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Untangling Complex Systems, page 93
Engine (based on the Fuzzy rules), and a Defuzzifier (based on the Fuzzy sets of the output variable). Graph b,
on the right, shows the Fuzzy rules that, like patches, cover and describe the non-linear relation (represented
by the winding black curve) between input and output variables.
rules that involve the Fuzzy sets activated by the crisp input values. Finally, the Defuzzifier is based
on the Fuzzy sets of the output variables, and it transforms the collection of the output Fuzzy sets,
activated by the rules, in crisp output values (see graph a of Figure 13.18). A complete FLS is an inferential tool to make predictions and take decisions about the nonlinear relation it refers to.
There are two main strategies to develop artificial intelligence: one is writing human-like intel-
ligent programs running on computers or special-purpose hardware, and the other is neuromorphic
engineering. For the first strategy, computer scientists are writing algorithms that can learn, ana-
lyze extensive data, and recognize patterns. At the same time, psychologists, biologists, and social
scientists are giving information on human sensations, emotions, and intuitions. The merger of the
two contributions provides algorithms that can easily communicate with us. Among the most prom-
ising algorithms, there are the artificial neural networks (remember Chapter 10, when we learned
these algorithms for predicting chaotic time series) (Castelvecchi 2016). Recently, a program called
AlphaGo Zero, based on an artificial neural network that learns through trial and error, has mas-
tered the game of Go without any human data or guidance, and it has outperformed the skills of the
best human players (Silver et al., 2017).
The second strategy for implementing Artificial Intelligence is neuromorphic engineering (the
term was coined by California Institute of Technology electrical engineer Carver Mead in the
1980s). It implements surrogates of neurons through non-biological systems either for neuro-
prosthesis (Donoghue 2008) or to devise brain-like computing machines. Brain-like computing
machines will exhibit the peculiar performances of human intelligence, such as learning, recog-
nizing variable patterns, and computing with words as some programs have commenced doing.
However, it is expected that brain-like computers will have the advantage of requiring much less
power and occupying much less space than our best electronic supercomputers. Surrogates of neu-
rons for brain-like computers can be implemented by using different strategies. A first strategy
exploits conventional passive and active circuit elements, either analog (Mead 1989; Service 2014)
or digital (Merolla et al. 2014). A second one grounds on circuits made of two-terminal devices
with multiply-valued internal states that can be tuned in non-volatile or quasi-stable manners,
keeping track of the devices’ past dynamics (Ha and Ramanathan 2011; Di Ventra and Pershin
2013). A third approach uses phase-change devices wherein the actual membrane potential of the
artificial neuron is stored in the form of the phase configuration of a chalcogenide-based mate-
rial, which can undergo phase transition on a nanosecond timescale and at the nanometric level
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Untangling Complex Systems
(Tuma et al. 2016). Finally, a fourth methodology trusts in nonlinear chemical systems that can
communicate through either continuous mass exchange (Marek and Stuchl 1975) or electro-
chemical linkages (Hohmann et al. 1999; Zlotnik et al. 2016) or mechanical/light-induced pulsed
release of chemicals (Taylor et al. 2015; Gentili et al. 2012; Horvath et al. 2012) or UV-visible
radiation (Gentili et al. 2017a).
Artificial Intelligence is suitable to fulfill a long-cherished aspiration of humanity: that of design-
ing machines that can help humans in both manual and mental activities, at the same time. Such
devices, called robots, are programmable and potentially able to carry out many actions peculiar to
humans (Russell and Norvig 1995). In fact, robots are usually able to perceive signals through sen-
sors, plan activities through internal cognitive processes, and act through actuators, as the human
nervous system does through its sensory cells, brain, and effector system. Traditionally, robots are
made of electronic circuits, computer software, rigid mechanical parts and electric motors, and are
designed primarily for accomplishing dangerous tasks, such as bombs disposal, deep ocean, and
planetary exploration. Recently, the idea of developing robots grounded on wetware rather than on
hardware is taking shape (Grančič and Štěpánek 2011; Hess and Ross 2017). A “Soft Robot,” also
called a “Chemical Robot,” is thought as a molecular assembly that reacts autonomously to its envi-
ronment through molecular sensors, makes decisions by its intrinsic chemical artificial neural net-
works, and performs actions upon its environment through molecular effectors (see Figure 13.19).
Of course, it needs fuels to conduct its operations. Therefore, it is necessary to have a metabolic
system inside of it. Chemical Robots could be easily miniaturized and implanted in living beings
to interplay with cells or organelles. They are expected to play as auxiliary elements of the immune
systems for biomedical applications (Hagiya et al. 2014).
13.3.1.8 Protein Computing
In Chapter 12, we have seen that the signaling network of a cell works as the “cellular brain.”
