Wait till you see what I got.. you're gonna be scared for a little bit until you aclamatize . It's gonna be the most dramatic moment in history ever.. bigger than religion, known science, or anything.. it's so huge you will not process it at first . You may not even be capable of reacting .. it's definitely going to gradually take over your mental space .
This is such a great example that we never really know what applications New fields of mathematics will have. Topology and Neuroscience who would've predicted that? 😆
There's also a subfield of Topology called Knot Theory that has some interesting applications in Biology/The Study of DNA and Physics Chapter 7 of this book covers them if you're interested: www.math.cuhk.edu.hk/course_builder/1920/math4900e/Adams--The%20Knot%20Book.pdf
I don’t mean to be rude, but with some knowledge of topology I think it becomes obvious it could have applications to neuroscience. Just think of graph theory. But this paper is maybe a more surprising connection!
Dissociatives might be even more interesting in that regard, since they tend to more drastically change the person's perception of where (and what) they are even at smaller doses. Also dissociatives like ketamine have much more extensive history of medical use which makes things much easier for obvious reasons.
The Semi-circular canals of the cochlea have a surface area that is in the shape of a torus. Considering that the detecting cells are on that surface, it would make a lot of sense that brains would use a torus as a grid system. For reference, the cochlea is used for orientation & inertial tracking.
Kind of seems natural to use a torus. It provides smooth paths on a surface, for most directions you can keep going a while before repeating. If you chose a sphere instead, you can only go one circle in any direction without changing directions before it repeats. So you cover a lot more ground/local sensors walking on a torus than on a sphere, in a given direction
This actually makes total sense because if you want to condense self awareness within a 3D environment, a torus would be a shape that can work no matter how much you move, travel, shift. You retain a sense of lateral and up and down movement while focusing on important things such as returning to where you parked when starting your hike, etc. It seems like a simplified way to approximate which direction you should be going or are going, in any given space.
You don't go far enough: it is about efficient use of finite number of neurons. Neurons modelling points of environment that are falling far behind you will be re-purposed to model details far ahead of you. This will be the case whichever direction you go. Thus there will be a toroidal topology to the activity of the neurons.
@@Gounesh No, it makes sense the brain would evolve to a toroidal pattern of neuron activity in modelling movement through the environment, though I'm not sure Ben framed the reason particularly well. Evolution would require efficient use of of a finite number of neurons to model the environment arond you. As you move around, neurons which were modelling points that are falling far behind you should not sit idle entirely unused. Instead, they are repurposed to model points that are far ahead of you. Happens in all directions you move. Hence a toroidal patter of activity. You might suppose a sphere would serve the same purpose just as well. Not going to try to explain why not in a youtube comment.
This is very cool! I love how topology is finding so many applications these days. If I could chime in with an observation as a mathematician, I think the presence of the torus is actually unsurprising here, since the universal cover of a torus is the Euclidean plane (or perhaps more relevantly, the torus is what you get if you quotient the plane by a lattice - reflecting what you mentioned about neurons tracking lattice points). What this basically means is that your brain uses ‘Pac-Man’ coordinates - put that way it seems like the only real option for faithfully mapping the data of a potentially unbounded plane into a finite space.
The one time I smoked DMT 10 years ago, I was taken to the center of a Torus. There are a lot of phenomenological, aesthetic and symbolic information associated with this, but that is not important to this video topic. I've always felt that this experience was of a 4D version of the inside of the Flower Of Life pattern, which is fundamentally a hexagonal pattern. Ever since that time, I've been obsessed with tiling patterns, particularly Japanese and Islamic. Thank you for this video, it was very educational and helped explain some of the neuroscience behind my experience. I've always felt that the psychedelic experience is an inner journey, and any places you go and "spirits" you meet are different aspects of an evolutionary or contemporary framework which you are normally not capable of interfacing with.
This is so fascinating! You're the second person here who reported seeing a torus during a psychedelic trip. Hallucinogens are well known for bringing the unconscious mind into awareness, but perhaps these trips also offer glimpses into the fundamental workings of the brain itself?
