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: The following properties are characteristic of an excitable system: 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).



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