converted to html for fun and practice
31 slides, almost all here available as gifs
one 12 min video not available but 3 slides sample it
Background
The idea of excitability is that a small but not too small stimulus provokes big local reaction, which later fades back to original quiescent steady state attractor. Repetitively excitable media are commonplace in physiology and in other areas of biology. They are also known in chemistry.
If excitation is quick enough relative to recovery and if its maximum is big enough relative to threshold stimulus, then excitation propagates as a shock front. This can be made explicitly quantitative in terms of the partial differential eqns of local reaction mechanism and spatial coupling by electric fields or molecular diffusion.
The propagating pulse has a characteristic speed. It is absorbed at boundaries, so one stimulus ==> one outgoing wave, then all is quiet again. this is, for example, how your heart works, once a second, and how part of your brain work.
However there is another solution to the equations in which activity has angular rather than translational time invariance . This is a rotating vortex mode of activity, and it continues forever in idealized cases. I called this dissipative structure a "rotor" because I thought thats all it does, just rotate. It has a characteristic size and a characteristic rotation period, both of which are very simply estimated from the basic physics or the essential parameters of a model equation. In my lab here at UofA we discovered that it generally does lots more interesting stuff than just rotate, except in the special case that I thought was generic when first naming these things. I'll come back to that shortly.
As the rotor spins, a spiral shaped shock wave radiates away like this: a pair of them in a chemically excitable medium.
In 3d media, this vortex point becomes a vortex line or filament. It typically closes in a ring. The spiral radiating from it is a scroll ring in 3D. It looks like this cut open in a computer,and like this in the chemically excitable medium cut open at the glass wall of a test-tube. Here it is again in the chemical medium as a completely closed ring not touching the glass wall.
I called these structures organzing centers because they are the typical centers from which reactions throughout the whole volume of medium are periodically organised in time and space
About the time this last picture was taken, while I was working at Los Alamos in T-10 as guest of George Bell, I fancied there might be qualitatively distinct kinds of rings: no known chemical constraint forbids such rings to link and to knot. There would have to be certain geometrical peculiarities about the concentration fields, but nothing forbids them. The chemically realistic possibilities classify into a sort of periodic table of organizing centers. But that was just topological theory without any dynamics, and already a few years old.
The next step is to enquire which of them if any are stable. The familiar scroll ring, for example, is not: it shrinks by a certain area per second, proportional to the diffusion coefficient, and finally snuffs out. Maybe none of them are stable. There are mechanisms by which skew filaments could cross-connect and then pass through one another. This would obviously change the linking and knotting, so all might finally go the way of simple disjoint closed rings to extinction.
But mystical intuition told me that some of the topologically exotic alternatives would not be able to shrink all the way because their filaments would cross and snag on one another at inconvenient angles that dont allow reconnection. Then we would have a taxonomy of distinct particle-like solutions to the field equations of excitable media, and presumably the corresponding phenomena in nature would appear as persistent organzing centers of diverse sorts that cruise through the otherwise quiescent medium like elementary particles.
How to find out? There have been a number of publications over the years in which the reaction diffusion equation is treated analytically to derive the laws of motion of vortex filaments in excitable media. The most ambitious are by Keener in 1988, by Biktashev in 1994, MIkhailov 1995, and by Gabbay in press. They all treat a limiting case of vanishingly slight curvature and twist of the isolated filament. Though people who haven't read those papers with care often claim more for them, none can illuminate the case of compact stable organizing centers or even predict their existence, because organizing centers are stabilized by forces not considered in the limiting case analysis.
(I think the main collection of these explicitly-ignored factors concerns not departure from linearity, nor even the spontaneous curvature&twist-independent filament motions called meander, but the geometry of the collision interface surface that typically lies tangent to the filament rather than far from it as required in mathematical analyses of isolated-filament motions. This positioning results from from the rotor period depending on local curvature and twist of the filament, so that some segments of filament spit out waves faster than others, pushing the collision interface onto the slow segments, where they push that segment outward.)
