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Introduction
We could start writing this book by saying, with several other authors, that
the brain is the most powerful and complex information processing device
known, whether naturally developed or created artificially. Although we
fully agree with this statement, in doing so we would be misleading the
reader, in the sense that the present book basically aims to formalize the
knowledge concerning brain physiology accumulated over the past few
decades. Instead of merely describing the complexity of the cerebral struc-
ture or presenting a collection of commentaries and reviews of interesting
experimental results, we take into account novel achievements in quantum
information and quantum computation, and avail ourselves of recently de-
veloped mathematical tools.
Neuroscience was born in the 19
th
century with the works of Paul Broc-
ca. However, this fledgling field experienced a boom only in recent times,
following the development of powerful non-invasive techniques for prob-
ing the neural circuitry supporting the complex cognitive functions of the
human brain. Although sophisticated mathematical models and physical
theories are the basic tools behind the conceptual foundations and analyti-
cal implementation of these modern techniques, to the best of our
knowledge no effort was made to formalize the actual knowledge about
brain function into a coherent theoretical framework incorporating the re-
cent developments in mathematical and physical science. Addressing this
lack was our first motivation in writing this book. The adequacy of Fuzzy
Sets and Fuzzy Logic for such a purpose was realized by us in the pioneer-
ing days when these powerful ideas were first introduced by Prof. Lotfy
Zadeh. Since the very beginning we chose Fuzzy Formal Language theory
(FFL) as the fundamental formalism to implement this task. According to
modern neuroscience, chains of chemical transactions, called Signal
Transduction Pathways (stp), are the basis for explaining brain function. In
our approach, fuzzy sentences, supported by a fuzzy grammar G, provide
formal descriptions of biochemical transactions, or stps, within and be-
tween neurons. Soft Computing is characterized as the set of FFL tech-
niques used by the brain to solve both control and cognitive tasks.
The famously strange properties of quantum mechanics have been pro-
posed to explain complex cerebral functions since Penrose proposed that
2 Introduction
consciousness is the result of quantum transactions. But consideration of
these properties is also giving rise to new theoretical computational tech-
niques for qualitatively and quantitatively enhancing information pro-
cessing, techniques which are fast becoming a practical reality with the re-
cent experiments in quantum computation employing both Ion Trap and
MNR procedures. Thus, quantum properties of ions or molecules are used
to create quantum gates by way of unitary transformations that form the
logical basis of quantum computation. Nonlocal properties theoretically al-
low for massively parallel processing that promises to greatly enhance the
computational power of distributed processing systems like the brain.
Moreover, recent experimental data from neuroscience support the hypoth-
esis that certain neural structures, like the dendritic spine, are specialized
to function as Quantum Computing Devices, taking advantage of the quan-
tum mechanical properties of specific stps. These kinds of considerations
inform our view of the brain herein, which we describe as a Quantum Dis-
tributed Intelligent Processing System.
Neuroscience is demonstrating the existence of inherited or innate cog-
nitive modules for language, arithmetic, biology, physics, and music. This
knowledge is expected to dramatically influence the way we approach
modern education in the strongly competitive technological society of the
present day. This is our second main motivation in writing this book. It is a
well-established fact that current methods for teaching arithmetic are influ-
enced by our cultural belief that numbers are abstract entities living a life
of their own and that numerical knowledge distinguishes man from the an-
imals. This is in striking contrast with the experimental evidence of the ex-
istence of number sense in animals and in the human infant. But if man is
phylogenetically endowed with arithmetic neural circuits, why is learning
(and teaching) mathematics so stressful?
Recent experimental results demonstrate that learning by observing is a
powerful way whereby animals may quickly acquire the competence to
solve many kinds of problems, by simply watching another animal discov-
ering how to arrive at solutions to these challenges. However, as these
same experiments also show, observing an incomplete portion of the entire
sequence of discovery is insufficient for this type of learning; moreover,
partial observation also precludes the animal from discovering the best so-
lution on its own. Perhaps because traditional mathematical teaching does
not take into account the innate human capability for arithmetic, our teach-
ers demonstrate a suboptimal means of doing arithmetic. This not only
causes initial distress to the pupil, but perhaps may lead to a permanent
dislike or fear of mathematics. If this assumption turns out to be correct,
then actual knowledge about the arithmetic brain, as formalized in this
1.1 Why and How has Arithmetic Cognition Evolved? 3
book, may be used to plan a better and happier way of teaching not only
arithmetic but mathematics as a whole.
For the above reasons, we believe that neuroscientists will profit from
taking notice of our proposals in developing powerful neural models to
guide their experiments and to better interpret their results. By doing so,
they will describe their discoveries in a more comprehensible way, and for
a broader audience. Mathematicians could discover an entire new field, not
only for applications, but also for new developments in their own area,
which could feedback in the near future to a new wave of discoveries
about brain function. The emerging field of quantum computation will
profit from understanding that quantum information processing is a tech-
nique discovered by natural selection some millions of year ago. This, we
suppose, makes a strong case for the hope of creating artificial quantum
computers in the near future. The understanding of the brain as a quantum
processing intelligent system, together with the creation of artificial quan-
tum devices, will also expand the field for experimental physics to test the
strange properties of quantum mechanics. This makes the present work in-
teresting to the theoretical physicist.
But above all, we hope teachers and all other related professionals may
profit from understanding how the brain they have to teach operates, in or-
der to easy the thorny pathway that youngsters have to travel throughout
their schools years. Most especially, we hope that understanding the func-
tions and plasticity of arithmetic neural circuits may help educators make
learning arithmetic less stressful.
To accomplish such purposes, this book is organized as follows.
Chapter 1 is devoted to a review of the literature, showing that:
1. several species of animals identify and manipulate the cardinality of sets
of elements important to their survival;
2. the human infant has a capability for recognizing and operating on small
quantities;
3. the brain is organized to handle quantities and to have a number sense.
The opening chapter also introduces the hypothesis that human arithme-
tic capability is the result of the mutual influence of genetically inherited
and culturally transmitted information.
Chapter 2 introduces the basic concepts of Fuzzy Formal Languages and
Signal Transduction Pathways, and how the physiology of the neuron may
be understood, taking into account these concepts. In this context, a gram-
matical hierarchy is introduced and some basic theorems are provided. The
synaptic transactions are described as processes supported by a Self-
Controlled Grammar G, and become dependent on both the quantity and
the quality of the chemicals involved. The expression of the languages
4 Introduction
L(G) supported by G is subject to constraints imposed by limited re-
sources available in the external environment. The neuron is therefore
formalized as a symbolic processing machine, in contrast to the numerical
approach of classic neural network theory. It is also shown that the McCul-
loch-Pitts neuron is a special case of our model, if the number of sentences
is considered, and not their symbolic structure.
Chapter 3 discusses the brain as a Distributed Intelligent Processing
System (DIPS), whose agents are specialized in handling different subsets
of the language L(G) supported by the grammar G. The ontogeny of the
brain is formalized by a specialized subset of L(G) obeying the rules dis-
covered by biology for the functioning of a set of genes called homeobox
genes. Reasoning is proposed to be the result of transactions among a
complex set of agents. Learning is assumed to obey a set of rules control-
ling both the creation of new agents by specifying expressible subsets of
L(G), as well as communication among agents, by setting the amount of
resources available for synaptic transactions. Learning, in such a context,
is viewed as an evolutionary process, by means of which an initial set of
models is set to evolve under the influence of the (material and cultural)
resources provided by the external environment.
Chaper 4 introduces the basics of quantum computation and quantum in-
formation, and discusses how molecular transactions at the cellular level
may implement such concepts. We propose that the brain uses a quantum
computational strategy, allowing microstate entanglement of several spa-
tially distributed processing units, thereby providing a supplementary
communication channel to overcome possible mismatches. This channel
also provides instantaneous binding of the informational content being
processed in such units.
Chapter 5 describes dendritic spines (DS) as specialized synaptic struc-
tures for trapping Ca
2+
ions and as very plastic structures involved in both
rapid (imprinting) and slow (reaction to environmental changes) learning.
DS are assumed to have quantum computing capability and some calcula-
tions are presented to support such a claim. A DS model of the Deutsch-
Josza algorithm is presented and used to show how quantum computation
may be used by the brain for pattern-recognition purposes.
Chapter 6 introduces the logic and basic assumptions of memetics, pre-
senting its definition and proposing memetics as a potential answer to the
problem of brain evolution. The human capacity for dealing with complex
mathematical processes is proposed to be the result of a meme-gene inter-
action. A simulation study illustrates how knowledge spreads in a small
community, when information was provided by means of ―broadcasting‖
and ―mail‖ systems. In this context, evolution of mathematical knowledge
1.1 Why and How has Arithmetic Cognition Evolved? 5
in human culture is proposed to be the result of a meme-gene coevolution
of brain size and complexity.
Chapter 7 proposes a new class of fuzzy numbers, called K Fuzzy Num-
bers (KFN), to model arithmetical knowledge in the brain of animals and
humans. Three populations of agents - Controllers; Accumulators and
Quantifiers - are assumed to handle KFN. Also, the evolution of the ac-
cumulating function (changing from a monotonic to a periodic function),
and of the classifying function, are proposed as the means whereby KFN
evolved to create Crisp Base Numbers (CBN) like our decimal system.
This may explain the innate human capacity to create complex number
theories and arithmetic.
Chapter 8 describes the experimental results on performance in solving
arithmetic calculations by adults and children while having their EEG ac-
tivity recorded. The experimental data show:
1. a clear-cut distinction between genders, males being faster than females
in providing an equally correct answer;
2. a quick learning characterized by a calculation time decrease, which is
dependent on the order of problem presentation: and
3. a number size dependence of the calculation time that differed for chil-
dren and adults.
These data are interpreted in the framework of the model proposed in
Chap. 7.
Chapter 9 deals with the capacity for learning arithmetic among brain
damaged children with low and normal IQs. The results clearly demon-
strate that:
1. widely distributed bilateral parietal lesions reduce the children‘s arith-
metic learning capability;
2. left frontal lesions may dissociate the capability of handling and operat-
ing ordinal and cardinal numbers; and
3. brain plasticity allows children to overcome most of their arithmetic
learning problems.
Also, the results corroborate the conclusions about the organization of the
arithmetic neural circuits discussed in chapter 8.
Finally, chapter 10 assumes that learning arithmetic implies the devel-
opment of inherited CBN circuits under the guidance of teachers. This
process is proposed to follow an orderly pathway that must begin with the
construction of many different circuits in distinct cerebral areas, a process
triggered by questions posed in a set of problems of increasing complexity.
Based on experimental results from the literature on ―learning by ob-
serving,‖ we argue that, when the evolutionary nature of mathematical
6 Introduction
learning is not taken into account, the development of arithmetic
knowledge is seriously compromised, rendering instruction less effective.
This book owes a lot to the many children and adults who volunteered
in the experiments we conducted, wherein we probed many human cogni-
tive functions. We thank them very much for the brain time they shared
with us. We are also indebted to the parents of these normal and disabled
children, who understood that, through research, their youngsters pave the
way for a better education for succeeding generations. We must also
acknowledge the participation of many teachers and other persons in-
volved in the instruction of these of children. We would like to stress the
special participation of people from Colégio Clip – Guarulhos and APAE-
Jundiai, where we collected most of the experimental data discussed in
Chaps. 8 and 9.
The authors are also indebted to several colleagues and friends who,
through discussions and invaluable comments, greatly contributed to the
final result. Among them we especially acknowledge Francisco Antonio
Bezerra Coutinho, Fernando Gomide, Witold Pedrycz and Janusz
Kapcrzyk. We are also grateful for the technical support provided by Brian
J. Flanagan, Marcos Paulo Rebelo, Mateus Fuini, Ednilson Rodela e Fabio
Luiz Picceli Luchini. But above all, we are in very special debt to Cassia
Medea, who provided us with the illustrations used throughout the book.
Other friends, from the Discipline of Medical Informatics and
LIM01/HCFMUSP of the University of São Paulo, provided the right envi-
ronment, which greatly stimulated the production of this book.
1.1 Why and How has Arithmetic Cognition Evolved? 7
1 Quantification and Calculation in Nature
Several species of animals identify and manipulate the cardinality of sets
of elements important to their survival. The human infant has the capabil-
ity to recognize and manipulate small quantities. These facts will be used
to support the hypothesis that specific neural circuits have evolved in na-
ture to mediate these capabilities and that these circuits constitute the
foundations for the further development of mathematics by modern man.
1.1 Why and How has Arithmetic Cognition Evolved?
When the English philosopher John Locke got in touch with the Indians
from the fierce tribe of the Tupinambás (who lived in what is now the
State of São Paulo, Brazil, and were well known for their ferocity and can-
nibalism), he not only escaped violent death but even noted that their lan-
guage lacked names for numbers above five (Fig. 1.1). When the Tupi-
nambás went beyond five they simply showed their fingers and the fingers
of others (Butterworth, 1999). As a matter of fact, the Tupi language has
names for the first four numbers, whether used as cardinals or ordinals
(Navarro, 1998). After four, quantities were referred to by saying ―nhã‖
and showing the corresponding number of fingers. The number ten was re-
ferred to as ―my hands‖ and twenty by ―my hands and feet.‖
The fact that these Native Americans never developed a full language
for numbers reflects, first, the sheer lack of necessity – trade was not their
cup of tea. In the first place, quantities above five in general did not de-
serve special status and were denoted by ―many.‖ Secondly, our innate
counting capability is similar to that enjoyed by several animals (e.g.,
Dehaene, 1997; Gallistel and Gelman, 2000; Shettleworth, 1998).
We are, therefore, born with a capacity to enumerate objects, which is
strictly limited to some items, and we share this genetically determined
characteristic with more ―primitive‖ animals. Several empirical studies
have demonstrated that the counting capacity is innate in rats and pigeons,
and there is even an observable and remarkable competence in human in-
fants for simple arithmetic (Wynn, 1998). This genetically determined
8 1 Quantification and Calculation in Nature
arithmetical competence has been studied by several authors since the be-
ginning of the 20th century.
Fig. 1.1. The Tupinambá Number System: How far is this place? Five moons
walking.
Some interesting experiments in which human subjects are asked to
enumerate objects have shown that enumerating a collection of items is
fast when there are one, two or three items, but starts slowing drastically
beyond four. In addition, errors begin to accumulate at the same point
(Dehane, 1991; Fayol, 1996; Gallistel and Gelman, 1991, 2000). It was
hypothesized that ―mental operations with verbally or visually presented
digits depends on a mapping to mental magnitudes that seems to obey We-
ber‘s law,‖ which proposes a logarithmic rather than a linear relation be-
tween ―numbers‖ and magnitudes (Dehaene, 2003; Dehaene et al. 1998;
Gallistel and Gelman, 1991, 2000 ; McCloskey et al. 1991; Nieder et al.
2002). To better explain why numbers below five are quickly identified at
the almost the same speed, it was proposed that counting in this condition
1.2 The Numerical Competence of Animals 9
is performed in blocks, a process called subtizing in the literature (Butter-
worth, 1999, Dehane, 1997; Fink et al. 2001). It takes about five to six-
tenth of a second to identify a set of three dots, about the time it takes to
read a word aloud or to identify a familiar face, and this time slowly in-
creases from 1 to 3 dots.
How and why have arithmetic cognitive abilities evolved? It is tempting
to answer this question with the obvious notion that the larger the brain the
greater its owner‘s capacity to adapt and survive in aggressive and/or rap-
idly changing environments. But one could then argue that, in addition to
the size of the brain, its configuration and neuronal specialization have
roles to play. But is there a cerebral region responsible for mathematical
thinking? The first experiments, carried out in the early 1980s, demon-
strated higher cerebral activity in the inferior parietal cortex as well as
multiple regions of the prefrontal cortex in the course of numerical per-
formance tasks. Recent experiments with functional magnetic resonance
demonstrated that several other cerebral areas are activated during mental
calculations (Butterworth, 1999, Dehane et al. 1998; Göbel et al. 2001;
Nieder et al. 2002; Sawamura et al. 2002, Zorzi et al. 2002). It is now ac-
cepted that the inferior parietal region is important for the quantification of
cardinalities, and the representation of relative number magnitudes. The
extended prefrontal cortex is, in turn, responsible for sequential ordering of
operations, control over their execution, error correction, inhibition of ver-
bal responses, etc. Therefore, arithmetic neural circuits are now believed to
be widely distributed over several brain areas.
1.2 The Numerical Competence of Animals
There is a good deal of evidence that many animals, from pigeons to rats to
chimpanzees, have a certain, if limited, numerical competence. Apart from
some famous hoaxes, like the well-known case of ‗clever Hans‘ (the horse
that was supposedly able to perform some calculations), many experiments
have demonstrated that there is an innate numerical competence, which
evolved along the phylogenetic scale, increasing dramatically with human
beings. Although most experiments with animals required extensive train-
ing, numerically relevant behavior has also been observed in the wild.
In the 1950s and 1960s, some animal psychologists from Columbia
University (Mechner, 1958 and Dehaene, 1997) performed a series of ex-
periments with starving rats, which were then induced to press a lever a
certain number of times to get a given amount of food (Fig. 1.2). They
demonstrated that the rats not only learned to press the level to receive the
10 1 Quantification and Calculation in Nature
food but also that their responses were close to the expected ones. The dif-
ferences between the expected and the actual number of lever pressings in-
creased with the size of the number. The variance of the distribution of the
actual number of lever pressings also increased with the size of the number
of lever pressings (Fig. 1.3). Other recent experiments resulted in similar
results (Platt and Johnson, 1971; Gallistel and Gelman, 2000).
Fig. 1.2.a The Rat Number System: press the lever x times to get food (adapted
from Mechner experiments described in Dehaene, 1997)
These experiments demonstrated that the numerical competence of
small-brained animals is surprisingly good but limited to small quantities.
As shown in Table 1.1, the distances between the desired and the actual
number of presses, as well as the width of the distributions, increased in
proportion to their mode. This trial-to-trial variability in the accuracy with
which the animals approximated target numbers was proportional to the
1.2 The Numerical Competence of Animals 11
magnitude of the target, even for a number as small as four (Gallistel and
Gelman 2000).
Fig. 1.2.b Variance of Lever Pressing
A fundamental question in cognitive science is whether animals can rep-
resent numerosity and use numerical representations computationally. In a
recent experiment, Brannon and Terrace (1998) showed that Rhesus mon-
keys represent the numerosity of visual stimuli and detect their ordinal dis-
parity.
Table 1.1. Lever pressing statistics
D #: pressings
A #: pressings
D – A
Variance
4
4.5
0.5
1.25
8
9.0
1.0
6.25
12
13.5
1.5
12.25
16
18.0
2.0
16.00
D#: Desired number of lever pressings
A#: Actual number of lever pressings
V: Variance of distribution in actual number of lever pressings
Two monkeys were first trained to respond to exemplars of the numer-
osities 1–4 in ascending numerical order. As a control for non-numerical
cues, exemplars were varied in size, shape and color. The monkeys were
later tested without reward, on their ability to order stimulus pairs com-
posed of the novel numerosities 5–9. Both monkeys responded to ascend-
12 1 Quantification and Calculation in Nature
ing order in the novel numerosities, demonstrating that Rhesus monkeys
represent the numbers 1–9 on an ordinal scale.
In the next year, Carpenter et al.(1999) reported neurons located in the
monkey motor cortex that are specifically sensitive to the order of presen-
tation of visual stimuli, providing the physiological substrate for the results
of Brannon and Terrace (1998).
In another experiment, Nieder et al. (2002) demonstrated the existence
of a specific set of neurons in the lateral prefrontal cortex of monkeys that
were tuned for quantity, irrespective of the exact physical appearance of
the training displays. The tuning curves of those neurons form overlapping
filters, which may explain why behavioral discrimination improves with
increasing numerical distance and why discrimination of two quantities
with equal numerical distance worsens as their numerical size increases.
But the more impressive results come from experiments with chimpan-
zees, particularly in the work by Woodruff and Premack (1981). These ex-
periments demonstrated that chimpanzees are able to perform rather so-
phisticated abstract computation. They are able, among other things, to
conclude that one-quarter of a pie is to a whole pie as one-quarter of a
glass of milk is to a full glass of milk. The authors showed, in addition,
that chimpanzees could even mentally combine two fractions. If they had
to add one-quarter apple ‗plus‘ one-half glass of milk, and they had to
choose between a full disc or three-quarter disc, the animals correctly
chose the latter, and the result was statistically significant.
