The “Eleven Dimensional” Brain? Topology of Neural Networks

By Neuroskeptic | July 17, 2017 3:21 pm

Last month, a neuroscience paper appeared that triggered a maelstrom of media hype:

The Human Brain Can Create Structures in Up to 11 Dimensions

The human brain sees the world as an 11-dimensional multiverse

Scientists find mysterious shapes and structures in the brain with up to ELEVEN dimensions

The paper, published in Frontiers in Computational Neuroscience, comes from the lab of Henry Markram, one of the world’s most powerful neuroscientists. As well as being head of the Blue Brain Project at the École Polytechnique Fédérale de Lausanne, Markram founded the 1 billion Human Brain Project and co-founder of scientific publishing giant Frontiers.

The new paper is fascinating. But the headlines were completely misleading: this paper has nothing to do with multiverses and very little to do with anything 11-dimensional.

The paper is actually all about “cliques” of neurons. A clique is simply a group of neurons, each of which is connected to all of the others. Here’s a diagram (graph) showing a hypothetical clique of 5 neurons:

5cliqueIf these were real neurons, they would of course occupy 3-dimensional space, just like everything else in the universe. Now, mathematicians refer to a clique of 5 neurons as having a dimension of 4. This doesn’t mean that five connected neurons occupy some mysterious fourth dimension. It just means that the graph of the connections could be depicted as a 4-dimensional object. But we could equally well depict it as a 2-dimensional diagram (seen above.)

In general, a clique containing n nodes has a dimension of n-1. The nodes could be neurons or anything else.

So where did the wacky multi-dimensional media coverage come from? It seems to have been prompted by the hype-filled Frontiers press release on this paper, called “Blue Brain Team Discovers a Multi-Dimensional Universe in Brain Networks.”

If you look beyond the headlines, the paper is actually pretty interesting, although it raises more questions than it answers. The big claim is that neural networks in the brain contain an unexpectedly high number of cliques, including large ones with a dimension of 7 (not 11, though.)


A caveat is that most of the work in this paper wasn’t done in actual human brains, but rather in a computer simulation of a small chunk of rat cerebral cortex. The virtual neural network features 31,000 neurons and 8 million connections. The authors say that the simulation “closely resembles” the real rat cortex in many ways, but still, it’s a simulation. The paper does contain some work on real neural networks (from rats and C. elegans worms) which confirms the presence of lots of large cliques.

So what does the cliquey nature of nervous system mean? That remains to be seen. The authors suggest that cliques could be very important. Indeed, that they might be a unifying principle of the brain:

We conjecture that a stimulus may be processed by binding neurons into cliques of increasingly higher dimension, as a specific class of cell assemblies, possibly to represent features of the stimulus, and by binding these cliques into cavities of increasing complexity, possibly to represent the associations between the features.

I wonder however whether cliques and cavities might just be a side effect of the fact that neurons don’t connect to each other randomly throughout the brain, but are contrained by e.g. spatial proximity?

If a group of neurons are physically close together, they are more likely to synapse with each other than with neurons far away, and so a clique could be formed just by proximity. Artzy-Randrup et al. (h/t) made this point back in 2004 in response to claims of complex network structures in the C. elegans nervous system:

Neurons are spatially aggregated and connections among neurons have a tendency to form in local clusters (15). Two neighboring neurons have a greater chance of forming a connection than two distant neurons at opposite ends of the network.

The authors of the new paper did consider a control condition, a simulation based on “distance-dependent connection probabilities between the different morphological types of neurons.” They report that the ‘real’ rat cortex simulation showed more cliques than the control simulation. This suggests that the abundance of cliques is not just a product of proximity. But remember, this was all a simulation.

  • Uncle Al

    Why do we see no non-planar skeletal graphs re Kuratowski’s theorem and subgraphs homeomorphic to K5 or K3,3? It’s not even a Schlegel diagram. “Thinking Machines Corp,” Seymour Cray, massively parallel computing architectures overall teach us that minimal length dense connectivity derives from polytopes.

    a computer simulation of a small chunk of rat cerebral cortex.” That renders it third order BS: 1) dicing rat brains for structure, 2) simulating structure in software, then 3) cherry-picking hypothesis-supportive curve fits. Call it “non-classical gravitation” to join a very select club of shameless mathematical fudge artists.

