New strategies, new circuitries, new mathematics

I came upon an MIT News article about the work of Ila Fiete who studies brain functions, like the neurological processes that govern navigational reasoning about our surroundings. Fiete uses computational and mathematical tools. Her interest in biology, and her respect for the “aesthetic to thinking mathematically,” (as she put it) led her to systems biology, where computational and mathematical analyses and modeling, are applied to the understanding of complex biological systems. She did most of her PhD research at MIT,

….where she studied how the brain uses incoming signals of the velocity of head movement to control eye position. For example, if we want to keep our gaze fixed on a particular location while our head is moving, the brain must continuously calculate and adjust the amount of tension needed in the muscles surrounding the eyes, to compensate for the movement of the head.”

Later, at the University of California at Santa Barbara, Fiete began working on grid cells, a system of neurons I have written about on more than one occasion. These cells, located in the entorhinal cortex of the brain, actually create a grid-like, neural representation of the space around us that allows us to know where we are. Grid cell firings correspond to points on the ground that are the vertices of an equilateral triangular grid. It happens with surprising regularity.

In a Dec 2014 article in Neuron, Nobel laureate Neil Burgess said the following:

Grid cell firing provides a spectacular example of internally generated structure, both individually and in the almost crystalline organization of the firing patterns of different grid cells. A similarly strong organization is seen in the relative tuning of head-direction cells. This strong internal structure is reminiscent of Kantian ideas regarding the necessity of an innate spatial structure with which to understand the spatial organization of the world.

I have written on this topic before:
The mathematical nature of self-locating and
Grid cells and time cells in rats, continuity, and the monkey’s mind

At MIT again, Fiete has continued to explore her PhD thesis topic, specifically, how the brain maintains neural representations of the head’s direction, where it is pointed, at any given time. Now, in a paper published in Nature, she explains how she identified a brain circuit in mice that produces a one-dimensional ring of neural activity that acts like a compass. It allows the brain to calculate the direction of the head, with respect to the external world, at any given moment.

MIT’s report on that paper explains her approach:

Trying to show that a data cloud has a simple shape, like a ring, is a bit like watching a school of fish. By tracking one or two sardines, you might not see a pattern. But if you could map all of the sardines, and transform the noisy dataset into points representing the positions of the whole school of sardines over time, and where each fish is relative to its neighbors, a pattern would emerge. This model would reveal a ring shape, a simple shape formed by the activity of hundreds of individual fish.

Fiete, who is also an associate professor in MIT’s Department of Brain and Cognitive Sciences, used a similar approach, called topological modeling, to transform the activity of large populations of noisy neurons into a data cloud in the shape of a ring.

More than one aspect of these studies impresses me. The approach is interesting. Topological modeling, in some sense, is a shape-directed computation rather than a purely numerical one. And the simplicity of the ring impresses me. The brain somehow isolates, from a flood of sensory data, the variables that produce this simple representation of the head’s position. And the value of a circle, something we think of as a purely abstract idealization, is something the body seems to know, constructing it, as it does, outside what we call the mind.

“There are no degree markings in the external world; our current head direction has to be extracted, computed, and estimated by the brain,” explains Ila Fiete, … “The approaches we used allowed us to demonstrate the emergence of a low-dimensional concept, essentially an abstract compass in the brain.”

This abstract compass, according to the researchers, is a one-dimensional ring that represents the current direction of the head relative to the external world.

Their method for characterizing the shape of the data cloud allowed Fiete and colleagues to identify the variable that the circuit was devoted to representing.

My first reaction to this story was that it was beautiful. On the one hand, Fiete seems to be making increasingly creative use of the mathematics for which she has always had an affinity. On the other hand that simple ring, or compass, that tells us which way we are looking, highlights the presence of inherent mathematical tools that just exist in the body, in how brain processes just work. And the ring is stable, even through sleep.

Her lab also studies cognitive flexibility — the brain’s ability to perform so many different types of cognitive tasks.

“How it is that we can repurpose the same circuits and flexibly use them to solve many different problems, and what are the neural codes that are amenable to that kind of reuse?” she says. “We’re also investigating the principles that allow the brain to hook multiple circuits together to solve new problems without a lot of reconfiguration.

It would not surprise me if, when viewed from a careful, neuro-scientific perspective, we would find some resemblance between the way the brain makes new use of already configured circuits and the way mathematicians consistently build novel mathematical structures with tested strategies that have built other structures, like ordered pairs, symmetries, compositions, closure properties, identities, homotopy and equivalence, etc….

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