The Human Brain Is Capable of Building Structures With Up to 11 Dimensions

To study the human brain, one of the most complex structures on earth, the team found that traditional mathematics were not effective. "The mathematics usually applied to study networks cannot detect the high-dimensional structures and spaces that we now see clearly,” Markram said.

So the researchers decided to use algebraic topology, a branch of mathematics concerned with the construction of algebraic invariants associated to topological spaces which serve to distinguish between them. The Blue Brain Project sought the assistance of mathematicians Kathryn Hess from EPFL and Ran Levi from Aberdeen University in applying this discipline.

"Algebraic topology is like a telescope and microscope at the same time. It can zoom into networks to find hidden structures - the trees in the forest - and see the empty spaces - the clearings - all at the same time," explained Hess.

Using this approach, the team experimented on a model of the neocortex published by the Blue Brain Project in 2015 and on real brain tissue in rats. This technique enabled them to study details of the brain's neural network both at the level of individual neurons and of the brain structure as a whole.

Cliques and cavities

By stimulating the virtual brain tissue, they found that collections of tightly connected neurons, called cliques, bound to enclose high-dimensional holes, called cavities. "The appearance of high-dimensional cavities when the brain is processing information means that the neurons in the network react to stimuli in an extremely organized manner," said Levi.

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"It is as if the brain reacts to a stimulus by building then razing a tower of multi-dimensional blocks, starting with rods (1D), then planks (2D), then cubes (3D), and then more complex geometries with 4D, 5D, etc. The progression of activity through the brain resembles a multi-dimensional sandcastle that materializes out of the sand and then disintegrates," added Levi. 

The more complex geometries went up to 11 dimensions. The researchers proposed that "the brain processes stimuli by forming increasingly complex functional cliques and cavities."

The study may also eventually be able to answer a question that has long evaded neuroscience; where the brain stores its memories. "They may be 'hiding' in high-dimensional cavities," Markram said.

This is not the first time researchers use algebraic topology to study the brain. A team led by Ann Sizemore at the University of Pennsylvania has been doing similar work with somewhat similar results.