How Squishy Math Is Revealing Doughnuts in the Brain

How Squishy Math Is Revealing Doughnuts in the Brain

Benjamin Adric Dunn, a data scientist at the Norwegian University of Science and Technology, shows me a picture of unevenly spaced dots arranged vaguely like the rocks at Stonehenge. The overall pattern is obvious to a human, at least. He says, “When we look at it, it’s clearly a circle.” An algorithm would have difficulty recognising this simple shape. “It often misses .”

the big picture

Many scientific processes use repetitions or loops. Scientists who need to find circular patterns in large numbers of data points will face problems due to the computer’s inability see these relationships. Data can be visualized as dots floating in space, much like stars in the night skies. A dot could represent a physical location, similar to the numbers for longitude or latitude that mark where a ship is located on the high seas. Genes can also be plotted in a mathematical space with many dimensions, sometimes hundreds of times. This allows for two genes with identical DNA sequences to be represented by adjacent points. The context will determine the significance of circular patterns within data. A ship’s position may have a circle, which could indicate that it has been lost. However, genetic data may have circles that indicate an evolutionary relationship.

These starry skies of data points can be too complex and high-dimensional for the naked eye to understand. Researchers need to have a set of instructions that can be understood by a computer in order to detect circles. Many data analysis techniques are based on linear algebra, which studies straight lines as well as flat planes. Researchers are turning to topological data analytics (TDA) to find loops. This offers a completely different perspective.

TDA is rooted in topology, a branch mathematics that studies flexible, flexible shapes. This is in contrast to linear algebra’s relatively rigid and simple structures. Topology is also known as rubber sheet geometry because its practitioners assume that all shapes can be arbitrarily flexible.

Topological data analysis is a way for mathematicians to create revealing shapes out of scattered dots, much like the formations of constellations from star clusters. Researchers start by using the data points to create complex structures that can span hundreds of dimensions. The resulting picture retains much of the original data, but is more tangible. These structures are examined from a topological perspective, which allows them to look for preserved features even when the scaffolding is bent or stretched.

Topology is useful in data analysis because it can reveal qualitative rather than quantitative properties. It identifies those aspects of the structure that are still valid despite random errors or noise in the underlying measurements. Noise can sometimes jiggle the underlying data but the topology is the same, revealing robust features of a system. Robert Ghrist, a University of Pennsylvania mathematician, says that there are many instances in real life where the data you receive is squishy. “So you have to use squishy mathematics

Scientists and mathematicians are now working together to discover unusual topological shapes in a wide range of data. These shapes can represent anything, from biological processes with daily rhythms to drug molecules’ structures. Brain structure is perhaps the most fascinating of these investigations. Topology is a method that mathematicians use to study how neurons react to stimuli and interact with their surroundings. Dunn, in collaboration with neuroscientists discovered that certain brain cells use a “torus”, the mathematical name for the doughnut’s surface, to map their environment.

Drawing of a dimensional torus appears similar to a blue doughnut sitting on a tan tabletop.
Credit: Jen Christiansen

Doughnuts and Coffee Cups

Topologists are skilled at manipulating rubber sheets to make a living. However, they take great care to keep the number of holes intact. They never make a hole new or close an existing one. A topologist cannot tell the difference between a doughnut or a coffee cup. They both have one hole.

Topologists categorize holes based upon their dimension. A closed loop, such as the number 0, is a one-dimensional hole. It is formed by gluing the ends of a 1-dimensional line. Taping the edges of a two-dimensional plan, such as a sheet paper, will produce something similar to a hollow ball. This hollow ball has a 2-dimensional hole.

One panel shows a line curving around into a circle. Another panel shows a sheet of paper wrapping up to form a hollow ball.
Credit: Jen Christiansen

Higher-dimensional shapes can have higher-dimensional holes. A three-dimensional hole can be formed by “closing up” an empty space (similar to a cube). This process is only visible from a four-dimensional perspective. It is not possible to see it from most people’s (or anyone’s) reach.

Some shapes can have multiple holes with different dimensions. For example, an inflatable ball with an attached handle which a child sits on and bounces on. The hollow center of the ball has a two-dimensional hole. The solid handle is a one-dimensional hole. Topology uses many precise methods to count holes within higher-dimensional shapes. This ability is useful in studying the brain’s neuronal activity.

Neuroscientist Olaf Sporns, Indiana University, sees the brain as a huge transportation network. The neurons and their connecting synapses are what build roads and infrastructure. These streets are driven by the brain’s chemical and electrical signals. Sporns states that the physical roads limit the traffic patterns you can observe dynamically from above. Traffic patterns change as we think and move.

A diagram of the brain might look like a collection points representing neurons. Some of them are connected with lines, which indicates a synapse among those particular neurons. This structure is called a graph by mathematicians: a collection nodes connected with edges. While the graph reduces the biological complexity of brains, it maintains the overall shape and form of the circuits. This is a common trade-off when creating a mathematical model. It weighs simplicity and analysis against usefulness.

