord. But instead it hears some third chord. Expected and unexpected chords have perpendicular representation, if I understood correctly.
Here you see a picture, and expects one interpretation or other. You have memory of both, but you get just one.
Possibly it doesn't apply, I do not know. I'm trying to understand it. The obvious step is to make a prediction from a theory, should interpretations oscillate, if it has something to do with perpendicularity of representation in neurons?
When I hear another chord instead of a predicted one, do prediction and sensations oscillate? I'm not quick enough to judge based on a subjective experience.
Really? I have no trouble seeing both at the same time. Nothing special about it, the angles of their respective faces are different enough that it doesn't feel like there's any interference at all.
I'm not really sure if I'm able to see both (or the three of them in the case of the 'Mother, Father and Daugther' figure), but at least I can switch stupidly fast.
Hmmm.. I tried to visualize them both at the same time.. it took some effort, but quickly "oscillating" between the two ended up settling (without a jittery oscillating feeling) on seeing both at the same time. Maybe my brain was playing meta tricks on me though?
I can "see" both at the same time, but only if I am not focusing on either. I think this conflict of focus is the real effect people are talking about.
I figured out how to change it at will eventually, if you close your eyes then open them and look at the bottom of the picture first it’s an old woman. Do the reverse and it’s a young woman. Eventually you can do that without the eye closing step but never would I say I could see both at once.
I had trouble at first too until I noticed the ear looking a little suspicious. If you create a diagonal obstruction from the top of the hat, to the nose, you are left will only the mother-in-law; the ear has now become an eye.
Once I'd seen it once, the mother-in-law is now prominent. I can still see the wife if I concisely choose to, but the mother-in-law is now the default, strange huh?
There was another recent article on applications of geometry to analyse neural mechanisms to encode context. It also mentioned a rotation/coiling geometry:
> The work could help reconcile two sides of an ongoing debate about whether short-term memories are maintained through constant, persistent representations or through dynamic neural codes that change over time. Instead of coming down on one side or the other, “our results show that basically they were both right,” Buschman said, with stable neurons achieving the former and switching neurons the latter. The combination of processes is useful because “it actually helps with preventing interference and doing this orthogonal rotation.”
This sounds like the early conservation of momentum / conservation of energy debates. (Not that they used those words back then.)
I don't remember where I came across this (was probably some pop neuroscience blog or maybe radiolab), but there was some theory about how memories seem subject to degredaton when you recall them a lot, and less so when you don't.
I guess that would sort of be like the opposite of DRAM - cells maintain state when undisturbed, but the "refresh" operation is lossy.
Yeah I'm thinking that's because our interpretation of reality and it's abstractions är falsy and that filter is applied every time we update the memory. Maybe then when we are learning a new subject through say reading our filter is minimal and every time we read the same info we combat our falsy interpretation of reality.
Quote (although it’s missing context if the full show):
> Yeah, I think it's really interesting. I think it's really interesting to think about why we do these things, why we misrecollect our past, how those kinds of reconstruction errors occur. And I think about it in my own personal life - I share my memories with my partner. And many of us who have partners, we have these sort of collaborative ways in which we recollect. But those collaborations often result in my incorporating information into my memories that were suggested by this individual, but I never experienced. And so I might have this vivid recollection of something that only my partner experienced because we've shared that information so often. And so that's how we can distort memories in the laboratory. We can just get individuals to try and reconstruct events over and over and over again. And with each reconstructive process, they become more and more confident that that event has occurred.
I'm under the anecdotal and subjective impression that I can do a "brain dump" describing a recently-experienced physical event. But it's a one-shot exercise. Close to read-once recall. The archived magnetic 9-track tape that when read becomes a take-up reel of backing and a pile of rust. The memories feel like they're degrading as recalled, like beach sand eroding under foot, and becoming "synthetic", made up. The dump is extremely sparse and patchy. Like a limits-of-perception vision experiment: "I have moderate confidence that I saw a flash towards upper left". Not "I went through the door and down the hall" but "low-confidence of a push with right shoulder, medium-confidence passing a paper curled out from the wall at waist height, and ... that's all I've got". But what shape curl? Where in the hall? You've whatever detail was available around the moment you recalled it, because moments later extra information recalled start tasting different, speculative fill-in-the-blanks untrustworthy.
How fascinating, I've experienced this myself to a large degree. I have a few songs that very vividly remind me of certain periods or points of my life. When I play them, I always feel like I'm scratching up the vinyl surface of the memory, and I lose a little bit each time. Rather disappointing :(
This maps wonderfully onto SVD, Neural networks and embeddings.
Word embeddings frequently encode particular traits in different 'regions' of a 256(ish) dimensional space. AFAIK, It is also why we think of element wise addition (merging) in neural networks as an efficient and relatively loss-less computation. The aggregation after attention step used in Transformers (GPT-3) fundamentally relies on this being true.
Although from my reading, there is an inherent assumption of sparsity in such situations. So, is it reasonable to assume that human neurons are also relatively sparse in how information is stored ?
I read the abstract and don't really get it. How is this different from saying that a group of neurons A is responsible for memory storage and a group of neurons B is responsible for sensory processing, and A != B? I think I'm misunderstanding this "rotation" concept.
