Untangling Complex Systems, page 87
one channel, the signals coming from the Red cones are summed (OR operator) to those coming
from the Green cones to compute the intensity of the light stimulus. In a second channel, the signals
from the Red and Green cones are subtracted (NOT operator) to compute the red-green component
of a stimulus. Finally, in a third channel, the sum of the Red and Green cone signals is subtracted
from the Blue cone signals to compute the blue-yellow variation in a stimulus. The signals in these
three channels are transmitted in distinct pathways. Whereas a bipolar cell encodes the visual infor-
mation in the value of a graded potential, which is analog and not discrete, a ganglion cell encodes
the information in the analog value of the frequency of firing action potentials. The information
encoded through action potentials is relayed more safely over long distances and can reach the
visual cortex (located in the back of the head). The risk of losing information by noise is minimized.
The visual cortex is partitioned into different areas: V1, V2, V3, V4, and V5. Each area is
divided into compartments. Recent studies (Johnson et al. 2001) reveals that the function of the
compartments is not unique, but Fuzzy. For example, the analysis of luminance and color is not
separated, but there is a continuum of cells, varying from cells that respond only to luminance, to
a few cells that do not respond to luminance at all. Other investigations (Friedmann et al. 2003)
Complex Systems
443
have revealed that in areas V1 and V2, orientation and color selectivity are not binary measures;
they are Fuzzy. In fact, cells vary continuously in their degree of tuning, and it is possible to
assign a membership function for orientation and another for color perception to each cell. The
extraction of information in the visual cortex is carried out with the same mechanism observed in
the neurons of the retina, i.e., by granulation of neurons in Fuzzy sets that integrate and abstract
information.
Based on this description, it might seem that human vision is deterministic, reproducible, objec-
tive, and universal. But this is not the case because human vision, like any other sensory perception,
depends on the physiological state of the perceiver, his/her past experiences, and each sensory sys-
tem is unique and not universal. Moreover, every human brain must deal with the uncertainty in the
perception. Under uncertainty, an efficient way of performing tasks is to represent knowledge with
probability distributions and acquire new knowledge by following the rules of probabilistic infer-
ence (Van Horn 2003; De Finetti et al. 1992). This consideration leads to the idea that the human
brain performs probabilistic reasoning, and human perception can be described as a subjective pro-
cess of Bayesian Probabilistic Inference (Ma et al. 2008; Mach 1984). The perception of a stimulus
S by a collection of cortical cells X will be, then, given by the posterior probability p( SM X ): M
M
M
p( X
) ( )
M SM p S
p( S
M
)
M X M =
p( X )
[12.39]
M
In equation [12.39], p( S )
(
)
( )
M is the prior probability, p X M SM is the likelihood, and p XM is the
plausibility. The plausibility is only a normalization factor. In agreement with the theory of Bayesian
probabilistic inference generalized in Fuzzy context (Coletti and Scozzafava 2004), the likelihood
may be identified with the deterministic Fuzzy information described earlier. The prior probability
comes from the knowledge of the regularities of the stimuli and represents the influence of the
brain on human perception. Human perception is a trade-off between the likelihood and the prior
probability (Kersten et al. 2004). The likelihood represents the deterministic and objective part of
the human perception. The prior probability represents its subjective contribution. In fact, it is rea-
sonable to assume that all the possible patterns of activity of cortical neurons of a specific area are
granulated in Fuzzy sets, whose number and shape depend on the context. Moreover, such Fuzzy
sets are labeled by distinct adjectives within our brain. The noisier and ambiguous are the features
of a stimulus, the more prior probability driven will be the perception, and the less reproducible and
universal will be the sensation.
Sometimes, we receive multimodal stimuli that interact with more than one sensory system.
Each activated sensory system produces its own mono-sensory Fuzzy information. Physiological
and behavioral experiments have shown that the brain integrates the mono-sensory perceptions
(Ernst and Banks 2002) to generate the final sensation. There are brain areas, such as the Superior
Colliculus (Stein and Meredith 1993), which contain distinct mono-sensory, as well as multisensory
neurons. Neurophysiological data have revealed influences among unimodal and multimodal brain
areas, as well (Driver and Spense 2000). Multisensory processing pieces signals of different modal-
ity if stimuli fall on the same or adjacent receptive fields (according to the “spatial rule”) and within
close temporal proximity (according to the “temporal rule”). Finally, multisensory processing forms
a total picture that differs from the sum of its unimodal contributions; a phenomenon called multi-
sensory enhancement in neuroscience (Stein and Meredith 1993; Deneve and Pouget 2004), or colli-
gation in the Information theory (Kåhre 2002). The principle of inverse effectiveness states that the
multisensory enhancement is stronger for weaker stimuli. Since sensory modalities are not equally
reliable, and their reliability can change with context, multisensory integration involves statistical
issues, and it is often assumed to be a Bayesian probabilistic inference (Pouget et al. 2002).
