Untangling Complex Systems, page 57
ξ
r
The Emergence of Order in Space
281
TABLE 9.2
Examples of Chemical and Electrochemical Waves Described by Equation [9.59]. The Data
have been Extracted from the References Indicated in the Table
Chemical/Electrochemical
Waves
ka
D
v
Squid giant axon
3 × 103 s−1
3.4 × 102 cm2/s
103 cm/s
(Cross and Hohenberg 1993)
Spreading Depression
8 × 10−2 s−1
8 × 10−5 cm2/s
5 × 10−3 cm/s
(Dahlem and Müller 2004)
Mechano-chemical waves in heart
3 × 102 s−1
0.6 cm2/s
13 cm/s
(Cross and Hohenberg 1993)
Calcium waves in fertilized mammal eggs
1.2 s−1
5.3 × 10−6 cm2/s
5 × 10−3 cm/s
(Whitaker 2006; Donahue and
Abercrombie 1987)
cAMP waves in aggregation stage of
10−2 s−1
4 × 10−6 cm2/s
2 × 10−4 cm/s
Dictyostelium discoideum (Cross and
Hohenberg 1993)
Muskrat’s biological invasion in Europe
0.15 year−1
15 km2/year
3 km/year
(Murray 2002)
BZ reaction
2.25 s−1
2 × 10−5 cm−2 s−1
0.014 cm/s
intuitively, thirty-one years before, by the German chemist Robert Luther (1906) who was a pioneer
in the study of chemical waves.19
In Table 9.2, there are examples of chemical waves whose speed of propagation is calculated by
using equation [9.59].
9.7.3 waves in our brain
It is clear that the brain is an electrochemical organ. Its electrical activity is studied either nonin-
vasively by electrodes placed along the scalp, which record the voltage over time and originate the
electroencephalograms (EEGs), or invasively by intracranial electrodes that collect electrocorti-
cograms (ECoGs). Both techniques have facilitated investigating the patterns of brain activity. It
has been observed that spatially and temporally organized activity among distributed populations
of cells often takes the form of synchronous rhythms. These collective electrical oscillations are
called “brain waves” (Gray 1994). The brain waves are sorted into five categories, the δ-, θ-, α-, β-, and γ- bands, depending on their characteristic frequencies and purposes. The α- (9–14 Hz), β-
(14–40 Hz), and γ- (40–100 Hz) frequency band oscillations all contribute to the neuronal underpin-
nings of attention, working memory and consciousness (Palva and Palva 2007). When the brain is
aroused and actively engaged in mental activities, it generates β waves. When our brain is involved
in particular complex tasks, it generates γ waves. On the other hand, when we have completed a task,
and we sit down to rest, we are often in a α state. Also, when we take time out to reflect or meditate,
we are usually in a α state. If we begin to dream and mentally relax, we are in a θ state (4–8 Hz).
We can maintain this state even when we perform repetitive tasks, and we are mentally disengaged
from them. The final brainwave state is δ (1.5–4 Hz). Deep dreamless sleep takes us down to the
lowest frequencies of the δ state. It is a well-known fact that humans dream in about 90-minute
cycles (Hartmann 1968). When the delta ( δ) brainwave frequencies increase into the frequency of
19 The original paper by Luther has been translated by Arnold et al. (1987) and commented by Showalter and Tyson (1987).
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theta ( θ) brainwaves, active dreaming takes place. Typically, when this occurs there is Rapid Eye
Movement (called REM sleep). The problem that the brain has to solve during sleep is how to inte-
grate memories of experiences that happened during the day with old memories. Scientists know
that waves of electrical activity, referred to as spindles, help to consolidate and integrate memories
during sleep. Spindles are active in the cerebral cortex in the time between dream sleep and deep
sleep. The spindle is a wave that begins in a portion of the cortex close to the ear, spirals through
the cortex towards the top of the back of the head and, then, on the forehead area before circling
back. It propagates with a speed of 2–5 m/s. The spindles strengthen connections among brain cells
in distant parts of the brain. For example, these circular waves of electrical activity may strengthen
the links between cells of the cortex that store memories of the sound with those storing memories
of the sight, et cetera (Muller et al. 2016).
Electrochemical waves are also involved in pathophysiological events, such as Spreading
Depression that is at the basis of a migraine (Dahlem and Müller 2004). A wave of Spreading
Depression propagates when the ionic gradients across the plasma membrane of neuronal cells
break down, leading to a massive efflux of potassium ions into the extracellular space and influx of
sodium, calcium, and chloride ions into the cells. During this phenomenon, single neurons remain
electrically silent, until the proper ionic gradients across the membrane are reestablished. A wave
of Spreading Depression propagates within the different cell layers of the brain with a speed of
3 mm/min, which is 200,000 times slower than the speed of an action potential propagating within
a neuron (Table 9.2).