So we have to look to numerical solutions of the reaction-diffusion equations in 3D. I tried the first of those as a cellular automaton in 3D on a home-built Apple. The several organizing centers proved stable, but then cellular automata are not much like continuous reaction diffusion systems so I did it over again as a partial differential equation on the Lab's CRAY 1. It produced a stable symmetrical trefoil-knotted scroll ring.
Further numerical experiments produced several more topologically distinct members of the periodic table. Now there is a zoo of them that seem stable. But they all look rather delicate. Their vortex tubes are closely pressed together like knotted party balloons, and they turn in such close contact like differential gear trains: any distortion might seriously spoil the perfection of this symmetric intimacy, resulting on filaments cross-connecting and topological "invariants" no longer preserved.
About this time we discovered, first by numerical experiments then in the chemical laboratory, that the vortex does not typically just sit there and spin, but moves around in quasi-periodic patterns . I called this "meander", demonstrated it for the first time in chemical media, and classified the varieties of meander, including what I called "hyper-meander": this is not quasi-periodic but seems not chaotic either. It is full of amazing temporal structure that you can analyze forever from its peculiarly rich fourier spectrum .
I began to worry that if vortex filaments started spontaneously
bouncing around like this, then the known stable compact organizing
centers would not likely stay as stable as appeared from our first
computations that happened to use NON-meandering rotors.
So 3 years ago I re-did the computations of a dozen stable organizing
centers with excitability parameters moved into the generic domain of
vigorous meander. As you can see from
this slide ,
Wavefronts encountered in slices through the volume of excitable medium
look pretty disorganized, and you never see the same thing twice.
Turning attention to the vortex rings that radiate these waves that you
saw sliced, you see thy are pretty wiggly.
The next slide shows three vortex
filaments entangled in the way of Borromean rings . They are not topologically linked,
but the filaments don't cross-connect or pass through each other, so
the rings dont come apart or shrink and vanish.
This slide show 3 rings each
linked once through each of the other two. This is another persistent
organizing center. At least, it has here gone on through 40 rotor
periods and while always wiggling and never repeating a configuration,
it shows no sign of systematic change.
Here is a trefoil knot like the
symmetric one first discovered in non-meandering media. Here it is
agitating itself with the most violent meander, which makes it bigger
as though by thermal expansion. Its filaments lash and bang around as
you can see in the 12 minute video, but it persists apparently forever,
at least through 70 rotor turns without systematic change.
(turn on the video from which these slides are snapshots)
You might expect occasionally to see filaments strike one another and
cross-connect and change topology. In some organizing centers they did.
But in half a dozen other kinds they dont. Those topologies seem to
persist forever. I call them not SOC's but POC's. Computations
indicate robustness as parameters change, and indicate persistence in
time, and stability against deliberate perturbations: kicked and bent
within limits, these things recover and resume business as usual in the
computer.
Does this mean they should be observable in the lab if they exist in
nature? Where to look? One place might be the human heart, which is a
reaction diffusion medium in which electric potential rather than
molecular concentrations are diffusing. The equations used for these
movles are in fact a simple caricature of the cardiac cell membrane. My
collaborators in a medical school did living dog experiments and found
the predicted rotors in the heart, and found that they constitute the
first stage of ventricular fibrillation and sudden cardiac death. In 3D
they found the predicted vortex filaments in heart muscle. And now
meander of those filaments has been reported from yet another lab. But
it is too hard to record electrically from living heart muscle at the
needed resolution in time and space.
So I turned to the chemically excitable medium. Here you can plainly
see 3D wave structures. It is hard to make out quite what they are,
though. Even having a video movie instead of just still snapshots, does
not make clear what's going on. Even having two videos from 6 degree
different angles for stereo viewing still doesnt help much. What is
really needed is a movie of VOLUMES in which the chemical concentration
is measured at each point in 3D. At leisure you can then use these
volumes of data to isolate the vortex filament and follow its
evolutions, the same way I did when solving PDE's 3Dly.