Dorsal brain cross-section shape is dotted with spheres, or nodes. Many are connected to other nodes with lines, or edges.
Credit: Jen Christiansen

The graph of neuronal connections looks like a web: neurons are plentiful and interwoven. In 2017 Kathryn Hess, a mathematician at the Swiss Federal Institute of Technology in Lausanne, tackled this complication by doing something that is initially surprising: she made the graph more complicated. She used data from the Blue Brain Project to analyze the activity of a rodent Neocortex, which is a part of the brain that is involved in higher-order functioning. The computer model includes representations of individual neurons that are connected via synapses to other simulated neurons. These links and when they will fire are determined by basic biological principles as well as experimental data from rodents.

The simulation can show traffic patterns in the brain, which is the activity of neurons in response to stimuli. The simulation can be paused to show scientists a freeze frame of which synapses are firing in response. This is different from the aerial view of the brain. This static image can be converted into a graph by showing the data points and the lines between them. Two neurons are connected if there is a synapse connecting them. Hess created a simple complex from this picture, which transforms the simple graph into an voluminous shape.

A simplicial complex is made from triangles with different dimensions. Three neurons transmitting signals from all three synapses formed the vertices in a hollow triangle, as shown in the Blue Brain graph. The mathematicians colored the hollow triangle with a solid two-dimensional triangle to make it a simpler complex. They also filled in larger clusters with connected neurons using higher-dimensional triangles analogs. A tetrahedron is a solid three-dimensional pyramid that has four triangular faces. It would fill in the group of four neurons firing together.

Hess and others saw eight neurons firing together, so the largest part of this simplicial complex was a 7-dimensional triangle. Many elements overlapped, creating a multidimensional sculpture. A triangle might be jut out from a tetrahedron to meet another triangle at one point. Mathematicians and scientists also examined a series freeze frames after simulating a gentle stroke by the rodent’s whiskers. Each of these maps was converted into a simple complex and the topology tools were used to analyze how it changed over time.

12 nodes floating in space. Many are connected to other nodes with edges. Some edges form triangles. One set forms a pyramid.
Credit: Jen Christiansen

The simplicial complexes grew quickly after the stimulus was given. They added pieces of larger dimensions until the sculpture reached a maximum of three or four dimensions depending on the stimulus. The whole thing quickly vanished. Hess states, “You have these increasingly complicated structures that are being made by the stimulus until it all collapses.”

To a topologist, three lines that connect to form a triangle are equal to a hollow circle. This is because one shape can be bent into another. The simple complexes Hess and her coworkers created from the simulated rodent brains were seven-dimensional. They can have holes in as many as seven dimensions. The number of holes per square inch increased with increasing shape, according to their analysis. The structure had a surprising number of holes, both two- and three-dimensional. This was far more than any random complex or one made from a different biological process. This specific pattern of holes indicated a high level organization in the neuronal responses; this complexity may be indicative of a fundamental feature of thought processes.

Stubborn Holes

More often, data are represented as isolated points floating in an abstract mathematical space with no predetermined connections. Mathematicians must figure out how to connect them in order to apply TDA. There are many ways to connect stars to a constellation. Mathematicians use persistent homology to find these implicit images. Topologists examine a series of simplicial complexes at different scales in order to identify the essential features of a data cloud.

To create the first simplicial complex they cast the broadest net possible, connecting each point to the other to form a dense network. This web is filled with solid forms, resulting in a high-dimensional simplicial complicated with few discernible features. Mathematicians must compare this complex to others created by connecting data at smaller scales. Next, they create a narrower net, connecting only the closest points. They now have a more sparse web that they use to create a second simplicial complex. This mesh has fewer data points so its simplicial complex has smaller dimensions. The researchers repeated the process using a number of smaller nets. Ranthony Edmonds from Ohio State University, a mathematician, said that at every scale, you will have a different picture of what the complex looks like.

Each simplicial complex is a possible configuration formed with the same scattered dots. Topologists study this spectrum and record the number of holes in each dimension. They are particularly interested in holes that persist across many scales. Some holes appear briefly and then disappear. But the persistent holes, those that persist through a variety of scales, point to the most important features of the data. TDA can reduce complex data to a list of stubborn holes in the same way that a JPEG file compresses an image. Ghrist states that TDA is a way to reduce the data to the most important parts, so that we can create something more manageable.

Sometimes, the holes that are identified this way can be interpreted in a direct manner. Jose Perea, a mathematician from Northeastern University, and a group of computational biologists used persistent homology to identify periodic biological processes that occur at regular intervals. Examples include the yeast metabolic cycle and the circadian clock of a mouse. Perea questions, “What is repetition or recurrence?” Geometrically, it should feel like you are traversing a loop in the space of what you’re seeing .”

.

Researchers have also been able to design new drugs using

TDA. These compounds can often be found by altering the molecular structures of existing drugs. The structure of molecules is complex and difficult to understand, even for machine learning algorithms. Computers must be able to use simplified representations of existing molecules in order to design new drugs. There are many ways to do this, but a team led by Guowei Wei of Michigan State University chose to reduce molecules to their “topological signatures.” This is the description of the chemical based on its topological characteristics–essentially the collection of information gained through persistent homology, such as the number of stubborn holes in each dimension.