Chimpanzees trained with Arabic digits can even identify two Arabic
digits (such as 2 and 3) and point to their sum (as 5) amidst other Arabic
digits (Dehaene et al. 1998). In addition to an impressive numerical com-
petence, chimpanzees also display a fairly good numerical memory span.
In a recent experiment, Kawi and Matsuzawa (2000) tested the memory
span and other numerical skills of a female chimpanzee called Ai. These
researchers designed their experiment based on the fact that humans can
easily memorize strings of codes such as phone numbers and postal codes
if they consist of up to seven items. They therefore tried to determine the
equivalent limit in chimpanzees. Their design demonstrated that chance
levels with three, four and five items were 50, 13, and 6%. The chimpan-
zees scored more than 90% with four items and about 65% with five items,
significantly above the chance in each case. The results indicate that chim-
panzees can remember the sequence of at least five numbers, the same as
preschool children.
Finally, let us briefly review the most recent research on the numerical
competence of primates, specifically, the search for a ‗cerebral seat‘ of
numerical representation in monkeys. In an elegant experiment, Sawamura
et al. (2002) showed that the anterior part of the parietal association area in
1.3 The Numerical Competence of Human Infants 13
the cerebral cortex is active in primates performing numerically based be-
havioral tasks. The authors required monkeys to select and perform
movement A five times, switch to movement B for five repetitions, and re-
turn to A, in a cyclical fashion. Cellular activity in the superior parietal
lobe reflected the number of self-movement executions. For the most part,
the number-selective activity was also specific for the type of movement.
The authors reported that this type of numerical representation of self-
action was seen less often in the inferior parietal lobe, and rarely in the
primary somato-sensory cortex. According to the authors, such activity in
the superior parietal lobe is useful for processing numerical information,
which is necessary to provide a foundation for the forthcoming motor se-
lection.
1.3 The Numerical Competence of Human Infants
It is now a well-established fact that in the first few months of life, human
infants can enumerate sets of entities and perform numerical computations.
The capacity to represent approximate numerosity, found in adult animals
and humans, develops in human infants prior to language and symbolic
counting (Wynn, 1998; Xu and Spelke, 2000). However, it remains an
open question in cognitive sciences, philosophy and psychology as to how
we acquire our knowledge of number.
One prevalent view is that the numerical abilities of humans arise from
general cognitive capacities not specific to number. However, a growing
number of researchers have been arguing that the current body of data
supports the thesis that humans possess a specialized mental mechanism
for number, one which we share with other species and which has evolved
through natural selection (Wynn, 1998). Recent experiments demonstrate
that infants represent cardinal values of small sets of objects (Feigenson et
al., 2002). In addition, as early as six months of age, infants discriminate
between large sets of objects on the basis of numerosities when other ex-
traneous variables are controlled, provided that the sets to be discriminated
differ by a large ratio (8 vs.16, e.g., but not 8 vs.12) (Xu and Spelke,
2000).
The classic experiments to assess numerosity awareness in children deal
with time spent looking at displayed objects (Starkey and Cooper, 1980
and Strauss and Curtis, 1981). This parameter has demonstrated good sen-
sitivity in quantifying children‘s interest, and shows that infants can dis-
criminate between different small numbers of entities (Wynn, 1998). Thus,
when repeatedly presented with displays of a given number of visual ob-
14 1 Quantification and Calculation in Nature
jects, infants become bored and spend less time looking at the displays. If
the number of objects displayed is changed, children regain interest and
look for longer periods.
OK! 1 + 1 = 2
No! 2 – 1 is not 2!
Fig. 1.3 Studying the Baby Number System: the baby looks longer when 2 – 1 =
2 than when 1 + 1 = 2
In addition to number sensitivity, small children are able to engage in
numerical computation. When presented with a 1 + 1 operation and a 2 – 1
operation in looking time experiments (Wynn, 1996), infants showed sig-
1.3 The Numerical Competence of Human Infants 15
nificant differences in their looking time with respect to results represent-
ing correct and incorrect answers (Fig. 1.3).
The ―greater than” and ―lesser than” relations between numbers consti-
tute another essential component of our number system. In a recent exper-
iment, Brannon (2002) showed that 11-month old infants successfully dis-
criminated, whereas 9-month old infants failed to discriminate, between
sequences of descending numerosities from sequences of increasing numerosi-
ties. This suggests that 11-month old infants can appreciate the ―greater
than‖ and ―lesser than‖ relationships between numerical values, a charac-
teristic that develops after nine months of age.
Wynn (1998) proposed that there exists a mental mechanism, dedicated
to representing and reasoning about number, that constitutes part of the in-
herent structure of the human mind. In addition, as we humans share this
numerical discrimination with a large set of warm-blooded vertebrates, it is
reasonable to conclude that such a characteristic has a strong adaptation
value, and has therefore quite likely evolved by natural selection — a con-
clusion borne out by many similarities between infants and animals‘ nu-
merical abilities: animals as well as human infants enumerate a wide range
of entities, including moving or stationary objects and events, presented
simultaneously or sequentially (Davis and Perusse, 1988; Gallistel, 1990;
Wynn, 1998).
A compelling piece of evidence regarding the nature and development
of numerical knowledge is the historical development and ontological dis-
cussion of the concept of zero. The so-called place-value system of nu-
meral notation took over 1500 years to introduce the notation for zero
(Ifrah, 1985, Joseph, 1990). It is arguably more than a coincidence that our
individual difficulties with the numeral zero is mirrored in the historical
development of that concept. The individual understanding of zero does
not follow the same pattern of development as children‘s understanding of
positive integer values (Wynn, 1998). It normally takes a long time before
infants appreciate that zero is a numerical value, even when children have
already learned what the word zero and its corresponding Arabic numeral
symbol stand for. In another example provided by Wynn (1998), pre-
school children will name ―one‖ as the smallest number, even when they
know that the word zero applies to ―no items.‖
In conclusion, there is strong evidence showing that small children can
enumerate different kinds of entities, and can compute the numerical out-
comes of operations on small numbers of entities. Also, accumulated ex-
perimental findings support the existence of a dedicated brain mechanism
specific to numerosities, which serves as a foundational core of numerical
knowledge for our sophisticated mathematical abilities.
16 1 Quantification and Calculation in Nature
1.4 A Brief Account from Neuroscience
The famous neurologist Oliver Sacks, in his book The Man Who Mistook
His Wife for a Hat (1985), wrote that ‗deficit‘ is the favorite word of neu-
rologists. By this he meant the deficiency and loss of capacity so pervasive
in neuropathologies. Indeed, several deficiencies are directly attributable to
lesions of more or less specific areas of the nervous system, and the sci-
ence of neurology owes a great deal to such deficits, or ‗experiments of na-
ture‘ in the words of Dehaene (1997).
The first scientist to associate deficits with function was Paul Brocca,
the famous French physician, who in the 19th century correlated language
impairment with lesions of a very specific area in the frontal lobe, later
named Brocca’s area after him. Gertzmann was the first to associate
arithmetical cognitive disability with brain damage, when he described the
case of a 52-year-old woman, who was admitted at the Wiener Psychias-
triche Klinik, complaining of difficulties with memory and writing ability.
Neurological examination showed calculation impairment, writing disabil-
ity, lack of recognition and orientation of her own body, and incorrect se-
lection and orientation of individual fingers or hands. This latter deficit is
called finger agnosia. This clinical picture was shown to be the result of a
very specific left parietal lobe lesion compromising finger movements,
writing and arithmetic. This complex of clinical manifestations, due to a
brain lesion, is now called Gertzmann Syndrome (Gertzmann, 1924, see al-
so a recent case report by Mayer et al. 1999). The current understanding of
the distribution of numerical processing in the brain is largely due to stud-
ies regarding dissociation of arithmetical capabilities resulting from le-
sions on many different areas.
Dissociation is an important phenomenon in cognitive neuroscience, and
refers to the fact that, following cerebral damage, one domain of compe-
tence becomes inaccessible while another remains largely intact. So, in
some patients, severe difficulties in reading Arabic numerals have been re-
ported. In other cases patients can display double dissociation. For in-
stance, one patient had the grammatical structure of numerals intact,
whereas for a second patient this faculty had deteriorated; in contrast, the
selection of individual words was deficient in the first and intact in the se-
cond. As a result, the first patient often replaced one numeral with another
but he never erred in the decomposition of a number. For instance, he was
able to read 681 as ―six hundred fifty-one‖, that is, the structure of the
string is correct except for the substitution of fifty for eighty. The second
patient, in contrast, never took 1 for ―two,‖ like the first, but misread 7,900
as ―seven thousand ninety‖. The conclusion of these clinical cases is that
1.4 A Brief Account from Neuroscience 17
perhaps some of the cerebral regions engaged in reading Arabic numerals
aloud contribute more heavily to number grammar, while others are more
concerned with accessing a mental lexicon for individual numeral words
(Dehaene, 1997).
The recent development of non-invasive techniques for the study of the
living human brain is bringing new light to bear on the neurosciences.
Brain mapping machines, like MRI (Magnetic Resonance Imaging), fMRI
(functional MRI), SPECT (Single-Photon Emission Computerized Tomog-
raphy), PET (Positron Emission Tomography), MEG (Magneto-
Encephalography), and EEG (Electro-Encephalography), all heavily de-
pendent on computer power, are changing the way scientists explore the
function of the brain. These new techniques are disclosing fundamental
functional contributions of frontal and parietal neurons to numerical pro-
cessing, as well as the existence of different neural circuits to deal with
both approximate and precise numerical reasoning.
But perhaps the greatest contributions of these techniques for the ques-
tions at hand consist in facilitating the study of the plasticity of neuronal
functioning in respect of numerical competence. The marriage of brain-
lesion studies and non-invasive techniques is showing that, while congeni-
tal lesions may be devastating in brain tissue loss, these kinds of damage
do not restrain children from building up new circuits at new places in or-
der to acquire numerical competence as complex as that attained by the
normal brain (e.g. Rocha et al. 2003b). These authors have followed up the
cognitive evolution of three congenitally brain damaged children who ex-
perienced delayed linguistic and arithmetical skills acquisition, described
in details in Chapter IX.
One of the children lost most of her left-frontal lobe, spoke her first
words at the age of five, and began to read and write simple words, as well
as master quantities above five, at the age of eleven. At the age of thirteen,
she started to perform very well at addition and subtraction — after she in-
vented a way to cope with her dissociation; she was able to quantify but
she had problems in (orally/or graphically) naming these quantities.
Another girl had her right parietal lobe damaged, and acquired language
at the normal age, but experienced early difficulties with arithmetic. At the
age of sixteen, she began to master complex calculations with up to three-
digit numbers.
The third case is that of a boy who lost his left-parietal lobe, started to
use his first phrases at the age of five years, began to understand quantities
above five at the age of seven, and to perform very simple calculations
(one-digit summation and subtraction) at the age of ten. At the age of four-
teen he started to master addition and subtraction up to three digits and
18 1 Quantification and Calculation in Nature
multiplication of up to two digits. Brain mapping clearly showed a reor-
ganization of neural circuits paralleling such developmental achievements.
Contrasting with these examples of severe brain damage and reasonable
performance, some lesions that go clinically undetected (even by the ma-
jority of those techniques described above) may drastically change the ac-
ademic life of many children, as in the case of developmental dyscalculia.
Around 5% of school children experience minor brain cell deaths that re-
sults in an abnormal EEG pattern (called Intermittent Rhythmic Delta Ac-
tivity), associated with one or more of the following symptoms: hyperac-
tivity, attention deficit, dyscalculia and dyslexia.
In conclusion, data gathered by the neurosciences appears to indicate
that mathematical reasoning is deeply seated in widely distributed and very
plastic neural circuits.
1.5 Stand Up and Count (and Get Smarter!)
It is difficult to be precise as to when our forebears evolved from the
chimpanzee-like, knuckle-based gait to the upright position and bipedal
gait of modern humans. This important change in the way our ancestors
walked had tremendous consequences for our brain development.
Recent molecular biology dating, based on DNA comparison and as-
suming a constant mutation clock, points to a time when our species first
diverged from our cousin chimpanzees between 5 and 7 million years ago
(Marks, 2002). However, as we share about 98.5% of our DNA with the
chimpanzees, it is difficult to account for the differences between us and
them, based only on the meager 1.5% of non-paring genetic material. We
will therefore concentrate on the fossil record, which points to important
skeletal transformations which occurred when the first members of our lin-
eage dropped down from the African trees, in order to try to understand
these differences.
Our genus is denominated Homo and it is now widely accepted that we
evolved from a line of other primates, called hominids, from whom we
split some 2.5 million years ago. Some twenty different species of homi-
nids are listed in the current literature (Tattersall and Schwartz, 2000).
What is important for our subject, however, are the implications of bipeda-
lism on brain development, including the consequent correlation between
bipedalism and the numerical competence of our ancestors.
Two basic models of human development can be identified in the cur-
rent literature (Tattersall and Schwartz, 2000). The first model is called
―linear‖ and it states that the anatomical characteristics of the hominids
1.5 Stand Up and Count (and Get Smarter!) 19
appeared once in the phylogenetic record and all the descendants built up-
on the basic features in a linear fashion. The second model, called the
―disordered‖ model, argues that hominid evolution happened through a set
of evolutionary diversification events, during which the anatomical charac-
teristics mixed up in a way we have only recently begun to comprehend.
What is important for our discussion is the very definition of hominid.
According to the specialized literature, the hominids are defined by the
structure of the face and teeth as well as by bipedalism. The recent discov-
ery of a candidate for our most ancient ancestor stirred debate within the
anthropological community on the definition of hominid; this was Sahe-
lanthropus tchadensis (Brunet, 2002), with an estimated age of 6 or 7 mil-
lion years. This was due to the fact that only the facial bones and some
teeth were found; the absence of post-cranial fossil remains renders the de-
termination of its gait difficult, with the consequence that the determina-
tion of that species as our forebear is highly controversial (Wolpoff et al.
2002).
In any case, when the first hominid dropped from the trees and started to
forage the African savanna, some important changes in its brain began a
long series of adaptations that culminated with our own brain. The most
important aspect of this primitive bipedalism was the liberation of the
hands, which, along with better gait control, demanded an improvement in
the neuro-motor systems. This process was perhaps the most important
event in human evolution so far as brain development is concerned, and re-
sulted in tool-making culture, some two million years ago. Also, the up-
right position put other strong demands on the evolution of the visual sys-
tem, including better control of eye movement. Free hands and smarter
eyes greatly improved the ability to correctly focus on complex collections
of objects and count them. This started a process which demanded more
intellectual processing, which in turn resulted in more sophisticated count-
ing processes.
The relatively sudden explosion in brain size occurred between Homo
erectus and Homo heidelbergensis, between 700 and 500 thousand years
ago, when the brain almost doubled in size, from around 500 cc to 1000
cc., and coincided with the discovery of fire. It is possible that cooking
preserved food for longer periods of time, which implies less frequent
hunting and more socialization. This sparked the beginning of the modern
human intellectual enterprise. First, the development of a lithic tool-
making industry increased the human power for hunting and raising crops.
Second, this development freed time for culture, initially represented by
religion and burial rituals, but soon complemented by painting in order to
record the history of the human activities (Fig. 1.4).
20 1 Quantification and Calculation in Nature
Fig. 1.4. Pedra Furada, 6–12 thousand years ago: all around the world, man be-
gun to paint cave walls
This history telling required quantification of many of these activities,
and primitive number representation started to appear as sequences of
marks denoting, e.g., the number of hunted animals. Trade, the exchange
of goods, soon developed into a common activity within and between
groups of people. The subsequent increase in the complexity of trade re-
quired the corresponding development of the capacity of number pro-
cessing to deal with it.
2.1 Molecular Neurobiology 21
2 The Cells of the Brain
The neuron is modeled as a fuzzy formal language processing device, in-
stead of a numerical processor as in classical neural nets. This is because
modern neuroscience shows that synaptic events involve complex chains
of chemical transactions triggered by the coupling of transmitters or neu-
romodulators to membrane receptors. These chains of chemical transac-
tions are called Signal Transduction Pathways (stps) and are responsible
for complex cellular processing, which may include gene reading. Assum-
ing the neuron is a fuzzy language processor sets the background for dis-
cussing the brain as a Distributed Intelligent Processing System (DIPS) in
the next chapter, and for understanding it as a quantum computer in Chaps.
4 and 5.
2.1 Molecular Neurobiology
Since Galvani‘s classical experiments, electrical membrane gradients and
their variations have played an important role in the understanding of cere-
bral physiology. The work of Hodgkin and Huxley in the first half of the
20
th
century, for which they were awarded a Nobel Prize, clearly identified
the main components governing the membrane‘s electrical behavior, and
formalized this behavior as equivalent to that of a dynamic system having
two stable, and one unstable, equilibrium state(s) (for further discussion of
this subject, see Rocha, 1992). Another promising approach to the under-
standing of brain function has been molecular neurobiology. Combining
measurement of brain activity and observation of concomitant behavior, in
both normal and genetically modified animals, brain scientists have been
able to understand the role of macromolecules in processes that support the
observed behavior (e.g., Smythies, 2002; Bickle, 2003). Cognitive and af-
fective molecular neurobiology are current research areas where biochemi-
cal properties of macromolecules, and the processes determined by their
interactions, have been demonstrated to correlate with cognitive and affec-
tive functions of the brain, which putatively support the observed behavior.
The construction of explanatory models based on molecular neurobiology
22 2 The Cells of the Brain
focuses on interacting biological macromolecules forming a functional
unity, the cell.
The individual cell can be considered as a cluster of signaling elements,
comprising the DNA, the mRNA, the proteins produced by them, and
functional ions, such as Na+, K+, and Ca++. The interaction of such sig-
naling elements forms ordered chains of chemical transactions, called
"signal transduction pathways" or stps, for short. Such stps include, there-
fore, signals that come from the outside (e.g., transmitters that bind to neu-
ronal membrane receptors, hormones that come in blood flow), all signals
that are operative inside the cell, and signals released by the cell to the out-
side (e.g., transmitters released into the synaptic space).
More importantly, we also consider ion movements across gates and
generating bioelectrical processes (as in the formation of action potentials
at the dendrites and soma of neurons) to be parts of stps. This is a useful
way to avoid the frequent dualism of electrical and chemical explanations
of cerebral processes. In fact, bioelectrical processes are composed of ionic
fluxes that can be analyzed into strings of discrete signals, such as the pas-
sage of Na+ and K+ ions in and out of the neuronal membrane through
specialized gates. Of course, such currents are measured in EEGs and
MEGs as continuously varying electromagnetic fields. The discrete and
continuous variants would be different forms of description of the same
phenomenon.
Cellular specialization is defined in terms of the set of stps realized in
the cell. DNA is the same in all cells of a given organism, but the protein
pool (the proteome) is different. The proteome in each cell comprises
thousands of different proteins, which are not identical in different tissues,
organs, and systems of the body, although the amino acids present in each
one are all determined by a DNA nucleotide sequence. Each pool is in fact
a small subset of the total combinatorial possibilities of the DNA, whose
composition depends on many factors (transcriptional and post-
translational modifications driven by developmental and environmental
forces, RNA splicing, etc.; see Bray et al. 2003) that direct genetic expres-
sion towards a subset of the total possibilities. Different types of neurons
are characterized by the specific stps they express under the control of both
genetic and environmental factors.
The gene is composed of two basic nucleotide strings (Fig. 2.1):
1. The code string: containing the nucleotide string encoding the amino
acid sequence of a given family of proteins. The reading of the code
string results in the synthesis of the mRNA, which is in charge of con-
veying the genetic information to the ribosomes, where polypeptide syn-
thesis occurs. The code string is composed of exons and introns — that
is to say, meaningful nucleotide sequences (exons) are mixed with
2.1 Molecular Neurobiology 23
meaningless nucleotide sequences (introns) in the same gene. The in-
trons are excised from the mRNA, after the DNA reading, by the action
of specific molecules, which are able to recognize its delimiting se-
quences. The remaining string of exons contains the protein code.
Therefore, DNA encoding is now assumed to be a non-linear process.
Biochemical sequences
Formal structure
Fig. 2.1. Signal Transduction Pathways (STPs): the sentences of the computa-
tional language used by neurons
2. The control string: containing the nucleotide string encoding the condi-
tions necessary to enable and to repress the code reading. The control
string is, in turn, assumed to be composed of the following substrings:
TATA box (promoter): a nucleotide sequence composed mainly
of thymine and adenine, which must be activated in order to enable
the code reading;
24 2 The Cells of the Brain
Inducer substrings: the activation of these nucleotide substrings
accelerates the code reading, enhancing the number of available
copies of its associated mRNA, and
Repressor substrings: the activation of these nucleotide sub-
strings represses the code reading, reducing the number of availa-
ble copies of its associated mRNA.