It's a good question. It looks like they actually specifically check for this and show that it's not two separate groups of neurons. Instead a subset of the neural population changes their representation of the input as it moves from sensory to memory, so it's more like a single group of neurons that represents current sensory and past memory information in two orthogonal directions.
In a simple example that I can think of it could just be a vector of <present, past> aka the current info could be encoded like [<2, 0>, <4, 0>] then rotated to ("y axis") [<0, 2>, <0, 4>] allowing you to write more "present" data to the original x dimension without overriding the past data.
If you're asking about the exact numbers here's a snippet from the xlsx document.
```
ABCD_mean ABCD_se ABCD_mean ABCD_se XYCD_mean XYCD_se XYCD_mean XYCD_se day neuron subject time
0 6.012574653 0.5990308106 6.181361381 0.5737310366 6.59759636 0.6419092978 6.795648346 0.5716884524 1 2 M496 -50
```
According to the article SEM neural activity, though this is way beyond my ability to interpret.
My simplified picture of what’s going on is something like this (if I’m understanding the paper correctly). Stimulus A starts out represented by the vector (1,1,1,1) and B by (-1,-1,-1,-1). Those are the sensory representations. Later A is represented by (1,1,-1,-1) and B by (-1,-1,1,1). Those are the memory representations. The last two component/neurons have “switched” their selectivity and rotated the encoding. The directions (1,1,1,1) and (1,1,-1,-1) are orthogonal, so you can store sensory info (A vs B in the present) along one and memory info (A vs B in the past) aling the other.
Maybe that is the origin of consciousness - when the processing becomes high level enough that the interference of past into present becomes sophisticated/multilayered enough to be useful.
The problem is suppose you have 4 neurons that need to understand a memory and present experience at the same time to make a decision (for example that you see a hot stove and memory that hot stoves hurt). The incoming neurons from memory and experience each have 4 neuron connections, which fire at some rate. Lets represent this as the firing rate of the neurons per second in a 4d vector:
experience: <1.0, 0, 0, 0> (1 pulse per sec on axis 0)
memory: <1.0, 0, 0, 0> (1 pulse per sec on axis 0)
If you "add" these together at the downstream neurons, you won't be able to tell which was a memory and which was sensation. A simplified explanation of how neurons work is by combining voltages from their incoming neurons. Example:
downstream sees: <2.0, 0, 0, 0>
upstream could be a memory with <1.0, ...> and experience <1.0, ...>, or memory <2.0, ...> experience <0.0, ....>, or memory <0.5, ....> and experience <1.5, ....>. There are many possible vectors that could "add" to produce the downstream effect, so it makes it harder for those neurons to "learn" the pattern.
As a math equivalence, if I ask "what two numbers sum to 10", there are many solutions (its impossible to disentangle the original numbers).
To make it easier to learn these patterns, what if we used only separate elements of the incoming vectors to represent this information (so the elements of memory and experience could be seperated)?
So some intermediate neurons can transform the representation. We can constructor orthogonal vectors (since the vectors above are sparse):
The "memory" must undergo a "rotation" which moves data into an "unused" portion that won't conflict with the experience neuron firing pattern.
Now downstream neurons can use the data from each (its effectively merging memory and experience without confusing the signal). There is only a single memory and a single experience that combined will give the firing pattern, so the pattern can be learned.
Due to the way linear algebra works, its possible to do this with more complex numbers along arbitrary axes in an n-dimensional space (instead of doing it with a single axis/neuron and all others being zero).
For a physical corollary, imagine two images super-imposed on each other. If they are very distinct, you might be able to infer what the two source images were, but if they are similar it would be difficult. Now imagine a "lenticular" image that clearly displays two images by printing them at orthogonal angles on the medium. You can easily determine what content belongs to which image, but only having a single "print" to store the data (this isn't a perfect anology, but it illustrates the idea):
The experiment was on mice, but the process has been observed elsewhere.
From the article:
> This use of orthogonal coding to separate and protect information in the brain has been seen before. For instance, when monkeys are preparing to move, neural activity in their motor cortex represents the potential movement but does so orthogonally to avoid interfering with signals driving actual commands to the muscles.
"Our findings reveal that the specific spectral tunings of the four cone types near optimally rotate the encoding of natural daylight in a principal component analysis (PCA)-like manner to yield one primary achromatic axis, two colour-opponent axes as well as a secondary UV-achromatic axis for prey capture."
Now, you want to use this sparse array to represent a note in a song. So you need every note to consistently map to a distinct* sparse array.
However, you also want to be able distinguish a note as being in one song or another. The representation should tell you not only that this is note A but note A in song X.
How might you do that? Well some portion of the ON bits could be held consistent for every A note and some could be used to represent specific contexts.
Stable and variable bits of you will.
Now if you look at two representations of the note A from two songs you'll see they're different. How different are they? Well you could just count the bits they have in common or not, or you can treat them as vectors. (Lines in high dimensional space) Then you can calculate the angle between those two lines. As that angle increases its easier to distinguish the two lines. They won't ever get to full "right angles" between them because of the shared stable bits, but they can be more or less orthogonal.
That's what's happening here. The brain is encoding notes in a way that it can both recognize A, but also recall it in different contexts.
*But not perfectly consistent, we use sparse representations because the brain is noisy and it's more energy efficient. Pretty close is good enough in the brain and you can encode a lot of values in 1000 choose 20 options.