What about 4 dimensions being a toroidal einstein-rosen bridge between (North) the event horizon of a black hole and the (South) inflation epoch, with the center of the column being the singularity (Vector equilibrium)?
That is so fascinating. I am really intrigued by the taurus now. I’ve always live Islamic tessellations which I found had an inherent spiritual quality without believing in Islam myself, and I’ve always loved MC Escher’s art. But I’ve never thought to connect the tauroid shape, with tesselations and neuroscience. Its all kind of spooky and mystical.
A toroidal structure makes perfect sense. Rather than think of ego(the animal) moving around in the model of the environment. Think of the model moving around ego, because ego is always in the center of its current environment. Ego must maintain an internal model of its surroundings. As ego moves forward through the environment, neurons that were once tracking details farther away will track details nearby. But behind ego, neurons that were tracking distant points are now tracking point to far back to bother keeping track of anymore. Those neurons no longer need to be used to model those points far behind, and it would be inifficient not to use them, so they are re-purposed to being tracking points far ahead. The brain does not have infinite neurons with which to map the environment around it. Neurons that were modelling details that have passed far behind ego may now be repurposed to model detail far ahead of ego. Whichever direction ego goes, we would see the same process, neurons mapping distant points ahead are now mapping nearby points, and distant points behind fall further behind and eventually "wrap around", re-purposed to model points distantly ahead. This is (loosely) reminiscent of old 2D computer games, like asteroid, which have a toroidal topology -- go off the top of the screen and appear at the bottom, go off the sides, and appeart at the other side.
I like how you framed this- the 'environment' moving around the ego and not vice versa like everyone else seems inclined to do. Fundamentally I don't think one can say whether it's you that moves around the universe, or if you're stationary and the universe moves around you; informationally they're equivalent. I'm still thinking about this video, but as others have pointed out, rather than framing this study as saying 'the brain' 'uses' a torus to navigate space, it could be that the universe, in the most basic information-theory sense shorn of all assumptions, has the proto-geometry of something like a horn torus. Evolution didn't create anything and it's not a survival/purposeful/efficiency based 'design' concocted by the so-called brain... in fact it's the structure of existence itself so how could thought and mentation be any different?
This is probably connected to more fundamental things like the tissue structure of the brain, in fact it probably goes much deeper, because in the first stages in the development of any individual animal, the cells also form a toroidal topology where the inside becomes the digestive system
Indeed. I bet we’d find in nature many such similar structures that map onto a torus or similar topologies. For example, if you picture a large oddly shaped oak tree and the way its branches uniquely spread out, then imagine how underground its root system also oddly spreads out to roughly match and counterbalance the upper branches - odds are there is some kind of chemical or bio electrical communication system that enables on a meta level the coordination of root-branch balance, and that the contents and “bits” of communication can be mapped onto a torus. Ditto for underground mycelia networks. And as the previous poster mentioned, how endothelial cells in an embryo form the gut. And to think that is the embryo of any vertebrate animal, from fish to humans to chickens to alligators. Biological networks pervade the bio and eco spheres. And if you generalize from particulars to the more emergent whole of each biological system, it makes total sense. Schools of fish form into toroidal or globular forms, as do flocks of birds. On more flat land, with its 2D limitations, the attractor forces in a system confine groups into flat flowing structures, like ant swarms and herds of bison stampeding across a plain. Each individual is free to roam in any direction, but there is a tendency of attraction to congeal as a whole for a seemingly larger overall purpose. So it’s not entirely unexpected that clumps of cells form toroids or hollow globules, as in cell colonies or frog xenobots. Ditto not only for the structure of the brain itself, but also for the more abstract toroidal clouds of communication that its member neural networks spontaneously form. Not so much in non biological clumps - piles of sand or puffs of smoke or clouds of water vapor formed from splashing waterfalls all do not maintain cohesion into a dynamic toroidal shape that dynamically reacts to what its member parts are communicating. Because those parts are not communicating anyway. Unlike in living systems where there is constant communication happening on many levels, from chemical to bio electrical to epigenetic to, as recently discovered, cellular bio electrical fields that guide and map tissue expression and regeneration, where such a field that programs an eye to form can be itself transplanted onto embryonic gut cells to instead form an eye there. Indeed. Biological network structures seen with the eye are only part of the story; there are also corresponding toroidal and other topologies that are continuously expressing in the chemical and bio electrical realms. From brains to mycelial networks to root systems to embryonic development, there is much more than meets the eye alone.