9.7.4 waves in our hearT
Another beautiful example of the important role played by chemical waves in nature is in our
heart. Our heart (see the left part of Figure 9.25) is a muscle that works continuously and pumps blood through the blood vessels of the circulatory system. Blood carries oxygen and nutrients
and assists the removal of metabolic wastes, such as carbon dioxide. The heart’s beating origi-
nates tiny electrical changes on the skin that are monitored by using electrodes placed on the
patient’s body. The signal that doctors record is called an electrocardiogram (see the right part of
Figure 9.25). Each beat of our heart originates locally, in the cells of the “sinus node” (located in the heart’s right atrium), which are in oscillatory regime, synchronized, and with a period of
roughly 0.8 seconds. Such oscillations arise from self-sustained periodic changes of the intracel-
lular calcium concentration, and, therefore, can be regarded as originated by calcium oscillators
(Nitsan et al. 2016). From the sinus node, an electrical signal spreads across the cells of our
heart’s left and right atria. In fact, outside the sinus node, the cells are excitable, and they transmit
the outgoing waves produced periodically in the sinus-node. The waves cause the atria to contract.
The right atrium has previously collected deoxygenated blood from veins, whereas the left atrium
has previously collected oxygenated blood via the pulmonary veins. The contraction of the atria,
marked by the P wave in the electrocardiogram (cf. Figure 9.25), pumps the blood through the open valves from the atria into both ventricles. The signal arrives at the Atrioventricular node near
the ventricles (Figure 9.25). Here it is slowed to allow our heart’s right and left ventricles to fill with blood. On the electrocardiogram, this step is represented by the horizontal segment between
the P and Q (Figure 9.25). Then, the signal is released and moves downwards, dividing into left and right bundle branches (this step corresponds to the Q wave in the electrocardiogram). As the
signal spreads across the cells of the ventricle walls, both ventricles contract, although not at the
same time. In fact, the left ventricle contracts slightly before than the right ventricle. On the elec-
trocardiogram, the R wave marks the contraction of the left ventricle, whereas the S wave marks
the contraction of the right ventricle. The contraction of the left ventricle pushes oxygenated
blood through the aortic valve to the rest of the body. The contraction of the right ventricle pushes
blood to our lungs for saturating it with oxygen. As the signal passes, the walls of the ventricles
relax and await the next wave. Such step is represented by the T wave on the electrocardiogram.
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Sinus node
R
Right atrium
Left atrium
(blood from the body)
(blood from the lungs)
Atrioventricular
P
T
node
Left ventricle
(blood to the body)
Q S
Right ventricle
0.8 seconds
(blood to the lungs)
FIGURE 9.25 Sketch of the human heart (on the left) and the characteristic profile of an electrocardiogram
for a healthy human heart (on the right).
The entire cycle, from P to T, repeats over and over. As it is shown in Table 9.2, the waves propagate at a speed of 13 cm/s, which is two orders of magnitude smaller than the propagation speed
of the electrochemical waves in a squid giant axon.
A recent study (Nitsan et al. 2016) has demonstrated that the communication among the cardiac
cells is not based only on electrochemical signals. In fact, there is also a mechanical communication
that is essential for converting electrical pacing into synchronized beating. As mechanical coupling
depends on the elastic properties of tissue, disruption of the normal mechanical environment can
impair this interaction. For example, if there exists even a small region with disordered features,
heterogeneities or defects, such as an unexcitable tissue, the excitation waves may be perturbed
in passing through that region, and they may break apart (Glass 2001). When an excitation wave
breaks, it leaves two free ends. These ends tend to curl up into spirals. Spiral waves have a larger
speed (remember the eikonal equation [9.46]). Eventually, the heart tissue will oscillate at a higher
frequency. This effect is thought to cause a heart disorder known as “paroxysmal tachycardia,”
when the frequency of the heartbeat increases by a factor of ten (Cross and Hohenberg 1993).
Another cardiac pathology, fibrillation, is thought to involve an inhomogeneous excitable medium
in which an excitation target wave breaks up to form many spirals. The net result is the presence of
many different structures vibrating asynchronously in a chaotic way.
9.7.5 calcium waves
Chemical waves of Ca+2 are relevant not only for the control of rapid and frequently repeated
responses such as heartbeat, neurotransmitter release, and muscle contraction. They are apparently
ubiquitous. They are present in somatic cells and sex cells. Calcium waves are known to trigger
transformations of the cell cortex (that is the layer of proteins on the inner face of the plasma mem-
brane of the cell) and cytoplasm, as well as to stimulate many enzymatic and metabolic processes
(Whitaker 2006). For example, the activation of eggs by sperm is accompanied by a significant
transient in intracellular calcium concentration. The calcium wave initiates at the point of the sperm
entry and crosses the egg as a tsunami-like wave at a speed of about 5–50 μm/s (Table 9.2 reports the values relative to mammal eggs of 100 μm crossed by calcium waves in ~2 s) up to reach the
antipode of the egg. The large calcium signal triggers a reorganization of the entire egg cortex and
cytoplasm (Sardet et al. 1998). There is evidence that calcium signals are important in all three
relevant stages of embryogenesis: in embryonic axis formation (anterior-posterior, dorsoventral and
left-right axes), in coordinated cell migrations forming tissues (like in gastrulation forming gut and
in neurulation generating the spinal cord), and in organogenesis (once the overall body plan is laid
out, local differentiation gives rise to organs) (Whitaker 2006).
