1951-2: Belousov, Turing, Hodgkin&Huxley
 
48 years ago these three made discoveries that still interest lots of people
today in contexts remote from the original intent. As with Aristotle, Galileo,
Copernicus, Curie, and so on, their novel contributions have been told from one
writer to the next over so many generations that one needs to read the originals
to find out what they actually said. Here's my version, adapted as best I can to 
present context:


Alan Turing's reaction-diffusion instability of 1952 was an immediate
hit with theorists. 

[ I didn't read the original with care until 1971. By this time
it had already started getting Citations, and these continued doubling 
very 9 years up to the current 100 per year. This is roughly the exponential 
inflation rate of journal literature in the biophysical-biomedical areas I read: 
9%/year in the 5000+ titles I have filed since 1980. For comparison, the overall 
USA scientific literature has grown 2.7%/year since 1980.]

The first Turing-pattern example was contrived in the lab 2 decades later:  
in 1974 Flicker and Ross interpreted Liesegang patterns in such terms (see Ross 
et al 1995). Two more decades later Castets et al (1990) then Lengyel et al
(1993) contrived more convincing laboratory illustrations of the Turing
mechanism. In these a 2-dimensional array of concentration peaks about 1/4 mm
apart arises over several hours, starting from uniformity. Ouyang and Swinney 
(1991) first showed that this chemical reaction really is showing a Turing
bifurcation in the sense that no threshold kick is needed to get it
started: the pattern arises just as Turing foresaw, spontaneously from
infinitesimal 'noise' in a uniformly featureless area. This occurs when
and only when conditions are made permissive (a certain temperature
range). Bumps fade back into uniformity when conditions are made
non-permissive. The only technicality left unchecked, that could finish
confirming this as an example of Turing's mechanism, is that when the
diffusion coefficients become equal, pattern should not spontaneously
arise. No one doubts this, so far as I know, so chemistry now has its long-sought 
example. I think there is still no confirmed biological example.  

Turing's 1952 contribution was two-fold: a surprisingly counter-intuitive
concept, and its demonstration by numerical example and by mathematical
analysis. He showed that the uniform state can be unstable to infinitesimal
perturbations. He illustrated this using the general non-linear
reaction-diffusion equation, in the special case that in the absence of
diffusion, its reaction rests at an attracting steady state, and that
each rate of synthesis and degradation additionally depends in a
certain way on the local concentrations of those species, and that
their diffusion coefficients are unequal in a certain way. Then spatial
inhomogeneity can arise spontaneously from prior uniformity, needing no
finite kick. He thus made us aware that molecular diffusion can power
not only the intuitively expected trend toward homogeneity: in certain
situations it also powers the opposite trend, toward spatially periodic 
bunching up of concentrations.

There are, by the way, plenty of other kinds of spatial instability known today, 
e.g., the reaction-diffusion instabilities of the complex Ginsberg-Landau model
representing continuous fields of coupled oscillators. These can spontaneously 
go turbulent in instructive ways, as Yoshiki Kuramoto pointed out in the mid 1970s.

Well before Turing's paper, the reaction-diffusion equation was
accepted by physiologists as the key dynamical principle underlying
electrical propagation in nerve cells. Hodgkin and Huxley based their
work in this tradition.  Does Turing's discovery about
that equation's behavior mean we might expect spontaneous pattern-
formation in nervous tissue? In nerve membrane the diffusion
coefficients do indeed differ, as required. But in the right way? In
nerve membrane all the diffusion coefficients but one are zero. That
alone precludes Turing patterns. Moreover that one is the diffusion of 
activating electric potential, not of any of the
nominal inhibitors: the wrong ones are zero, presenting the starkest
possible opposite to Turing's requirements. Correspondingly, nothing like 
Turing's instability has turned up in context of action potential
propagation, neither theoretically nor in reality, so far as I recall. This
does not preclude the apparition of Turing patterns in nervous tissue on the
basis of some other kinds of interaction ... for example maybe ocular dominance
columns in the visual cortex could be analyzed in such terms ... but a lot of
wiring and the chemical properties of synapses would have to be invoked, not
just the ion channel kinetics and electric potential diffusion that underlie 
action potential propagation.