@Escher and @psionic111 Thank you for your eloquence on a tricky subject. So I get we share interests, and glastular torriodal structure enables ontology to recapitulate phylogeny, and the field for hybrids and chimaeras is wide open. Mike Levin needs to be clearer, right? His slides are confusing. There's also much to learn from Paul Stammetts, Nick Lane, Sarah Walker and the regius professor of Chemistry at Glasgow , possibly Stephen Wolfram. Are these name all familiar to you both?
As a postdoc woring in the field of cognitive neuroscience I can attest your content is amazing. Touching and exploring concepts that I'm only vaguely familiar in such a clear way. Great work!
Keep up the good work 👍high quality content from the cutting edge of neuro is pretty rare, since people in the know dont generally have the time, interest, or the skills to do effective scientific communication. I'm impressed you are able to find the time to make such high quality videos while pursuing your dissertation!
You don't even need the PCA to see that it has toroidal topology- You can see how it arises from the periodic nature of the grid cells. Pac man also lives in a topological torus :)
I can't begin to imagine how did the brain develop such complex architecture. It all looks so chaotic when you just look at populations of neurons and yet the neural activity behind this apparent chaos, is extremely well organised and efficient. Also, although we are beginning to uncover these mechanisms, we're really only scratching the surface.
i'm taking an evolution of the human brain class currently which goes over the development of brain structure, function and connectivity. it really humbles me to know how complexly yet elegantly the brain has evolved, and how much painstaking research has been done over the years to, as you say, only scratch the surface. it really is a breathtaking thing to behold.
Fourier analysis - and it's analogies in the fuzzy logic of thought - is a great way to demonstrate how we see the patterns in the noise. In thought it is infinitely scalable to find a match. One of my favourite examples is faces found in clouds.
This channel is extraordinary. Next 2 minutes papers. Not seeing quality novel information like this in many other places. Keep it up the algorithm finding you is inevitable.
@@ArtemKirsanov If it's an open research topic, people will have to be more comfortable with unknowns. We are only just beginning to understand this subject
@@ArtemKirsanov It's not a problem! The problem was more with me having very little previous understanding of this subject matter. Your presentation of the material was really engaging!
There's a structure in music theory called the Tonnetz, which arranges every major and minor chord together in a grid of triangles. It can be arranged as a torus, I wonder if it actually exists somewhere in the brain
Here’s another connection you might want to explore: quantum mechanical descriptions of Bain activity. More specifically the idea that all ideas are existing in a superposition state and only a small number collapse as the prefrontal tap cortex observes it. Collapses are heavily influenced by external stimuli in that seeing a familiar face causes all the ideas associated with that face to begin collapsing in conjunction with other relative information existing at the same time. 6:30 The torus is topographically equivalent to a two-dimensional field that wraps around on itself in two dimensions. Like the video screen of “Asteroid” where the bottom edge is continuous with the top edge and the left edge is continuous with the right edge. The “Asteroid” game world is a torus flattened out on a TV screen. The brain virtualizes and simulates reality. Visual information is two dimensional. 7:45 Language is meaning encoded into a physical pattern. Each neuron encodes meaning in the pattern of activations sent to each other. Language processing occurs subconsciously between every neuron that communicates with each other. “Colors” are symbolic patterns that represent wavelengths of light, normally. This is why you can perceive colors in the absence of light (hallucinations and synesthesia) or perceive no color in the presence of light (blindness). The same concept applies to all the other sensory (physical) perceptions (psychological). So, if nobody is around to hear it, a tree falling on a mime in a forest makes no sound. It does produce patterns of air compressions that radiate from the commotions. However, with no device to capture those patterns and translate them to “sound”, no sound is created.
