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.
[0]: https://stevenson.lab.uconn.edu/scaling/ [1]: https://www.nature.com/articles/nn.3776 [2]: https://doi.org/10.1016/j.conb.2015.04.003 [3]: https://doi.org/10.1016/j.conb.2019.02.002 [4]: https://arxiv.org/abs/2104.00145 [5]: https://doi.org/10.1016/j.neuron.2017.05.025
[0]: https://www.biorxiv.org/content/10.1101/214262v1.abstract
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.
For an abstract perspective, try Sheldon Axler's Linear Algebra Done Right.
For a more concrete perspective, Gilbert Strang's lectures: https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D
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.
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.
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.
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.
Smartphones eg. measure six dimensions of freedome, including rotation about every axis. 3 for location, 3 for orientation.
this has very little to do with synapses.
The 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)
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!
In the visual cortex, neurons are arranged in layers of 2D sheets, so that perhaps gives an extra dimension to fit connections between layers.
If I 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.
You can multiplex in frequency and time. I'm not sure if neurons do it, but it's certainly possible with computer networks.