How the Mathematics of Algebraic Topology Is Revolutionizing Brain Science
The human connectome is the network of links between different parts of the brain. These links are mapped out by the brain’s white matter—bundles of nerve cell projections called axons that connect the nerve cell bodies that make up gray matter.
The conventional view of the brain is that the gray matter is primarily involved in information processing and cognition, while white matter transmits information between different parts of the brain. The structure of white matter—the connectome—is essentially the brain’s wiring diagram.
This structure is poorly understood, but there are several high-profile projects to study it. This work shows that the connectome is much more complex than originally thought. The human brain contains some 1010 neurons linked by 1014 synaptic connections. Mapping the way this link together is a tricky business, not least because the structure of the network depends on the resolution at which it is examined.
This work is also uncovering evidence that the white matter plays a much more significant role than first thought in learning and coordinating the brain’s activity. But exactly how this role is linked to the structure is not known.
So understanding this structure over vastly different scales is one of the great challenges of neuroscience; but one that is hindered by a lack of appropriate mathematical tools.
Today, that looks set to change thanks to the mathematical field of algebraic topology, which neurologists are gradually coming to grips with for the first time. This discipline has traditionally been an arcane pursuit for classifying spaces and shapes. Now Ann Sizemore at the University of Pennsylvania and a few pals show how it is beginning to revolutionize our understanding of the connectome.
In pursuing their art, algebraic topologists set themselves the challenging goal of finding symmetries in topological spaces at different scales.
In mathematics, a symmetry is anything that is invariant as the viewpoint changes. So the shape of a square persists unchanged as it rotates through 90 degrees—this is one type of symmetry.
But some mathematical structures have symmetries that persist across scales. These are known as persistent homologies, and the search for them is turning out to be a key in understanding the connectome.
Neurologists have long known that certain cognitive functions make use of various neural nodes that are distributed across the brain. How these nodes are connected by white matter is one of the central questions for connectome projects.
Neurologists study white matter fibers by looking at how water diffuses along their length. A technique known as diffusion spectrum imaging can then reveal the pathways for this diffusion and hence the structure of the white matter.
To find out more, Sizemore and co measured the brains of eight healthy adults. This allowed them to look for the same structures in all of them. In particular, the team looked at the links between 83 different regions of the brain that are known to be involved in cognitive systems, such as the auditory system, the visual system, the somatosensory system involved in touch, pressure, pain, and so on.
Having constructed a wiring diagram in this way, Sizemore and co applied the techniques of algebraic topology to study its structure. This threw up some important insights.
To start with, it revealed that certain groups of nodes are “all-to-all connected”—in other words, each node in the group is connected to all the others, forming a structure called a clique. All the cognitive systems are made up of cliques of containing different numbers of nodes.
But the analysis revealed another important group of topological structures as well. These are closed loops called cycles in which one node connects to another, which connects to another and then to another, and so on, until the cycle is completed when the final node connects to the first.
This creates a neural circuit that can carry information around the brain and allow feedback loops to act, perhaps in the formation of memories and in controlling behavior. Sizemore and co say their analysis reveals a wide range of cycles of different sizes.
While cliques tend to exist within specific parts of the brain, such as the cortex, cycles span different regions, linking wildly different regions with different functions. “These cycles link regions of early and late evolutionary origin in long loops, underscoring their unique role in controlling brain function,” say Sizemore and co.
Another important difference between cliques and cycles is their density. Because cliques represent all-to-all connected nodes, they are dense structures. By contrast, loop-like cycles are relatively diffuse. Indeed, one way to characterize them is by the absence of links between the parts of the brain they encompass.
In essence, cycles define cavities in the connectome across a wide range of scales. And the work of Sizemore and co shows that these cavities play a significant role. “These results offer a first demonstration that techniques from algebraic topology offer a novel perspective on structural connectomics, highlighting loop-like paths as crucial features in the human brain’s structural architecture,” say the team.
That’s fascinating work revealing how algebraic topology is making an important contribution toward a better understanding of the connectome. Like all good science, this work raises as many questions as it answers. One suggestion is that cycles could allow a much broader repertoire of cognitive computations than is possible in other network architectures. But what kind of computations would these be?
And the neural networks that AI systems depend on draw their inspiration from the structure of brain. Now that new structures are emerging through this kind of analysis, how will the AI community incorporate these discoveries and exploit algebraic topology is their work?
This is clearly an exciting time to be an algebraic topologist.
Ref: arxiv.org/abs/1608.03520 : Closures and Cavities in the Human Connectome
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