From this paragraph, we understand that human vision is extraordinarily complex. Its complexity
is magnified by the uniqueness of each sensory system, the dependence of the sensory action on the
444
Untangling Complex Systems
physiological state of the perceiver, and his/her past experiences. Probably, the human power of recogniz-
ing variable patterns derives from the granulations of neurons and their patterns of activity in Fuzzy sets.
12.4.3 emergenT ProPerTies
The Complexity of a natural system can be estimated by the degree of difficulty in predicting its
properties when the features of the system’s parts are given. In fact, any Complex System is a net-
work that exhibits one or more collective properties that are said emergent because they come to
light, as a whole.26
Complexity ( C) derives from a combination of three features: Multiplicity ( M), Interconnection
( Ic) and Integration ( Ig) (Lehn 2013):
C ∝ M Ic Ig [12.40]
Many and often diverse nodes (Multiplicity), which are strongly interconnected (Interconnection),
exhibit emergent properties because they integrate their features (Integration). This statement is
valid especially when the systems are in out-of-equilibrium conditions. The symbol © expresses a
peculiar combination of the three parameters: M, Ic, and Ig. Complexity and emergent properties can be encountered along a hierarchy of levels: for instance, at the molecular, supramolecular, and
cellular levels. But also passing from cells to tissues, from tissues to organisms, and from organisms
to societies and ecosystems. At each level, novel properties emerge that do not exist at lower levels.
In the history of philosophy and science, different taxonomies of the emergent properties have
been proposed (Corning 2002; Clayton and Davies 2006; Bar-Yam 2004). Here, I offer a new tax-
onomy based on the structures of the networks.
1.
Regular and random networks show emergent properties in both equilibrium and out-of-
equilibrium conditions. 27 At equilibrium, the emergent properties are not affected signifi-
cantly by the removal or the addition of a few nodes. Examples are the phases of matter
(such as solids, liquids, and gases). Other cases are the pressure and temperature of a
macroscopic phase.
In out-of-equilibrium conditions, examples of emergent properties of regular and ran-
dom networks are all those phenomena of self-organization, in time and space, which we
have studied in the previous chapters of this book. For instance, the predator and prey
dynamics, the optimal price of a good in a free market,28 oscillating reactions, chemi-
cal waves, Turing structures, periodic precipitations, convection in fluids, et cetera. These
kinds of emergent properties are sensitive to the removal or addition of a few nodes. In
fact, large out-of-equilibrium systems can self-organize into highly interactive, critical
states where minor perturbations may lead to events, called avalanches, spanning all sizes,
up to that of the entire system. This feature is described by the Sandpile model (Bak and
Paczuski 1995). Let us consider a pile of sand on a table, where sand is added slowly.
When the pile becomes steep, it reaches a stationary state. In fact, the amount of sand
26 The word “emergence” comes from the Latin verb emergere that is composed by the preposition ex that means “out” and the verb mergere that means “to dip, to immerse, to sink.”
27 Machines, software, and sentences in natural language can be described as instances of regular networks exhibiting emergent properties as a whole. Their emergent properties depend on the delicate structure of the network and are
strongly affected by the removal of just one or more nodes. The nodes, with known properties, work together predictably, and the whole does or means (if we refer to sentences) what is designed to do or mean.
28 Adam Smith used the term “invisible hand” to describe the emergent property of trade and market exchange, which spontaneously channel self-interest toward socially desirable ends.
Complex Systems
445
added is balanced, on average, by the amount of sand leaving the system along the edges
of the table. This stationary state is critical. In fact, a single grain of sand might cause an
avalanche involving the entire pile. Natural Complex Systems exhibit a kind of emergent
property called Self-Organized Criticality (SOC): they have periods of stasis interrupted
by intermittent bursts of activity. Since Complex Systems are noisy, the occurrence of a
burst of action cannot be predicted, and it is hard to be reproduced. At most, what is pre-
dictable and reproducible is the statistical distribution of these avalanches, which usually
follows a power-law. Thus, if an experiment is repeated, with slightly different random
noise, the resulting outcome is entirely different.
