# So why is topology important?

September 20, 2017 8:40 AM Subscribe

Understanding the brain using topology: the Blue Brain project.

You may also enjoy the Defining Topology through Interviews series on the same blog (note the links to individual entries in the list at the bottom).

*The brain is one of the main mysteries of medical science so this is huge! For me the most exciting part of this is that the researchers build their theory through the lens of Algebraic Topology and I will try to explain the main players in their game here.*You may also enjoy the Defining Topology through Interviews series on the same blog (note the links to individual entries in the list at the bottom).

It's been said, often, the map is not the terrain.

posted by marycatherine at 9:48 AM on September 20 [1 favorite]

posted by marycatherine at 9:48 AM on September 20 [1 favorite]

When you think the human brain is the most marvellous thing in the whole universe, don't forget which organ is telling you that.

posted by Multicellular Exothermic at 11:14 AM on September 20 [4 favorites]

posted by Multicellular Exothermic at 11:14 AM on September 20 [4 favorites]

My son has been hooked on the PBS Infinite Series videos, and they just finished a related sequence:

Your Brain as Math #1

Simplicial complexes - Your Brain as Math #2

Your Mind is Eight-Dimensional - Your Brain as Math #3

It's a really, really good show in general.

posted by RedOrGreen at 11:35 AM on September 20 [4 favorites]

Your Brain as Math #1

Simplicial complexes - Your Brain as Math #2

Your Mind is Eight-Dimensional - Your Brain as Math #3

It's a really, really good show in general.

posted by RedOrGreen at 11:35 AM on September 20 [4 favorites]

*one of the neurosurgeons I work with has a PhD in mathematics*

A vampire? I mean how is this even possible in a human time frame? Feeling very dumb today.

I'm not loving the explanations of topology in the "Interviews" so far, all good and correct but seems to be the surface view and not connected to some deeper essence. (see what I did there:-) But Topology is certainly a huge leap past arithmetic and formulas into the real thing that is math.

But good post, we need more serious math on the blue, we should deconstruct Grothendieck, no really there are goats.

posted by sammyo at 1:00 PM on September 20 [3 favorites]

*The researchers found that over time, simplices of higher and higher dimension were born in abundance, as synaptic communication increased and information flowed between neurons. Then suddenly all cliques*

**vanished**, the brain had finished processing the new information.I think, some law like conservation of energy is violated there. Has anyone penned or video taped the notation for the physical properties of the "information"?

posted by marycatherine at 1:01 PM on September 20

So why is topology important? (Besides leading to beautiful illustrations.) I can try...please correct this if you see any point of error!

The researchers have noticed that the directed graphs of connected firing neurons display a certain pattern after stimulation: after the third Betti numbers reach a maximum level the activity drops off. (Presumably some higher Betti numbers also show up and drop off as well.) Betti numbers are the total numbers of cavities of a given dimension.

It was crucial to realize that the key measurement is the number of cavities, which does not include the simplices themselves--those latter, which correspond to the little complete graphs, are assumed to be "filled in," but collections of them can surround a cavity. n-simplices can be glued together to make the skin of an (n+1)-dimensional hole.

Topology is useful since it gives you algorithms that allow you to count those cavities just by knowing all the individual edges and their directed flows.

Tldr; topology (surprisingly?) gives a concise way to describe the observed types of patterns of neuron reaction to stimulus. (Presumably those patterns happened in the researchers' brain when they recognized those patterns.) Maybe other important brain events will correspond to other specific patterns--and now we know a good type of pattern to look for!

posted by TreeRooster at 1:39 PM on September 20

The researchers have noticed that the directed graphs of connected firing neurons display a certain pattern after stimulation: after the third Betti numbers reach a maximum level the activity drops off. (Presumably some higher Betti numbers also show up and drop off as well.) Betti numbers are the total numbers of cavities of a given dimension.

It was crucial to realize that the key measurement is the number of cavities, which does not include the simplices themselves--those latter, which correspond to the little complete graphs, are assumed to be "filled in," but collections of them can surround a cavity. n-simplices can be glued together to make the skin of an (n+1)-dimensional hole.

*its just that the directed graph of neuron chains looks exactly like the graph of a simplicial decomposition (chop it into triangles, tetrahedra,etc.) of a higher dimensional shape. Cavities = bits of the graph that correspond to shapes that are closed manifolds but not simplices.*

So there isn't anySo there isn't any

*real*topology,Topology is useful since it gives you algorithms that allow you to count those cavities just by knowing all the individual edges and their directed flows.

Tldr; topology (surprisingly?) gives a concise way to describe the observed types of patterns of neuron reaction to stimulus. (Presumably those patterns happened in the researchers' brain when they recognized those patterns.) Maybe other important brain events will correspond to other specific patterns--and now we know a good type of pattern to look for!

posted by TreeRooster at 1:39 PM on September 20

*I think, some law like conservation of energy is violated there.*

Well, the brain isn't a closed system.

posted by thelonius at 1:54 PM on September 20

*a closed system*

Neither is the universe, of which human body "and" mind of complex organic --of which glucose and lipids &tc + inorganic molecular interactions + energy released (heat, for example) or converted at the junction between neurons. It's a space.

My problem is that my daughter took her bio notes to college. So the terms "vanished" and "information" just strike me as expletives, disjunctive concepts, in an exercise purporting a meaningful understanding of brain topology. What good is a map without water features for instance?

It's a little troublesome to me and perhaps symptomatic of how I interpret the unreliable benefits of so-called medical chemistry, or experiments with synthetic compounds.

posted by marycatherine at 2:53 PM on September 20

*The framework works without parameters (for example there is no measurement of distance between neurons in the model)*

That said, how meaningful is a representation of "more or less" dense cliques and cavities (that should not be explained as growth of neurons), valence of cliques, or possibly the strength or weakness of

*attraction*between receptors found neuron cells? Do neurons move to "close" synaptic "circuits"?

posted by marycatherine at 3:27 PM on September 20

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posted by TedW at 8:51 AM on September 20