The neurons of the “cellular brain” are proteins. Many thousands of proteins, functionally con-
nected to each other, carry information from the membrane to the genetic network or directly to
the metabolic network (Bray 1995). Such system of interacting proteins acts as a neural network;
the information of extracellular stimuli is collected by receptor proteins and transduced in the
activities and concentrations of the inner cell proteins. The activity of any protein depends on
its structure. Originally, it was taught that the interaction between proteins and substrates could
be described only by the lock and key paradigm (Conrad 1992). Each type of enzyme works as a
specific key, and its substrate is the lock. The fitting of the key to the lock is a sophisticated pat-
tern recognition operation. The nonlinear process of recognition trusts in the three-dimensional
structure of the protein. Nowadays, we are aware that many proteins or portions of proteins are
structurally disordered in their native, functional states (Tompa 2010). This situation means that,
quite often, a protein cannot be assumed to be just one rigid key. Instead, it might be described as a
Chemical robot
Environment
Molecular
Molecular
sensors
Artificial
effectors
Environment
neural
networks
Metabolic unit
Fuel
Waste
FIGURE 13.19 Essential modules for the design of a Chemical Robot: (I) Molecular sensors; (II) Chemical
Artificial Neural Networks; (III) Molecular Effectors, and (IV) a Metabolic Unit.
How to Untangle Complex Systems?
477
pretty flexible key because the macromolecule is characterized by a dynamic disorder. Otherwise,
it can be conceived as a bunch of keys because it exists as an ensemble of conformations. Static and
dynamic structural disorder provides computational power to proteins because different conforma-
tions can work in parallel.
Some proteins are multifunctional. They exhibit phenomena of moonlighting. In analogy to
moonlighting people who have multiple jobs, moonlighting proteins perform multiple autonomous,
often unrelated, functions (Huberts and van der Klei 2010). An example of moonlighting protein
is phosphoglucose isomerase (PGI), a protein that is both a cytoplasmic enzyme in glycolysis and
an extracellular cytokine and growth factor. In other words, it can participate in both metabolic and
signaling cellular events. In PGI, the enzyme active site is made of loops and helices from many
parts of the protein, so that the active site domain is not separate from its receptor-binding domain
(Jeffery 2014). At present, it is still difficult to assess how abundant moonlighting proteins are.
However, moonlighting is a phenomenon that illustrates nature’s ingenuity, and it can be exploited
for computational reasons.
13.3.1.9 Amorphous Computing
In Chapter 9, we were stunned to investigate morphogenetic processes. For example, it is astonishing to observe how the cells in an embryo self-organize to form an organism. Under the point of
view of Natural Computing, an embryo is a collection of cells that compute in parallel, based on a
single program written in their DNA, and produces a globally coherent result.
The effort of mimicking natural morphogenetic processes for computational reasons has
been named as Amorphous Computing by a team working at MIT in 1996 (Abelson et al. 2000).
An amorphous computing system is an extensive collection of identically-programmed computa-
tional particles, sprinkled irregularly on a surface or mixed in a volume. They interact only locally
because they communicate within a fixed distance, which is large compared to the dimension of the
elements but small compared to the size of the entire system. The single elements compute asyn-
chronously. The challenge of the amorphous computing is to write programs that when executed
on each agent give rise to self-organization phenomena. The most promising way for implementing
amorphous computing is by using cells. The goal is to organize cells into precise patterns that func-
tion as actuators or transform them in programmable delivery vehicles for pharmaceuticals or in
chemical factories for the secretion of specific chemical components.
13.3.1.10 Building Models of Complex Systems: ODEs, Boolean Networks,
and Fuzzy Cognitive Maps
Any Complex System involves many molecular components. Its behavior is intractable from the
computational point of view. In fact, we cannot use the principles of quantum physics to predict the
phenomenon accurately. Therefore, we need to develop a model. Such a model must have the struc-
ture of a network. The construction of a model is a multi-stage procedure as demonstrated by Le
Novère (2015) for molecular and gene networks. The first stage requires the collection of data that
are representative of the phenomenon we want to describe. The next step is to infer from the experi-
mental data the biological entities that must be included in the model. The difficulty is to understand
the components that are fundamental to the process we want to represent. The third stage consists in
defining the interactions between the selected components, specifying the directionality and prop-
erties of the relationships. The model is complete by making a list of the constraints that represent
the context of the analysis. Essential constraints are the initial conditions; they usually ground on
conservation laws. After building the model, we need to test how it evolves, and if it reproduces
the experimental evidence. For running the model, we can choose different methodologies. One
methodology consists in writing the Ordinary Differential Equations (ODEs) for each component
of the network and in solving the ODEs by numerical integration (read Appendices A and E for
more details). Of course, we need to know the values of the parameters appearing in the equa-
tions (sometimes the values of the parameters are inferred by fitting the experimental time series).