Other examples of emergent properties of regular and random networks are the fish
schooling and birds flocking, on the one hand, and phase transitions, on the other. A school
of fish or a flock of birds exhibits an emergent collective and decentralized intelligent behav-
ior that is called swarm intelligence (Reynolds 1987; Bonabeau et al. 1999). It has been
demonstrated by simulation that the collective behavior derives from the interactions among
the individuals that follow a few simple rules based on local information. For instance, the
behavior of a flock of birds can be reproduced by assigning three simple rules to every
agent. The first is the alignment rule: a bird looks at its neighbor birds and assumes a veloc-
ity (regarding module, direction and versus) that is close to the mean speed of its nearest
group mates. The second is the cohere rule: after alignment, a bird takes a small step in the
direction that the center of mass of birds takes. The third is the separate rule: a bird should
always avoid any collision. This kind of behavior reminds that of matter at any phase tran-
sition, called criticality among statistical physicists (Christensen and Moloney 2005). The
phase transition occurs when an external agent finely tunes specific external parameters to
particular values. At a phase transition, the many microscopic constituents of material give
rise to macroscopic phenomena that can be understood by considering the forces exerting
among the single particles. Whereas the actions of living beings, in a flock of birds or school
of fish, are information-based, those of particles in a material are force-driven.
TRY EXERCISE 12.11
2. Social insect colonies have features of Scale-free networks. The connections among nest-
mates are nonrandomly distributed for most colony functions (Fewell 2003). A few key
individuals disseminate information to many more nestmates than do others; they play like
hubs. The most obvious hub is the queen: she does not centrally control all the colony func-
tions but, in honeybees, she secretes a pheromone that represses reproduction in workers
and maintains colony cohesion. Essential hubs are also present within worker task groups:
they are the scouts or dancers. Such vital individuals communicate most of the information
about resource location and availability and maintain the cohesion of the group that goes
out to forage. The removal of hubs can disrupt the system severely, whereas the loss of any
of the vast majority of workers would have little effect.
3. In modular networks, each module can have an emergent property, and their cooperative
action gives rise to one or more synergistic effects. Examples are the symbiotic relation-
ships that can be encountered in ecology. For instance, more than 1.2 billion years ago, a
cyanobacterium took up residence within a eukaryotic host (Gould 2012). This event gave
rise to algae that contain a photosynthetic organelle (plastid), which is remnant of the cya-
nobacterium and mitochondria, which are organelles derived from the integration of other
prokaryotes early in eukaryotic evolution. The endosymbionts optimize both the respira-
tion and the photosynthesis by synergy. “The whole is something over and above its parts,
and not just the sum of them all…” as alleged, more than 2000 years ago, by Aristotle in its
philosophical treatise titled Metaphysics. In the presence of synergistic effects, 2 + 2 does
not make 4, but more.
446
Untangling Complex Systems
4. In a hierarchical network, each level has an emergent property. Examples of hierarchical
networks are the living beings. Life is the emergent property of the network as a whole
(Goldenfeld and Woese 2011). A living being’s isolated molecular constituents, such as
phospholipids, water, salts, DNA, RNA, proteins, and so on, can never show life. Only, if
we consider all the constituents organized in the dynamic hierarchical structure of a cell,
we can observe the fantastic phenomenon of life. In a hierarchical network, we have both
upward and downward causation. Upward causation is when the features of lower levels
rule the emergent properties of the higher levels. Downward causation is the opposite.
The properties of higher levels influence those of the lower levels. I report a few examples
(Noble 2006). A mother and her environment transmit to the genes of her embryo adverse
or favorable influences. The heart of an athlete shows different gene expression patterns
from those of a sedentary person. Hormones released by endocrine glands and circulating
in the blood system can influence events inside the cells. The act of sexual reproduction
ends in fertilizing an egg cell. And so on. It is the highly dynamic, heterogeneous, orga-
nized, fractal-like, structure of chromatin (see Chapter 11) that marks the intersection of
upward and downward causation (Davies 2012). In fact, its structure and behavior are
influenced by both the genes it contains and the macroscopic forces acting on it from
the rest of the cell and the cell’s environment. The possibility of having both upward and
downward causation gives living beings the power of influencing their environment, but
also of adapting to it. Living beings and their societies are Complex Adaptive Systems
(Miller and Page 2007).
12.5 KEY QUESTIONS
• Make examples of Complex Systems.
• Which are the Natural Complexity Challenges?
• Present the challenges in the field of Computational Complexity.
• What is the link between Natural Complexity and Computational Complexity?
• What is the fundamental and unavoidable limit we will always encounter in the description
of Complex Systems even if it were demonstrated that NP = P?
• Which are the essential properties of Complex Systems?
• Which are the parameters that characterize the structure of a network?
• Present the features of the model networks.
• Which are the most common network models in nature and why?
• Which are the essential factors maintaining the Earth out-of-equilibrium?
• Describe the thermodynamic properties of thermal radiation.