The control of DNA reading is the way nature found to control the
number of proteins involved in each stp. Activation of gene reading de-
pends on the binding of specific proteins to their control strings. The pro-
teins are in turn specified by their genetic code strings.
In this context, the set P of proteins encoded by a set D of genes or
DNA (RNA) molecules may be organized into different ordered sets of
chemical interactions, each one defining a biochemical chain or pathway,
called a signal transduction pathway. Sets of these stps are associated with
functional biochemical systems supporting complex cellular activities such
as cellular metabolism, excitability, reproduction, etc. Some proteins act as
triggers (P
x+0
) of these biochemical chains (Fig. 2.1), while other subsets of
proteins (P
x
..., P
x+i
) are activated at intermediary steps or as end products
(P
x+t
) of the corresponding pathway. Early genes control the expression of
other downstream genes (Fig. 2.1) that specify the sets of proteins (P
x+0
,
P
x+i
, P
x+t
) participating in a given stp. These stps are activated by outside
signals P
x+0
binding to membrane receptors (e.g., the glutamate receptors).
Some of the proteins (P
x+i
) enter the nucleus to control the gene reading,
while other P
x
s remain in the cytoplasm to control cellular events. Finally,
other proteins (P
x+t
) are exported to act on other cells or upon the environ-
ment.
From a formal point of view, a stp is a serial-ordered set of transactions:
p
x+0
p
x+i
p
x+k
p
x+t
(2.1)
where p
x
stands for a given protein x, and denotes some effect induced
by p
x
upon p
x+1
due, for example, to some energy transference to, or struc-
tural modification of, p
x+1
. For instance, the glutamate binding to its
metabotropic receptor is proposed to expose an internal site of this receptor
that splits a G-protein into its components G- and G-, each one of which
then controls a different chain of biochemical events. Because p
x
act over
different p
y
s, which in turn may control different biochemical transactions,
the order imposed upon any stp is a partial ordering. In some instances, the
activation of a p
x+1
may be dependent also on the action of proteins p
y,
p
z
besides p
x
, that is:
p
y
p
x
p
z
p
x+1
(2.2)
2.1 Molecular Neurobiology 25
In this condition, it may be said that p
x
acts upon p
x+1
in the context
(background) provided by p
y,
p
z
.
Biological remark 2.0. Note that an stp is not an ordinary chemical re-
action, when two or more components react in order to generate one or
more new chemical components. An stp is a sequence of energy transfer-
ences that convey some information to modulate the expression of a bio-
chemical process supporting life. In such a way, it deserves a special kind
of mathematical modeling, other than that classically applied to formal de-
scriptions of the dynamics of chemical reactions.
The activity of any cell C at any moment is the result of a well orches-
trated activation of many different stps. This may be described formally as:
C = {P
o
, P
n
, P
t
, B, D}
(2.3)
Where:
1. P
o
is the set of initial chemical signals p
x+0
;
2. P
n
is the set of intermediate chemical signals p
x+i
;
3. P
t
is the set of terminal chemical signals p
x+t
;
4. D is the set of genes encoding P = Po
n
P
t
, and
5. B is the set of biochemical transactions of the type p
y
p
x
p
z
p
y
p
x+i
, p
z
denoting the interaction among the proteins p
y
p
x
p
z
which activates p
x+i
in the context provided by p
y
, p
z
such that
6. the biochemical chain stp(p
o
, p
j
) is characterized by the ordered set of
transactions required to activate p
j
t
whenever p
o
i
is availa-
ble, that is
stp(p
o
, p
j
) = p p
x+i
p
x+t
(2.4)
Two different stp
i
, stp
j
are said to be a hierarchy if they share a subset of
products P
c
which are terminal products P
t
of stp
i
and initial products P
o
of
stp
j
. In this condition, stp
i
is said to regulate stp
j
. Because early genes may
coordinate the expression of different sets of downstream genes, then D
becomes a hierarchical encoding H of P* = P
i
P
n
P
t
. H organizes fami-
lies {stp
j
}
j = 1 to n
of stps into a partially ordered set of biochemical chains
(Fig. 2) supporting life. In the context of this paper, a Genetic Network
(GN) is a set of these hierarchical stps supporting a given cellular activity:
GN = ({stp
j
}
j = 1 to n
, H)
(2.5a)
Where H may, for instance, be defined as:
H
k
ij
j
stpmin
(2.5b)
26 2 The Cells of the Brain
but many other kinds of hierarchies may also be proposed.
Biological remark 2.1. Central to the notions of stps and Genetic Net-
works (Fig. 2.2) is the concept of ambiguity in the molecular interactions
in the chain of chemical reactions. Despite the specificity of many en-
zymes, many important proteins are involved in many stps, and in many
instances the degree of molecular interaction is extremely variable, as in
the case of transmitter and receptor binding, promoter binding to the
TATA box, degree of protein phosphorylation, etc.
Fig. 2.2. A genetic network defining a stp: a coherent set of genes is activated in
order to produce all chemicals of a stp.
One of the major challenges in the age of ―Function Genomics‖ is to in-
fer regulatory interactions between genes from experimental data collected
from micro-array experiments. Genome expression analysis involves the
use of oligonucleotide or cDNA microarrays to measure, in a parallel fash-
ion, the mRNA levels of as many genes as possible in a genome. Many
techniques are being developed to analyze these experimental measure-
ments in order to disclose the main gene interactions in a given moment.
Among these techniques, the Genetic Network has been used as a formal-
ism to represent causalities among, and to reason about, these gene interac-
2.2 Fuzzy Formal Languages 27
tions. Again, many mathematical tools are being used to develop different
kinds of genetic networks: boolean modeling (e.g., Akutsu et al. 2003);
abductive reasoning (e.g., Zupan et al. 2003); multi-criterion optimization
(e.g., van Smorerem et al. 2003), etc. Each of these techniques may be
used to build H in Eq. 2.5.a.
From the present point of view, any GN is a structured set of infor-
mation necessary to specify when and how one stp, or a group of stps, is to
be expressed. A way of describing any GN is to provide a direct graph
(Fig. 2.2), where the nodes represents elements of P* and the arcs describe
the relations between pairs p
i
, p
j
P*. Different types of functions may be
associated with these arcs to describe the restrictions controlling the flow
of information in the GN.
2.2 Fuzzy Formal Languages
Definition 2.1. A grammar G (e.g.; Chomsky, 1955; Mizumoto et al.
1973; Negoita and Ralescu, 1975; Rocha et al. 1980; Searls; 1992, 2002) is
a structure defined as
G = {V
o
, V
n
, V
t
, P}
(2.6)
where:
1. V
o
is a set of initial or starting symbols;
2. V
t
is a set of terminal symbols;
3. V
n
is a set of non-terminal symbols, and
4. P is a set of rewriting rules defined as
p: s
i
s
j
, p
P, , ,s
i
, s
j
V
o
V
n
V
t
(2.7)
In other words, p rewrites s
i
as s
j
in the context defined by and s
i
, For
the sake of simplicity, we denote
V
+
= V
o
V
n
, V
#
= V
n
V
t
, V
&
= V
o
V
t
and V
*
=V
o
V
n
V
t
(2.8)
V
*
is supposed here to include the empty symbol .
The derivation chain d(s
i
, s
j
) of the s
i
, s
j
V
*
is the ordered set of pro-
ductions required to transform the symbol s
i
V
s
into s
j
. In other words
d(s
i
, s
j
) = s
i
s
k
s
l
s
j
(2.9)
28 2 The Cells of the Brain
The grammar defined so far is called a type 0 grammar or G
0
. Certain
restrictions can be made on the nature of the productions of a grammar to
give other types of grammars (Hopcroft and Ullman, 1969).
A language L(G) supported by G is the set of all n derivation chains
d(s
o
, s
t
), s
o
V
o
and s
t
V
t
, that is
L(G) = {d(s
o
, s
t
) = s
o
s
i
s
t
| s
o
V
o
and s
t
V
t
}
(2.10)
The processing of any derivation chain d(s
o
, s
t
) is a partially sequentially
ordered set of rewriting operations, each one involving the following steps:
1. Matching: a symbol at the left-hand side (e.g. s
k
) of a prospective re-
writing rule (e.g., s
k
s
j
)
is matched (s
i
s
k
) to the symbols of the
string s
i
being processed. If this matching succeeds, then
2. Rewriting: the matched s
i
is substituted by the right-hand side of the
accepted rewriting rule s
i
s
k
s
j
,
and finally:
3. Acceptance: the membership degree V
t
(s
i
) of s
i
to V
t
is evaluated. If s
i
is accepted as belonging to V
t
, that is, if s
i
= s
t
V
t
the rewriting pro-
cess is stopped, and d(s
o
, s
t
) is assumed to be a well formed formula of
L(G).
These steps describe the following:
2.1.a. Matching: a protein (symbol) s
k
occurring on the left-hand side of
a prospective biochemical reaction (or, rewriting rule) s
k
s
j
,
is matched
(s
i
s
k
) to the symbol (protein) s
i
being processed (activated). The degree
of matching (s
i
s
k
) is a measure of the chemical affinity (Tuszynski and
Kurzynski 2003, pp 245-246) between s
i
, s
k
. In this way it may be defined
as:
(s
i
s
k
) = {q(s
i
s
k
) / [min (q(s
i
), q(s
K
))]} = { * exp (- / T )}
K = exp (-
0
/ RT)
that is, as a function of , the thermodynamic force or chemical affinity
connected to the independent variable s
i
, where is the Boltzmann con-
stant;
0
is the free energy, T is the temperature in Kelvin degree; R is
the gas constant; q(s
i
), q(s
K
), q(s
i
s
k
) are the number of initial molecules
(symbols) s
i
, s
K
and of s
i
s
k
, respectively. The function is defined such
that:
2.2 Fuzzy Formal Languages 29
(s
i
s
k
) 1 if q(s
i
s
k
) min (
q(s
i
), q(s
k
)) and (s
i
s
k
) 0 if
q(s
i
s
k
) 0
Now, if (s
i
s
k
) > 0, then
2.1.b. Rewriting: the matched s
i
triggers an action over s
k
to activate s
j
s
i
s
k
s
j
and finally:
2.1.c. Acceptance: if s
j
is either a molecule being exported outside the
cell to act as a trigger of another stp in other cells, or an intermediary
product of another stp in the same cell, then s
i
is accepted as belonging to
V
t
;
the rewriting process is stopped, and d(s
o
, s
t
) is assumed to be a well
formed formula of L(G).
Because a given s
k
produced by a given d(s
o
, s
k
) may be involved in
more than one stp, it may be the case that it is an end product for one stp
and an intermediate signal for another. For instance, ATP is a product in
the case of respiratory stps, and intermediate components in many other
biochemical processes, e.g., those illustrated in Fig. 2.
Along this line of reasoning, let the following be defined:
2.1.d: The degree of similarity (s
i
, s
k
) of two strings s
i
, s
k
is a map-
ping in the Cartesian product space V
s
V
n
V
t
to the closed interval
[0,1] such that:
V
o
V
n
V
t
0, 1]
(s
i
, s
k
) 0 if s
i
s
k
, otherwise 0 < (s
i
, s
k
)
2.1.e: The degree of acceptance (s
j
,V
t
) of s
k
as belonging to V
t
is cal-
culated as the maximum degree of similarity (s
j
, s
t
) of s
j
with the strings s
t
V
t
. In other words:
(s
j
, V
t
) = max (s
j
, s
t
) V
t
Now, let s
k
V* be a chain which is defined over a set of characters
given by A = {a
1
..., a
n
}, called a fuzzy alphabet (Sadegh-Zadeh, 2000) by
means of a set of concatenation rules F:
F: (A U )
l
V*
(2.13)
such that each s
k
V* is
s
k
= { a
i
... a
k
... a
m
}
(2.14)
30 2 The Cells of the Brain
or it is a concatenation of characters of A with the maximum length
equal to l, and
(A)
n
0, 1
(2.15)
measures the acceptability that a
l
replaces a
k
such that
(a
k
, a
l
) 1 if s
k
= {a
i
... a
k
... a
m
} then s
l
= {a
i
... a
l
... a
m
}
(2.16)
In this context (a
k
, a
l
) is defined as a function
that is
((A)
n
)
l
0, 1
(2.17
In such a condition, A and F define the words s
k
composing the sentences
s
o
accepted by L(G) if there exists d(s
o
, s
t
) such that (s
t
, V
t
) 1.
Biological remark 2.2.The set of nucleotides composing the DNA and
RNA, and the set of 20 amino acids forming the proteins, are examples of
fuzzy alphabets. In this line of reasoning, s
k
V* may be considered words
composing the stp sentences of L(G) about the biochemical transactions
supporting life. Genes are words encoding the proteins and proteins are al-
so words used to compose the stps. In addition, genes are composed of two
different substrings: the code substring containing the description of one or
a family of proteins, and the control substring encoding when and how the
gene is to be read. In such a way, Fuzzy Formal Language Theory is used
here to deal with GN, in a context different from that used by Searls (1992,
2002). For instance, instead of focusing attention on the structure of the
nucleotide sequences (words) in DNA, we discuss the dynamics of gene
hierarchies and their role in specifying and organizing the (sentences) stps.
Most of what follows deals with the stp sentence dynamics, rather than the
structure of both the genetic and proteomic words.
2.3 Ambiguity
Fuzzy languages exhibit distinctive properties compared to crisp languages
because, given s
i
V
*
, many s
k
V
*
may exist for which (s
i
s
k
) > 0.
This means that many derivation chains d(s
o
, s
k
) may rewrite s
o
into many
different s
t
V
t
, depending on the ambiguity of any among its rewriting
steps s
i
s
k
s
j
. The ambiguity of L(G) depends on how many derivation
strings d(s
o
, s
t
) exist for the same s
k
V
o
resulting in different s
j
V
t
.
Therefore, Fuzzy Grammars are naturally ambiguous and the amount of
2.3 Ambiguity 31
ambiguity, (G), of a given grammar G is strongly related to the cardinali-
ties of V
o
, V
i
, V
t
and to the distribution of the actual values of (s
i
, s
k
) in
its state space V
s
V
n
V
t
.
The usefulness of a fuzzy grammar G is dependent on how (G) is con-
strained by the resources available to process G. These constraints may be
imposed upon, for instance, the dictionary A generating the words s
k
of
L(G), in such a way that q(s
k
) becomes dependent on the availability q(a
i
)
of a
i
s
k.
It may also be dependent on any control imposed upon F in Eq.
2 13. In this context of, the ambiguity (G) of G is assumed to be depend-
ent on:
1. the total number or quantity, q(s
i
), of available copies of s
i
V
*
: at least
one copy of s
i
V
*
has to be available to trigger each possible deriva-
tion chain d(s
i
, s
j
| s
k
), supported by s
k
s
j
(s
i
s
k
) > 0,
2. the total number, q(s
k
) of copies of s
k
V
*
: at least one copy of s
k
has to be available to allow the rule s
k
s
j
(s
i
s
k
) > 0 to be
used, and
3. the total number, q(s
j
) of copies of s
j
V
*
: at least one copy of s
j
has to
be available to allow s
i
to be rewritten into s
j
.
In this context:
Definition 2.2.a: The possibility (d(s
i
, s
j
) | H) that the derivation
chain d(s
i
, s
j
| s
k
) supported by s
i
s
k
s
j
is used to rewrite s
i
into s
j
, un-
der the constraints imposed by the environment H is a function of (s
i
s
k
)
and the chemical affinity associated to s
j
(Tuszynski and Kurzynski, 2003,
pp 249-250). In this way, it is calculated as:
(d(s
i
, s
j
| s
k
) | H) = (s
i
s
k
) [q(s
j
) / q(s
i
)* q(s
k
)]
[q(s
j
) / q(s
i
)* q(s
k
)] = * exp (- / T)
= exp (-
0
/ RT)
(2.18)
where n is the cardinality of V
n
, and is a T-norm (see Pedrycz and
Gomide, 1998, about the concept of S and T-norms. The max and min
functions are examples of S and T-norms, respectively).
In such conditions, (d(s
i
, s
j
| s
k
) | H) may assume any value in the
closed interval [0, 1].
For the sake of simplicity, (s
i
s
k
) is denoted as (s
i
, s
k
) in the rest of
the paper.
32 2 The Cells of the Brain
Definition 2.2.b: The possibility
k
(d(s
o
, s
t
)) of a given
d(s
o
, s
t
| s
k
) = s
o
s
k
s
l
s
t
including a defined s
k
s
l
, is determined by the weakest rewriting
step. In other words:
(d(s
i
, s
j
| s
k
) | H ) =
n
k 1
{(s
i
s
k
) [q(s
j
)/q(s
i
)* q(s
k
)]})
(2.19a)
where is an S-norm, it will be assumed as the min operator in the rest of
this chapter.
Definition 2.2.c: The possibility:
(d(s
o
, s
t
)) of d(s
o
, s
t
) = s
o
s
t
is
(d(s
o
, s
t
)) = max (d(s
o
, s
t
)| k)
(2.19b)
because ambiguity in G creates many alternative pathways d(s
o
, s
t
| s
k
)
whereby one can derive s
t
from s
o
. A given d(s
o
, s
t
) is said to be expressible
if (d(s
o
, s
t
)) > 0.
Proposition 2.1. The constraints imposed by H define a unique distribu-
tion (d(s
o
, s
t
) | H)) over V
*
such that the mean possibility d(s
o
,
s
t
))>
(d(s
o
, s
t
))> =
1/m
m
j 1
(d(s
o
, s
t
))
(2.20)
is subject to
d(s
o
, s
t
))> , 0 < < 1
(2.21)
where m is the cardinality of V
*
.
Proof: It follows from Definition 1, in order to guarantee the existence
of at least one d(s
o
, s
t
| s
k
) such that (d(s
o
, s
t
| s
k
) | H) > ; thus:
(d(s
o
,s
t
| s
k
) | H ) > 0.
Biological remark 2.3. H is assumed here to stand for the set of Earth‘s
conditions supporting the existence of a set of living beings defined by G.
The restrictions imposed by H may be upon the availability of:
1. the elements of the alphabet A: that is, upon the availability of nucleo-
tides, amino acids, etc;
2.3 Ambiguity 33
2. the energy to support the chemical STP transactions: that is, upon
the energy required for protein phosphorylation;
3. other chemicals, such as ions, required to support STP transactions:
that is, upon the availability of ions such as Ca
2+
used as a second mes-
senger; or Na
+
, K
+
and Cl
-
, specifying the electrical cell environment or
its pH, etc.
Life is characterized by chemical changes in an inevitable progression,
which is not due to blind chance. The chemical sequence is from a reduc-
ing, to an ever increasing, oxidizing environment wherein organisms
struggle to retain reduced chemicals. Most of this dynamics was deter-
mined by the environmental conditions of primeval earth, when bio-
diversity began to blow up (Williams and Fraústo da Silva, 2003). The re-
strictions imposed by H are one of the driving forces of evolution, the oth-
er one being the incompleteness or the genetic information specifying any
living being (Barbieri, 2003). Environmental randomness and d(s
i
, s
j
) am-
biguity will conjoin forces to drives life evolution.
Since the focus of the present chapter is on the informational contents of
the biochemical transactions supported by the stp sentences, given a cer-
tain d(s
o
, s
t
) supported by the grammar G in H, and s
i
s
k
s
j
, let the fol-
lowing to be defined:
1. entropy: h(d(s
o
, s
t
| s
k
) | H) as
h(d(s
o
, s
t
) | H) =
- ( (d(s
o
, s
t
) log (d(s
o
, s
t
) - (1- (d(s
o
, s
t
) ) log (1- (d(s
o
, s
t
))
(2.22)
2. mean ambiguity: < d(s
o
, s
t
) | H)> as
<(d(s
o
, s
t
) | H)> = - (<(d(s
o
, s
t
))> log<(d(s
o
, s
t
))>
- (1 - <(d(s
o
, s
t
))>) log(1- <d(s
o
, s
t
))>))
(2.23)
3. expressiveness: (d(s
o
, s
t
| s
k
) | H) of d(s
o
, s
t
| s
k
) as
(d(s
o
, s
t
| s
k
)) = < (d(s
o
, s
t
))> - h(d(s
o
, s
t
| s
k
) | H)
(2.24)
In this context, the expressiveness of L(G | H) is calculated as
(L(G | H)) =
HnG
i
|
1
(d
i
(s
o
, s
t
| s
k
))
(2.25)
where n
G|H
is the number of d
i
(s
o
, s
t
| s
k
) for which (d(s
o
, s
t
| s
k
) | H) >0.
