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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: 2) by opinionated_science on Thursday August 25 2016, @06:37PM

    by opinionated_science (4031) on Thursday August 25 2016, @06:37PM (#393125)

    add the exponents....1010 and 1014

  • (Score: 0) by Anonymous Coward on Thursday August 25 2016, @06:47PM

    by Anonymous Coward on Thursday August 25 2016, @06:47PM (#393129)

    Add AT to the list of stuff I need to catch up on.

  • (Score: 4, Insightful) by PizzaRollPlinkett on Thursday August 25 2016, @07:47PM

    by PizzaRollPlinkett (4512) on Thursday August 25 2016, @07:47PM (#393159)

    I wish someone would write a book that explained the WHY behind algebraic topology. All the books I've seen tell the HOW in mind-numbing detail, but never explain why they're doing anything or what it's for. I needed two years to figure out the basic "why" of algebraic topology. I could explain in a clear paragraph what took me years to figure out. Why can't anyone do that in an introductory book? Authors act like you should already know everything they're telling you. Even one brief sentence explaining why something is being introduced, or why you need it, or why it's important can help clarify everything. If algebraic topology is going to be important to neuroscience (don't bother reading the article, it's content free), algebraic topologists ought to do more to teach their discipline to beginners. It's an absorbing, fascinating world that few people even know exists. It's a blend of geometry, algebra, and pure thought. It's a tour of how the human mind creates abstractions. And yet no one can seem to explain it coherently.

    If you're ever thinking about learning algebraic topology, learn group theory first. Very, very well. Algebraic topology is mainly a whole lot of group theory.

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    • (Score: 2) by dingus on Thursday August 25 2016, @08:52PM

      by dingus (5224) on Thursday August 25 2016, @08:52PM (#393178)

      This is a problem with most math books IMO. Only a few really teach well.

      • (Score: 0) by Anonymous Coward on Thursday August 25 2016, @08:58PM

        by Anonymous Coward on Thursday August 25 2016, @08:58PM (#393181)

        i suppose everybody climbing mount mathematics arrives bruised, marked and sometimes maimed, so i guess no one is going to install a lift (or just share a map of the most daring pitfalls) 'cause you'd look better then the rest, arriving at the summit (smart AND good-looking? that's a NO-NO).

      • (Score: 2) by PizzaRollPlinkett on Friday August 26 2016, @02:12PM

        by PizzaRollPlinkett (4512) on Friday August 26 2016, @02:12PM (#393480)

        I couldn't agree more. Most math books are abysmal. Given math's terrible reputation, and the difficulty of attracting people to it, you'd think the math community as a whole would write more popular books. Fraenkel's "Love and Math" a few years ago was really good (although not really in the areas I know anything about), but books like that are few and far between. I thought he might be the "David Greene of higher math" but he hasn't done anything else since then. I guess popular math books don't sell.

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    • (Score: 1) by khallow on Thursday August 25 2016, @08:59PM

      by khallow (3766) Subscriber Badge on Thursday August 25 2016, @08:59PM (#393183) Journal

      And yet no one can seem to explain it coherently.

      Just before, you wrote:

      It's a blend of geometry, algebra, and pure thought.

      That would be why. Making really hard and complex subjects understandable with a few paragraphs is not a common skill.

    • (Score: 4, Interesting) by aliks on Thursday August 25 2016, @09:07PM

      by aliks (357) on Thursday August 25 2016, @09:07PM (#393188)

      There is a tremendous amount of machinery (tools and techniques) in the world of algebra, not just groups, but rings, fields vector spaces (and modules if you like).

      Before the 1950s this machinery was not available to geometers and topologists. With the growth of algebraic topology, mathematicians found new ways to associate symmetries with the underlying structure of "shapes". This gave access to algebra techniques and hey presto, some intractable problems were solvable.

      If you want a classic example of the power of these techniques, just look up Bott periodicity - it turns out that increasing the number of spatial dimensions does not always increase complexity. The dimensions fall into blocks of 8 so that an 11 dimensional space has properties that closely resemble 3 dimensional space and so on.

      Very fundamental stuff - but beware - Michael Atiyah has a warning about trusting too much to algebra -

      Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.'

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    • (Score: 2) by Fnord666 on Friday August 26 2016, @03:53AM

      by Fnord666 (652) on Friday August 26 2016, @03:53AM (#393327) Homepage

      I wish someone would write a book that explained the WHY behind algebraic topology. All the books I've seen tell the HOW in mind-numbing detail, but never explain why they're doing anything or what it's for. I needed two years to figure out the basic "why" of algebraic topology. I could explain in a clear paragraph what took me years to figure out.

      Thanks for volunteering! I will be happy to proof read it for you.

    • (Score: 1, Interesting) by Anonymous Coward on Friday August 26 2016, @08:37AM

      by Anonymous Coward on Friday August 26 2016, @08:37AM (#393402)

      I could explain in a clear paragraph what took me years to figure out

      I looked for that paragraph but didn't find it.

      • (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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  • (Score: 2) by zugedneb on Friday August 26 2016, @03:58PM

    by zugedneb (4556) on Friday August 26 2016, @03:58PM (#393535)

    and some other deep statistics with some self organizing maps and recursive stuff...
    well, whatever... abstract algebra is fine too...

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