Is this related to Penrose / Hameroff OrcOR theory at all? Still trying to understand that one. I think it should be pretty obvious that there is a quantum connection in some form even though many people don't seem to think so.
@@conceptflow I’m not talking about a quantum connection. I’m talking about the similarity in mechanisms. The pattern of cause and effect are congruent.
A torus is not such a surprising structure though... If you had to represent a Pac-Man level, a torus is what you would get: when you exit on one side, you loop back to the opposite side. So, if the brain is mapping a 2D surface as a surface where the x-axis loops back on itself and the y-axis does the same, then you get a torus. So the interesting thing is whether having the 2 dimensions loop has an intrinsic advantage or is it just a minor artifact from evolution that the brain has to deal with?
Pacman famously lives on a torus. A plane that cycles in both directions is a torus. There's a finite working memory for connected space, so overwriting receding points with approaching points is basically a circular buffer
My guess it that the Taurus emerges because of our eyes, more specifically the fact that we are merging visual information from 2 and the shape seems to fit the paths
The torus is used for making procedural materials tileable in x and y direction. also used in fusion reactors. It is a pretty good shape to use for mapping coordinates from 3d to 2d
I've always had a dreamy obsession with Tori because of their mathmatical implications - so happy to see that this idea gets its connection to observable data. Thank you so much for your contribution and work, I will watch your videos however you choose to design them.
A Torus just means that it's an intrinsically 2D structure. A torus is what you get when you take a 2D map, roll it up into a tube by connecting the top/bottom edges, then bend the tube around and connect the left/right edges. Or it's effectively what you get when playing one of those old infinitely scrolling 2D map dungeon crawler games where if you go far enough in any one direction, you end up back where you started. Saying that the brain uses 2D toroidal coordinates for navigating is interesting, and suggests that such creatures should struggle badly to cope with a 3D maze. Which we probably do struggle with. It's probably an evolved structure for navigating across the mostly 2D surface of our planet.
I’m a mathematician, and although it’s only a hunch, seeing this makes me think that a miracle might just have happened here. You say that tori don’t occur often in the real world, but they play a fundamental role in the theory of elliptic curves, which yield some of the most beautiful results in all mathematics. A natural way to obtain a torus is to consider a game of Pac-Man. Pac-Man lives in a square screen where if he travels up, he ends up on the bottom screen (and vice-versa), and if he travels to the left, he ends up on the right side of the screen (and vice-versa). Mathematical parlance, we say that the top and bottom edges of Pac-Man‘s world are “identified“ with one another; we say the same thing about the left and right edges of the screen. The space you obtain I making these identifications is a torus. instead of the Pac-Man screen, let’s do this identification procedure to a 1 meter by 1 meter square. Pick a point (x,y) inside the square, then, let p and q be numbers >1. If we allow a particle to move up or down by steps of length 1/p and allow the particle to move left or right in steps of length 1/q, Observe that as we allowed to point to travel through all possible combinations of four directional movements at the scale, we will get a grid, or “lattice”. Choosing a different starting point and a different set of step scales will get you a different lattice. You don’t need to restrict yourself to moving purely in x-axis and y-axis. You can use any lattice of points generated by two finite length linearly independent directions. The Weierstrass elliptic function f(z) can be defined as a function which accepts an input point on a torus. Here, z is a complex number (it has both an x and a y coordinate). f(z) is periodic in two different directions; this corresponds to the two identifications used in constructing a torus. If you graph the point P(z) = (f(z), f’(z)), where f’ is the derivative of f, you will get the type of curve known as an elliptic curve. If we allow z to move along a lattice in the torus, as z varies, P(z) will jump around the elliptic curve. Tori naturally arise when we talk about two-dimensional lattices. Every 2-dimensional grid can be placed inside a torus, resulting in what is called a finite subgroup of the torus. Likewise, every finite subgroup of a torus corresponds to a 2-dimensional lattice. In this way, the torus forms what is known as a moduli space for lattices, in that it indexes/parameterizes the space of all possible lattices. These equivalences were established in the first half of the 19th century by the mathematicians Abel and Jacobi. In this respect, the emergence of a torus from the