34 2 The Cells of the Brain
Note that entropy here is a measure of both the matching (s
i
s
k
) uncer-
tainty and the uncertainty in word frequency, because from Eq. 2.18 (d(s
i
,
s
j
)) is a function of both (s
i
, s
k
) and q(s
i
), q(s
k
), q(s
i
).
Theorem 2.1. (L(G | H)) n
G|H
bits if 0.5 in Eq. 2.21.
Proof: If (d(s
o
, s
t
))> = 0.5 then
(d(s
o
, s
t
| s
k
)) = 1 - h(d(s
o
, s
t
| s
k
) | H)
and
(L(G|H)) = n
G|H
-
H|nG
1k
h
i
(d(s
o
, s
t
| s
k
) | H)
Since
h(d(s
o
, s
t
| s
k
) | H) 0 if (d
i
(s
o
, s
t
| s
k
)) 1 or 0
then
h
i
(d(s
o
, s
t
| s
k
) | H) < n
G|H
Therefore
(L(G | H)) n
G|H
bits.
as the number d
i
(s
o
, s
t
| s
k
) | (d
i
(s
o
, s
t
| s
k
)) 1 or 0 increases, while main-
taining 0.5.
This is accomplished if for each (d
i
(s
o
, s
t
| s
k
)) 1 there is another
(d
i
(s
o
, s
t
| s
k
‘
)) 0 such that
(d
i
(s
o
, s
t
| s
k
)) + (d
i
(s
o
, s
t
| s
k
‘
)) 1
Corollary 1: (L(G | H)) bits, 0 as 1 or 0 in Eq. 2.21.
Proof: If (d
i
(s
o
, s
t
))> 1 or 0 then
(d(s
o
, s
t
| s
k
)) = - h(d(s
o
, s
t
| s
k
) | H), 0
and
(L(G | H)) = * n
G| H
-
HnG
k
|
1
h
i
(d(s
o
, s
t
| s
k
) | H)
Since
h(d(s
o
, s
t
| s
k
) | H) 1 if (d
i
(s
o
, s
t
| s
k
)) 0.5,
otherwise
h(d(s
o
, s
t
| s
k
) | H) 0.
Then
2.3 Ambiguity 35
HnG
k
|
1
h
i
(d(s
o
, s
t
| s
k
) | H) 0
and
(L(G | H)) = * n
G|H
-
such that
(L(G | H)) * bits, 0
Biological remark 2.4. The expressiveness (L(G | H)) of a given lan-
guage L(G | H) supported by G in H is a key issue in understanding adap-
tation and evolution. Evolution and adaptation are allowed in those cases
where L(G | H) is expressible. In contrast, extinction will imply the expres-
siveness is greatly reduced. The actual value of (L(G | H)) is mostly de-
termined by the restrictions imposed by H, and determining ((d(s
i
, s
j
) |
H)). The restrictions imposed by H are among the driving forces of evolu-
tion, because they control what the expressible d(s
o
, s
t
| s
k
) are that shape a
living being. The incompleteness of the genetic code defines the possible
expressible d(s
o
, s
t
| s
k
)s. Because of this, ((d(s
i
, s
j
) | H)) becomes a gen-
eral formalization of H in Eq. 2.5.a.
Definition 2.3: The space S in the environment H is said to be a pro-
cessing space for the fuzzy grammar G if it guarantees (L(G | H)) > 0 that
is
V
t
(H | S) = {s
t
V
t
| ((d(s
o
, s
t
) | H,S ) 1)}
If ((d(s
k
, s
j
) | H) is inhomogeneous in S, the processing space S can
be partitioned into a set of n subspaces {S
m
}
m=1 to n
that become specialized
spaces for processing defined sets of derivations chains d(s
k
, s
j
) of G.
Therefore, each S
m
may become a specialized space for processing defined
(d(s
o
, s
t
| s
k
).
Biological remark 2.5: The concept of processing space S is not to be
mistaken with the geographic location l of H, but it must be assumed as a
defined physical subspace, corresponding, for example, to that occupied
by an organism‘s body. In this context, S
m
, S
n
will denote two distinct ele-
mental bodies, or cells.
Definition 2.3.a: Two processing spaces S
m
, S
n
for the fuzzy grammar
G are said to be compatible S
m
S
n
processing spaces of G if and only if
((d(s
i
, s
j
) | H, S
m
), ((d(s
i
, s
j
) | H, S
n
) are such that
V
t
(H | S
m
) V
t
(H | S
n
) .
36 2 The Cells of the Brain
Definition 2.3.b: The degree of compatibility (S
m
, S
n
) of S
m
, S
n
as pro-
cessing spaces of G in H is measured by the similarity (V
t
(H | S
m
), V
t
(H |
S
n
)) between V
t
(H | S
m
) and V
t
(H | S
n
) such that
(S
m
, S
n
) = (V
t
(H | S
m
), V
t
((H | S
n
)) = c
i
,
j
max (c
i
, c
y
)
where c
i,j
is the cardinality of V
t
(H | S
m
) V
t
(H | S
n
), and c
i
, c
j
are the car-
dinality of V
t
(H | S
m
) and V
t
(H | S
n
), respectively.
Definition 2.3.c: If the transcription s
i
s
j
is performed at the
time t
i
at the processing space S
m
of G, the result of this transcription will
be available at the time t
j
= t
i
+ at the processing space S
n
of G, where
is the finite time restriction imposed by H. Also, let
<(d(s
i
, s
i
) | H, S
m
, S
n
, )>
be the mean value of
(d(s
i
, s
i
) | H, S
m
,
S
n
, (t))
of moving all s
i
V* from S
m
to S
n
in H.
Biological remark 2.6. The temporal restrictions imposed by H are
mainly those associated with the dynamics of the intra and intercellular
transportation systems in charge of moving proteins inside and between
cells, and DNA and RNA between cell compartments.
2.4 The Hierarchy of Fuzzy Grammars
According to Searls (2002), the dynamics of both genes and proteins must
be modeled by grammar above the level of the context sensitive (Type 1
grammar - Hopcroft and Ullman, 1969) grammars in the Chomskyan hier-
archy. This is one of the reasons for introducing the following hierarchy of
the Recursive Enumerable Grammars:
1. Replicating Grammars: to formalize properties and consequences of
DNA duplication;
2. Self-controlled Grammars: to provide the tools to control grammar am-
biguity and to improve adaptability; and
3. Recombinant Grammars: to formalize properties and consequences of
the sexual reproduction to life evolution.
Definition 2.4. Replicating Grammars G
®:
Let there be the grammar
G
®
= {V
o
, D V
n
, V
t
, R P, )
(2.26)
where
2.4 The Hierarchy of Fuzzy Grammars 37
D = {s
d
D | (d(s
d
, s
m
) ) > 0 for each s
y
V
*
}
(2.27)
and
R = { s
d
s
1
s
i
s
d
, s
d
D}
(2.28)
D is said to encode the symbols of G under the rules R. Both D and R de-
fines de genetic G describing V*, that is
G
®
= (D, R, V*)
(2.29)
The genetic G allows G
®
to be copied or duplicated from the processing
space S
m
to S
n
in H if for all s
d
D such that (d(s
d
, s
d
) | H
m
) 1 there
exists such that (d(s
d
, s
d
) | H, S
m
, S
n
, ) 1.
Biological remark 2.7.a. D is the set of genes encoding the genetic in-
formation of a given living being. Any gene s
d
is composed by two sub-
strings, one of the encoding one or more proteins s
k
of this living being,
and the other being part of the gene reading control mechanism. In this
context, ((d(s
d
, s
d
) | H, S
m
)) describes the constraints over the process of
gene copying and ((d(s
d
, s
d
) | H, S
m
, S
n
, )) describes the constraints
over the mitosis and meiosis at the cellular level. In the same line of rea-
soning, ((d(s
d
, s
k
) | H, S
m
)) depends on the constraints imposed upon the
gene reading in the process of protein making, whereas the constraints on
protein making are encoded by ((d(s
d
, s
d
) | H, S
m
, S
m
, )) in the same
processing space (cell) S
m
.
Let D(H | S
i
), D(H | S
m
) be the original and copied fuzzy sets of s
d
s of
G
®
. Since the ambiguity of G
®
, may produce s
d
‘
D(H | S
m
) as a copy of
s
d
D(H | S
m
), whenever s
d
‘
, s
d
) > .5, the fidelity
G
®
(H | S
m
, S
n
) of the
duplication of G
®
from S
m
into S
n
in H is defined here as
G
®
(H | S
m
, S
n
) =
1
5.0
max
( * d max (c
i
, c
j
))
(2.30)
where d is the cardinality of D
(H | S
m
) D
(H | S
n
), and c
i
, c
j
are, respec-
tively, the cardinalities D
(H | S
m
) and D
(H | S
n
). The actual values of d,
c
i
, c
j
are dependent on both
((d(s
d
, s
d
) | H, S
m
)) and ((d(s
d
, s
d
) | H, S
m
, S
n
, ))
because
G
®
(H | S
m
, S
n
) 1
iff (d(s
d
, s
d
) | H, S
m
, S
n
, ) * (d(s
d
, s
d
) | H, S
m
) 1
38 2 The Cells of the Brain
Thus it may be assumed that
G
®
(H | S
m
, S
n
) = (d(s
d
, s
d
) | H, S
m
, S
n
, ) * (d(s
d
, s
d
) | H, S
m
)
(2.30b)
Biological remark 2.7.b. Gene mutation is defined when a gene copy s
d
‘
of s
d
is such that (s
d
‘
, s
d
) 0. The set D of G
®
corresponds to the ge-
nome of any living being, and the notion of gene mutation in this paper is a
consequence of the ambiguity of G
®
, in addition to possible errors in gene
copying. Therefore, gene mutation is dependent on the behavior of (d(s
d
,
s
d
) | H, S
m
). In this context, the rate of mutations is inversely related to
G
®
(S
i
, S
j
| H), because d in Eq. 2.30 will decrease as the number of muta-
tion augments. Also, the genetic information (D, S
n
) will be modified in
respect to (D, S
m
) if (d(s
d
, s
d
) | H, S
m
, S
n
, ) is changed. In this context,
the rate of possible alterations of (D, S
m
) is inversely related to
G
®
(S
i
, S
j
|
H).
Definition 2.5. Self-controlled Grammar: Let there be a replicating
grammar
G
©
= {V
o
, V
n
, C V
t
, E P, )
(2.32)
C is the set of s
t
V
t
that controls the amount of copies q(s
i
); q(s
j
)
or the
matching capability μ(s
i
, s
j
) of s
i
, s
j
V
*
, that is
{s
c
V
t
| [(q(s
i
); q(s
j
)
= g(q(s
c
)), μ(s
i
, s
j
) = z(q(s
c
)), s
i
, s
j
V
*
]}
(2.33)
A given s
c
C exerts its control over the q(s
j
) of s
j
V* by:
1. promoting the decoding of s
d
: in this case
d(s
d
, s
j
) = s
d
s
c
s
j
, or
2. enhancing or inhibiting the decoding of s
d
: in this case s
c
substitute
more (or less) efficiently another d(s
k
, s
l
) of d(s
d
, s
j
):
d(s
d
, s
j
) = s
d
sk
d
s
l
s
j
d(s
d
, s
j
) = s
d
s
c
s
m
s
j
3. changing μ(s
i
, s
k
) of any intermediate s
i
s
j
d(s
d
, s
j
):
in this case s
c
becomes an intermediate step of d(s
i
, s
k
) of d(s
d
, s
j
)
d(s
i
, s
j
) = s
i
s
c
s
j
As a matter of fact, the above actions may be exerted over any d(s
o
, s
j
)
supported by G
©
.
Biological remark 2.8. In the case of genes, the actions in a and b are
those ascribed to the promoters, enhancers and inhibitors of the DNA read-
ing, whereas that of action d is that exerted by polymerase A. In the case of
2.4 The Hierarchy of Fuzzy Grammars 39
any other stp, the actions a and b correspond to the triggers, facilitators and
inhibitors of any of its steps, whereas the action d corresponds to that of
those enzymes phosphorylating the proteins in this stp. The control al-
lowed by C over (d(s
i
, s
j
)) is the main tool Nature has invented in order to
enhance adaptation over an increasing number of different environments
Hs. This is because C allows the organism S
k
to try to maintain ((d(s
i
,
s
j
)| H, S
k
)) as constant as possible, where the restrictions imposed by the
different H‘s would make survival unlikely.
Definition 5.a. For each
d(s
o
, s
j
| s
k
) = s
o
s
i
s
k
s
j
(2.34)
there exists a set of genes describing their symbols
D = {s
d
D | d(s
d
, s
k
) for each s
k
in Eq. 34}
(2.35)
the control set C of D
C = {s
c
C | (q(s
d
)
=
g(q(s
c
))}
(2.36)
and the set of genes specifying C
E = {s
e
D | d(s
e
, s
c
) for each s
c
C}
(2.37)
The set E is called here the set of early genes, each of them controlling the
expression of a given c C, which in turn controls the expression of the
descriptive genes in D in charge of specifying the other elements s
k
V
*
.
Eqs. 34 to 37 comprise a hierarchical fuzzy tree whose purpose is to guar-
antee the existence of all s
k
involved in a given d(s
o
, s
j
).
Proposition 2.2: The set of early genes E specifies the root node set of a
fuzzy tree GN
i
and are connected to those first order of intermediary nodes
are associated to C, which in turn are linked to the set of nodes represent-
ing D. The D nodes are finally linked to the terminal nodes representing
each s
k
involved in d(s
o
, s
t
). GN
i
is called here the genetic network control-
ling d(s
o
, s
t
). N
i
specifies the grammar of such control, which is considered
to be a sub-grammar of G
©
.
Proof: This is a consequence from the fact that E is in charge of control-
ling of the expression of D and thus of the dynamics of a serial ordered
chain of rewriting rules s
i
s
k
Therefore, E imposes a hierarchy
H over d(s
o
, s
t
).
In this context, the genetic G
©
supported by D, R may be considered as
a family of GN
i
each one controlling one of the m families {d(s
o
, s
t
)}
k=1 to n
of stps supported by G
©
and specifying a given cellular function
m
:
40 2 The Cells of the Brain
G
©
= {GN
i
}
i = 1 to m
(2.38)
The resources constraints imposed by H over G
©
implies
q(s
d
) 1 for each s
d
GN
i
(2.39)
and the number of copies q(GN
i
) of GN
i
to be maintained as minimum as
possible
1 q(N
i
) < for each GN
i
G
©
, 2
(2.40)
Biological remark 2.9. All living beings (some viruses being perhaps the
only exceptions), are examples of the self-controlled grammar G
©
generat-
ed from the replicating grammar G
®
supported by DNA. Viruses are con-
sidered an exception because they use the control machinery of the infect-
ed cell. In this paper, self-control will be proved to be a very important
tool to both increase adaptation and influencing the evolution of the spe-
cies. The complexity of G
©
greatly increases in respect to that of G
©
be-
cause the structure of each GN
i
G
©
describes a sub-grammar specifying
how and when any stp is produced and organized as a specific combination
of proteins belonging to V
t
.
Definition 2.6. Recombinant Grammars: Let there be the self-controlled
grammar
G
= {V
o
, D
V
n
, V
t
, R, M P, )
(2.41)
where M P is a set of rules allowing:
D
to be divided into new D
♀
, D
♂
, such that q(N
i
) 1 at D
♀
, D
♂
and
two copies:
G
♀
(S
m
) = {V
o
, D
♀
V
n
, V
t
, R, M P, q(N
i
) 1
N
i
G
)
G
♂
(S
m
)
= {V
o
, D
♂
V
n
, V
t
, R, M P, q(N
i
) 1
N
i
G
)
(2.42)
of G
(S
m
)
at S
m
are move to two different sub-processing spaces of S
m
♀
,
S
m
♂
in H, and the recombination G
♀
(S
m
) G
♂
(S
n
) of G
♀
(S
m
), G
♂
(S
n
)
from two different sub-processing spaces S
m
♀
, S
n
♂
into G
(S
s
) such that
D
s
= D
m
♀
D
n
♂
and
G
= {N
i
}
i = 1
to c‘
{N
i
}
i =1
to c‘‘
,
c‘, c‘‘ c the cardinality of G
i
Biological remark 2.10.The recombinant grammars G
i
generated from
the DNA replicating grammar G
®
support sexual reproduction and greatly
2.5 Fuzzy Languages, Distributed Processing and Biological Diversity 41
facilitates the increase of the complexity of G
®
required to promote com-
plex animals, such as man, in evolution. The steps in 1 are the main steps
of meiosis, and those in 2 are the main events of fecundation. G
♀
(S
m
) and
G
♂
(S
m
) are the female and male gametes, respectively.
For the sake of simplicity, G will denote in the rest of this article any
grammar G
®
, G
©
or G
, such that any of these special notations will be
used only if necessary to stress properties not shared by all these grammar
types.
2.5 Fuzzy Languages, Distributed Processing and
Biological Diversity
4.1. Definition: A language L(G) supported by G is composed by the
set of all d(s
o
, s
t
) converting s
o
V
o
into s
t
V
t
and can be defined as
L(G) = {d(s
o
, s
t
) | s
o
V
o
, s
t
V
t
}
(2.43)
Any d(s
o
, s
t
),d(s
o
, s
t
´
) are said to be equivalent interpretations of s
o
in L(G),
that is d(s
o
, s
t
) d(s
o
, s
t
´
), if (s
t
, s
t
´
)1
Biological remark 2.11. Each d(s
o
, s
t
| H,S ) is supposed to describe a
given stp in the real cell.
Definition 2.7. The hypothetical L(G) is said to be an expressible lan-
guage Ł(G | H, S) in the space S if and only if there exists at least one H
providing ((d(s
o
, s
t
) | H, S)) such that (d(s
o
, s
t
for at least one
s
o
V
o
. In this context
Ł(G | H, S) = {d(s
o
, s
t
) | (d(s
o
, s
t
H, S
V
o
(L | H, S) = {s
o
V
o
| (d(s
o
, s
t
H, S for V
t
}
V
t
(L | H, S) = {s
t
V
t
| (d(s
o
, s
t
H,S for all s
o
V
o
(L | H, S)}
V
n
(L | H, S) = {s
i
V
n
| s
i
is part of at least oned(s
o
, s
t
of Ł(G | H, S)}
V
o
(L | H, S) is the set of all expressible s
o
V
o
and V
t
(L | H, S) is the
set of expression of L(G) at S, given H. Ł(G | H, S) is composed of all
expressible derivations chains d(s
o
, s
t
for which (d(s
o
, s
t
H, S
If S
m
, S
n
S are processing subspaces Ł(G | H, S), such that
V
t
(L | S
m
) V
o
(L | S
n
) , V
o
(L | S
m
) V
t
(L | S
n
)
(2.44)
42 2 The Cells of the Brain
then the processing of Ł(G | H, S) is said to be distributed over {S
m
, S
n
} or
Ł(G | H) is said to be a distributed processing language.
Fig. 2.3. The brain as a distributed G processor: different types of neurons are
specialized to process different subsets of L(G).
Remark 2.12.a. The several expressed languages Ł(G | H, S) of the
grammar G, supported by the DNA replicating grammar G,
correspond to
the different types of living beings existing or having existed on earth. In
this context, Ł(G | H, S
i
) the language characterizing a type of living entity
S
i
. In the case of unicellular organisms S
u
or colonies of S
u
s, the processing
space S
u
is unique. Now, if S = {S
m
k
}
k=1 to n
then Ł(G | H , {S
m
k
}
k=1 to n
) is a
multi-cellular organism S
m
, composed of n types of cells and expressing n
types of languages Ł(G | H, {S
k
}) of those supported by G. These different
types of cells are part of the different organs in a multi-organ being. Each
organ will contain many cells of each type; thus, each organ express a fam-
ily of languages Ł(G | H, {{S
k,m
}
k=1 to n
}
m=1 to r
), where n is the number of
types of its cells and m is the number of each of these cell types. The brain
B is one of these organs and it is constituted of a subset of these processing
spaces S
i,k
; that is, B = {{S
i,k
}
i=1 to n
}
k=1 to m
is composed of m neurons of n
types. In this context Ł(G | H, B) is the knowledge an animal may use to
survive in H.
