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posted by martyb on Thursday August 25 2016, @06:33PM   Printer-friendly
from the something-to-think-about dept.

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.

[...] 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.

I had always hoped algebraic topology would finally unlock the secrets to untangling my fishing line, but figuring out how the brain works is useful, too.

arXiv.org hosts both an abstract and Full Paper (pdf).


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  • (Score: 3, Informative) by PizzaRollPlinkett on Friday August 26 2016, @02:19PM

    by PizzaRollPlinkett (4512) on Friday August 26 2016, @02:19PM (#393488)

    I seriously thought about adding it, but I haven't looked at it much in a few months, and might write something that makes no sense.

    Algebraic topology is the study of whether two topological spaces are the same or not. To figure that out from the spaces themselves is not easy, so the idea is to use the idea of partitions (factor groups, quotient groups) from group theory to partition the space so that you're only dealing with the essential invariants of the space. You can bring the considerable weight of group theory to bear on this problem, since once you've made the partition, you have a group. If two partitions are isomorphic, then the original spaces are topologically the same. You can't just map a topological space into a group, however, since they're two different categories. You need a functor (note that C++ borrowed the word but uses it in a very different context!) to turn a topological space into a group. Because you're looking at groups, you need to know a ton about group theory. There are many ways in group theory to tell if groups are isomorphic. Creating these groups from geometric-like topological spaces is hard, so the idea of approximating geometric spaces using triangle-like objects called simplexes (arranged together in complexes by using a few rules as to how they can be joined) makes it easier. Simplexes are interesting objects in themselves, and even more abstract objects have been made which do the same thing for less geometric applications.

    That's just a brief summary.

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