  • Arturo Tozzi cns

    Even if this paper does not clarify it, the brain IS multidimensional. For a review, see the recently puiblished:

    • Costa Vakalopoulos

      The pimple on my nose is four dimensional as well, but what we need is an theory of dimensions of neuron networks from which other properties emerge i.e. what sort of algoryth,s differentiate mind fromsay gut motility

      • Arturo Tozzi cns

        Dear Costa, if you read my paper, you will realize that the pimp on your nose is three-dimensional, and that the theory does exist…

        • Costa Vakalopoulos

          Impressive. Still teaching myself the math and physics to do these papers justice. The pimple though is 4 dimensional as it changes over time.

  • smut clyde

    My thoughts on the paper into three parts are divided:
    1. As Uncle Al sez, it’s “third order BS”. The authors’ model was

    reconstructed from experimental data, guided by biological principles
    of organization, and iteratively refined until validated

    — that is to say, whatever observational data they have was edited, extended and structured by their theoretical expectations. They are using their connectivity calculations to tell us the implications of those expectations, while pretending that they are statements about reality. It is a classic map / territory confusion, or if you prefer a different metaphor, they are feeding us cookbook photographs which they pretend to be food.

    2. The paper talks about the flow of information in the brain. “Information” is a term. It has a meaning, it is not just a synonym for “connections”. In practice the authors never measured “information”, and how could they? I have no idea what information was passing through the reviewers’ brains when they let this past.

    3. If the press release led to a furore of churnalism and bullshit headlines, it’s not because it was misunderstood, it’s because Markram wanted that result. He is hardly a naif in the ways of science journalism.

    4. Reading between the lines, I wonder if the Human Brain Project is desperately in search of headlines. As if Markram needs something to justify the billion-Euro investment that poured into the HBP sinkhole. So he laboured and brought forth a mouse-brain.

    OK, that’s four parts. My bad.

    • Neuroskeptic

      They did look at the C. elegans nervous system and at a small number of rat cortical neurons using patch-clamp.

      They found lots of cliques in these datasets too. But the C. elegans example is weird because this organism has no brain, it has 302 neurons spread throughout the body and a few ganglia or “nuclei” which probably account for many of the cliques.

      It would be surprising if the C. elegans nervous system didn’t contain cliques. But this says little about the mammalian cortex.

  • 9eyedeel

    “We conjecture that a stimulus may be processed by binding neurons into cliques of increasingly higher dimension, as a specific class of cell assemblies, possibly to represent features of the stimulus, and by binding these cliques into cavities of increasing complexity, possibly to represent the associations between the features.”

    This reminds me of Wittgenstein’s bit about whether the form/structure of a sentence has to reflect the form/structure of the item that the sentence is about.

    Too much fun ahead on that one.

  • joseph2237

    To those of us you believe that the universe IS made up of eleven dimensions and is otherwise great news someone has confirmed that by illustrating eleven dimensions in the thinking process.

  • Lee Rudolph

    Uncle Al asks “Why do we see no non-planar skeletal graphs re Kuratowski’s theorem and subgraphs homeomorphic to K5 or K3,3?”

    But we do, in the diagrams above! Indeed, any “clique” (so called by graph theorists, workers in what J. H. C. Whitehead famously derided as “the slums of topology”) of 5 or more nodes is a “non-planar skeletal graph”—indeed, Reimann et al.’s rightmost figure is (assuming that the neuronal connections in its upper portion are in fact realized according to the schematic depiction in its lower portion) a clique with 8 nodes, which is what non-slummy topologists like Whitehead (and me…) would call “the 1-skeleton of a 7-simplex”.

  • stargene

    The apparent existence of such neuronal cliques seems to parallel
    the more general modularities associated with biological processes,
    processing and evolution. Eg: See
    I’m imagining that various cliques and sub-cliques might possess
    plasticity and capacity to alter connectivities, within and between

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Neuroskeptic is a British neuroscientist who takes a skeptical look at his own field, and beyond. His blog offers a look at the latest developments in neuroscience, psychiatry and psychology through a critical lens.


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