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Untangling Complex Systems
Alternatively, we may use Boolean network models. In a Boolean network, a variable can take two
states (ON or OFF, +1 or −1) and the edges are causal connections. An excitatory or positive (+1)
edge from node A to node B means A causally increases B from −1 to +1. An inhibitory or negative
(−1) edge from node A to B means A causally decreases B from +1 to −1.
TRY EXERCISES 13.2 AND 13.3
In certain situations, the use of discrete variables can be a crude approximation. In these cases, it is
worth considering Fuzzy logic models. The electrical engineer Bart Kosko (1986) proposed Fuzzy
Cognitive Maps (FCMs) for representing causal reasoning. The idea came after the political scientist
Robert Axelrod (1976) introduced cognitive maps for representing social scientific knowledge. Cognitive
maps are Boolean networks where nodes are variable concepts (like social stability), and edges are causal
relationships among them. Since causality admits of degrees, and degrees are vague if expressed through
words, Kosko had the brilliant idea of introducing continuous values of causality and variables. Since
their invention, FCMs have been used in many fields for dealing with Complex Systems (Glykas 2010). In
a FCM, each edge is associated with a number (i.e., weight), included in the [−1, +1] interval, which quan-
tifies the degree of causal relationship. Values of −1 and +1 represent entirely negative and positive causal
effects, respectively. A value of 0 denotes no causal effect. Traditionally, the values of the weights are
fixed by experts, and they quantify the strength of the relationships, which often are expressed through
natural language fuzzy terms, such as “strong,” “medium,” “weak.” More recently, the weights are fixed
through learning algorithms, such as artificial neural networks and genetic algorithms.
Any FCM is described by a square matrix, called “connection matrix,” which stores all the
weight values for edges between the variables defining rows and columns. A system with N nodes
will be represented by N × N connection matrix C. A FCM is useful for predicting the dynamics of the Complex System it models. We start from a vector N ×1 of initial values for all the variables:
V( t). Usually, they range in the normalized interval [0, 1]. Then, we select a function f ( x) that transforms the product of the connection matrix and the initial values vector, C × V( t), in a vector of values for the nodes at an instant later, V( t + )
1 = f ( C × V( t)). Three most commonly used transfor-
mation functions are (Stach et al. 2005):
1. The Heaviside function:
,
0 x ≤ 0
f ( x) =
[13.4]
,
1 x > 0
2. The trivalent function:
− ,
1 x ≤ −0.5
f ( x) = ,
0 −0.5 < x < 0.5 [13.5]
+ ,1 x ≥ 0.
5
3. The logistic function:
1
f ( x) =
[13.6]
1+ e− Dx
where D is a parameter that determines the shape of f ( x).
The possible dynamic scenarios that can be achieved by FCMs depend on the transformation
function that is chosen. If we select a discrete transformation function, such as (1), the dynamics can
evolve to a fixed point or a limit cycle. If, on the other hand, we select a continuous transformation
function, such as (3), even chaotic attractors can be found.
How to Untangle Complex Systems?
479
13.3.1.11 Agent-Based Modeling
Some biological processes having living beings as protagonists require agent-based modeling to
be understood and reproduced. An agent is an autonomous individual element that exhibits the
capacity to achieve a goal or affect an outcome. Often, an agent operates within a population and
interacts with other agents as well as a more passive environment. Many homogeneous agents,
having limited individual capabilities, acting in parallel, without a centralized control, can exhibit
a collective intelligent behavior called “Swarm Intelligence,” which is better than the intelligence
of the individuals. Examples of swarm intelligence are fish schooling and birds flocking, which
we read in Chapter 12. Other examples are either ants that find the shortest path between their
nest and a good source of food, bees that find the best source of nectar within the range of their
hive, or termites that build their outstanding nests. Social insects, such as ants, termites, and bees
communicate through stigmergy. The term stigmergy was introduced by the French zoologist
Pierre-Paul Grassé (1959) from the Greek words στίγμα (“sign”) and έργον (“work,” “action”).
It refers to the notion that the action of a social insect leaves signs in the environment, signs that
it and other insects sense, and that determine and incite their subsequent actions. For example,
ants look for food sources by wandering randomly and laying pheromones in their trails. When
an ant finds a suitable food source, it returns to its nest. Other ants setting out to seek food sense
the pheromone laid down by their precursor, and this influences the path they take up. The col-
lective behavior that emerges is autocatalytic (i.e., a behavior controlled by positive feedback)
because the more the ants follow a trail, the more attractive that trail becomes for being followed.