Brain loops

The most interesting application of TDA may be at the simplest level of brain organization–a single kind of neuron. In 2014 John O’Keefe and research partners May-Britt Moser and Edvard Moser received the Nobel Prize in medicine for discovering, respectively, place cells and grid cells, types of neurons that activate when an animal is in specific locations. Carina Curto, a Pennsylvania State University mathematician, said that they act as sensors for location.

A rat’s brain light up when it is located in different places in its environment. Neuroscientists chose one grid cell to study in order to determine the relationship between grid cells and rat’s location. They created a floor model on a computer and marked the location of the rat each time the cell activated. The rat moved around the square box freely, creating a pattern of repeating dots, which mathematicians call a hexagonal lattice. The lattice contained all locations where each grid cell lit up. They did this with multiple grid cells, marking them each with a different color. Each grid cell was marked with a different color. The dots that corresponded to each grid cell had the exact same geometric pattern, but were offset from each other, covering the box like busy tile.

7 hexagonal tiles hold the same pattern of color dots. Some dots cross tile boundaries, creating a larger repeating pattern.
Credit: Jen Christiansen; Source: “What Can Topology Tell Us about the Neural Code?” by Carina Curto, in Bulletin of the American Mathematical Society, Vol. 54, No. 1; January 2017 (reference)

Neuroscientists were interested in understanding how grid cells represented spatial locations. In essence, they were trying to find the template that produced the hexagonal pattern. Imagine a rubber stamp with different cartoon characters printed on it. The stamp will form a line as you roll it out. An image of Mickey Mouse will appear along the line at regular intervals. All of these images were taken from the same spot on the original rubber stamp. It’s easy enough to imagine rolling out a stamp. But the reverse question is more difficult: How do you create the template stamp from the pattern it created?

Four neighboring red dots formed corners of a slanted rectangle, known as a parallelogram, by tiling colored dots that represented where the rat was at each grid cell firing. Like the repeated images of Mickey Mouse’s face, all the red dots in the same color corresponded with a single grid cell. Topologists then identified all the red dots and folded the parallelogram into a doughnut shape using an operation called “gluing”. They first glued two opposing sides of each parallelogram together, creating a cylinder with two red dot on the top and one at the bottom. The cylinder was bent and the ends were glued together to make a torus. The four red corners of each parallelogram are now a single point on a doughnut. The torus will have exactly one dot for each other color. A torus, like the circular stamp, is the correct map that shows how grid cells correspond to the floor of the box.

Parallelogram with a dot at each of 4 corners is rolled, forming a tube with 2 dots. Tube is curved into a torus with 1 dot.
Credit: Jen Christiansen

Neuroscientists were able to see this pattern while the rat was running around in a box. It was more difficult to see the pattern when the rat moved about other test fields, such as a bicycle wheel with spokes or a central hub. Scientists were not certain about the map, though each grid cell was firing in multiple positions. The structure of the dots was not obvious.

Red dots in a square form a clearly regular pattern. Red dots in wheel shape do not.
Credit: Jen Christiansen

In a February 2022 Nature paper, a team of mathematicians and neuroscientists, including Dunn, used grid cells to test a theory called continuous attractor networks, which predicts that certain neurons are wired together in a specific pattern–and the pattern does not change even if the animal is in a different situation. Researchers needed to test the theory that continuous attractor networks are true. They needed to determine if grid cells form a torus regardless of the environment in which they are found. They were looking for tori in messy neurological information–the perfect job to do for TDA.

This time, instead marking the positions of a single grid cell firing, the researchers looked at the collective activity of a network of grid cells. They recorded the state of the network at regular intervals using a string consisting of 0s or 1s. This indicated whether each grid cell was active. This long string, from a mathematician’s perspective, is a point in a very large-dimensional space. The researchers were actually accumulating high-dimensional data points as they recorded the state of each system at different moments. These points show how grid cell activation patterns change over time. However, the data is too complex to be viewed with the naked eye.

Using standard techniques to simplify data, the team calculated the persistent homology of system by connecting data points at different scales and looking at the resulting simplicial complexes. The data formed a torus as the rat ran around a box. The real test came when researchers used data derived from a rat that was running around a wheel-shaped field. It again created a torus, much to their delight.

Same square and wheel patterns as previous graphic. Evoking earlier parallelogram example, both map to a torus with 1 dot.
Credit: Jen Christiansen

Researchers were even able collect data from a sleepy–or dreaming–rat. They found a torus which was a shape that remained constant regardless of the environment or state of being. This supports the theory that continuous attractor networks are possible. The shape of a doughnut seems to be intrinsically linked to the way grid cells represent space.

Many of these applications of topological analysis are possible only because of the development of new computational tools. Vidit Nanda, a mathematician from the University of Oxford, says that none of this would have been possible if people hadn’t started to build algorithms. “If it’s not efficient, if it doesn’t scale well, then nobody will want to use it .”

, no matter how beautiful or complex the theory.

Topology is a rapidly growing field. It was once considered a quaint, but amusing, branch in mathematics. “The applications are getting stronger and stronger,” says Gunnar Carlsson, a mathematician at S

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