2.6 Knowledge Adaptation and Evolution 43
The constraint (d(s
o
, s
t
H, S 1 implies that (s
t
V
t
) is bounded in-
to a limited interval or
(s
t
V
t
)
t
s
t
V
t
(L | S), where
t
t
& 0
t
0.5
(2.45)
This means that V
t
(L | H, S) is the
t
cut level set of V
t
(Pedrycz and
Gomide, 1998) having at least one s
o
V
o
and one s
t
V
t
such that
(d(s
i
, s
t
H, S= 1. Because of this, each Ł(G | H, S) is uniquely identi-
fied by its characteristic set
V
t
@
(L | H, S) =
1
max
{s
t
V
t
| (d(s
o
, s
t
H, S
t
1,
(2.46)
the maximum non-empty
m
cut level set of V
t
. The power of Ł(G | H, S)
is then the number p of those d
i
(s
o
, s
t
(d(s
o
, s
t
m
1 which support
V
t
(L | H, S)
@
.
The similarity of two languages (Ł(G | H, S), Ł(G | H‘, S‘)) is calculat-
ed as
(Ł(G | H, S), Ł(G | H‘, S‘)) = l
i
,
j
max (l
i
, l
j
)
(2.47)
where l
i
,
j
is the cardinality of V
t
@
(Ł | H, S ) V
t
@
(Ł | H‘, S‘ ) and l
i
, l
j
are
the cardinalities of V
t
@
(Ł | H, S ) and V
t
@
(Ł | H‘, S‘), respectively.
Biological remark 2.12.b. Any Ł(G | H, S) is supposed to support a
family of similar, but not identical, living beings Ł(G H, S
i
) as a conse-
quence of the ambiguity in G. This diversity is a very important issue in
understanding evolution and speciation.
2.6 Knowledge Adaptation and Evolution
The ambiguity of G, together with the restrictions imposed by H over
((d(s
i
s
j
) | H)) will be used here to model the evolution of the expressed
languages Ł(G | H, S) supported by the grammar G in the attempt to under-
stand how learning is accomplished by the brain B as a distributed proces-
sor of G.
Theorem 2. 2. Given Ł(G | H, S) as a expressed language of G in H and
Ł(G | H‘, S‘) as a possible expression of G in H‘ if |
H´
– 0.5 | |
H
– 0.5 |
then Ł(G | H‘, S‘) is adapted to H‘.
44 2 The Cells of the Brain
Proof: From corollary 2.1, (Ł(G| H‘, S‘)) decreases and Ł(G| H‘, S‘)
becomes inexpressible as
H
0 or 1, and theorem 2.1 shows that (Ł(G |
H‘, S‘)) increases as
H
0.5.
Hence if |
H´
– 0.5 | |
H
– 0.5 | then Ł(G | H‘, S‘) is as expressible in
H‘ as fit in H or more expressible in H‘ than Ł(G | H, S) in H.
But if Ł(G | H, S) is already expressed in H then Ł(G | H‘, S‘) is also ex-
pressed in H‘.
Therefore Ł(G | H‘, S‘) is adapted to, or fits in, H‘.
Corollary 2.2. If (Ł(G | H, S), Ł(G | H‘, S‘)) = 1 - , then Ł(G |
H, S) is as fit in H‘ as in H. Otherwise Ł(G | H‘, S‘) is a new language and
fitter in H‘ than in H. Therefore, Ł(G | H‘, S‘) is an evolution of Ł(G | H,
S) in H‘.
Proof: From Eq. 2.46, if
(Ł(G | H, S), Ł(G | H‘, S‘)) = 1 - ,
then Ł(G | H, S) and Ł(G | H‘, S‘) are the same language, and Ł(G | H, S) is
therefore adapted to H´. Otherwise Ł(G | H‘, S‘) is a new language in re-
spect to Ł(G | H, S). Therefore Ł(G | H‘, S‘) is an evolution of Ł(G | H, S)
in H‘.
Theorem 2.3. The fitness A(Ł(G | H)) of Ł(G | H) in a given environ-
ment H‘ is a function of < (V
i
| H )> of V
n
.
Proof: From Eq. 2.23:
< (V
n
| H)> = - (<(d(s
i
, s
j
))> log <(d(s
i
, s
j
))>
- (1- <(d(s
i
, s
j
))>) log (1- < d(s
i
, s
j
))>)) V
n
such that < (V
i
| H )> increases if the cardinality of V
n
is increased and
<(d(s
i
, s
j
))> 0.5.
In such a condition, if (d(s
o
, s
t
) | H) is likely that (d(s
o
, s
t
) | H‘)
because
(d(s
o
, s
t
)) = max min(d(s
o
, s
k
) ..., d(s
i
, s
j
), d(s
m
, s
n
), d(s
r
, s
s
) ..., d(s
l
, s
t
))) V
n
and it is likely that here exists d(s
i
, s
j
) ,d(s
m
, s
p
), d(s
r
, s
j
) in H and
(d(s
i
, s
j
),H‘) 1.
Therefore, the greater < (V
i
)> is, the greater is the number of envi-
ronments H‘ for which all (or at least almost all)(d(s
o
, s
t
) | H)) com-
posing Ł(G | H) are maintained as (d(s
o
, s
t
) | H‘)) In such a condi-
tion:
(Ł(G | H, S),Ł(G | H‘, S‘)) = 1 - ,
2.6 Knowledge Adaptation and Evolution 45
Hence, the greater < (V
i
)> is, the greater is the number of environ-
ments H‘ for which Ł(G | H, S) is adaptable.
Theorem 2. 4. The capacity E(Ł(G | H)) of Ł(G | H) to evolve in a given
environment H‘ is function of < (V
t
| H )> of V
t
.
Proof: From Eq. 2.23
< (V
t
| H )> = - (<(d(s
l
, s
t
))> log <(d(s
l
, s
t
))> -
(1- <(d(s
l
, s
t
))>) log (1- < d(s
l
, s
t
))>)) V
t
such that < (V
t
| H )> increases if the cardinality of V
t
is increased and
<d(s
i
, s
t
))> 0.5.
In such a condition, if
(d(s
o
, s
t
) | H) and (d(s
o
‘
, s
t
‘
) | H) 0.5
it is likely that
(d(s
o
, s
t
) | H) 0.5and (d(s
o
‘
, s
t
‘
) | H)
and
(Ł(G | H, S), Ł(G | H‘, S‘)) = 1 - , 0.5.
Therefore, Ł(G | H‘, S‘) tends to be an evolution of Ł(G | H, S).
Hence, the greater < (V
t
)> is, the greater is the number of those envi-
ronments H‘ where Ł(G | H, S) evolves.
Remark 2.13. Theorem 2.2 and its corollary show how a given brain
may adapt or evolve a given knowledge Ł(G | H, B) in response to a
changing environment H‘ by maintaining or improving those restrictions
(
H‘
1) upon G
®
, required to keep Ł(G | H‘, B) expressible in the new
conditions H‘. Theorems 2.3 and 2.4 set the conditions in which this oc-
curs. Therefore, the greater the ambiguity of a given genetic grammar, the
greater is the genetic plasticity for adaptation to changing environments.
Similarly, the conditions for the rejection of Ł(G | H, B) as an adequate
knowledge of new environments are derived from the same theorems. A
given Ł(G | H, B) will be rejected if restrictions imposed by the new condi-
tions H‘ upon G reduces the expressiveness of Ł(G | H, B) in H‘ because
H‘
0 or 1.
46 2 The Cells of the Brain
2.7 Self-Controlled Expression
Life would be unlikely without the mechanisms of control exerted mainly
by the cellular apparatus of enzymatic repair of nucleic acids. Also, protein
synthesis is dependent on external sources of both energy and amino acids,
and many stps are in charge of controlling the cellular level of these raw
materials used to produce other cell components. Therefore, the expres-
siveness of any genetic language is very much dependent on the mecha-
nism of self-control promoted by a set proteins C over any d(s
o
, s
t
), that ac-
cording to definition 2.5 are in charge of either:
Controlling the production or the quantity q(s
j
) of proteins s
j
d(s
o
, s
t
)
that is:
q(s
j
)
= g(q(s
c
)), s
c
C;
or, specifying the chemical compatibility μ(s
i
, s
j
) among proteins of a giv-
en stp, that is
μ(s
i
, s
j
) = z(q(s
c
)), s
i
, s
j
d(s
o
, s
t
), s
c
C
In this way, the control Ł(C | G, H, S) supported by G is exerted by
changing (d(s
i
, s
j
)) and therefore by controlling ((d(s
i
, s
j
) | H, S)). This
control is dependent on the expressiveness (Ł (C | G, H, S)) of all those
d(s
o
, s
c
) producing s
c
C and result in modifying the original S into a con-
trolled space S
©
. In other words:
Ł (C, | G, H, S) Ł(G | H, S) Ł(G | H, S
©
)
(2.47)
The degree of control promoted by Ł(C | G, H, S) is dependent on
(Ł (C | G, H, S)).
Theorem 2.5. The control Ł (C, G | H, S) maximize (Ł (G | H, S)) if it
promotes
|
H,S
©
– 0.5 | |
H,S
– 0.5 |
and for all (d
i
(s
o
, s
t
) | H, S
©
) it promotes (d
j
(s
o
, s
t
) | H, S
©
)
such that
(d
i
(s
o
, s
t
) | H, S
©
) (d
j
(s
o
, s
t
) | H, S
©
) = 1
Proof: It is a consequence from theorems 2.1 and 2.2 and their corollar-
ies.
Theorem 2.6. Given <(G | H, S)>, <(G | H‘, S)> as the mean ambi-
guity of G expressed in H, H‘, then
| <(G | H, S)> - <(G | H‘, S)> | (Ł (C, G | H, S))
Proof: Given d(s
o
, s
t
) controlled by s
c
C then
2.8 Speeding up Brain Processing 47
(d(s
l
, s
t
)) = (..., q(s
c
) | H, S)
and in the case of a linear control
| (d(s
l
, s
t
) | H, C ) - (d(s
l
, s
t
) | H, C ) | q(s
c
)
But
q(s
c
) (d(s
o
, s
c
))
and
| (d(s
l
, s
t
) | H,C ) - (d(s
l
, s
t
) | H, C ) | (d(s
o
, s
c
))
Also, the number of individually controlled d(s
o
, s
t
) by s
c
C is de-
pendent on the cardinality of C. Therefore
| <(G | H, S)> - <(G | H‘, S)> | (Ł (C, G | H, S))
Remark 2.14. Self-controlled languages have better odds of keeping
their expressiveness in a greater number of environments than any other
replicating grammar G because of their capacity to keep their within ad-
equate boundaries. The neuron is a self-controlled grammar processing
space, whereas the brain is in charge of controlling the expression of a
self-controlled grammar in a multi-organ organism. Because of this, theo-
rems 2.5 and 2.6, combined with theorems 2.3 and 2.4, specify the condi-
tions to improve the success of adaptation and/or evolution of a given
knowledge Ł (G | H, B). In other words, these theorems set the basic rules
of the learning strategies discussed in this book.
2.8 Speeding up Brain Processing
Different alphabets A are used to compose the elements s
i
, s
j
of d(s
i
, s
j
)
(Eqs. 2.13 to 17 and Remark 2.1). One of these alphabets is supported by
ionic currents due to the movement of sodium (Na
+
) and potassium (K
+
)
ions (Fig. 2.4) across the cell membrane. Na
+
is predominant outside of the
cell, whereas K
+
is mostly an intracellular ion. These ions move across the
membrane through specific channels. The K
+
channel is relatively opened
(its conductance g
k
is relatively high) and the Na
+
is almost entirely closed
(its conductance g
Na
is almost zero) if the neuron is not activated. The elec-
trochemical equilibrium E
K
potential of K
+
is around –90mV and E
Na
ap-
proaches +40 mV. This means that, at rest, the K current is directed to the
extra-cellular space and the Na current is toward the inside of the cell. This
generates an electrical gradient EM across the membrane due to the differ-
ent ionic concentrations. EM approaches E
K
because the g
k
> g
Na
.
48 2 The Cells of the Brain
The channel state is controlled by both the actual value of the membrane
potential EM and molecules composing many stps. Most of the dynamics
of this membrane electrochemical ionic system is described by a set of
equations proposed by Hodgkin and Huxley (1952) in the first half of the
20
th
century (Fig. 2.4). These equations describe a dynamic system having
two stable equilibrium points (E
K
and E
Na
) and an unstable equilibrium
point characterizing a bifurcation (see Fig. 2.4). The actual topology of the
ionic system state space is determined by both the number and the states of
the ionic channels, i.e., it is under the control of many stps.
On the one hand, it is assumed that g
Na
is kept very low at the dendrites
such that i
Na
cannot achieve high values. Because of this, whenever a
transmitter (s
o
) attaches to and opens the Na channel (Fig. 2.4 and Fig. 2.5)
it generates a small i
Na
that promotes a reduction of EM. This EM variation
enhances g
K
and consequently i
K
what tends to counterbalance the i
Na
en-
hancement. Because of this, the state point moves around in the E
K
surface
in the ionic state space. This produces a local variation of EM, called local
response s
d
, that recodes molecular sign s
o
into a electrical sign s
d
.
Fig. 2.4. The source of the electrical code
2.8 Speeding up Brain Processing 49
On the other hand, g
Na
is allowed to reach high values at the axonic
membrane, such that now the opening of the Na channel induced by the
EM variation at the dendrite may achieve the adequate value for moving
the state point to the bifurcation zone. If this unstable zone is reached, the
state point jumps from the E
K
to the E
Na
surface and then returns back to
E
K
. This rapid EM variation is called a spike, and it is strong enough to
promote the same dynamic alterations in the nearby Na and K channels.
Fig. 2.5. The neuron: the chemical machinery
The spike is able to travel along the axon toward its terminal buttons,
where the Na and K movements release the calcium ions Ca
+2
from their
storage sites. The released Ca
+2
activates intracellular contractile proteins
(CP in Fig. 2.5) that are able to move the pre-synaptic vesicules toward the
cell membrane, in order to release a package of transmitter (s
t
) over the
next neuron. In this way, the axonic spike train (s
a
) recodes the local re-
sponse s
d
and it is recoded by s
t
(transmitter) released at the pre-synaptic
terminal button. In this way, the derivation chain d(s
o
, s
t
) describes the
neuronal transactions triggered by the release of the transmitter (s
o
) by the
presynaptic cell, that results into another release of transmitter (s
t
) by the
postsynpatic cell.
The axonic electrical recoding of the dendritic activity is necessary to
speed up chemical message exchange between neurons. The axonic elec-
tric code is dependent on the state space topography. Different types of en-
coding are used by the brain, each one corresponding to a specific state
space topography (Rocha, 1997). The movement of molecules in the axon
is slow and controlled by the so-called axonic transportation system, com-
50 2 The Cells of the Brain
posed of specialized contractile proteins and activated by means of energy
released from special molecules like ATP, which stores chemical energy in
phosphate bonds. This axonic transport moves the transmitter or its precur-
sors, produced at the cell body, to the terminal buttons where the transmit-
ters are stored in vesicles. In this way, whenever necessary, chemical in-
formation at the dendrites or cell body is recoded into a train of spikes that
are quickly transmitted along the axon to trigger the release of chemical in-
formation stored at the terminal button (Fig. 2.5).
2.9 The Chemical Talk at the Synapse
Contrary to the simplistic view of a numerically modulated synapse pro-
posed in neural net theory, the complex physiology of the synapse is here
formalized by means of a self-controlled grammar G and the expressed
languages Ł(G | H, S
i
) supported H in each neuronal processing space S
i
(Fig. 2.6).
The molecular talk at the synapse is supported by chemical transactions
in both directions, i.e., from the pre- to the post-synaptic neuron and vice-
versa. Also, the chemicals released from the pre-synaptic neuron may or
may not be linked to pre-synaptic spike firing. Many molecules are stored
at the pre-synaptic terminal button and released by local molecular pro-
cesses (e.g., a retrograde signal from the post-synaptic cell: Fig. 2.6), in
addition to the transmitters that are released as a consequence of the chem-
ical and electrochemical processing at the pre-synaptic cell body (Fig. 2.5).
Chemicals released from the pre-synaptic cell can act as initial triggers
(d(s
o
, s
t
)) of post-synaptic derivation chains d(s
o
, s
t
) which may or may not
involve DNA reading (Fig. 2.6). DNA reading may be part of the control
of ionic gates or may be promoted by some other molecule involved in the
synaptic machinery. If this is the case, then d(s
o
, s
t
) may be part of a learn-
ing process, as in the line discussed by Rocha (1992), or even in the line of
a numerically modulated synapse, as proposed in standard neural nets the-
ory.
However, the final product s
t
of d(s
o
, s
t
) may also be exported (retro-
grade signal in Fig. 2.6) and used by the pre-synaptic neuron as initial trig-
ger of other pre-synaptic d(s
o
, s
t
). The retrograde signal may even be trans-
ported toward the cell body to control the pre-synaptic DNA reading (Fig.
2.5 and 2.6). Rocha (1997) discussed the role played by this post-synaptic
control of the pre-synaptic DNA reading. For instance, dual talk between
pre and post-synaptic neurons is very important during the stage of embri-
2.9 The Chemical Talk at the Synapse 51
ogenesis as well as during adult life, in controlling either the neurons‘ vi-
tality or death (apoptosis).
Fig. 2.6. Synaptic chemical talk: a complex set of chemical transactions take
place at the synapse.
Finally, the exported pre or post-synaptic s
t
may be sequestered by other
neighboring cells (even glial cells) and may trigger other sets of d(s
o
, s
t
) in
the neighborhood, initiating multiple bilateral or multilateral talks among
different types of cells. In this way, the local processing achieves a com-
plexity which depends on the expressiveness L(G | H) of G, and the local-
ity of this processing is mainly determined by the temporal restrictions
imposed by <(G | H, S
i
, S
j
, )> in distributing s
t
in the neighborhood.
52 2 The Cells of the Brain
2.10 Summary
The neuron is formalized here as a device handling a subset Ł(G | H, S
i
) of
the formal language Ł(G | H), supported by a grammar G in the environ-
ment H. Each derivation chain d(s
o
, s
t
) of Ł(G | H, S
i
) describes an ordered
set of chemical transactions or stp that characterizes a neuronal function.
The expressiveness of d(s
o
, s
t
) is measured by the possibility (d(s
o
, s
t
) | H,
G) of their symbol rewritings (or chemical transactions), and it is depend-
ent on both the resource restrictions imposed by H and the self-control
Ł(C, | G, H, S) allowed by G. The brain B is then formalized here by the
set Ł(G | H, B) of the languages Ł(G | H, B), expressed by each type S
i
of
neuron composing it. The learnability of any d(s
o
, s
t
) in Ł(G | H, S
i
) is,
therefore, determined by both the restrictions imposed by H and the total
expressiveness (Ł (C, G | H, B) of the self-controlling languages of Ł(G |
H, B). The classical learning procedures used in neural net theories may
apply to the neuron described here, for (d(s
o
, s
t
) | H, G) is dependent on
the available quantities of the symbols s
i
d(s
o
, s
t
). Moreover, new learn-
ing procedures may be developed too, exploring the symbolic properties of
Ł(G | H, B).
3.1 Distributed Intelligent Processing Systems 53
3 Brain: A Distributed Intelligent Processing
System
The brain is characterized as a Distributed Intelligent Processor of a Dis-
tributed Fuzzy Formal Language. This type of modeling takes into consid-
eration recent findings concerning the physiology of the brain, as disclosed
by many different brain mapping techniques, such as PET, fMRI, EEG
mapping, etc., which allow cognitive functions to be studied in both nor-
mal and disabled human beings. The formalization introduced in this chap-
ter provides the theoretical background required for the understanding of
the brain as a complex computational device handling numerical, symbol-
ic, and quantum calculations. Each neuron will be considered as a pro-
cessing space of a subset of derivations supported by a grammar G. In this
way, each neuron will be considered as a specialized processing subspace
of G and the result of any cerebral processing will be assumed to be the re-
sult of the distributed processing of G by a collection of neurons recruited
for such a purpose.
