The Mind-Boggling Math That (Maybe) Mapped the Brain in 11 Dimensions

Kathryn Hess can’t tell the difference between a coffee mug and a bagel. That’s the old joke anyway. Hess, a researcher at the Swiss Federal Institute of Technology, is one of the world’s leading thinkers in the field of algebraic topology—in super simplified terms, the mathematics of rubbery shapes. It uses algebra to attack the following question: If given two geometric objects, can you deform one to another without making any cuts? The answer, when it comes to bagels and coffee mugs, is yes, yes you can. (They only have one hole apiece, lol.)

If that all sounds annoyingly abstract, well, it kind of is. Algebraic topologists have lived almost exclusively in multidimensional universes of their own calculation for decades. It’s only recently that pure mathematicians like Hess have begun applying their way of seeing the world to more applied, real-world problems.

If you can call understanding the dynamics of a virtual rat brain a real-world problem. In a multimillion-dollar supercomputer in a building on the same campus where Hess has spent 25 years stretching and shrinking geometric objects in her mind, lives one of the most detailed digital reconstructions of brain tissue ever built. Representing 55 distinct types of neurons and 36 million synapses all firing in a space the size of pinhead, the simulation is the brainchild of Henry Markram.

Markram and Hess met through a mutual researcher friend 12 years ago, right around the time Markram was launching Blue Brain—the Swiss institute’s ambitious bid to build a complete, simulated brain, starting with the rat. Over the next decade, as Markram began feeding terabytes of data into an IBM supercomputer and reconstructing a collection of neurons in the sensory cortex, he and Hess continued to meet and discuss how they might use her specialized knowledge to understand what he was creating. “It became clearer and clearer algebraic topology could help you see things you just can’t see with flat mathematics,” says Markram. But Hess didn’t officially join the project until 2015, when it met (and some would say failed) its first big public test.

In October of that year, Markram led an international team of neuroscientists in unveiling the first Blue Brain results: a simulation of 31,000 connected rat neurons that responded with waves of coordinated electricity in response to an artificial stimulus. The long awaited, 36-page paper published in Cell was not greeted as the unequivocal success Markram expected. Instead, it further polarized a research community already divided by the audacity of his prophesizing and the insane amount of money behind the project.

Two years before, the European Union had awarded Markram $1.3 billion to spend the next decade building a computerized human brain. But not long after, hundreds of EU scientists revolted against that initiative, the Human Brain Project. In the summer of 2015, they penned an open letter questioning the scientific value of the project and threatening to boycott unless it was reformed. Two independent reviews agreed with the critics, and the Human Brain Project downgraded Markram’s involvement. It was into this turbulent atmosphere that Blue Brain announced its modest progress on its bit of simulated rat cortex.

While some saw potential for the in silico brain slice to open up novel analysis pipelines, many others viewed the paper as proof that the idea was a colossal waste of money. Moritz, Helmstaedter, director of the Max Planck Institute for Brain Research, said at the time that the paper confirmed his worst suspicions about the project: “There are no real findings. Putting together lots and lots of data does not create new science.” Blue Brain’s utility to neuroscientists still remained far from clear.

Emergent Effects

But Kathryn Hess is no neuroscientist. Instead of a meaningless pile of data, she saw in Markram’s results an obvious place to apply her abstract math goggles. “Topology is really the mathematics of connectivity in some sense,” she says. “It’s particularly good at taking local information and integrating it to see what global structures emerge.”

For the last two years she’s been converting Blue Brain’s virtual network of connected neurons and translating them into geometric shapes that can then be analyzed systematically. Two connected neurons look like a line segment. Three look like a flat, filled-in triangle. Four look like a solid pyramid. More connections are represented by higher dimensional shapes—and while our brains can’t imagine them, mathematics can describe them.

Using this framework, Hess and her collaborators took the complex structure of the digital brain slice and mapped it across as many as 11 dimensions. It allowed them to take random-looking waves of firing neurons and, according to Hess, watch a highly coordinated pattern emerge. “There’s a drive toward a greater and greater degree of organization as the wave of activity moves through the rat brain,” she says. “At first it’s just pairs, just the edges light up. Then they coordinate more and more, building increasingly complex structures before it all collapses.”

Blue Brain

In some ways this isn’t exactly new information. Scientists already know that there’s a relationship between how connected neurons are and how signals spread through them. And they also know that connectivity isn’t everything—the strength of the connection between any pair of neurons is just as important in determining the functional organization of a network. Hess’s analysis hasn’t yet taken synaptic weight into account, though she says it’s something she hopes to do in the future. She and Markram published the first results of their decade-in-the-making collaboration yesterday in Frontiers in Computational Neurobiology.

So far, it does not have scientists racing to join them in their brand new field of neurotopology. Most of the researchers WIRED contacted said the paper was too dense, jargony, or outside their scope for them to comment. Chris Eliasmith, a theoretical neuroscientist at the University of Waterloo, didn’t find much to be interested about in his first reading: “It strikes me as an over-stated and minor contribution,” he wrote in an email. Peter Latham, a neuroscientist at the University College London went one step further, saying that it “reads a bit like an Alan Sokal hoax”—referring to the NYU mathematical physicist who got a bogus article about quantum physics as a “social construct” published in an academic journal. And both researchers were quick to point out that Frontiers journals are published by a company with a history of questionable editorial independence. Its founders? Henry Markram and his neuroscientist wife Kamila Markram.

Algebraic Applications

Hess says they tried to shop around the idea to other neuroscience journals first, to avoid the conflict of interest—but didn’t have much luck, since most editors they spoke to had never heard of algebraic topology. If that’s the real reason they couldn’t land the paper, though, it may be a problem unique to neuroscience. Dozens of other fields that are starting to capitalize on the unique way topologists see the world, as more mathematicians like Hess start putting aside theoretical problems for applied ones. Just last month she published a different paper in Nature Communications where she put on her topology goggles to look at a totally different problem: very small holes.

Nanoporous materials are super useful for all sorts of industries—from gas separation to chemical storage to medicine. And the performance of these materials depends on the shape of their pores, something that’s really difficult to quantify. So when scientists are looking for new materials to do certain jobs, they rely almost entirely on visual inspection of the more than 3 million nanoporous materials out there. Hess used algebraic topology to quantify the similarity of pore structures instead, assigning a sort of geometric fingerprint to each one. It’s a computational method chemical engineers can now use to find exactly what they need without having to stare into a microscope for days on end.

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Other topologists have used similar mathematical frameworks to accomplish practical goals: to predict how well cancer treatments work based on the shape of tumors, or design factory robots to move efficiently without smashing into each other. These are fields where massive amounts of data have proven incomprehensible to humans—until the algorithms stepped in. It’s possible that neuroscience may someday see the world through topology-tinted glasses. But they’re only just beginning to collect datasets that could even be viewed through that lens, and Markram’s might not be one of them.

He may still be light years away from delivering on a unified theory of the brain, but as neuroscience’s futurist-in-chief, Markram is as on-brand now as ever. His signature style is to present ideas too speculative for most scientists to countenance and then find ways to test them despite (and often in spite of) the haters. His latest hypothesis: that those patterns of increasingly complex neuronal structures represent ever richer and more interesting responses to stimuli. He thinks it’s how people learn. Maybe even where they store memories. To find out, Argonne National Laboratory outside of Chicago, Illinois gave him 100 million core hours on their super-super computer to run a year-long simulation to see how those patterns change and evolve over time. At the end of 2017 Markram will pass off that mountain of data to Hess. Then it will be up to the math to decide.