superimposition of many different 2-d grids is a mathematical inevitability. This stuff is most important in number theory. Even though any two lattices can be transformed into one another by a combinations of stretches and rotations (a.k.a, linear transformations), if you place restrictions on the kind of transformations that are allowed, you get different families of lattices, the relations between which end up having quite a bit of significant when studying behaviors of square roots, cube roots, and other numbers of interest. From a neurobiological perspective, I agree that it’s absolutely fascinating to consider how the vertebrate brain evolved toward these mathematical principles. It would be interesting to see if a torus also emerges when the experiments are performed on cephalopods, spiders, insects, and other invertebrates. Of course, you wouldn’t be able to prove that it isn’t convergent evolution, but, assuming it isn’t, it has really interesting implications for the evolution of spatial perception in animals. A single grid wouldn’t be of much use in determining an animal’s *absolute* position, but I think it would work rather well to determine the animal’s position with respect to the set of points on the grid, which would be great for a simple animal which fed by grazing on the bacterial mats that blanketed the ocean floor during the Precambrian epoch. More grids would mean more accuracy of motion, and once you hit a certain critical mass of grid density and if you throw eyes into the mix, animal-animal predation become feasible. Anyhow, I’m just speculating here. But, again, this is fascinating stuff. Definitely made my day!
I think a torus makes sense for the task at hand. Handwaving by working in x and y instead of frequency and phase, the brain needs a finite way to represent location in what feels like an infinite and flat world. Since we only are ever working with a small local patch anyway, we can cheat the ability to "always move left" by wrapping the x axis back around on itself. And we cheat the ability to "always move forward" by wrapping the y axis back on itself. It could/"should" have been a sphere, but eh, it works well enough as is. Without reading the paper it's still not entirely clear how the topology of the points shown corresponds to these frequency/phase firing patches, but I think my view probably adapts fine. TLDR: by definition all manifolds are locally Euclidian, and there are only so many ways to glue the boundary of a compact region to itself.
It could not be a sphere, try forming a square into a sphere, youll have to collapse whole edges to points thereby loosing information, a thorus does not have the same definition
It seems like a reasonable hypothesis that a torus is ideal for mapping 3D location specifically BECAUSE real-world spaces are not often tori... Forces the mapping to be abstract rather than direct... and thus more universal.
This makes sense to me. You can't have an infinite number of neurons to represent all of space, so while traversing long distances, neurons will eventually have to be reused. To do this continuously and homogeneously (i.e. without any "edges", and ensuring that all locations are treated in the same way), the best model is a torus with the flat metric, respecting the local flatness of Earth. [EDIT] That actually makes me wonder if migratory birds have grid cells that fire on a spherical manifold -- to help them better navigate over distances for which the curvature of the Earth becomes significant.
A 2D plane with X/Y wrapping (reaching top takes you to bottom, reaching left takes you to right edge) is topologically identical to the surface of a Torus. So maybe it just represents a looping grid?
My ignorant opinion is that the torus is very intuitive. Color perception is an in vivo dimensionality reduction of the stimulation of retinal color receptors to one dimension, but unlike light perception, color perception forms a ring, yielding the color wheel. In two dimensions you get a torus, which works out conveniently with typical 3D NLDR techniques. I'm surprised that i haven't encountered any variants of popular NLDR techniques for non-euclidean spaces, like the surface of a torus, for example. Though I guess all ANNs involve this.
Dude. I've had so many dreams where I am moving along a Torus. I didn't even realize what it was till I saw this video, and in my dreams whenever my brain is like.. going between areas that aren't immediately connected, as if I were teleporting between them, like if I were going from a room immediately to the outside, often I'll be in that Torus shape with snippets of the areas I am 'teleporting' across represented as 2D pictures on the Torus. And when I make it to the environment I had in mind, it transforms back into a typical environment and the torus disappeared. That is so weird.
I thoroughly enjoy learning and following your contents, ofc, I have only music and sound engineering background and somehow I still can engage with the complex concept. I am glad you make these topics the way they are. Keep up the wonderful work!