Now here is a curious twist on this story. Going back to chemistry,
there are reactions whose dynamics resembles the electric events in
nerve membranes. One such reaction that has been familiar for
centuries already is the burning of a gunpowder fuse. Another (which
brings in lots more physics than just local reaction and diffusion) is
the burning of a candle (see Presentation 9), and another the flame front
in a combustible gas mixture. Others, in gelled water solution, have
two additional nice features: the only transport mechanism is
diffusion (there is no convective motion, for example, nor radiation),
and a restorative reaction behind the front regenerates excitability,
much as happens in cell membranes. Such reaction-diffusion media are
remarkably similar to nerve membrane in their dynamics and intheir
support for propagating wave patterns.
 
The oxalate/permanganate example mentioned by Luther in 1906 seems not 
one such. That experiment seems nowhere described in adequate
detail; no one living today seems sure what people saw at Luther's
lecture. The reaction has been studied with care in well-stirred reactors,
but the phenomena Luther reported in unstirred liquid have never been confirmed.
Contemporary attempts to obtain propagation from the published recipe
produce only a sharp color transition between the top and bottom layers
of liquid. This creeps a short way briefly, and not at a steady speed,
then the colors all become the same all over. Possibly something better
could be developed along these lines with modern techniques, but no one
has yet reported success.

Forty-five years after Luther, Belousov reported the beginnings of
a reaction that would propagate in homogeneous liquid solution... but that
behavior wasn't discovered in it until yet another nineteen years passed. In 
1951 it only oscillated, like Bray's peroxide oscillator of 1921. Referee 
objection to Belousov's 1951 and 1958 attempts to report on this
oscillation has been variously remembered, sometimes unkindly. Another
possible interpretation of establishment reluctance at that time is that the 
objection may have been just the same as voiced in the prior literature 
of the Bray reaction. The issue was not so much whether a reaction can oscillate,
because there were accepted examples, and Alfred Lotka had even made simple 
theoretical models in 1920 showing that it could happen in homogeneous
solution if certain idealized conditions were met.  But does this
particular instance oscillate by the mechanisms claimed, i.e., by homogeneous
chemical reaction alone, with no spatial gradients or particulate nuclei 
suspended in the solution? Both Bray's and Belousov's in fact do, but in the
absence of clear laboratory proof, that seemed hard to swallow in 1951 and 1958.  
 
In any case, this reaction could only be said to support "pattern formation" in
the sense that oscillations were not quite synchronous everywhere unless it was
well stirred. This is nothing like Turing's instability. Then in early 1970 
Zaikin and Zhabotinsky showed in modified Belousov soup a propagating
chemical pulse strikingly similar to the action potential. Here again, there were
similar precedents, e.g., the gas-phase propagating 'action potentials'
of phosphorous oxidation explored by Lord Rayleigh in 1921. But like many things
that appear before the audience has been prepared for them, that was
forgotten until Z+Z's aqueous version emerged in a more favorable climate
of interest.

A further twist: it was discovered in Belousov-Zhabotinsky soup later in
1970, then understood by computing from the reaction-diffusion partial
differential equations of nerve membrane in 1972, that there is another kind
of reaction-diffusion mechanism for pattern formation, independent of 
Belousov's oscillations and of Turing's instability. This new mechanism works
even if all diffusion coefficients but the excitor's are zero or equal to
the excitor's, impossible conditions for Turing patterns. And the
reaction need not oscillate, an impossible condition for Kuramoto's instability. 
Moreover, this new kind of pattern
formation does not occur spontaneously: it requires a finite starter
kick with a particular spatial structure. Its immediate result is, like
Turing's, a pattern in space, but it is not initially periodic in space.
It is periodic in time,  but not because of any pre-existing oscillation.
It is a rotation of concentration gradients in a tiny disk
(1/4 mm diameter in the first laboratory case) located anywhere within
a wide area of mostly uniform concentrations with no periodic aspect, 
initially. Its period is about 100 times longer than it takes the
chemical (or neural) medium to excite. A heart cell, for example,
excites electrically in about 1 msec, so this mode of behavior in a
sheet of heart cells has a period near 100 msec, the characteristic
period of fibrillation onset just before sudden cardiac death.