3.1 Distributed Intelligent Processing Systems
The following is proposed:
Definition 3.0: A natural or artificial intelligent entity is a system able
to efficiently find new solutions to new problems. But the existence of
problems suggests both goals and a lack of strategies (plans, models) for
efficiently using available tools in order to achieve those goals. To have
goals means to have motivation (appetite) to do (obtain) something (e.g.,
survive). Knowledge of failure in achieving goals demands tools to match
performance and goals; matching, in turn, requires memory to store the da-
ta used in the evaluation process. Efficiency implies doing a good job un-
der cost and time constraints; effectiveness is evaluated by success in sur-
vival. Time and cost efficiency is more likely to be achieved if new
solutions may be built up from reorganization and/or generalization of old
models — from reselection of tools and capabilities, etc. But whenever
necessary, true innovation must be achieved.
54 3 Brain: A Distributed Intelligent Processing System
Intelligence is a very real property of certain systems, called Distributed
Intelligent Processing Systems (DIPS), formed by collections of loosely
interacting specialized agents. Agents specialize in data collection (sen-
sors), problem solving (experts), data communication (channels), acting
upon the surrounding environment (effectors), etc. Intelligence is then ap-
proached in terms of a society of communicating specialized experts and
the brain is then an example of a natural DIPS (Chandrasekaran, 1981;
Davis and Smith, 1983; Ferber, 1999; Fox, 1981; Hewittt and Inman,
1991; Knight, 1997; Lesser, 1991; Maunsell and Ferrera, 1995; Rocha,
1992; 1997, Rocha et al, 2001).
DIPS reasoning is the cooperative activity among an optimally decen-
tralized and loosely coupled collection of experts that may provide the so-
lution of a problem. Decentralized means that both control and data are
logically and often spatially distributed; there is neither global control nor
global data storage. The programming intends to build models in which the
control structure emerges as a pattern of passing messages among the
agents being modeled. Task distribution is an interactive process between
an agent with a task to be executed and a group of other agents that may be
able to execute the task.
In this kind of system, intelligence is a function of the types of agents
composing the system, but also of the means and purposes with which the-
se agents are used. Intelligence becomes dependent on both the behavior of
the specialized agents that are in charge of solving specific tasks and,
above all, on the versatility of the relations shared by these specialized
agents — or, the plasticity of the commitments to actions among these
agents. Of course, the complexity of the tasks solvable by a DIPS deter-
mines the number of agents to be enrolled in their solution.
For instance, the task of counting (Rocha and Massad, 2002, 2003a) re-
cruits a set of sensory agents in order to visually inspect the environment
and to identify the elements to be counted (Fig. 3.1.). Counting tasks also
need to recruit motor control agents in charge of positioning the eyes over
the elements to be counted as well as controlling the fingers to point to, or
otherwise mark, the identified elements. Also, other agents are in charge of
accumulating the number of identified elements and encoding the results
into words or numerals. Learning to count, therefore, implies the control of
DNA reading in order to create specialized agents for accumulating and
classifying data, as well as for controlling the communication resources
among these agents at the synaptic level (Rocha and Massad, 2003a). Neu-
ronal specialization is assumed here to be the consequence of the control
exerted over G
(d)
to define the language L(G
(d)
| H, n
i
) to be expressed by
the neuron n
i
and learning is assumed to be controlled through specific
stps or d(s
o
, s
t
)s (Rocha, 1997).
3.2 Distributed Processed Languages 55
Fig. 3.1. Counting as a distributed task: different neurons take charge of the
different tasks in the counting process.
There is one important respect in which our model differs from tradi-
tional neural network models. In the latter kind of model, all neurons are
considered to be identical machines, which differ only in their input-output
relations. In our approach, neurons are considered to be internally struc-
tured by thousands of stps, formalized by the d(s
o
, s
t
) supported by a self-
controlled grammar G
(d)
. The language Ł(G
(d)
(H,S)) expressed by neu-
rons‘ cells, as discussed in Chapter 2, specialize them as agents that inter-
act dynamically in a DIPS.
3.2 Distributed Processed Languages
In the present context, the set of expressed languages {Ł(G(H,S
c
))}
c=1 to n
is
supported by a grammar G
(d)
.
Definition 3.1: Distributed Processed Grammars (DPG) are a self-
controlled grammar of the type
G
(d)
= {V
o
, O D V
n
, I V
t
, S P, }
where
56 3 Brain: A Distributed Intelligent Processing System
a) O is a set of genes in charge of specifying the expression of each
cellular language Ł(G
(d)
(H, S
c
)) of {Ł(G
(d)
(H, S
c
))}
c=1 to n
in their
corresponding type of processing spaces S
c
;
b) I is the set of symbols (inducers) used to specify each Ł(G
(d)
(H,
S
c
)) such that d(s
o
, s
i
| H, S, O, ), s
o
O, s
i
I
c) S is a set of rules, such that for d(s
d
, s
t
| H, S, ) in Ł(G
(d)
(H, S
c
))
q(s
d
) = (q(s
i
)) and/or = (q(s
i
))
(d(s
d
, s
t
| H,S, )) 1 iff
m
< <
n ,
s
d
D
(3.1)
d) there exist Ł(G
(d)
(H, S
i
)), Ł(G
(d)
(H, S
j
)) Ł(G
(d)
(H, S
c
)) such that
V
t
(L | S
i
, H) V
o
(L | S
j
, H) ≠ ;
V
o
(L | S
i
, H) V
t
(L | S
j
, H) ≠
(3.2)
Definition 3.2: G
(d)
is said to be a hierarchical distributed grammar
whenever there exists Ł(G(H, S
b
)) Ł(G(H, S
c
)) such that
V
t
(L | S
b
, H) V
o
(L | S
c
, H) ≠
V
o
(L | S
b
, H) V
t
(L | S
c
, H) ≠
For any other Ł(G(H, S
c
)) in {Ł(G(H, S
c
))}
c=1 to n
(3.3)
Definition 3.3: The organ O(G
(d)
(H, S
e
)) is the family of n processing
spaces S
c
, each S
c
= {(S
c, d
)}
d=1 to m
composed of m cells c
cj
in charge of
processing a given Ł(G
(d)
(H, S
c
)) of a distributed G
(d)
, that is
O(G
(d)
(H, S
e
)) = {Ł(G
(d)
(H, S
c,d
))}
d=1 to mc
}
c=1 to n
(3.4)
such that
V
t
(L | H, S
c, i
) V
o
(L | H, S
c, j
)
V
o
(L | S
c, i
) V
t
(L | H, S
c, j
)
(3.5)
For the sake of simplicity, from now on G
(d)
will be denoted by G.
3.2 Distributed Processed Languages 57
The size s of O(G(H, S
j
)) is its total number of processing subspaces,
that is
s =
n
c
cm
1
(3.6)
If (Ł(G(H, S
c, i
)), Ł(G(H, S
c, j
))) 1 for any i, j then O(G(H, S
c
)) is
called a simple organ, otherwise it is called a complex organ.
Definition 3.4: The multi-organ organism O(G(H, S
j
)) is composed by a
family of organs O(G(H, S
e
)) that is O(G(H, S
j
)) = {O(G(H, S
e
))}
e=1 to l.
Remark 3.1: The grammar defines a multi-organ organism O(G(H, S
j
))
and each cell type S
i
is in charge of processing a subset Ł(G(H, S
c
)) of the
language Ł(G(H, S
i
))}
i=1 to n
supported by G. In this line of reasoning, each
cell of an organ is assumed to be the processing space of a specific set of
the chemical language defined by the grammar specified by the genome of
the plant or animal. An organ may have just one type of cell, and called a
simple organ, or it may have a family of different cells, each one handling
a different set of signal transduction pathways for expressing a specific
subset of the genetic language. The process of specifying the family of
languages Ł(G(H, S
c
))
of type O(G(H, S)) is called embriogenesis, and it is
described by the language Ł(O(H, S)), supported by O D V
n
and S
P. The genes in O are called homeobox genes. These genes are in charge of
when and for how long a specific set of genes are activated to build a given
organ. The symbols of I correspond to the promoters of early genes E con-
trolled by O. Each homeobox gene controls the build up of a organ or a
part of the organism by controlling the expression of the early genes re-
quired to specify Ł(G(H, S
c
)). The set S P is composed by those rules
governing the activation of the homeobox genes and the control of these
genes upon the early genes.
Theorem 3.1: The mutations of a homeobox gene of O changes the
structure of the descendant O(G(H, S
j+n
)) of O
i
(G(H, S
j
)) supported by a
given set of expressible languages {{Ł(G(H, S
c, d
))}
d=1 to k
}
c=1 to l
either be-
cause (1) they change the cardinality of 0, modifying the number l of or-
gans of O(G(H, S
j
)) because they create new processing spaces or cells for
a new subset Ł(G(H, S
n, d
)) of the genetic language defined by G; or
(2) because they alter I and consequently change and the size s of a given
organ O(G(H, S
e
)) by changing the number d of a given type of processing
space or cell S
c, d
.
Proof: Given any s
o
O such that
d(s
o
, s
i
| H, S, O, ) , s
o
O , s
i
I
58 3 Brain: A Distributed Intelligent Processing System
d(s
d
, s
t
| H, S ) , s
d
D, q(s
d
) = (s
i
), s
t
V
t
(Ł (G(H, S
c
))
a mutation of s
o
O into s
o‘
O results in
d(s
o‘
, s
j
| H, S, O, ) , s
o
O , s
j
I
d(s
d‘
, s
t‘
| H, S) , s
d‘
D, q(s
d‘
) = ( s
j
), s
t‘
V
t
(Ł (G(H, S
c
))
But, if
a) (s
i
, s
j
) 0, then it is expected that (s
d
, s
d‘
) 0,
such that
(S
i
, S
j
) 0,
and
1) a new organ O(G(H, S
e‘
)) is created if s
o
O, s
o’
O
are maintained as expressed genes, then
2) O(G(H, S
e
)) is removed if s
o
O, s
o’
O result in un-
expressed genes;
3) otherwise O(G(H, S
e
)) will be replaced by O(G(H, S
e‘
));
b)(s
i
, s
j
) 0,5 < < 1, then it is expected that (s
d
, s
d‘
) < 1
and
‘ = (q(s
j
)) = (q(s
i
))
(d(s
d
, s
t
| H, S, ‘)) (d(s
d
, s
t
| H, S, ))
{Ł(G(H, S
c, d
))}
d=1 to k
{Ł(G(H, S
c, d
))}
i=1 to k‘
, k k‘
such that new types of cells may be incorporated into O(G(H,S
e
)) and/or
the size s in Eq. 3.6 is augmented.
Corollary 3.1: Mutations of O belonging to G(H, S) greatly contribute
to the creation of new species.
Proof: As a consequence of the fact that any mutation (s
o
, s
o’
) 0
of O results in changes of the structure of O(G(H, S)), by adding, remov-
ing or changing a given organ O(G(H, S
e
)).
Remark 3.2: The meaning of mutation in this book is that of any
change of the DNA/RNA nucleotide sequence, during any copying pro-
cess, due to the grammar ambiguity. The mutation of homeobox genes
may result in the genesis of a distinct new species, because it may result in
the expression of new sets of languages expressed by G or a huge increase
in the processing capacity of a given organ. These are mechanisms that are
important in explaining the increases in complexity of the nervous system.
3.3 The Nervous System 59
3.3 The Nervous System
Evolution has differentiated animals from other organisms by providing
them with a Nervous System, or O(G(H, S
b
)), that assumed the task of
controlling the languages Ł(G(H,S
c
)) expressed by the other organs of the
animal, in order to better adapt it to a changing environment H.
Definition 3.5: If O(G(H, S
b
)) is a complex organ and
V
t
(L | H, S
b
) V
o
(L | H, S
c
) and V
o
(L | H, S
b
) V
t
(L | H, S
c
)
for all other O(G(H, S
c
)) of O(G(H, S)) then O(G(H, S
b
)) is called the con-
troller (or the Nervous System) of O(G(H, S)).
Definition 3.6: The language Ł(G(H, S
b, s
)) describing the synaptic
transactions at O(G(H, S
b
)) is composed of all expressible
d(s
t
, s
o
| G, H, S
c, i,
S
d, j
) ), s
t
V
t
(L | H, S
c, i
) , s
o
V
o
(L | H, S
d, j
))
derivation chains associating two neurons (n
c, i
and n
d, j
) in charge of pro-
cessing
Ł(G(H, S
c, i
)), Ł(G(H, S
d, j
)) O(G(H, S
b
)),
respectively. Thus, for 0 < < 1
Ł(G(H, S
b,s
)) =
d(s
t
, s
o
|G, H, S
c, i,
S
d, j
) | (d(s
t
, s
o
| H, S
c, i,
S
d, j,
)) ,
(3.7)
Remark 3.3: The dynamics of Ł(G(H, S
b, s
)) are responsible for most of
the computational power of the brain as a DIPS because they define the
power of the neuronal enrollment in any kind of processing. Also, most of
the classical types of learning, such as Hebbian learning, conditioning, etc.,
supporting connectionist theories involving traditional neural nets, are
easily modeled using those G properties that are dependent on the numeri-
cal restriction imposed by
((d(s
t
, s
o
| H, S
b
)))
But new types of learning procedures may profit from the symbolic
properties of G.
Definition 3.7: O(G(H, S
b
)) is called an intelligent distributed control-
ling organ of G(H, S) if
(d(s
t
, s
o
| H, S
b
)) , | - 0.5| 0
for all
d(s
t
, s
o
| H, S
b
) Ł(G(H, S
b, s
)).
Remark 3.4: Ł(G(H, S
b, s
)) describes both the classically described
chemical interaction at the synaptic level, as well as all the chemical trans-
actions between the pre- and post-synaptic neurons required for maintain-
60 3 Brain: A Distributed Intelligent Processing System
ing the viability of these cells (Rocha, 1997). Also, the conditions imposed
in Definition 3.6 guarantee the necessary freedom for any agent to enroll
itself with other, different sets of agents in the attempt to solve distinct
tasks, and so to render the brain a DIPS.
Theorem 3.2: The augmentation of the cardinality of O increases the
types of neurons and/or changes in the composition of I augmenting the
number of neurons; are required to increase the complexity of Ł(G(H,S
b
)).
Proof: As a consequence from Theorem 3.1 and Definition 3.4.
Remark 3.5: The organ in definitions 3.4, 3.5 and 3.6 is a nervous sys-
tem NS that is initially composed of groups of cells distributed over gan-
glia located at different body sites. Evolution increased the complexity of
the NS, and centralized most of the ganglia into the CNS (Central Nervous
System). As proposed by Rocha (1997) and Rocha et al. (2001) the CNS
(or brain) is a DIPS whose intelligence depends on both the type of its neu-
rons and the way these neurons are combined to solve complex tasks. Ac-
cording to Definition 3.6, the brain is an intelligent distributed processor of
the expressed language Ł(G(H, S
b
)) supported by G(H, S) because Ł(G(H,
S
b
)) is dependent on both the specialized neurons processing Ł(G(H, S
c, i
)),
Ł(G(H, S
d, j
)) and on (d(s
t
, s
o
| G, H, S
c, i,
S
d, j,
) S
c
that is mostly de-
pendent on the restrictions ((d(s
t
, s
o
| G, H, S
b
) )) imposed by H over S
b
.
3.4 The Brain
In such a context:
Definition 3.8: The brain is an organ O(G| H, S
b
)) operating a distribut-
ed grammar
G = {S
V
o =
V
o
, V
n
, M
V
t =
V
t
, P | S
V
o
= ; M
V
t
= }
where:
a) S is a set of initial symbols sensed in H by means of special senso-
ry spaces S
s
, and
b) M is a set of terminal symbols acting over H by means of special
motor spaces S
m
, such that:
c) d(s
o
, s
t
), s
o
S and s
t
M: describes sensory-motor pro-
cessing;
d) d(s
o
, s
t
), s
o
V
o
and s
t
V
t
, describe internal processing;
e) s
o
V
o
and s
t
M then d(s
o
, s
t
) describes volitional action over
H similar to d(s
o
, s
t
); and if
3.4 The Brain 61
f) s
o
S and s
t
V
t
then d(s
o
, s
t
) describes a memory about H,
similar to d(s
o
, s
t
) of the organism O(G(H, S));
Remark 3.6: In this context, the brain is viewed as a distributed proces-
sor of the genetic grammar defined by G, composed of at least three super-
families of neurons, namely Sensory Systems, Effector (mainly Motor) Sys-
tems and Internal Processing Systems. Many different types of neurons n
will be part of these superfamilies, each one of them characterized by a
given subset Ł(G(H, S
n
)) of all expressible languages supported by G in
H.
Definition 3.9: The knowledge K of O(G (H, S)) about H is
K = {d(s
o
, s
t
) | d(s
i
, s
j
) d(s
o
, s
t
) ∩ d(s
o
, s
t
) }
(3.8)
Remark 3.7: Each item of knowledge K is then shared by O(G(H, S
b
))
considered here to be described by a subset Ł(G(H, S
k
)) of the language
supported by G in H that have a non-empty intersection with the sensory-
motor language Ł(G(H, S
m
)) = {d(s
o
, s
t
) | s
o
S and s
t
M} promoting
the survival of O(G(H, S)) in H.
Definition 3.10: Learning by observing is the process of acquiring
knowledge d(s
o
, s
t
) K about H from the observation d(s
o
, s
t
) of H.
Remark 3.8: This type of learning is therefore the result of the process-
es that modify the expression of the languages used by the neurons of a
given O(G(H, S
b
)). This may be achieved either by changing the neuronal
enrollment by modifying (d(s
t
, s
o
| H, S
c, i,
S
d, j,
)), or changing the lan-
guage Ł(G(H, S
k, n
)) expressed by a set of neurons S
k, n
, in order to create a
new specialized agent.
Proposition 3.1: DIPS reasoning involves different types of agents be-
cause it entails:
a) defining a goal to be achieved: selecting a need to be fulfilled im-
plies the existence of agents to detect the system‘s actual needs;
b) determining the DIPS’ actual state: performing a global evaluation
of the main ongoing activities, implying the existence of agents to
monitor the agent enrollments supporting those ongoing activities;
c) planning the means of achieving goals: selecting and organizing
available tools judged to be adequate for solving problems, which
implies the existence of agents able to recruit and organize other
agents based on the actual state and past experience;
d) retaining useful information: recalling past experiences in order to
orient planning and to store data about ongoing planned activities,
62 3 Brain: A Distributed Intelligent Processing System
implying the existence of agents specialized to memorize such piec-
es of information;
e) utilizing a set of disparate tools: developing tools adequate to dif-
ferent tasks which may be modified or combined for new purposes,
implying the existence of different agents specialized in handling or
modify tools or creating new ones;
f) evaluating the performance in achieving the desired goal: compar-
ing what is being done with what was planned, implying the exist-
ence of agents able to evaluate the progress (or lack thereof) of the
planned action;
g) optimizing communication: promoting message exchange between
all the agents to (1) announce the task; (2) support its solution by
those enrolled agents; and (3) alter enrollment if necessary. All of
this implies the capability of each agent to relocate its communica-
tion resources whenever necessary.
Proof: It follows from definitions 3.0 and 3.7.
Remark 3.9: Reasoning supported by O(G | H, S
b
)) involves different
types of neurons (Fig. 3.2) in charge of:
a) goal definition: or, detecting a need to be fulfilled, such as when the
limbic neurons detect basic necessities like food, water, etc., or
when frontal cells guide environmental searches;
b) attention control: or, monitoring actual agent activities, such as
when frontal and parietal cells control limbic and sub-cortical agents
involved with the control of cortical activity;
c) planning: or, selecting and organizing available agents judged to be
adequate to solve the problem, as for example frontal recruiting
agents described in the literature as executive agents (see also Chap.
5);
d) memorizing useful information: or, encoding and recalling both ret-
rospective (hypocampus) and prospective memories (frontal lobe);
e) implementing tools: or, monitoring (sensory systems) and acting
(motor systems) upon the environment, storing knowledge (e.g. in
semantic memory), etc;
f) evaluating performance: or, verifying if needs are being fulfilled,
via the emotional agents located in the limbic system and the frontal
neurons which recruit them; and
g) communicating resource information: as provided by transactions
supported by Ł(G(H, S
b, s
)) (see definition 3.6).