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9.7.6 cAMP waves: The case of dicTyosTelium discoideum
The chemical waves of another compound, 3′,5′-cyclic adenosine monophosphate (cAMP), are
responsible for aggregation and morphogenesis of a simple organism, the slime mold Dictyostelium
discoideum ( Dd) (Dormann et al. 1998). Slime molds live as single amoebae in the upper layers of
the soil and leaf litter in the eastern Northern Hemisphere and eastern Asia. They feed on bacteria
and divide. Starvation, caused by scarce bacteria and a crowded environment, triggers a survival
strategy: the single amoebae aggregate to form a mass of 104–105 cells (see Figure 9.26). In the
aggregated state or mound, the cells cooperate, specialize, and start to differentiate into a num-
ber of different cell types. The mound elongates, falls over and forms a cylindrically shaped slug
(after ~15 h). The slug has a distinct polarity with a tip at the anterior end that guides its movement.
Food finding is enhanced in the slug that acts as an interacting assembly of sensors and effectors,
gathering and analyzing more information about the world than could a single amoeba. These sen-
sory integration systems transduce physical and chemical signals into social cues, which amplify
or attenuate group responses. The slug has a photo- and thermo-tactic power and can migrate to the
surface of the soil. Finally, on the soil, the slug transforms into a fruiting body (up to 4 mm high)
consisting of a stalk supporting a spore mass. The spores that are dormant cells, under suitable
conditions, germinate into new amoebae and the cycle, requiring 24 h at room temperature, can
start again.
The developmental program of Dd is controlled by the interplay between the shape and dynam-
ics of cAMP waves and the cAMP chemotactic cell movement of the cells (Dormann and Weijer
2001). In the aggregation stage, it has been demonstrated that the chemotactic power of an amoeba
grounds on a transmembrane cAMP receptor that can assume two states: an active state promoting
the intracellular synthesis of cAMP and working at low extracellular [cAMP], and a desensitized
state working when the extracellular [cAMP] is very high and inhibiting a further production of
cAMP (Halloy et al. 1998). When stimulated with cAMP, Dd cells, suffering starvation, respond by
synthesizing and secreting more cAMP, which results in non-dissipating cAMP waves. Such waves,
propagating at a speed of ~102–103 μm/min (cf. Table 9.2), guide the aggregation of the individual amoebae (Gregor et al. 2010). It has been shown (Dormann and Weijer 2001) that cAMP waves are
also involved in the slug’s movement. The waves are initiated periodically in the anterior part of the
slug tip and propagate backward at a speed of ~30 μm/min. They reflect the coordinated periodic
Amoebae
Aggregation
duction Spores
Repro
Aggregation
streams (6 h)
Aggregation
Fruiting
body (24 h)
Differentiation
Mound (9 h)
Differentiation
Differentiation
Differentiation
Early
culminate (20 h)
Slug (15 h)
Tipped
mound (12 h)
FIGURE 9.26 Dictyostelium discoideum ( Dd) life cycle.
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285
movement behavior of the cells in the slug. Moreover, also orange light stimulates the slug’s tip to
release cAMP, which may play a role in its photo-tactic response (Miura and Siegert 2000). In its
final metamorphosis, the cells at the slug’s tip migrate down through the center of the aggregate and
initiate stalk formation. They die and transform in sturdy cellulose walls to hold up the spherical
ball full of spores (Figure 9.26). The final fruiting body consists of about 20% stalk cell and 80%
spore cells. This social stage is remarkable because it involves altruism: the stalk cells die to sup-
port the spore cells. It is vulnerable to cheaters. Cheating is a social action that takes place in the
context of cooperative acts that the cheaters somehow violate. In Dd, the expected social contract
is that the frequency of each genetically distinct clone among the spores will be the same as it was
in the original mixture of aggregated cells. The same should be true in the stalk tissue. If this is not
the case, the dominant clone has cheated the minority clone by getting more than its fair share into
spores. Therefore, it is evident that this property makes Dd a great model even for social evolution.
However, it is worth stating that it is not possible to attribute any conscious awareness to cheating
in Dd that lacks any nervous system. In humans, it is different because cheating is value-based and
assumes a certain awareness of the moral grounds of actions (Strassmann and Queller 2011).
9.7.7 sPreading of sPecies, ePidemics and … fads
