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"Homological scaffolds of brain functional networks" appeared in the Journal of the Royal Society Interface (it was accepted on October 3, 2014). The authors, an Anglo-Italian group led by Giovanni Petri (Turin) and Paul Expert (London), use topology and in particular persistent homology to characterize the connectivity properties of functional networks in the human brain. They begin by explaining how persistent homology works for a network (a graph) in which the links have weights.
This application concerns 1-dimensional persistent homology, and measures the way the connectivity of the graph, as characterized by the number of "holes," varies as weaker (less weighty) links are added in. A hole, here, is collection of links which form a 1-cycle (a closed path), and that hole is considered closed if it can be filled in by triangles (2-simplexes) of links. [Two-dimensional persistent homology would be similarly defined with 2-cycles made of triangles, filled in if they can be spanned by tetrahedra (3-simplexes) of links; and so on for higher dimensions. The idea is to amalgamate sets of nodes which are maximally connected.] In the following illustrations, taken and adapted from Petri et al., there are 12 nodes, labeled $a$ to $l$, and links can have weights between 1 and 10.
The first image shows the network with only the highest-weight links. There is one cycle, with vertices $bci$; since it is a triangle it counts as being filled in. In the second image, nodes of weight 8 have been added. An open cycle $abcd$ has been formed.
The third image includes links with weights 5 and 6. A new open cycle has been formed: $efghi$, while $abcd$ "persists" as open. Nodes of weight 4 appear in the fourth image. The cycle $efghi$ is filled in by the triangles $efh$, $fgh$ and $ehi$, while $abcd$ persists.
Finally a link of weight 1 joins $a$ to $c$ and forms the two triangles $abc$ and $acd$, filling in the cycle $abcd$. There are no open cycles left.
The authors define the persistence of a cycle to be the difference between the weight at which it is "born" and the weight at which it "dies;" So the cycle $abcd$ in the illustration, born at 8 and filled in at 1, would have persistence 7. They use this concept to define a new graph, which they call the persistence homology scaffold. They keep all the edges which are part of a cycle at any point in the process (an edge may be part of several cycles); and assign to each of them as a weight the sum of all the persistences of the cycles they participate in. "A large total persistence for a link in the persistence scaffold implies that the local structure around that link is very weak when compared with the weight of the link, highlighting the link as a locally strong bridge."
As an example of the kind of information this construction can reveal, the authors present a comparison of brain functional connectivity, between subjects who were injected with a placebo (saline solution) and those whose shot contained 2mg of the magic-mushroom hallucinogenic psilocybin. The article reproduces a "simplified visualization" of the contrasting persistence scaffolds.
Persistence scaffolds for subjects injected with a placebo (a) and with psilocybin (b). Larger image. Image reproduced from Petri et al., published by the Royal Society under the terms of the Creative Commons Attribution License.
The authors remark: "Note that the proportion of heavy links between communities is much higher (and very different) in the psilocybin group, suggesting greater integration." And also "We can speculate on the implications of such an organization. One possible by-product of this greater communication across the whole brain is the phenomenon of synaesthesia which is often reported in conjunction with the psychedelic state." (Synaesthesia is sense-mixing, for example the association of specific colors with musical tonalities or with smells).
This image was picked up as the Science Graphic of the Week by Brandon Keim in Wired, under the heading: "How Magic Mushrooms Rearrange Your Brain." Keim: "In recent years, a focus on brain structures and regions has given way to an emphasis on neurological networks: how cells and regions interact, with consciousness shaped not by any given set of brain regions, but by their interplay. Understanding the networks, however, is no easy task, and researchers are developing ever more sophisticated ways of characterizing them. [The authors' approach] involves not simply networks but networks of networks. Perhaps some aspects of consciousness arise from these meta-networks." He then describes the experiment, and quotes Petri: "Investigating psychedelia wasn't the direct purpose of the experiment. Rather, psilocybin makes for an ideal test system: It's a sure-fire way of altering consciousness." He also reports that, according to Petri, "the network depiction above is still a simplified abstraction, with the analysis mapped onto a circular, two-dimensional scaffold. A truer way of visualizing it, he said, would be in three dimensions, with connections between networks forming a sponge-like topography."