Definition 3.11: G(g) = {V
g
, V
n
, V
e,
P} G is the grammar supporting
the processing of the goal g where:
3.4 The Brain 63
a) Ł(G(H, S
g
)) is the language expressed by the agents S
g
in charge of
defining the goals g of O(G | H, S
b
));
b) Ł(G(H, S
e
)) is the language expressed by the agents S
e
in charge of
verifying if the goals g of O(G | H, S
b
)) are being fulfilled;
c) V
g
= {s
t
d(s
o
, s
t
) | [ d(s
o
, s
t
) Ł(G (H, S
g
)), (d(s
o
, s
t
)) 1]}
and
d) V
e
= {s
t
d(s
o
, s
t
) | [d(s
o
, s
t
) Ł(G(H, S
e
)) , (d(s
o
, s
t
)) 1]}
In this context:
Proposition 3.2: The DIPS reasoning R(G(g) | H, S
b
)) about g is
R(G(g) | H, S
b
)) =
{d(s
o
, s
t
) | [ (d(s
o
, s
t
) ) 1 s
o
V
g
, s
t
Ł(G(H, S
e
)]}
Also:
a) acceptance V
g
of such a reasoning is
V
g
= {s
t
Ł(G(H, S
e
)
| d(s
o
, s
t
) ) 1};
b) refutation ~V
g
of such a reasoning is
~V
g
= {s
t
Ł(G (H, S
e
)
| (d(s
o
, s
t
) ) 0};
c) degree of acceptance (R(G(g) | H, S
b
))) of R(G(g) | H, S
b
)) as a so-
lution for g is
R (G (g) | H, S
b
))) = (V
g
V
g
) / V
g
)
d) degree of refutation ~ (R(G(g) | H, S
b
))) of (R(G(g) | H, S
b
)) as a
solution for g is
~ (R(G(g) | H, S
b
))) = (~V
g
V
g
) / V
g
.
Proof: follows from proposition 3.1 and definition 3.11.
Remark 3.10: From Definition 3.9 & Proposition 3.2, (R(G(g) | H, S
b
)) is
supported by the knowledge
K = {d(s
o
, s
t
) | [(d(s
o
, s
t
) )1 and s
o
V
g
, s
t
V
g
]}.
Let this be denoted by R(G(g) | H, K, S
b
)).
64 3 Brain: A Distributed Intelligent Processing System
Communication resources
provided by Ł(G(H, S
b,s
))
Agent specialization supported by Ł(G(H, S
c
))
Fig. 3.2. Neural agents supporting reasoning: reasoning is the result of the in-
teraction among a complex set of specialized neurons.
3.5 Brain Communication Channels
Communication among DIPS‘ agents is established by means of two
main strategies:
a) Mail addressing: both the sending and the receiving agents know
themselves; that is to say they have the capacity to address messages
3.5 Brain Communication Channels 65
specifically to each other. It its the case of the many neural tracts
that specifically innervate defined cerebral areas or nuclei; and
b) Blackboard posting: agents deliver messages that are not specifical-
ly addressed to another defined agent, but to those interested in the
subject. This is clear in the case of hormones released in the blood
stream, which act upon cells whose physiologies require these hor-
mones. It is also the case of the neuromodulators that are broadly re-
leased over large areas of the brain by means of a very spread axo-
nic net. Neuromodulators have different binding (sites where
matching occurs) and effector (rewriting) sites. It is also the case of
those broad neural circuits controlling brain reactivity, as in the case
of the waking, sleep and arousal states. These distinct communica-
tion strategies play different roles in brain processing and learning:
Hormones are assumed to be general messages released by a group
of agents and broadcasted by means of the blood stream to influence
many parallel and independent processes in order to coordinate their ac-
tions. It may be a useful process to broadly spread information about the
system‘s state and/or goals to be achieved;
Neuromodulators or neuropeptides are used to coordinate the activi-
ties of neural systems operating in a parallel fashion and/or according to
some general hierarchy. Modulators will also be very useful in control-
ling learning by means of many different strategies, because they may
be used to control synaptic specification, growth, stabilization and/or
death. For instance, they may be used to promote the growth of the syn-
apses among well succeeded agents and to reduce communication re-
sources among neurons that cooperated in a failed attempt to reach a
specified goal; and
Neurotransmitters are used either as general messages broadly re-
leased in the brain by means of very spread neural circuits (e.g. those re-
lated with arousal and sleep control) or as local information in those
more specific neural tracts (e.g., cortical-spinal and spinal-cortical cir-
cuits controlling the muscles). In the first case, neurotransmitters will set
the general operating conditions for defined groups of agents, such as
specifying the type of axonic code to be used; the set of useful messag-
es, etc. In the second case, neurotransmitters will be part of the actual
processing and because of this they will be under the control of learning
mechanisms related with the plasticity of the commitment of agents in
solving tasks. Changes in the amount of communication resources (e.g.,
transmitter) modify the capacity of any agent to enroll in the solution of
a given task.
66 3 Brain: A Distributed Intelligent Processing System
3.6 The Basics of DIPS Learning
According to Definition 3.0, learning is the tool used by any intelligent
system to solve the problem when a defined goal g is not achieved. This
section is devoted to discussing this subject.
Theorem 3.3: R(G(g) | H, K, S
b
)) has to be reviewed whenever
(V
g
, V
g
| H, S
b
) .5.
Proof: As consequence from the fact that the difficulty in satisfying g
increases as
(V
g
, V
g
| H, S
b
) 0.5.
Corollary 3.3: The learning of a new R(G(g) | H, K, S
b
)) is necessary if
(V
g
, V
g
| H, S
b
) .5 for and given R(G(g) | H, K
o
, S
b
))
Proof: As a consequence of the fact that R(G(g) | H, K
o
, S
b
)) does not
provide a solution for g as (V
g
, V
g
| H, S
b
) .5.
Theorem 3.4: If R(G(g) | H, K
o,
S
b
)) has to be reviewed, then DIPS
learning modifies ((d(s
i
, s
j
) | H, S
b
))) by changing (d(s
i
, s
j
) | H,S
b
)
and/or creating new s
n
V
n
V
g
in order to guarantee a new
K = {d(s
o
, s
t
) | [((d(s
o
, s
t
)) 1) and (s
o
V
g
, s
t
V
g
)]},
such that (V
g
, V
g
| H, S
b
) 1.
Proof: Let it be supposed that a given knowledge K
o
, developed under
0
((d(s
i
, s
j
) | H, S
b
))), supports a given reasoning R
0
(G(g) | H, K
o
, S
b
)) as
an attempt to achieve the goal g. Also, let it be assumed that
(V
g
, V
g
| H, S
b
) 0.5.
In this condition
V
g
= {s
t
d(s
o
, s
t
) | d(s
o
, s
t
) Ł(G(H, S
g
)) | (d(s
o
, s
t
)) > .5 + , 0}
for some d(s
o
, s
t
), because (d(s
i
, s
j
) ) .5 + , for d(s
i
, s
j
) d(s
o
, s
t
).
Now, from Definition 2.1:
1))|),(),(),(),((()),((
1
Hqqqfd
sssssss
kijki
n
k
ji
But, from Definition 2.5 for a self-controlled grammar and Eq. 2.34 de-
fining such a control
q(s
i
) and/or q(s
k
) and/or q(s
j
) =g(q(s
c
)) and/or (s
i
, s
k
)=g(q(s
c
))
3.6 The Basics of DIPS Learning 67
Now, if q(s
c
) is also a function of (V
g
, V
g
| H, S
b
), then it is possible to
change (d(s
i
, s
j
) | H, S
b
) to increase (V
g
, V
g
| H, S
b
). This implies that
reorganizing knowledge K
o
into K to update R
0
(G(g) | H, K
o,
S
b
)), and this
is done by changing (d(s
o
, s
t
)).
If s
n
in d(s
d
, s
n
) is an unexpressed symbol of G then q(s
n
) = 0 because
(d(s
o
, s
n
|
H, S
b
)) = 0. But according to Eq. 2.47 it is possible to imple-
ment a control
not K
o
Ł(G | H, S
b
) Ł(G | H, S
b
©
)
to obtain
(d(s
o
, s
n
|
H, S
b
©
)) > 0 and q(s
n
) > 0
and s
n
is expressed and incorporated into V
n
Vg.
In these conditions, let
not K
o
0
((d(s
i
, s
j
) | H, S
b
))) ((d(s
i
, s
j
) | H, S
b
))),
then a new
K = {d(s
o
,s
t
) | [(d(s
o
,s
t
)) 1] and [s
o
V
n
Vg, s
t
V
g
]
is developed under
0
((d(s
i
, s
j
) | H, S
b
))) to support R(G(g) | H, K, S
b
)).
Theorem 3.5: The learning capability of a DIPS is determined by the
ambiguity
<(G | H, S
b
)> of G(H, S
b
)
and by the control capability
(L(C |H, S
b
)) of Ł (C | G, H, S
b
).
Proof: On the one hand, the expressiveness
(L(C | H, S
b
)) of Ł (C | G, H, S
b
)
determines how much control over
0
((d(s
i
, s
j
) | H, S
b
))) may be imple-
mented according to Eqs. 2.34 and 2.47, and this limits the changes that
can be promoted over any (d(s
i
, s
j
)) supported by G(H, S
b
), whether s
i
belongs to D V
n
or not.
On the other hand, the ambiguity of <(G | H, S
b
)> determines the ca-
pacity for turning on the expression of d(s
d
, s
n
) to generate a new expres-
sion s
n
V
n
V
g
.
Therefore, both < (G | H, S
b
) > and (L(C | H, S
b
)) determines what is
learnable by a given O(G | H, S
b
)).
Remark 3.11: Theorem 3.4 specifies two different conditions for modi-
fying a given knowledge K. The first one changes the association among a
68 3 Brain: A Distributed Intelligent Processing System
defined set of specialized agents because it implies modifying the possibil-
ity of the already expressed d(s
i
, s
j
)s, whereas the second strategy implies
creating a new specialization because it will change the set of symbols V*.
The first strategy (K
0
K
0‘
) is more a process of optimizing or adapting a
pre-existent knowledge K
0
to new conditions. It is a sort of knowledge re-
engineering. The second approach (K
0
K
1
) is more a process of
knowledge evolution by means of which brand new solutions to new prob-
lems arise. The change of V* may be achieved by merely promoting the
expression of an already existing gene induced by the changing environ-
ment, which poses new questions, or by means of a mutation creating a
new gene to bring about a new s
n
V
n
Vg. Mutation requires at least
one generation to be accomplished because it has to occur first at the level
of gametes to be later expressed at the level of somatic (neural) cells.
However, mutation may also be hypothesized to occur during mitosis, cre-
ating a new cell, most probably during embriogenesis or even during post-
natal life. In the first case, different beings sharing the same genetics will
have different learning capabilities, although such differences cannot be
large as it is not expected that the mutation rate is high. In the second case,
very new specialization may result from demands of the changing envi-
ronment that must first of all promote the creation of new cells. The pro-
cess can be improved if the mutation rate is increased in such a condition,
to guarantee that these new cells S
n
will express new languages L(G | H,
S
n
)). This may be the case if some genes s
n
D are specially designed to
generate high V* variability, as has been discovered in the cases of genes
governing the immune cells or chemical receptors at the olfactory cells and
recently proposed to be also at work in the brain (Arshavsky, 2002). In any
case, when new cells are created in post-natal life, the new knowledge K
1
has to be re-discovered by each individual or has to be copied by epigenet-
ic means from those brains which first attained it. This is the role played
by memetics.
A clear example of knowledge evolution is the human transformation of
fuzzy quantification (K
0
) capability, which is shared with the animals, into
the crisp numbers (K
1
) supporting modern arithmetic (Fig. 1). Our key
proposition in this book (see Chap. 4) is that the creation of crisp numbers
is achieved by changing the specialization of some cells in the brain, which
may be accomplished by changing the expression of some genes already
existing in D V
n
of many animals and in charge of defining certain ionic
channels and some topographic molecular markers guiding axonic growth.
This proposition is supported by the fact that neurons of some primates are
specialized in identifying numbers above 5 (see Chaps. 1 and 7) and by the
fact that man invented crisp numbers on at least in four different occasions
3.6 The Basics of DIPS Learning 69
(Fig. 3.3) in four non-overlapping civilizations (e.g., Ifrah, 1985, Joseph,
1990).
Fig. 3.3. Number, evolution, and neural circuits: whenever culture pushed, the
brain improved its number of neural circuits.
It will be assumed here that demands imposed by increasing trade ac-
tivities have pushed the expression of such genes, whenever the complexi-
ty of the human society required it. This could explain why our Tupinam-
bás (Fig. 1.1) were still using primitive number systems when the
Portuguese people arrived in Brasil, as the result of a huge effort to in-
crease their commercial trade with India. Once crisp numbers are created
by a human culture they may be transmitted from generation to generation
of this culture or to others by means of memetic reproduction as discussed
in Chap. 6. This is because possibility cannot be ruled out, here, that the
evolution of the mathematical human capability is taking profit of meme-
gene co-evolution to create new s
n
D V
n
.
70 3 Brain: A Distributed Intelligent Processing System
3.7 Evolutionary Learning
To learn is to model the observable world in order to understand it (Fer-
ber, 1999, Rocha, 1982 a, b). A model is a set of relations between data or
evidence obtained with a set of instruments, and actions performed by a set
of acting (motor) agents. Understanding requires that the model to fulfill
some defined purpose (or goal), which may be a simple survival task, or
complex intellectual processing. The goal is set because some agents de-
tect some need for resources or data to accomplish a defined task. From a
general point of view, to understand is, therefore, to provide a set of ade-
quate responses in order to adapt the system to the surrounding world, or
in other words, to maintain its identity in a changing environment. In such
a line of reasoning:
Definition 3.12: Modeling is characterized as:
a) Detecting a motivation to act: the set H of agents in charge of
monitoring the actual conditions of the DIPS agents, detect a neces-
sity q(s
n
) of resource or data to support ongoing activity
d(s
o
, s
t
)= s
o
... s
n
... s
t
.
In this context, the set N of necessities evaluated by H is:
{q(s
n
) of s
n
V
#
|(s
o
... s
n
s
n
+1
...s
t
)}, and
(s
n
, s
n
+1
) 0;
b) Setting a goal g
definable over H: another set G of agents uses in-
formation about N to select a set of final states given by V
g
V
t
according to definition 3.11 and proposition 3.2, as those states to
be attained in order to satisfy the query posed by N. Thus:
g: N x V
t
(0,1) and g: N
V
g
Once V
g
is determined, it is necessary to define the set of d(s
o
, s
t
)
supported by the grammar G(g) = {V
n
, V
g
,
P} that fulfill g because
(d(s
o
,s
t
)) 1, s
o
V
n
, s
t
V
g
. Thus, according to proposition
3.2, this is accomplished by
R(G(g)|H, K, S
b
)) = {d(s
o
, s
t
) | [(d(s
o
, s
t
) ) 1] s
o
V
g
, s
t
V
g
;
c) Setting the relevance
(s
t
) of each s
t
V
g
: since V
g
is a fuzzy set,
not all s
t
are supposed to have the same relevance to satisfy N;
The actual value of (s
t
) is set as a function of the amount q(s
n
) of
resource or data needed. Thus:
(s
t
) = (q(s
n
)).
3.7 Evolutionary Learning 71
This is the task of the set H of agents in charge of evaluating the
attainability of g in definition 3.11;
d) Collecting a set F of facts, evidences or data about g
from H with a
set of sensory agents S: this process S of sensory information col-
lecting is: S : S x H x F x g (0,1) that is, obtaining a set F of
measures about H with the set of sensory agents S. The set of evi-
dences F can always be described by a the language Ł (G | H, S),
that is S : S x H x g Ł (G | H, S). In general, F is a redundant set
of measurements performed by S. In other words, there are many
similar d(s
o
, s
t
) (G | H, S) in F. Let F be the non-redundant set of
evidence associated to F. Thus F F;
e) Recognizing the sensory images I(o
i
) of objects o
i
of interest in H as
sets of relations between the collected pieces of evidence or facts.
The process of sensory recognition R is to obtain the set of relations
I(o
i
) among the measurements in the non-redundant E associated to
an object o
i
of H. To recognize R is
R: (F)
n
x I(o
i
) x g (0,1)
Recognition, therefore, consists in the result of the calculations per-
formed by a set of agents C specialized in classifying the objects o
i
U according to the set I(o) of relations between the measures m
i
about these o
i
. I(o
i
) is therefore described by Ł (G | H, R):
R: (F)
n
x g Ł (G | H, C);
f) Analyzing the possible behavior B(o) of the identified objects I(o
i
):
this analysis of B is performed by a set of agents B having the ade-
quate set of tools for such an analysis:
B: I(o) x g x B(o) [0,1] or B: I(o) x g Ł (G | H, B);
g) Planning P(g) how to reach g given B(o) and the available tools T:
that is, to assess how attainable g is from I(o
i
) given resources T. To
plan is P : B(o
i
) x g x P x T (0,1), and it is the result of the calcu-
lations performed by a collection of agents P using defined rules.
The possible plans are provided by Ł (G | H, P). Thus: P: B(o) x g
Ł (G | H, T);
h) Making a decision to act A(g) over H using a set of (motor) agents
M: this decision D is based upon P about the selection of the best
agents to act over H in order to achieve g and is the duty of a set of
specialized agents D. To make a decision is
A : P(g) x D x M (0,1) or A : P(g) x D Ł(G | H, M)
Ł(G | H, M) = {d(s
o
, s
t
) | [s
o
V
g
, s
t
M
]}
72 3 Brain: A Distributed Intelligent Processing System
i) Evaluating E(g) if the executed actions A(g) fulfilled the desired g:
the executed actions d(s
o
, s
t
) are supposed to result in the required
q(s
n
) of s
n
defining s
t
V
g
. Thus E: A x H x g (0,1) such that
d(s
o
, s
t
) d(s
t
, s
n
), d(s
o
, s
t
) R(G(g) | H, K, S
b
)), and:
d(s
t
, s
n
) Ł (G | H, E) E : A x H V
g
V
g
=
{s
n
V
g
| (d(s
t
, s
n
)) 1}
In this way, it is possible to calculate both the model‘s acceptance
and refutation (definition 3.2) as
(R(G(g)) | H, K, S
b
)) = (V
g
V
g
) / V
g
, ~(R(G(g) | H, S
b
))
= ( ~V
g
V
g
) / V
g
)
respectively. Such an evaluation is the duty of a set of specialized
agents E, such that: E : V
g
x V
g
Ł(G | H, E)
i) The attainability of s
t
V
g
is encoded by two strings d(s
t
, s
r
),
d(s
t
,s
p
) produced by two different sets evaluating sets E
r
, E
p
of
agents calculating the reward (s
r
) and punishment (s
p
) of s
t
as be-
longing to V
g
respectively, such that given as a T-norm
Ł(G | H, E
r
) = {d(s
t
, s
r
) | [(d(s
t
, s
r
)) = (d(s
t
, s
n
)) (s
t
)]}
Ł(G | H, E
p
) = {d(s
t
, s
p
) | [ (d(s
t
, s
p
)) = ((1 - (d(s
t
, s
n
))) (s
t
)]}
In this condition, the adequacy (M
0
(g) | H, K, S
b
) of M
0
(g) to fulfill
the goal g given H, K, S
b
is
(M
0
(g) | H, K, S
b
) = (d(s
t
, s
p
)) - (d(s
t
, s
r
))
In such a condition:
M
0
(g) adapts O(G| H, S
b
) to H according to g if
(M
0
(g) | H, K, S
b
) 1
and M
0
(g) has to be a rejected as solution of g in H if
(M
0
(g) | H, K, S
b
) -1.
Remark 3.12: The necessities of the DIPS agents are of two kinds: tool
maintenance and data processing (Fig. 3.4). The first type of necessity is
directly related with the house keeping of each agent. For instance, in the
case of the neuron, the cell has to deal with energy and structural compo-
nent supplies to keep itself alive. The second kind of necessity is directly
related with the activities the agent is involved in. Whenever it accepts a
task the cell may have need of specific types of data. In the case of neu-
rons, this may imply sensing the environment in specific searches.
The first kind of necessity is in general periodic and most of the struc-
ture of the models M
s
required to accomplish survival tasks are hardwired
3.7 Evolutionary Learning 73
or learned in the first stages of DIPS development. In the case of animals,
these tasks are related to feeding, sheltering, looking for sexual partners,
taking care of offspring, etc., and the neural circuits implementing such
tasks are mostly organized at the level of the limbic system involve special
neurons or receptors to monitor many body parameters like blood pressure,
temperature, oxygen supply, sugar, salt, etc. Or they may be involved in
organizing sexual, parental, and social behaviors, etc.
Eating
The model
Its neural circuit
Counting
74 3 Brain: A Distributed Intelligent Processing System
Fig. 4. Modeling by DIPS: to create a model implies having a set of specialized
agents able to implement it.
