Doughnuts in the Brain

By Rswilcox – Own work, CC BY-SA 4.0,

The illustration above is of a tokamak. That is a device scientists have been working with for decades to make nuclear fusion possible. So far surprisingly few positive results but it may still work in the end. Part of the way it works is to concentrate a electromagnetic field with external coils formed into a toroidal shape.

In How Squishy Math Is Revealing Doughnuts in the Brain by Kelsey Houston-Edwards writes of some new neuroscience research has “found that certain brain cells use a torus, the mathematical name for the surface of a doughnut, to map their environment”.

Typically the brain is represented as flattened diagram showing connections between neurons somewhat like a circuit diagram. The brain actually exists in three physical dimensions. Accounting to this new research when the temporal firings are mapped geometrical forms emerge that are up to seven dimensions. These are not actual physical dimensions but mathematical dimensions required to express the complexity of the patterns.

Immediately after receiving the stimulus, the simplicial complexes grew like a massive Lego construction, adding in pieces of higher and higher dimensions until the sculpture reached the maximum of three or four dimensions, depending on the stimulus. Then the whole thing rapidly disappeared. “You have these increasingly complex structures that are being created by the stimulus until it just all collapses,” Hess says.

In this topological analysis, the firing data gets mapped temporally and spatially at different levels of granularity. “‘At every scale you’re going to have a different snapshot of what that complex looks like,’ says Ranthony Edmonds, a mathematician at the Ohio State University”.

Topologists study this spectrum of shapes—recording, in particular, the number of holes in each dimension. They are especially interested in holes that persist through many different scales. Some holes briefly appear and then disappear, but the stubborn holes—those that survive through a range of scales—point to the most essential features of the data. TDA can thus reduce a complex mess of data to a simple list of stubborn holes, in much the way that a JPEG photo file compresses an image. “It’s a way of paring down the data to the stuff that really matters so that we have something much more workable,” Ghrist says.

Sometimes the holes identified in this way have direct interpretations. Mathematician Jose Perea of Northeastern University and a team of computational biologists used persistent homology to find periodic biological processes—those that repeat at regular intervals. Examples include the metabolic cycle of yeast or a mouse’s circadian clock. “What is recurrence or repetition?” Perea asks. “Geometrically it should be like you’re traversing some sort of loop in the space of the thing that you’re looking at.

Is any chance that the “holes”, especially the persistent ones, might be where electromagnetic fields generated by the firings are concentrated? Maybe the “fusion” of consciousness isn’t found directly in the firings but in the holes where the neurons don’t fire.

Just another broad speculation. 🙂

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