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A poster (@Mr_Phil_Osophy ) encountered an article on the network topology of the brain:
Unfortunately, he confused the dimensionality of neuron interconnections with physical dimensionality - and posted it in the Physics->"Beyond the Standard Model thread". Apparently, his misguided speculation was not well received by moderation and the thread was soon deleted.
However, the dimensionality of network topology is an interesting Computer Science topic where it bears directly on the design of multi-core processing. It is discussed extensively in this paper:
http://prod.sandia.gov/techlib/access-control.cgi/2008/080069.pdf
Since that's probably a bit heady for some posters and lurkers, I have resurrected parts of that original thread here.
As I explained to @Mr_Phil_Osophy, the article in question deals with topological dimensions, not physical ones.
https://www.sciencealert.com/science-discovers-human-brain-works-up-to-11-dimensions
For example, if I have 4 nodes and each one is connected to every other, I have a topological pyramid, a 3-dimensional structure. It doesn't matter whether those four nodes are on the same plane.
Similarly, if I have 12 nodes and each one is connected to every other one, I have a total of 66 connection lines and an 11-dimensional structure. But that's 11 topological dimensions, not 11 physical dimensions.
The topology of a computer network is often a key indicator of its capabilities. Moving from pyramids to cubes, consider: 8 processors arranged as the vertices of a cube, each with a direct communications path to three of the other processes in the same way that the edges of a cube connect each corner point to three other corner points.
In such a topology, there are many data processing operations (such as sorting or FFT's) where this arrangement of processors can cooperate very efficiently. And if that topology is expanded to 2048 processors in with the topology of an 11-dimensional hyper-cube, then those same data processing can be efficiently shared among 2048 processors instead of just 8.
As an example, here is a link to an article describing a highly efficient sort algorithm based on just such a structure:
http://oldweb.ltu.bg/jmsd/files/articles/20/20-22_D_Gichev.pdf
I should also add, that although this article was published in 2002, sort algorithms based on the cubic topology have been in the literature since at least the 1980's.
Our protagonist, @Mr_Phil_Osophy, then expressed some confusion over how such a topology would support the types of applications required by the brain - prompting this explanation:
------------------
You are confusing hardware and software. I'm not sure what the topological dimensionality of a Turing machine would be. You could probably design one that is limited to 2 dimensions. But that would not keep it from processing 3D, 4D, 5D, or more data structures. That would only be a matter of its programming.
What this article (https://www.sciencealert.com/science-discovers-human-brain-works-up-to-11-dimensions) is describing is the interconnection of neurons in the brain - and more particularly, in the human frontal cortex. In the simplest of cases (which is not a realistic case), every neuron would be connected to a single information bus line. That would allow every neuron to talk to every other neuron, but only one pair could exchange data at a time. That would be a 1D topology. It could still perform the same operations as an 11-D topology, but it would take much, much longer.
Here is the topology of the vertices of a 5-D cube, presented in 2-D form:
In the brain, each of those colored dots would represent a neuron and each line would represent direct data path between two neurons. With the 2-D representation, you might have problems with crossing lines "shorting out", but add just a scratch of the third dimension to it and this 5-D topology can easily be contained. Of course, when it is flatten out like this, it is hardly recognizable as any sort of cube.
But it has the same interconnections as the points of a 5-D cube. Thus it is topologically a 5-D hyper-cube.
--------------------
Finally, @Mr_Phil_Osophy inquired as to the meaning of the term "clique" as used in the article of interest.
That article talks about neuron "groups" forming cliques. A bit of interpretation is needed there. I believe the "groups" they are talking about are physical grouping - not just topological. So they are saying that in many cases, neurons that are physically close to each other are also tightly interconnected with each other and less tightly connected to more distant neurons. And they are calling those "cliques".
In the process of computing the topological dimensionality of the connections, they discovered these cliques by themselves rated high topological dimensionality (say 5D or 6D). These cliques were then connected to many, many other cliques in a topology that compounded the dimensionality, thus adding another 5D or 6D to the total.
Imagine four cubes, each a 3-D clique. Then connect each corner point of each cube to the corner points on two other cubes. That would give you the 5-D topology in the figure I posted earlier.
