Mathematically Built Brain: The Example of Grid Cells, Incarnating Algebraic Geometry.

Understanding how the cognitive functions of the brain arise from its basic physiological components has been the final frontier in logic and rational science for thousands of years. (As I tried to explain yesterday, the superstitious religious fanatics tried their best to bury all of science, and the scientific mindset, the essence of humanity; they nearly succeeded!)

The 2014 Nobel was given to John O’Keefe (a “half”!), the rest jointly to May-Britt Moser and Edvard I. Moser “for their discoveries of cells that constitute a positioning system in the brain.” I will develop here the philosophical viewpoint, which is broader (O’Keefe’s career was steered by the influence of Hebb, the famous psychologist, who got the idea of the outside patterns imprinting the neurocircuitry of the brain).

Here is Hebb: “Let us assume that the persistence or repetition of a reverberatory activity (or “trace”) tends to induce lasting cellular changes that add to its stability.[…] When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.”

Well it turns out that evolution has had even more imagination than that. I will even propose Patrice’s Neural Theory, a vast generalization.

Galileo famously said the language of nature was written in mathematics. It turns out that it is much more than that. Our brain is mathematically organized. What Descartes consciously discovered, a coordinate frame in which to set-up calculus, is automatically generated in the brain. This is the meaning of grid cells.

Grid cells are neurons that fire when an animal moving of its own free will traverses a set of small regions (firing fields) which are roughly equal in size and arranged in a periodic triangular array that covers all of the available environment. They were discovered in 2005 by a couple (literally) of Norwegian researchers, the Mosers, and rewarded by the Nobel Prize in 2014 (shared with O’Keefe, from London, who invented the basic experimental technique, and discovered “place cells”)

Once set, navigation can be done in the dark, blinded. Scientists’ discovery that rodents, bats and nonhuman primates have a system in the brain for so-called “dead reckoning navigation”… “Dead reckoning” refers to the ability to navigate without external cues. The term comes from ship navigation. A crew will “take a sighting” via cues such as the stars or landmarks to determine where the ship is on a map. Then, when the ship moves, ‘dead reckons’ to update location on the map paying attention to speed and direction. The Greco-Romans already had such systems, with little paddled wheels counting the distance covered over the sea. It turns out that ‘dead reckoning’ is enabled by the grid cell system, inside the brain.

Kate Jeffery, a professor of behavioural neuroscience at University College London puts it this way:

“The importance of grid cells lies in the apparently minor detail that the patches of firing (called ‘firing fields’) produced by the cells are evenly spaced. That this makes a pretty pattern is nice, but not so important in itself – what is startling is that the cell somehow ‘knows’ how far (say) 30 cm is – it must do, or it wouldn’t be able to fire in correctly spaced places. This even spacing of firing fields is something that couldn’t possibly have arisen from building up a web of stimulus associations over the life of the animal, because 30 cm (or whatever) isn’t an intrinsic property of most environments, and therefore can’t come through the senses – it must come from inside the rat, through some distance-measuring capability such as counting footsteps, or measuring the speed with which the world flows past the senses. In other words, metric information is inherent in the brain, wired into the grid cells as it were, regardless of its prior experience. This was a surprising and dramatic discovery. Studies of other animals, including humans, have revealed place, head direction and grid cells in these species too, so this seems to be a general (and thus important) phenomenon and not just a strange quirk of the lab rat.”

We should have looked for Plato’s cave. It turned out that this cave has been built, is being built inside our heads all along! This cave is built-in two ways: automatically (grid cells) and as a response to the environment, by.us, from the outside, from the environment, in.

(So it matters what our brain experienced before to mold afterwards what comes in anew from the outside! No experience is a neutral experience!)

That cave is both a topology (what’s near and what’s not, the logic of place), and a basic geometry (the grid and its grid cells). To have a grid built automatically is the equivalent of having a reference frame in mathematics. It makes sense if one wants to make mathematics!

And not just mathematics, but even Infinitesimal Calculus! It is indeed clear that animals such as dogs have a mastery of calculus: experiences have shown this, and anybody with a dog throwing a stick sideways in water will see the dog running along the shore a bit, and then jump in the water, so as to minimize the time to reach the stick, a typical calculus problem. Dogs can do calculus, because they can make algebraic geometry in their brains, having a reference frame made of these grid cells! (If they had no grid cells, they would not be able to do calculus.)

Thus Descartes rediscovered, consciously, something which had been found, evolved and calculated by evolution half a billion years ago (or more!). The reference frame, also known now as the neuronal grid cell system, is basic to all of mechanics, even Poincare’-Lorentz Relativity.  (An open question: Quantum Physics uses even more general reference systems, Hilbert spaces; I will therefore predict that the brain has also that sort of organization!)

The world is not as astonishingly understandable, as Einstein would have it. Neuronal grid cell studies show that we are the world. Understanding the world is understanding ourselves.

The world is not just written in mathematical language, as Galileo found out. We are made mathematically. We think mathematically, because we are made of math. We are mathematics.

We are not just looking at shadows in a cave, as Plato would have it. And the cave was not given to us by the gods, as Socrates had it. We are the cave, we, and our personal history, built it.

Any new experience, idea or emotion, taught or experienced, is another brick in that wall of perception and analysis, we better consider it carefully, before indulging in it. Call that the Principle of Mental Precaution But that Principle extends also to what we chose NOT to experience, which can be just as bad, if not worse.

You are not just what you think. You mentally are what you were submitted to, and what you decided to submit to. Fate is written in mathematical patterns, one theorem made out of neurons, their axons, dendrites and supporting glial cells, at a time.

Such theorems are written with the physics of minds, just as sturdy as the physics of stars. Just as hopeful, just as ominous.

Plato thought mathematics were “forms”, out there, outside of the physical world. This is not what science is finding. There are not “forms” out there, and physics, nature, somewhere else. Our minds are literally made of math.

So here is my theory:

“Whatever exists in mathematics exists in the brain. And reciprocally.”

Patrice Ayme’

ooOOoo

I do so hope that this essay from Patrice fires you up as it did me. If it leaves you with questions, then offer them to me as a comment to this post and I will take it upon myself to have Patrice answer them.

Finally, did you pick up on the fact that it isn’t just our human brains that are mathematical organs, it applies to the brains of dogs as well!