I was asked a question on Quora about a recent study that talked about high-dimensional ‘structures’ in the brain. It has been receiving an inordinate amount of hype, partly as a result of the journal’s own blog. Their headline reads:
‘Blue Brain Team Discovers a Multi-Dimensional Universe in Brain Networks’
As if the reference to a ‘universe’ weren’t bad enough, the last author, Henry Markram, says the following:
“We found a world that we had never imagined”.
The following passage in the blog post doubles-down on the conflation:
“If 4D worlds stretch our imagination, worlds with 5, 6 or more dimensions are too complex for most of us to comprehend.”
As will soon be clear, using words like ‘universe’ and ‘world’ in conjunction with the word ‘dimension’ creates a false impression that these researchers are dealing with spatial dimensions and/or how the brain represents them. This is simply not the case.
This is the question I was asked:
What exactly are the recently discovered multidimensional geometrical objects in the neuronal networks of the brain?
Here is what I wrote:
In this particular case the hype has gotten so out of control that the truth may already be irretrievably buried in mindless nonsense.
The key message is this: the word ‘dimension’ in this paper has nothing to do with the dimensions of space.
Here’s the paper that is receiving all the hype about higher dimensions:
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
It came out recently in the journal Frontiers in Computational Neuroscience. The authors employ somewhat complex ideas from graph theory to analyze connectivity among neurons in a small segment of rat neocortex.
This is where the authors of the paper talk about ‘dimension’:
“Networks are often analyzed in terms of groups of nodes that are all-to-all connected, known as cliques. The number of neurons in a clique determines its size, or more formally, its dimension.” [Italics in original.]
So dimension here just refers to the number of neurons that are connected in an all-to-all network. In the area of rat neocortex they studied, they found that 11-neuron cliques were common.
The other concept they talk about is ‘cavities’:
“The manner in which directed cliques bind together can be represented geometrically. When directed cliques bind appropriately by sharing neurons, and without forming a larger clique due to missing connections, they form cavities (“holes,” “voids”) in this geometric representation, with high-dimensional cavities forming when high-dimensional (large) cliques bind together.”
The word dimension has a variety of meanings in mathematics and science. The idea of spatial dimension is most common: when we refer to 3D movies, this is the kind of dimension we are thinking of. The space we are familiar with has 3 dimensions, which we can think of in terms of X, Y, and Z coordinates, or in terms of up-down, left-right, and front-back directions.
The dimension of a network has nothing to do with spatial dimension. Instead, it has more to do with the number of degrees of freedom in a system. In physics, the number of degrees of freedom is the number of independent parameters or quantities that uniquely define a system.
So really, this paper is just talking about the statistics of local neuronal connectivity. They describe their findings as surprising when compared to other statistical models. All this means is that certain connectivity patterns are more common that one might expect under certain ‘random’ models. That’s what they mean here when they compare their results to ‘null models’:
“The numbers of high-dimensional cliques and cavities found in the reconstruction are also far higher than in null models, even in those closely resembling the biology-based reconstructed microcircuit, but with some of the biological constraints released. We verified the existence of high-dimensional directed simplices in actual neocortical tissue.”
Outside of the narrow community of computational neuroscientists who use graph theory, these results are interesting but hardly ground-breaking. Moreover, as far as I can tell these findings have no definitive functional implications. (There are some implications for network synchrony, but in my opinion synchrony has itself not been clearly linked with higher-level concepts of function.)
Neural network modelers assume all kinds of connectivity patterns than deviate from pure ‘randomness’, so such findings aren’t particularly surprising.
So I am baffled by the hype this research is getting. It strikes me that extremely lazy science journalism has collided with opportunistic PR practices.
Given what I’ve explained, I hope it’s clear that these kinds of headlines are profoundly — almost maliciously— misleading:
In fact, given that the brain contains around 80–100 billion neurons, we might consider 11 dimensions to be rather low for sub-networks, if we remind ourselves that dimension in this case simply means the number of neurons that are connected to each other.