For the last two years I've been trying to do that, and it finally
worked last summer. When teaching and all that resumed in August the project
slowed down quite a lot, but I can show you where it had got to and
where it will shortly resume from. Meanwhile I published the current
mere-instrumentation-without-answering-new-science-questions
stage in Dec issue of CHAOS, and I have a few reprints here, together
with related backgound papers. Here's how it works.
It is almost the same as CAT scanning of people's insides by Xrays in
the hospital. The idea is that you have a transparent object and you
project a shadow of it on the wall. Suppose the object is only 2
dimensional, let's say a horizontal a slice through a standing human
body or through a volume of chemically excitable medium. The slice is
shown here in an
8-walled room . Really more like
100 walls are needed for proper resolution, but the principle is the
same in this 8-wall example. The object consists of empty space plus
this dense blob. Projected onto one wall, this 2D object becomes a 1D
projection, a distribution of optical density along this line, the
wall. You get a different distribution when you project on a different
wall. You do all the walls. Now throw away the original object and
just leave this empty octagonal area surrounded by the 8 shadows. Can
we reconstruct the object from the shadows? Sure, even from just these
three shadowns explictly shown here. This shadow tells us there was
density somewhere along this corridor. This shadow tells us it was
along this corridor. They have only this region in common, so that must
be where the blob was. This next shadow confirms that inference and
improves the resolution. And so on. If there were many blobs of various
densities, they are all reconstructed in superposition. If the object
was a continuum, its 2D density distribution is reconstructed better
and better as you increase the number of walls, and increase the
bitwise density discrimination of the detector, and increase the
resolution of detector pixels along the wall.
Here is a 2D test case.
I made this 2D object, went through a numerical simulation of this
procedure using 180 walls, and obtained this backprojected reconstruction
from 180 edgewise 1D shadows. Of course
there are some tricks. It doesn't really work this smoothly just from
what I hurriedly sketched, but that was the essential idea.
That takes care of this one horizontal slice. To do 3D you simply
replicate this setup vertically slice by slice. I do it with a 640 x
480 charge-coupled device for picking up blue light projected in a
collimated beam through the transparent excitable medium. The object
rotates on a vertical skewer in front of the CCD camera, so the camera
serves in succession for all the walls, typically 50-100. And the
shadow on each wall is a vertical stack of hundreds of lines, not just
one line, in other words it is a 2-dim shadow, but we reconstruct from
each of its constituent layers independently.
Needless to say, this required quite a lot of attention to numerical
technique; we wrote all the code from scratch in C. This was debugged
and polished mostly by Zoltan Szilagyi, an undergraduate in EECS. The
aim is that the reported optical density as a function of x,y,z
coordinates should be linear in the actual OD of the specimen. Here is
a result testing the reliability of that map on one test object for
which we could know exactly the original OD. It is not perfectly
linear, but good enough for my purposes:
linear map test case
Now here is a 3D test case, an steel ball
bearing to check for perfect
symmetry of the output. Then to check its handling of translucency I cut
two mm-size flakes of transparent
neutral density filter, each a fraction of a mm thick, each
attenuating light 2-fold, and rotated them on the vertical skewer in
the blue beam. A 3D reconstruction appeared in the workstation's RAM,
from which you can display any view with any volume coloring for the
measured optical density. Here is one view of a pair of flakes . It has some
edge artifacts due to refraction of light through the corners of the
transparent plastic flakes, but it got the sizes and most of the
optical densities right.
This is a piece of jet
fuel filter with holes in hexagonal array 140 microns apart to
test optical resolution of the microscope we built around the rotating
skewer. The lens arrangements were contrived by Mark Gallagher, a
student in optical sciences.