The evolutionary reason that may come to my mind is that our brain should be able to fast track its position on flat surfaces, mountains, dunes, and even underwater (in a completely 3d environment) seamelessly. Probably with different models would be much more expensive in energy to switch from being on a flat surface and than start climbing a mountain or a tree, or to jump into water from a cliff and then swim. Torus also is a better geometrical model to track the original position and be able to come back. It may even explain better why we move in circle if we travel across long distances in space like the desert for instance.
Thank you so so so very much. I have had this intuition for some time but have never been able to put it into words. You have helped me visualize and understand something I have been feeling was present but never had any evidence/understanding to confirm. Your channel is a gem. Please keep it up, definitely my favorite channel on RU-vid. I also have found great results in using your zettelkasten method/adapting it to my own brain.
This because the holodeck of our simulated universe is also a torus shape, which has a projection of what we experience as "real life" placed on its surface.
This is the first vid of yours I saw and I subscribed. I really liked how approachable you made this paper! The visuals and explanations were great. My only thing I'd say I found jarring was the jump-cuts. The way the camera angle or zoom level slightly changed when you did a jump-cut made it really distracting to me. I would try to do fewer jump cuts or at least keep the camera in the same position when you do them so they become less noticeable.
Grid cells are located in entorhinal cortex. They also form a torus. The trajetory of the grid cells are corresponsive to the trajetory of place cells which are located in hippocampus. -- This is what I learn from this lesson.
We're at the center of the toroid. This is why when something is directly above us, we duck and look up sideways. It puts the thing above us into the toroid. It looks like it is why we are oblivious to what is above and directly below us. There is no need for a perfect sphere since we do not have a clear or easy view of up and down except as we tilt our heads. Since we have to tilt them, only a toroid is needed.
So that's why sometimes i'll dream of something happening in some location but it looks nothing like it does in real life (still feels normal tho). It's the same grid cells firing in a different mental context.
A couple people have said that a torus kind of makes sense, like an old arcade game. But it still feels pretty abstract to me. Did they try measuring the torus? What size square does that cover in the real world, and how does that compare to the size of a lattice cell? Do the vectors on the torus have components from multiple modules, or is there a different torus for each module? When the lattice cells of two grid cells are the same size but the grids are rotated relative to each other, are they considered the same or a different module? Are the torus dimensions always a whole numbered multiple of the lattice dimensions, or are there seams in the lattice sometimes?
aaaaa i love your video！could you please introduce more about the application of topological spaces in the study of brain activity？？really looking forward to it！！
1) I would need to be sure this wasn't an artifact of the 'maze' just like the rectangle vs circle problem that arose from earlier research. 2) given that dead ends are not useful, it would make sense for loops to produce a taurus in the data, but that doesn't mean it's not a globe of hexagons being modeled. 3) So I'm not exactly sure what this research is telling us.
nice video and good explanation, but the misconception here is that the brain doesn't simply " maps " space with a torus pattern but actually create 3D reality (or space as you say) in a torus form by collapsing the "quantum wave function" in to defined 3d particles.
If I had to guess, I'd bet on the possibility that the structure in question is not just any torus, but a very specific one -- Fibonnaci torus. Why? For the exactly same reason why snail shells, leaves, and galaxies encode Fibonnaci numbers -- conservation of energy (in the sense of least energy expenditure). As the saying goes in physics, "Nature is lazy" (a.k.a. the Principle of Least Action), so evolution (of both biological and physical systems... which are *fundamentally* the same thing) is all about optimizing energy expenditure, and it would make perfect sense for a system (whether it's a brain or a galaxy) to expend the least amount of energy to encode something (whether spatial coordinate or shape), and the least amount of energy will be expended if the representation of a thing is as close as possible to the very thing it represents. I think that this discovery is yet another strong indication that this universe is a hypertorus. Undoubtedly, AI (neural networks) research has already taken notice of this discovery. It's now up to physics to catch up with neural science.
