Natural Patterns
Lectures 5 & 6 : waves
Periodic oscillations, such
as those produced by a clock or by a pendulum can be thought
of as defining a rhythm
or a temporalpattern.
In general, such oscillations will not last forever:
the pendulum will stop due to friction;
the spring of a mechanical clock will have to be rewound,
or the battery of an electric watch
will eventually need to be replaced. The temporal patterns
we will be concerned with are those
observed in systems far from equilibrium.
Such systems
are driven away from their equilibrium
state by some external forcing. In fact, most of the
patterns we discussed so far appear
in systems far from equilibrium. For instance, sand ripples
appear because the sand is subjected
to the force of the wind. Convection cells are created
in a thin layer of oil at the bottom
of a frying pan because heat is provided to the pan by the
stove underneath.
These two lectures are devoted to
oscillatory systems driven far from equilibrium. These are
important to study, in particular
because they can exhibit rich dynamical behaviors, such as chaos
or space-time disorder. We first
start with a few examples of oscillatory systems and then discuss
spatial effects, which are typical
of oscillatory pattern forming systems.
Periodic oscillations in chemical systems
It took many years for scientists to
accept the idea that oscillatory chemical
reactions not only
exist but also play an important
role in our understanding of essential phenomena such as heart
failure. The Belousov-Zhabotinsky
reaction is a striking example of an oscillatory chemical reaction.
Periodic oscillations are sustained
due to a feedback process in which the
reaction produces its
own catalyst. These oscillations
do not last forever of course since the system eventually runs
out of reactants, unless the latter
are provided at a regular rate.
Excitable media
Temporal oscillations also occur in
excitable
media. To understand the concept of excitability,
consider a forest fire. We start
from a forest with fully grown trees. If the forest burns, trees will
be destroyed. Fortunately, new
trees will soon start growing and after many years the forest will
be more or less back to its original
state. It will then have undergone a full cycle. If a new fire is
set, this pattern will repeat itself
and we will then have periodic oscillations of the forest between
a burnt state and a "recovered"
state in which all of the trees are fully grown again. Such oscillations
may appear similar to those of
a pendulum or of a clock, except that:
-
they would not exist in the absence
of a fire
-
they would not start unless the initial
fire is big enough to make the forest burn
-
once oscillations are triggered by
a fire, only one cycle is observed, after which the forest
comes back to its original state
by itself
The following properties are characteristic
of an excitable system:
-
the system only responds to excitations
of large enough size
-
when there is a response, it is big
(i.e. has dramatic or at least noticeable effects)
-
the system eventually comes back to
its resting state
Other examples of excitable systems
are the heart muscle (see below) and nerve axons.
Spatial effects: waves
At modern sporting events, supporters
of one team sometimes demonstrate their enthusiasm
by conspiring to create a "human
wave" in the bleachers. Assume that such a wave goes around
the stadium a couple of times and
think of the motion of each supporter, as a function of time.
The fan stands up and sits down
in a periodic fashion, and therefore follows a rhythm.
This simple phenomenon illustrates
that a spatially extended system
which exhibits temporal
oscillations can sustain traveling
waves. The human wave is a one-dimensional
phenomenon.
In two spatial dimensions, traveling
waves can take the form of target patterns
or of spiral waves.
In particular, target patterns and
spiral waves are observed in chemical reactions, as illustrated in the
following web sites:
Moreover, the dynamics of such objects
can be reproduced by numerical simulations of generic
models
of reaction-diffusion equations.
Rhythms of life
Breathing, circadian rhythms, or the
beating of our hearts are natural rhythms of our lives. The
heart is a muscle whose tissue
acts as an excitable medium. Contraction
of this muscle, leading
to the pumping of blood, is triggered
by an electric wave which propagates across the heart. This
wave is sent by the sinoatrial
node, which acts as a pacemaker, thereby creating a target pattern.
Ventricular fibrillation
occurs when different parts of the heart lose their synchrony. This phenomenon
is due to the appearance of three-dimensional
spiral waves across the muscle, which first
impose a
faster rhythm on the heart (this
corresponds to cardiac arrhythmia) and then lead to uncoordinated
electrical activity called ventricular
fibrillation. This condition quickly provokes sudden cardiac
death if not arrested. Numerical
simulations and models
confirm this description of ventricular
fibrillation.
Waves in bacterial colonies
Wave patterns are also observed in
colonies
of bacteria. These single-cell organisms often
communicate
by emitting some chemical, called
a chemoattractant.
Other bacteria can then move in the direction
in which the concentration of this
chemoattractant increases the fastest, thereby leading to cell
aggregation. This phenomenon is
called chemotaxis.
In some way, this is a mechanism to make a
group of unicellular organisms
behave collectively. A striking example of pattern formation linked
to the emission of a chemoattractant
occurs in colonies of the slime mold Dictyostelium discoideum.
When conditions (supply of nutrients,
moisture, ...) become critical, these unicellular organisms
gather into a mound, which then
develops a "fruiting body". The latter will disperse spores which
can survive in harsh conditions.
At the beginning of this aggregation process, cells emit chemoattractants
which lead to the appearance of
target
and spiral waves in the colony.
Bioconvection
provides another example of wave patterns. The pattern photographed on
the right is due to bioconvection of a bacterium called Bacillus subtilis.
This pattern is a one-dimensional wave pattern, which appears in the meniscus
created by the agar plate shown on the figure. The insert shows how the
pattern intensity varies along the white line drawn on the figure. |
From N. Mendelson & J.Lega, J. Bact. 180,
3285-3294 (1998).
|
Back to UNVR
195 A