Finally, its time to put the chemically excitable medium in. Here new
problems are encountered. First of all, the chemical solution convects
while it rotates, and that has to be stopped, so it must be gelled in a
way that doesn't interfere with optics or reaction or diffusion. With
that solved after a few months, then the CO2 gas byproduct of the
reaction is trapped, so it makes bubbles which first of all make
chemical gradients, secondly distort the gel, and thirdly serve as very
high power negative lenses. So I had to reinvent the chemistryby
finding a new substrate that doesn't make bubbles. With that solved
after another year, then the gel refracts light as it rotates, so it
must be confined in a perfectly centered cylinder, and that must be
placed in a liquid bath of identical refractive index. But the glass
wall of the containing cylinder has a different refractive index, and
no matter how thin we made it with help from the local glass shop, it
badly distorts the shadows and ruins the reconstruction. Scott Caudle,
an undergrad in Engineering Physics found that teflon has exactly the
RI of my gel, and located very thin cylindrical teflon jackets. All
other plastics are more like glass, by the way: teflon is a pretty
amazing exception.
The whole arrangement is now like this flow chart Now everything is
all set to look for 3D waves in a rotating cylinder of excitable gel.
This shows the first successful
2-dimensional slice through a scroll wave. The rotor, where the
vortex filament penetrates this plane in the interior of the gel, is
about here. In the next slide the color coding is different, but in
both it is proportional to ferroin concentration in that plane of the
gel. Here you have 3 vortex
filaments perpendicularly puncturing the selected plane, two making
clockwise scrolls, and one counterclockwise.
Finally the next slide shows in perspective a view of the 3D volume of ferroin concentrations
reconstructed to include the teflon cylinder. The cylinder wall is 0.4
mm thick surrounding its 9 mm diameter of chemical gel, and this chunk
of it is 2 mm high. Inside you see a segment of a scroll wave. Its
vortex filament runs vertically parallel to the axis of the cylinder
because of the way we induced the chemical initial conditions.
But the filament is seldom oriented so tidily. In this reconstruction,
the rectangular box is an
angled cut-out from within another cylinder's volume. It is 3.5 mm
square and 1 mm thick. That short direction is actually horizontal and
continually precessing, perpendicular to the vertical rotation axle of
the skewer and of the teflon cylinder. In the top panel you see the
blue excited region of the chemicals and you see the red recovered
region. Because this volume is displayed opaquely, the colors are only
on the 3 flat surfaces exposed. In the middle panel we made the red
material transparent and left only the blue excited material so you can
see inside the box. Notice the black line segment, isolated in the
bottom panel. That is the vortex filament, I mean a 1 mm segment along
it. The filament is a tilted chord across the disk cross-section of the
cylinder: it continues 2 mm in each direction beyond this cutout until
it hits the curved wall of the cylinder.
The numerical procedure for extracting the filament from concentration
data is under development by Gang Chen, a graduate student in my
Department.
Later on we will presumably see these black filaments closing in rings,
then try to contrive initial conditions from which linked and knotted
filaments would arise. Scott Caudle is adjusting the new chemical
recipe to maximize its fortuitous photosensitivity: my hope is that I
can then contrive an interesting sequence of light patterns to evoke
topologically interesting filaments. In the numerical experiments
reported recently in Physica D I did it by using polynomial fcns of two
complex numbers to produce concentration distributions in a 4-space,
which I then projected stereographically into 3-space for numerical
IC. But in the real 3D laboratory I'll have to think of another way.
The aim in any case is to see if such organizing centers persist in the
laboratory. They do in a wide variety of computational
reaction-diffusion scenarios that are thought to be very similar to the
real world.
At the present stage the new tomographic microscope works with a new
photosensitive bubble- free excitable gel, so I have a project that is
almost guaranteed to fly. I will be looking for a postdoc copilot to
fly it with me.
Summary:
1) Topological alternatives to known scroll ring
Discovered possible reconnection mechanism
2) Numerical solutions of R+D ==> symmetric SOC variety:
Reconnections don't happen if skew angle big enough
3) Discovered meander and hyper-meander and parameter plane:
SOC's likely delicate
4) Numerical solutions still persist:
robust enough, long lasting enough, should be observable in lab
5) No way to observe with any clarity
6) Optical tomography of strictly home-brew sort out of the lab junk-box
