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What is a topology and why is it in my neuroscience?!
Time to go back to math class and into a world where a coffee mug and a donut are the same thing. If you ignore distances and shapes, and instead focus on continuity and relations, a donut can easily be morphed into a coffee mug, making them “topologically equivalent”. The hole is the most important part since that is where there is a lack of continuity in the points.
Fig 1: If we focus on what’s important in topology, holes in shapes, then any shapes that can be molded into one another are equivalent. So a coffee mug is the same thing as a donut.
Algebraic topology is a field of mathematics that, in many forms, describes relations and simplifies operations. In the last decade or so, topology has become a major tool in analyzing and fainting a better understanding of large data sets. While most tools from the science of understanding networks focus solely on local properties based on pair to pair connections, topological tools reveal the global properties that emerge from structures of higher dimensions in the data than what can be revealed from only looking at data in a traditional way like measuring distance between data points.
Fig 2: How can we describe the data we see? If this is what our data showed, traditional methods would not pick up on its underlying shape (a circle). From being able to see that our data forms a circle and thus has a hole in the middle, we learn a lot about where points in space can and cannot exist. This could tell us that in wiring a neural network there are connections that cannot exist.
When theorems and definitions that come from the abstract field of topology are applied to studying the data sets of other fields, it is referred to as computational topology. Computational topology consists of algorithmic methods which investigate and find meaningful simplifications of high-dimensional data in a quantitative manner. One might think of it as a tool for understanding shapes and surfaces in data structures. This may seem pretty far removed from other fields like neuroscience, but this field of mathematics offers a framework for analytically describing and understanding brain function.
When theorems and definitions that come from the abstract field of topology are applied to studying the data sets of other fields, it is referred to as computational topology. Computational topology consists of algorithmic methods which investigate and find meaningful simplifications of high-dimensional data in a quantitative manner. One might think of it as a tool for understanding shapes and surfaces in data structures. This may seem pretty far removed from other fields like neuroscience, but this field of mathematics offers a framework for analytically describing and understanding brain function.
What the funk is neuro-topology
Since the early 20th century, Ramon y Cajal’s drawings of the vast, intricate networks of neurons have presented a view into the complex network of the brain. Neuroscientists have since been trying to formalize and capture the information contained in the structure of these neural networks. By trying to understand the structures neurons form, topology has long been a central concept in neuroscience, even though studying information that neurons relay through computational topology has been only recently developed. Neuroscientists can carefully record electrical activity of neurons at each point in time in these neural networks, with the goal of understanding the pattern of connections and relevant dynamics that can hopefully lead us to be able to predict when a neuron fires. However, analyzing the pairwise connections between two neurons in this way only captures part of the underlying network of information flow of the brain. These neural networks become increasingly complicated as the number of cells increase and thus in the process of understanding the human brain, the study of neuronal networks using mathematical tools has become increasingly important. The human brain has about 86 billion neurons that receive, conduct, and send signals. There is a clear missing link in research between neural network structure and emergent function. For instance, in many diseases like Alzheimer’s, changes in structure of neuronal connectivity can cause a disruption of function in the brain. Topology provides a formal and quantitative way to capture structural and functional aspects of the organization of the nervous system and how they engender cognition.
Fig 3: The structure of neural networks. Mapping neurons to other neurons they connect to reveals what shape they can make. Here the red dots are the somas, and the shapes below diagram the connectivity of this network. Different structures could be involved in different processes or serve different functions.
Here is an example of how this works
In order to better understand how scientists are using computational topology, we can look to how neuroscientists have used topology to understand how the brain encodes which places we have been in an environment.
The hippocampus is a region of the brain known for its role in spatial navigation. As animals navigate through mazes, this area is highly active. It has been thought that the hippocampus is a good topological template: it cares more about connectivity between cells as it builds a representation of the places the animal has been than it cares about the actual distances and angles between points of the maze.
How does the brain build this representation of a space it is moving through without exact distances and angles? The information is transmitted by neurons in a particular order and forms a cognitive map. These neurons have no direct access to the physical environment, the only information they receive is in the temporal pattern of the action potential activity in neurons(spike trains). In topology, sequence, relations between locations, and continuity are far more important than geometry. Thus instead of having to hardcode every physical space that could exist, the brain could use a much more efficient topological neural code.
Fig 4: Imagine every environment we walk into, we have an arbitrary number of shapes that cover the general area. In topology their exact size doesn’t matter, thus we could cover any space. We could understand where we are by the relationship of each shape to each other mapping to a physical distance in space.
Our brain does not have access to the outside world- the brain doesn’t know what a meter or a 90 degree angle is- but it could communicate to each neuron where it has been without regard to measured distance. It could do this by covering every environment in arbitrary shapes (in this example, circles). The amount or size, or really, exact shape doesn’t matter in topology as long as it is continuous like a circle.
Fig 5: A way neurons could build a code to tell us where we are in space. Neurons that fire in an ensemble could be marked with a zero. This could mark a relation in space to where we exist. 1110 could mean that we exist in an overlapped space between 3 neurons but not in a third shape.
A code could be built by a string of neurons firing together, such as in figure 5. This code could map back down to physical space. If different neurons’ activity coincide in a moment, like in the figure, we could build a code where a neuron is given a 1 if it fires an action potential and a 0 if it does not. This produces a codeword like 1110, indicating that the first three neurons fired together, but the last neuron did not. 1110 could explain that there exists a point where three circles overlap but not a fourth. What is meaningful here is that the neurons are not encoding distances, but rather relations. That is why it is thought that the hippocampus is a place that uses topology in action.[3]
This outlines a relatively simple example of topology in neuroscience, and more complex ideas of hidden structures are currently being studied. Talking about holes, shapes, and classifications seems abstract, but topology appears to play a fundamental role in the way the brain is organized and the underlying neural code. Currently, scientists have shown a mathematical framework for using topological data analysis for space and memory encoding, as well as healthy versus diseased brain function. Understanding network connectivity shows how changes in the intrinsic structures within the neural network appear to accompany neurological and psychological disorders and can even be used as diagnostic markers for conditions such as Alzherimers, strokes, and autism[1]. Researchers are also using topological tools to understand states of consciousness, sensory neuroscience, and brain arteries’ morphological properties across lifespan[2]. At a time where there is still much to learn about how the brain works, topology has proved a way to study the numerous questions in neuroscience that are still unanswered. The full potential of using network theory in neuroscience is yet to be tapped. While it is currently being used for mainly descriptive purposes of simpler networks in a moment in time (static networks), understanding how specific circuits in the brain connect with other circuits or evolve over time form promising areas of future work.
Advances in computational topology have greatly aided the search for understanding how neurons encode the world around us. It has also provided an inspiration for how even the most abstract mathematics can serve purpose to applications in science. Neuroscience is an interdisciplinary field drawing talent from all different subjects, and as neuroscience finds new links to other fields, our understanding of how the brain works can become more complete.
References:
[1] Expert, Paul, et al. “Editorial: Topological Neuroscience.” Network Neuroscience, vol. 3, no.
3, July 2019, pp. 653–55. PubMed Central, doi:10.1162/netn_e_00096.
[2] Stolz, Bernadette. Computational Topology in Neuroscience. p. 77.
[3] Curto, Carina. “What Can Topology Tell Us about the Neural Code?” Bulletin of the American
Mathematical Society, vol. 54, no. 1, Sept. 2016, pp. 63–78. DOI.org (Crossref),
doi:10.1090/bull/1554.
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