Dana Stevens reviewed "The Imitation Game," the latest film about Alan Turing, for Slate on November 26, 2014. "The true life story of Alan Turing is much stranger, sadder and more troubling than the version of it on view in The Imitation Game, Morton Tyldum's handsome but overlaundered biopic." Besides regretting that the movie's "carefully calibrated uplift" skirts away from "the grim facts of Turing's final days," Stevens criticizes its intellectual emptiness. "This film about one of the last century's smartest humans at times treats its own audience like a classroom of remedial learners." "What new insights did Turing bring to the field of codebreaking, and just how did his invention lay the groundwork for modern digital technology? We never get the chance to even try to understand what's going on in Turing's head as he scribbles and tinkers away. ... This abdication from any attempt to convey the substance of Turin'g innovations does the film's late hero a disservice by turning his genius into a hazy abstraction." Stevens recommends "this essay on Turing by a 17-year-old British mathematician" as "comprehensible to someone who can't reliably calculate a tip."
My colleague Dennis Sullivan, in Rome to receive the Balzan Prize, was interviewed by Francesco Vaccarino (one of the authors of "Homological Scaffolds," above) for La Stampa; interview published November 26, 2014. Vaccarino briefly sketches Sullivan's career, including his interest in algebraic topology ("I am looking for algebraic structures which would be the grammar and the syntax that govern our manipulations of the entities grasped by our topological and geometric intuition"), in dynamical systems and in the deveopment of an algebraic and topological theory for the analysis of three-dimensional fluid flow. At the end, Vaccarino remarks on how highly specialized this research is, and that yet, as Sullivan observes, "mathematics is a naturally 'human' activity. Children, even before they start school, are almost all little mathematicians: they are curious about numbers and about geometrical shapes. Then they go to school and learn 'something' that gets called mathematics and that turns them away from the actual discipline. I'm not judging the teachers, who are overburdened and try to do the best they can. I do think that we should find a new way of teaching mathematics, focused on examples and on the understanding of how things are, and not based on notionism. Instead, today, it's like studying the PlayStation manual without ever playing! Mathematics is so intrinsically human and universal that perhaps it should be taught by specialists, just as is done for music." [My translations -TP]
This item is not new, but I think it is worth mentioning. In 2011, Russian television (Rossiya 1) produced a documentary about Grigori Perelman, the mathematician who proved the Poincaré conjecture, refused the Fields Medal and turned down a $1 million Clay Millenium prize. It is now available on YouTube, with English subtitles. Nika Strizhak wrote, and Mikhail Mikheev directed the video, which states its goal early on: "Who is this man and what happened with him in mathematics and in life?" Looking for the answer, the crew interviewed about a dozen mathematicians around the world who knew Perelman or were familiar with his work. Missing, unfortunately, are Richard Hamilton (Columbia), whose work on the Ricci flow was the indispensable first stage of Perelman's mission, Shing-Tung Yau (Harvard), whose effort to claim part of the proof for Chinese mathematics temporarily derailed recognition of Perelman's achievement and may have contributed to his embitterment, and Perelman himself, who is only heard through his apartment door: "Everything I wanted to say, I have already said. Good bye."
The video was made for a Russian audience, and gives a valuable Russian perspective on Perelman's life and education. The mathematical hothouse of his schools in St. Petersburg, the city-wide and country-wide competitions where, we are told, he could not stand being in second place. The grandeur of the Soviet school of mathematics, and the tale of its disintegration after the collapse of the USSR. Ludwig Fadeev: "In the late 80s we probably had the best institute in the world. Among the 110 members, 70 had doctorates in some field of mathematics. If you had a question you could always find someone who could answer it. Of the 70 doctorates, 40 of them left. Can you imagine the loss?" And Perelman's place in the pantheon of Russian ascetics and holy fools. As Fyodor Bogomolov remarks at the end: "Perelman is a national hero. A national hero.... They tried to buy him and failed." [Correction, 1/6/15: Fyodor Bogomolov's last name was initially incorrectly given as Bogomolny. Correction, 1/12/15: The degrees Fadeev mentions were not "PhD degrees" (as rendered in the video subtitles, and as initially repeated here), but Doktor Nauk, a significantly higher distinction.]
Tony Phillips
Stony Brook University
tony at math.sunysb.edu