From: scerir (scerir@libero.it)
Date: Mon Feb 05 2001 - 16:20:45 MST
Monika's site
http://www.math.fsu.edu/~mhurdal/
and, also, this article:
Jan 25th 2001 | The Economist
Topology and neurology
Your neurologist would love to flatten your brain. Not physically, you will
be glad to know, but mathematically. For, just as the exploration of the
earthıs surface (a sphere, more or less) required the creation of
appropriate flat maps, so, too, does the exploration of the surface of the
brain.
It is not that hard to convert a sphere into a plane, provided that you are
clear about the compromises you are prepared to make. The Mercator
projection, for example, preserves shapes and angular relationships at the
expense of areasso that the polar regions look far too large in relation to
the equatorial ones. Flattening out the convoluted surface of the brain
requires more compromises than that. But Monica Hurdal of Florida State
University thinks that she has made them in an acceptable way. At the recent
joint annual meetings of the American Mathematical Society and the
Mathematical Association of America in New Orleans, she presented her work
on ³quasi-conformal maps of the human brain from circle packings².
Brains do most of their information-processing on the surfaceor cortex, as
it is known neurologically. So species that, in the course of evolution,
have come to need more brain-power, have developed brains with creased,
convoluted cortexes that pack more surface area into the same volume of
skull. This makes it hard for researchers to see exactly which parts of the
cortex are doing what, and how they are related to one another. So, although
it is possible to put somebody into a brain scanner, set them a task, and
see which bits of their cortex light up with activity, areas that look
adjacent in an image produced by the scanner may, if they have a deep fold
between them, actually be a long way apart.
That is where a mapa flattened version of the brain showing exactly where
on its surface the activity is taking placewould come in handy. It would,
for preference, be a projection that preserves shapes, like Mercatorıs. And
this is indeed possible, according to Dr Hurdal, using a law of geometry
known as the Riemann mapping theorem that was first enunciated in 1854.
This theorem says that every continuous surface can be turned into a special
kind of flat map, called a conformal map. Such a map means that if you have
two angles on the original surface, and one is twice as large as the other,
the first angle will remain twice as large as the second even if the actual
number of degrees in each angle changes. This is not quite as good as
keeping every angle the same, as with Mercatorıs projection, but it is good
enough for neurologists who want to steer between bits of the cortex.
To prove her point, Dr Hurdal showed her audience a map of part of the
brain, called the cerebellum, of a hapless student who had been induced to
undergo no fewer than 27 brain scans. The upshot of this scanning was a
high-quality three-dimensional image, to a precision of one millimetre in
each direction. This level of precision allowed her to distinguish precisely
between the ³white² matter of the brainıs interior, and the ³grey² matter of
its cortex. The result was a network of 50,000 cortical points, each
connected to its nearest neighbours by a line. Dr Hurdal thus ended up with
a landscape, still convoluted, of triangles.
The hard part was flattening this network of triangles. But an older
theorem, going back to the ancient Greeks, states that you can always draw
three circles around the corners of a triangle so that each one just touches
the other two. Since any two of these circles will also belong to a
neighbouring triangle, if you have thousands of triangles in a flat plane,
you can have a perfect packing of that plane by thousands of circles.
Unfortunately, the triangles that represent the surface of a brain are not
in a flat plane, so the touching circles drawn in each of them will stick
out. But there will still, according to the Riemann theorem, be a unique way
to move all the points until they settle down with the circles into a
well-packed plane. The result is, to a good approximation, a conformal map.
Navigation, however, also requires a reference grid. On the earthıs surface,
this is derived from two fixed points (the poles) and a line (the equator).
In the cerebellum, the equivalents of these have to be bits of anatomy that
can be recognised in everybodyıs version of the structure. Fortunately,
although much about the cerebellum is mysterious, enough is known for such
features to have been identified.
Once brain-navigation becomes routine (with luck, without the need for 27
separate scans), it will be possible to track changes in a brain over time.
It will also be possible to compare the brain maps of individuals, even if
everyoneıs cortex is folded slightly differently. And that should lead to
some interesting voyages of neural discovery indeed.
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