Really interesting stuff! I think we would like to learn more about how exactly these sorts of graphs are made from the neuron activation info and stuff, that part was pretty hard to follow and vague
Hey Artem and friends, So, are grid cells a way for our brains to represent a toroidal magnetic field? I know plenty of other species use electromagnetic fields for navigation and it somewhat stands to reason that we would have an internal process to do this. I also found it quite interesting that these grid cells are arranged in a hexagonal pattern. Perhaps they represent a version of simplex space that can be used to triangulate relative position. Even more complex representations of stored sequences of spacetime and n-dimensional could seemingly be represented as higher dimensions of abstract noise and collapsed to a plane for measurement and observation using extrapolations of Ken Perlin's simplex noise. Am I missing something? Or is there some nuanced maths rule that disallows manifolds to be represented by simplex noise? If not, I feel like according to the manifold hypothesis this would be an excellent way to represent neural data and extract results. Not to mention that it seems like the underlying geometry is there in the neural physiology when it comes to relating abstract space to actual triangulation of physical spaces. I would love to hear some other thoughts on this so that it doesn't keep rattling around my mind at 7:30am
In one of your previous videos you mentioned that place neurons can be shown to fire in a roughly spherical location when the subject is forced to interact with 3D space. I wonder whether if when extending to 3 dimensional space we see a Clifford Torus, or a normal torus just extended into a fourth dimension representing height. If the former then it would indicate that height is processed separately from location in the hippocampus, whereas the latter could indicate that distance to the ground is a big part of figuring out where you are, and could help explain why people without training are so disoriented in zero-g environments.
I can't stop thinking about how reality is like sand box video game. Big bang=turning the game on. Probable locality=pieces that are/aren't in the field of view. Things don't exist until observed. Etc etc.
Imagine a point on a rectangular map. Draw a line from it that reaches the vertical edge. It reappears at the opposite edge and comes back into the map, returning to the original point. Now imagine a line drawn from the point, crossing the horizontal edge, reappearing at the opposite edge and returning to the point. The lines are tracing paths on a torus. We can imagine the pairs vertical edges and horizontal edges being folded around to meet their opposites, forming a torus. Two lines can trace paths meeting one another, but never intersecting anywhere. The two structures are topologically identical. Aren’t I clever!
a toroid (ring) is also at the heart of processor of quantum computing, it's also at the heart of hard problem of consciousness, and even the human body is a toroid with two orifices for input and output.
It makes complete sense when thinking about the perceptive distance of different directions around me. Straight forward is shorter than diagonal paths to the same forward distance.
Artem, I don't know this dimension reduction algorithm for sure but I'm wondering why natural selection would favor a coding system that will looks like a toroidal coordinates, if reduced to 3 dimensions. I mean, 3d is intuitive for humans, but for neurons in the brain, is just as arbitrary as 6 or 19d, no? I'm sorry if I'm asking a dumb question
This is actually a very great question!!! You see, a torus is characterised by its own intrinsic topological characteristics (it is two-dimensional, has two "1D" holes and one "2D" hole, or a cavity). I recommend this Quanta Magazine article, if you want to know more - www.quantamagazine.org/topology-101-how-mathematicians-study-holes-20210126/ These properties refer to the toroidal shape itself. But we can embed a torus in 3D space (for visualization), or 4D, or N dimensions. A torus will still remain the same torus. You can think of an ordinary straight line drawn on a piece of paper. It is a one-dimensional shape with no holes embedded in 2D space (paper). Now think of a line in three dimensions, like a laser beam. It is the same one-dimensional structure with no holes, but embedded in 3D space. And you put the line in N-dimensional space, but it will still intrinsically remain one-dimensional structure. Similarly, the torus maintains its properties no matter in how many dimensions (>= 3) it is embedded in. The authors just used the projection to 3D for visualization purposes. But in fact, the quantification of the topology (to test whether it is actually a torus or not) was done in 6 dimensions ( www.nature.com/articles/s41586-021-04268-7/figures/7 ) So you are right, there is nothing special about 3D for the brain itself P.S. If you are interested in how it could be done: this method is called "persistent cohomology" and it essentially refers to "blurring the vision" of the computer to uncover some structure behind the cloud of points - www.frontiersin.org/articles/10.3389/fncom.2021.616748/full Hopefully, that clears things up ;)
@@ArtemKirsanov Wow, I almost missed this amazing answer in the middle of the internet. I think now I understand: at the same time you can put your arm inside a donuts, you can also eat it on the side or on top without it falling apart, topologically speaking. By the way, you do an amazing job with the videos, and I personally think the format of the articles explained in detail makes better use of your knowledge and enthusiasm than general topics. In addition there are more general psychological neuroscience videos on RU-vid. thanks for the answer
What comes to mind is that it is a repeating tiled pattern (in 2d anyway), would seem to naturally map to a torus. Of course, this is high dimensional, but still.