Mr_Phil_Osophy said:
Unfortunately, he confused the dimensionality of neuron interconnections with physical dimensionality - and posted it in the Physics->"Beyond the Standard Model thread". Apparently, his misguided speculation was not well received by moderation and the thread was soon deleted.
However, the dimensionality of network topology is an interesting Computer Science topic where it bears directly on the design of multi-core processing. It is discussed extensively in this paper:
http://prod.sandia.gov/techlib/access-control.cgi/2008/080069.pdf
Since that's probably a bit heady for some posters and lurkers, I have resurrected parts of that original thread here.
As I explained to @Mr_Phil_Osophy, the article in question deals with topological dimensions, not physical ones.
https://www.sciencealert.com/science-discovers-human-brain-works-up-to-11-dimensions
For example, if I have 4 nodes and each one is connected to every other, I have a topological pyramid, a 3-dimensional structure. It doesn't matter whether those four nodes are on the same plane.
Similarly, if I have 12 nodes and each one is connected to every other one, I have a total of 66 connection lines and an 11-dimensional structure. But that's 11 topological dimensions, not 11 physical dimensions.
The topology of a computer network is often a key indicator of its capabilities. Moving from pyramids to cubes, consider: 8 processors arranged as the vertices of a cube, each with a direct communications path to three of the other processes in the same way that the edges of a cube connect each corner point to three other corner points.
In such a topology, there are many data processing operations (such as sorting or FFT's) where this arrangement of processors can cooperate very efficiently. And if that topology is expanded to 2048 processors in with the topology of an 11-dimensional hyper-cube, then those same data processing can be efficiently shared among 2048 processors instead of just 8.
As an example, here is a link to an article describing a highly efficient sort algorithm based on just such a structure:
http://oldweb.ltu.bg/jmsd/files/articles/20/20-22_D_Gichev.pdf
I should also add, that although this article was published in 2002, sort algorithms based on the cubic topology have been in the literature since at least the 1980's.
Our protagonist, @Mr_Phil_Osophy, then expressed some confusion over how such a topology would support the types of applications required by the brain - prompting this explanation:
------------------
You are confusing hardware and software. I'm not sure what the topological dimensionality of a Turing machine would be. You could probably design one that is limited to 2 dimensions. But that would not keep it from processing 3D, 4D, 5D, or more data structures. That would only be a matter of its programming.
What this article (https://www.sciencealert.com/science-discovers-human-brain-works-up-to-11-dimensions) is describing is the interconnection of neurons in the brain - and more particularly, in the human frontal cortex. In the simplest of cases (which is not a realistic case), every neuron would be connected to a single information bus line. That would allow every neuron to talk to every other neuron, but only one pair could exchange data at a time. That would be a 1D topology. It could still perform the same operations as an 11-D topology, but it would take much, much longer.
Here is the topology of the vertices of a 5-D cube, presented in 2-D form:
In the brain, each of those colored dots would represent a neuron and each line would represent direct data path between two neurons. With the 2-D representation, you might have problems with crossing lines "shorting out", but add just a scratch of the third dimension to it and this 5-D topology can easily be contained. Of course, when it is flatten out like this, it is hardly recognizable as any sort of cube.
But it has the same interconnections as the points of a 5-D cube. Thus it is topologically a 5-D hyper-cube.
--------------------
Finally, @Mr_Phil_Osophy inquired as to the meaning of the term "clique" as used in the article of interest.
That article talks about neuron "groups" forming cliques. A bit of interpretation is needed there. I believe the "groups" they are talking about are physical grouping - not just topological. So they are saying that in many cases, neurons that are physically close to each other are also tightly interconnected with each other and less tightly connected to more distant neurons. And they are calling those "cliques".
In the process of computing the topological dimensionality of the connections, they discovered these cliques by themselves rated high topological dimensionality (say 5D or 6D). These cliques were then connected to many, many other cliques in a topology that compounded the dimensionality, thus adding another 5D or 6D to the total.
Imagine four cubes, each a 3-D clique. Then connect each corner point of each cube to the corner points on two other cubes. That would give you the 5-D topology in the figure I posted earlier.