This little disk is called a "rotor". There may be many of them, located
more or less randomly, each like an independent particle. As it spins, each 
rotor excites the nearby material so it radiates an outgoing wave in the shape 
of a spiral (in 2D; or a scroll ring in 3D). The medium thus eventually fills 
with spatially and temporally periodic patterns, but this
is only a long-term byproduct of the fact that each independent rotor is spinning. 
Moreover this whole development is not the result of any 
instability, nor anything to do with Turing's
principle. Nor do rotors ever arise spontaneously from near-uniform
conditions. Their principle is entirely different, except that it does involve
diffusion and reactions.

(Rotors can, however, arise spontaneously from strongly patterned
conditions, e.g., sufficiently high-frequency wavetrains can develop rotors,
and so spiral waves, by an instability of propagation that results in patchy
erasure of segments of wavefront.)

Now what has any of this to do with Mind ? There seems to be a tradition
since Max Delbruck's "Evolutionary Epistemology" that 
books on "consciousness" should present wide-ranging surveys of
every mysterious and wonderful science (quantum mechanics, chaos,
relativity, number theory, thermodynamics, non-linear partial
differential equations ...) salted with allusions to cultural and
intellectual history and especially to neuropsychology. Certainly biological 
and biochemical pattern-formation are poorly understood, seem counter-intuitive,
and are often modelled in non-linear partial differential equations. But Al
has better reasons that that for bringing them into Presentation 5. Is it
because consciousness arises only in biological systems (so far as anyone can
possibly know), especially in brains,which necessarily have spatial
structure that arose from simpler antecedents? But here we encounter the
problem that Turing-like mechanisms have never really been shown to occur in
biological settings, as Al stresses.  Notice that Al cites the Lotka
(1925) discussion of consciousness, which stresses the
pan-psychic notions of Spinoza (which in turn may have come
from 18th-century European liberal understanding that every educated
Chinese considered "God" to mean "the conscious spirit of the
universe", the world being God's body.) I.e., maybe most spatio-temporal dynamics
is associated with at least some sort of flicker of consciousness. Is
that why we should first understand nonlinear dynamics and pattern formation?
Lotka may have thought so: he disowned fixation on the baroque complications 
of vertebrate brains as necessarily the best way to discover the essence of
consciousness. But most scientists today assume that only brains are aware. So
maybe we have two reasons for putting non-linear dynamics of pattern formation in
Presentation 5. It is especially suggestive that at least one PDE model of chemical 
pattern-formation seems related to another that describes neuroelectric propagation, 
that essential principle of brain dynamics without which verbalizable,
memory-connected consciousness vanishes.  

Comparable reactions and propagations have been found in several kinds
of brain tissue in recent years: these are the slow waves of
calcium-induced calcium release in glial cells (and the like for
"potassium" and for "nitrous oxide" in place of "calcium".) What role
these play in consciousness, if any, no one knows. They are slow on the
scale, not of thought processes, but of mood shifts. Rotors based on this
particular biociemistry have been observed in single heart cells and in 
frog eggs but not yet in brain tissues so far as I know.

Martin Boerlijst (1991, 1993) explored the pertinence of rotors and
spiral waves to speculations about the origin of life in context of
hypercycle theory, with some beautifully provokative illustrations of
possibilities.


Regarding Figure 5 and the attached 'homework problem', see
data, discussion, and the figure originals.


Some references mentioned:

Boerlijst, M.C. and Hogeweg,P. (1991) Spiral wave structure in
pre-biotic evolution, Physica D 48, 17-28; and (same year) in the
Artificial Life II symposium proceedings ed. Chris Langton.

Boerlijst,M.C. et al (1993) Evolutionary Consequences of Spiral Waves,
Proc.Roy.Soc. B 353, 15-18

Castets,V. et al (1990) Experimental Evidence of a Sustained Standing
Turing-Type Nonequilibrium Chemical Pattern, Phys. Rev. Lett. 64,
2953-2956

Lengyel,I. et al (1993) Transient Turing Structures in a
Gradient-Free Closed System, Science 259, 493-495

Ouyang,Q. and Swinney,H.L. (1991) Transition from a Uniform state to
Hexagonal and Striped Turing Patterns, Nature 352, 610-612

Rayleigh, (1921) Proc. Roy. Soc. A  99, 372-384

Ross,J. et al (1995) Experimental Evidence for Turing Structures, J.
Amer. Chem. Soc. 99, 10417-10419