Just think of those expanding and contracting toy balls. They arent tauroidal but theirs and tauroids centres share the same kind of empty space. I think something related to the distance between any 2 grid cells and the fact that they move signals as fast as they do allow some hacky things the way putting in and taking out energy from the toy ball expands and contracts it. Maybe despite us observing this signal shape it in fact crosses signals near instantly over the "empty" space and the tauroidal shape allows for MANY MANY paths to take to near any other grid cell
Very Interesting and excellent job presenting the results of this publication! I am wondering what would happen to any point representing neural activity on the toroidal manifold, when applying arbitrary rotations to the observer. How does the torus encode rotational symmetry? Is this the most efficient way to do it?
Grid cells map to a “virtual” environment (the torus) rather than any “real” external environment. That make sense, because it would require too many grid cells, and they would have to be remapped (requiring synaptic modification) if the environment changed. Anyway, that’s not how our brain works. We don’t work with a “realistic” map when going to the coffee shop. In fact, not even all the intermediate locations can be “visualizd” at once. We go out the door, then to the car, then to some intersection, then to Main st., then park in front of the shop. In other words, we use woking memory, loading relevant information about the next waypoint from long-term memory. Suppose that the grid cells encode current working memory information. First you "enter" the destination, then the starting point. Why the geometry of the torus? As one moves along the surface of the torus, new waypoints must be added. The PFC must access starts at the door, then adds the car, and so on, updating the grid with information from LTM relevant to the current and next grid location. This produces a seamless map between waypoints as the current location changes, allowing one to move to the next waypoint. When you are at the car, you don’t think of the coffee shop - you can’t even (usually) think of all the points in between, because the number of “chunks” of information (locations) in working memory is limited to about 4. This is why the torus must be an omni-directionally continuous field. Another important point: working memory duration is at most about 15 sec. That’s probably longer than the time taken to get from the car to the next waypoint. So the current grid map of local environment cues must be refreshed multiple times. This involve sparse spiking in cortical neurons (Miller et al, 2018, Memory 2.0) to re-activate grid cells periodically to maintain them in WM. How to do this so that all relevant neurons in the field are re-activated? Put them in a torus so that that a wave of activation travels around every 10-15 sec! This means there must be an oscillator to activate all the cells within that period. Theta rhythm, present during wake and in REMS, is ~8 to 10 per sec, and could is likely involved in that function as it is present in wake and REM sleep, and theta is generated in the hippocampus!
Try to move your head less and your gestures look less impulsive, flowing into each other, less abrupt. You will come across more confident and command the space around you better. Great Video !
The moment I saw this video, I knew it was going to hurt my brain; I get obsessed with watching science based videos but it hurts my soul that I'm not smart enough to understand most of what I see.
Keep watching and just look up the definitions for anything you don't know. I've taught myself a ton of physics that way within a few years and can at least hold my own in a conversation now.
So it’s basically a way to superimpose several different vector databases within one space by utilizing the different scales and dimensionalities of the grids? Did I get that right? If so - that is an impressive thing to happen by pure coincidence
When the n-dimensional point cloud is reduced to 3 dimensions. Why reduce to 3? What does it look like if you reduce it to 2? A naive assumption might be a circle, but I don't know the dimension reduction process, so it might not be a circle. I'd like to see other visualizations of the data. What about reducing it to 4 and making a moving image of 3 dimensions over time (the 4th). Repeat that last step changing which dimension is the time dimension. Or use the display method commonly applied to a Tesseract to depict 4 dimensions in two.