The second type of necessity is more sporadic and variable, because it is
mostly determined by the exploration of H and the attempts to make the
survival activities easier. The neural circuits implementing such tasks are
mostly organized at the neocortex level and involve many different types
of specialized cells to sense or manipulate the environment. Once a neces-
sity is detected, memory has to be scanned to provide information about
previously successful and unsuccessful actions, or about places, means,
etc., of fulfilling such a necessity. All these kind of information are used to
set each s
t
V
g
and its relevance (s
t
), as well as to direct the sensory
search for information. These data and those already in memory are then
used to plan and implement actions over H, which are in general motor ac-
tions whose results are matched against the defined goals in an attempt to
evaluate how much of the necessities were fulfilled.
Models may or may not involve real sensory search and motor action.
Sensory data may be simulated by recruiting sensory images I(o
i
) directly
from R instead of involving S in their generation. In the same sense, ac-
tions may be simulated motor actions or manipulations of other DIPS
agents. Because of this, it is possible to model imagined environments H.
Finally, the model‘s performance is evaluated by two different sets E
p
,
E
r
of agents in charge of evaluating its acceptance (E
r
) or rejection (E
p
) as
a solution posed by the problem characterized by N. The result of this
evaluation guides learning, because both s
r
and s
p
are neuromodulators
broadly released in the brain, in order to locally reward or punish the
agents enrolled in the solution of N (Rocha, 1982a, b). The details of the s
r
and s
p
are locally defined, because neuromodulators have different action
sites that are locally selected by the chemical environment of the local
agent.
Theorem 3.6: Let there be a model M
0
(g
0
), adapting O(G | H, S
b
)) to H
according to g
0
. Now assume a change from environment H to H´ such
that
(M
0
(g
0
) | H, K
0
, S
b
) > (M
0
(g
0
) | H‘, K
0
, S
b
).
Given 0 < < 1, if (M
0
(g
0
) | H‘, K
0
, S
b
) then it is decidable if
there exists M
1
(g
1
) as an evolution (M
0
(g
0
) M
1
(g
1
)) of M
0
(g
0
) that will
be better adapted to H‘ than M
0
(g
0
).
Proof: There are two conditions to be considered:
3.7 Evolutionary Learning 75
1) If (M
0
(g
0
) | H‘, K
0
, S
b
) < then M
0
(g
0
) must be rejected and for-
bidden to evolve in H‘, and
2) If (M
0
(g
0
) | H‘, K
0
, S
b
) , and since (M
0
(g
1
) | H‘, K
0
, S
b
) is de-
pendent on both (d(s
o
, s
t
)) and (s
t
), then it is possible to evaluate:
a) if there exists any s
t
whose relevance may be increased in order to
augment (M
0
(g) | H‘, S
b
) such that
(M
0
(g
1
) | H‘, K
0
, S
b
) > (M
0
(g) | H, K
0
, S
b
).
If this is the case, them a new goal g
1
is accepted to be a modifi-
cation (g
0
g
1
) of g
0
. The condition to accept g
0
g
1
is that
g
1
continues to fulfill N. This is a decision of agents N based on
the actual values of (d(s
t
, s
n
)), because if q(s
n
) > 0 then it is
possible to increase (s
t
) of s
t
involved in
d(s
t
, s
n
);
b) if there exists any s
t
whose (d(s
o
,s
t
)) may be increased by chang-
ing the processing of M
0
(g) in any step from definition 3.12d to
h, to augment (M
0
(g)|H‘, S
b
) such that according to theorem 3.5
then: (M
0
(g) | H‘, K
1
, S
b
) > (M
0
(g) | H, K
o
, S
b
);
c) if a new g
1
may be generated (g
0
g
1
) from g
o
by exchanging
some s
t
V
g
by some other new s
t‘
V
g
such that
(M
0
(g
1
) | H‘, K
0
, S
b
) > (M
0
(g) | H, K
0
, S
b
) because
(d(s
o
, s
t
‘
)) (s
t
‘
) > (d(s
o
, s
t
)) (s
t
)
Those s
t
V
g
selected to be exchanged must be those associat-
ed to (d(s
t
, s
n
)) 1, because if the contents of g
o
are modi-
fied, then steps in definitions 3.12d – k and in definition 3.11
must be processed again.
If at least one of the above strategies succeed in promoting
(M
0
(g
1
) | H‘, K
o
, S
b
) > (M
0
(g
o
) | H, K
o
, S
b
) or
(M
0
(g
1
)|H‘, K
1
, S
b
) > (M
0
(g
o
)|H, K
o
, S
b
)
then M
0
(g) will evolve into M
1
(g); the process of this transfor-
mation is called here evolutionary learning.
Corollary 3.6: Let there be a model M
0
(g
0
) adapting O(G | H, S
b
)) to H
according to g
0
. Let also g
1
be a transformation (g
0
g
1
) of g
0
because
the relevance of s
t
V
g
is increased and/or s
t
V
g
is changed.
If g
1
promotes
(d(s
t
, s
n
)) (s
t
) > (d(s
t
, s
n
)) (s
t
), as an S-norm
g
1
g
0
76 3 Brain: A Distributed Intelligent Processing System
then M
1
(g
1
) is an optimization of M
0
(g
0
).
Proof: From definition 3.11 and theorem 3.6.
Remark 3.13: Evolutionary learning has some distinctive properties:
a) First of all, it requires an initial knowledge K
o
characterized by a
set of m initial models to support the initial modeling M
0
(g
0
) of H,
guaranteeing the initial survival of O(G | H, S
b
)) in H;
b) This K
0
is hardwired in the brain during embriogenesis, guided by
phylogenetic information stored in {Ł(G(H, S
c
))}
c=1 to n
;
c) The evolution of this K
0
is guaranteed by the ambiguity of G, that
is, by the fact that (d(s
i
, s
j
) | H)) 0.5, because in such a condi-
tion, the restrictions imposed by a changing H can be easily mod-
eled by modifying M
0
(g
0
) according to the Theorems 3.4, 3.5 and
3.6;
d) It is a sequentially ordered process, such that what is learnable at
step a is very dependent on the evolution of K
0
up to this moment
K
0
.... K
a;
e) C the ability L(M
j
(g
j
)) to learn a new model M
j
(g
j
) is directly re-
lated to the similarity of this new model and any M
a
(g
a
) compos-
ing the actual knowledge K
a
; this similarity is ( g
j
, g
a
). Thus:
),())((
max
1
ggg
M
kj
a
K
j
j
L
f) the capacity of learning L(M
j
(g
j
)) is also dependent on the ambigu-
ity of Ł(G | H, S
b
)) that defines the plasticity of the neural enrol-
ment in the tasks required to solve M
j
(g
j
).
g) if M
j
(g
j
) may be learned as a modification of M
a
(g
a
), then it is said
to be an evolution of this latter model. Thus:
M
a+1
(g
a+1
) M
j
(g
j
)
h) the capacity of learning L(M
j
(g
j
)) of M
0
(g
j
) is also dependent on
the restrictions imposed by H on the resources required by any g
j
,
g
a
.
i) On the one hand, if resources are scarce, then competition may be
established between M
j
(g
j
) and M
0
(g
a
), such that either M
a
(g
a
)
M
a+1
(g
a+1
) does not occur, or M
a
(g
a
) is blocked if not extin-
guished.
j) On the other hand, if resources are plentiful, then M
a
(g
a
) and M
j
(g
j
) may reinforce each other in a symbiotic process, in order to have
better odds to survive in O(G| H, S
b
))
k) a model M
j
(g
j
) may be stored in a set of {O(G| H, S
b
))
i
}
i=1 to h
liv-
ing together such that it is learned by organism O(G
| H, S
b
))
s
from
3.8 Summary 77
another organism O(G | H, S
b
))
t
that has previously learned or dis-
covered M
a
(g
a
).
l) In this context, discovering M
j
(g
j
) is to learn it by itself, whereas
learning M
j
(g
j
) means to learn it from some other M
a
(g
a
).
3.8 Summary
The brain O(G(H, S
b
)) is defined herein as a DIPS in charge of pro-
cessing a distributed language {Ł(G(H, S
c, d
)}
d=1 to mc
}
c=1 to n
. Each type of
neuron S
c
composing O(G(H, S
b
)) is specialized in processing a given
Ł(G(H, S
c
)) supported by G, and the brain may contain up to m
c
of such
neurons. The language given by Ł(G(H, S
b, s
)) describes the synaptic trans-
actions d(s
t
, s
o
| G,H, S
c, i,
S
d, j
)) between any two neurons S
c, i,
S
d, j
of
O(G(H, S
b
)). The types of neuron and their quantity are initially defined by
the set of homeobox genes O of G during the embriogenesis of O(G(H,
S
b
)), but post-natal learning changes both the number n of the different
types of neurons and the number m
c
of such cells. This type of learning is
called here evolutionary learning to distinguish it from the classical con-
nectionist learning procedures, (d(s
t
, s
o
| G, H, S
c, i,
S
d, j
)). DIPS reasoning
R(G(g)| H, K, S
b
)) is defined by the set of transactions d(s
o
, s
t
) fulfilling a
goal g associated with a detected resource need. The conditions for the
evolution of R(G(g)| H, S
b
)) are set by both the ambiguity of <(G| H,
S
b
)> of G(H, S
b
) and by the control capability (L(C| H, S
b
)) of Ł (C,| G,
H, S
b
).
78 3 Brain: A Distributed Intelligent Processing System
3.8 Summary 79
80 4 Neural Computational Mechanisms Supporting Cognitive Processes
4 Neural Computational Mechanisms Supporting
Cognitive Processes
In the previous chapter we proposed that the brain is a Distributed Intelli-
gent Processing System (DIPS), with specialized agents for the tasks of
gathering information from the environment, manipulating the information
according to established goals, and implementing actions to reach those
goals. Each cell in the brain is conceived of as a specialized agent that ex-
presses a subset of the brain's language L(G).
The performance of cognitive functions requires both cooperation and
competition between specialized agents. Cooperation allows for the per-
formance of functions, composed of sub-functions, which are performed
by different units. Competition is a characteristic of selective processes
(e.g., "winner takes all" mechanisms) that can help to increase the efficien-
cy of the whole system in coping with environmental challenges. Both co-
operation and competition are limited by possible mismatches between the
subsets of the language handled by the different processing units.
An evolutionary strategy to avoid computational breakdown is proposed
here wherein the brain employs quantum computation. This strategy al-
lows microstate entanglement of several spatially distributed processing
units, thus providing a supplementary communication channel to overcome
possible mismatches. This channel also provides instantaneous binding of
the informational content being processed in such units (a property which
has been regarded as an essential ingredient for a theory of consciousness;
see Rocha et al., 2001).
4.1 Basic Concepts of Quantum Computation
Two of the most surprising properties of quantum systems are microstate
superposition and entanglement. Superposition is the coexistence of differ-
ent microstate values of the same particle at the same time. Superposed
states are reduced to a single state by the act of measurement or by other
kinds of interaction with the macro-environment, which are called deco-
herence. Entanglement is a strong microstate correlation between spatially
3.8 Summary 81
separated particles. It is an experimental finding still not explained in
terms of causal processes based on electromagnetic or gravitational forces
(the forces that operate at the macroscopic level). One interpretation of the
phenomenon is that entangled particles behave as a single entity, despite
their distributed spatial locations.
Quantum computation is a research area devoted to experimentally ma-
nipulating the superposition and entanglement of microstates, and to de-
veloping algorithms that could be implemented in such quantum systems.
Of course, a superposition of states cannot be directly manipulated, since
any interaction of the quantum system with the experimental apparatus re-
duces superposed states to a single one. However, the combination of su-
perposition and entanglement can be manipulated through clever devices.
While entangled, two or more particles have correlated, superposed states
(e.g., spin up and down). Therefore, by measuring one particle, and there-
by reducing its microstate to a single one (e.g., spin up), we gain
knowledge about the actual microstate of its spatially separated, entangled
partner(s) (e.g., spin down).
The above situation is equivalent to obtaining two bits from one. Quan-
tum computation uses the concept of a quantum bit (qubit), which is
equivalent to a superposition of orthogonal states in Hilbert space; i.e., n
qubits corresponds to 2
n
superposed states. Entanglement of two particles
generates two qubits, corresponding to four classes of possible states,
which are called Bell states (Zeilinger, 1998; see eq. 3 below).
A qubit can be in any state:
| 1 > + | 0 >
(4.1)
where and are complex numbers called amplitudes, subject to:
| |
2
+ | |
2
= 1
(4.2)
Measurement on the system | 1 > + | 0 > results in the qubit making
a probabilistic decision (Brassard, 1997; Brassard et al., 1998): with prob-
ability | |
2
, the qubit takes the value | 0 > and with complementary
probability | |
2
, it equals | 1 >.
Because a physical system of n qubits requires 2
n
complex numbers to
describe its state, two qubits can be in the Bell states:
82 4 Neural Computational Mechanisms Supporting Cognitive Processes
| 00 > + | 01 > + | 10 > + | 11 >
(4.3)
such that
| |
2
+ | |
2
+ | |
2
+ | |
2
= 1
(4.4)
Let a quantum computer QC be defined by Eqs. 4.3 and 4.4. A given in-
struction i
i
may be written on it by, e.g., changing its state to
| 00 > + | 01 > + | 10 > - | 11 >
(4.5)
that is, by modifying the condition of qubit | 11 >.
The change influences the entanglement of this qubit with its partner(s).
Quantum computation is based on such an interference of qubits, which
has been proved capable of performing like classical logical gates as well
as other, so-called quantum gates.
The only constraint on quantum gates is unitarity, which is based on the
constraint 4.2. Therefore, "any unitary matrix specifies a valid quantum
gate!" (Nielsen and Chuang, 2000,p. 18). An important operation is instan-
tiated in the Hadamard gate, whereby | 0 > is changed into ( | 0 > + | 1 >) /
2 and | 1 > is changed into ( | 0 > - | 1 >) / 2. Applying the Hadamard
gate twice to a qubit generates an output that is identical to its initial state.
According to Nielsen and Chuang (2000), a QC has five requirements:
a) two parts, one classical and one quantum; although the classical
part is not necessary, in practice it is useful in order to specify in binary
logic the inputs and outputs to and from the quantum part;
b) a suitable state space; for n qubits the corresponding state space is
a 2
n
-dimensional complex Hilbert space;
c) the ability to prepare states in a computational basis (as pure en-
tangled states);
d) the ability to act as quantum gates, preferably universal gates such
as the Hadamard and Controlled NOT (CNOT; an equivalent of the
classical XOR, or exclusive disjunction) gates; and
e) the ability to perform measurements in a computational basis.
Therefore, any proposal of biological quantum computation should be
able to demonstrate the possibility of a biological structure to fulfill such
requirements.
3.8 Summary 83
4.2 Cellular Processing
The cell is herein conceived of as a distributed stp system. The set of all
possible stp strings processed by a given cell S
i
is (see Definition 2.7):
Ł(G | H, S
i
) = { d(s
0
,s
t
) | ( d(s
0
,s
t
) | H, S
i
) 1 }
(4.6)
Any cellular function F(S
i
) is therefore described by means of a set of
coupled sentences d(s
0
,s
t
) of Ł(G | H, S
i
), in the same way that a text is
composed of a set of phrases. Thus,
F(S
i
) = Ł(G | H, S
i
)
(4.7)
The F(S
i
) biochemical process is catalytic, dependent on the action of
enzymes. Any enzyme is a protein having two or more active sites able to
promote interactions between other molecular elements, interactions that
would not spontaneously occur, or would occurs at very low rates, in the
absence of such proteins. A theory of cellular catalytic processes was pro-
posed by Monod, Changeux and Jacob (1963). Called the theory of allo-
steric mechanisms, it states that there are two possible configurations for
the active sites of a protein, called T (tense) and R (relaxed). The R state
has higher affinity with the substrate than the T state. The "all-or-none" al-
losteric rule says that all sites are in conformation T or in conformation R
(a case of exclusive disjunction; see Babloyantz, 1986). Therefore, if an ef-
fector acts upon an active site of a protein, changing it from T to R, then all
other sites will automatically change to R (and vice-versa regarding chang-
es from R to T).
The simplest case of an allosteric mechanism consists of a catalyst hav-
ing only two active sites. In this case, binding of one site with an effector
selects the configuration of both sites. If the first site changes from T to R,
the second site also changes to R, whereupon it may become able to bind
to a substrate and so change some property of that substrate. In sequence,
the substrate may act as an effector by binding to a second catalyst, and so
on, generating stp sentences and texts (for an update on allosteric mecha-
nisms in the brain, see Changeux and Dahene, 2000).
There are two, more complex, cases of allosteric mechanisms, which
will be mostly important for the formation of stp sentences and texts in the
brain. These are known as: (1) coincidence-detectors: proteins whose spa-
tial configurations are determined by a set of other molecules that have to
84 4 Neural Computational Mechanisms Supporting Cognitive Processes
bind it within a defined time interval after the first binding, as in the case
of the membrane NMDA receptor and the cytosolic adenylyl cyclase; and
also (2) second-order catalysts: proteins that control the phosphoryla-
tion—i.e., activation of sites—of the other proteins, illustrated by the fami-
ly of kinases.
Coincidence detectors are proteins that have 3 or more active sites, and
require the conjoint activation of at least two sites to generate a conforma-
tional change (transition from T to R, or vice versa in all sites).
Therefore, it is necessary that the binding of two or more effectors to
two or more sites of the protein, in a fixed time window (depending on the
properties of the protein), to generate the conformational change that, in
turn, will promote its binding to the substrate, activating a biological func-
tion (i.e., triggering the formation of new words, sentences and texts).
In the above sense, coincidence-detection (CD) is a kind of self-
organizing mechanism, by which states previously achieved in the system
control the emergence of a new state. As the new state is necessarily corre-
lated to the previously obtained ones, CD increases the coherence of the
system as a whole, while at the same time promoting an increase in diver-
sity. In other words, CD is a means whereby biological systems increase
their organization while also decreasing entropy. (the alternative would be
to increase organization by increasing redundancy, but in this case the
strategy would fail, since beyond a given limit entropy would also increase
and then the organization would decrease—a bell-shaped curve).
Resting state
Depolarized state
Glu binding
Glu binding
Figure 4.1 The NMDA channel: a complex molecule that operates as a coinci-
dence detecting device
CD can be exerted on signals arriving at the cell and on internal stps at
the cytoplasm. We exemplify the first kind with the NMDA receptor and
the second one with the protein adenylyl cyclase. The NMDA receptor is a
protein with several active sites, which binds to glutamate (Glu), glycyne
(Gly), zinc (Zn), etc., and to one effector site (controlling its ion channel)
that binds to magnesium (Mg; see Fig. 4.1). In the resting state, the protein
3.8 Summary 85
adopts the R conformation, the one in which the effector site is active and
bound to Mg. In this case, the ion channel inside the protein is blocked by
Mg. This situation changes when the membrane is depolarized by activa-
tion of another channel (e.g., the AMPA channel), and Glu binds to its site.
When these two events occur inside a temporal window (around 100 milli-
seconds), the protein changes from configuration R to T, and then the Mg
that was bound to the ion channel site (and blocking it) is removed, allow-
ing the entrance of Ca
2+
ions from the extracellular milieu (see Bliss and
Collingridge, 1993; Konig et al., 1996). Ca
2+
entry activates several new
stps inside the cell.
Definition 4.1: A CD stp is a sentence d([s
1
,...,s
c
],s
t
) having a complex
s
o
=
[s
1
,...,s
c
], and submitted to a temporal restriction such that
(d(s
o
, s
t
) | H, S
i
, ) 1,
if and only if the initial chemical transactions s
1
s
c
occurs
within the time window (see Definition 2.3).
Changeux and Dahene (2000) have argued that other kinds of ligand-
gated channels, such as the nicotinic acetylcholine receptor, can display
the properties of a temporal coincidence detector: ―a large majority of lig-
and-gated ion channels display desensitization and/or potentiation with ki-
netics which may be fitted by allosteric models. In addition, because of
their transmembrane disposition these receptors carry sites on both their
synaptic and cytoplasmic sides, letting the molecule integrate within a giv-
en time window multiple convergent pre-and post-synaptic signals. A time
coincidence detection mechanism may then be built from the discrete all-
or-none mechanism of the slow allosteric transitions‖.
There are eleven kinds of adenylyl cyclase (AC) enzymes in the mam-
malian brain, having the main function of generating cAMP from ATP.
They have been thought to be molecular coincidence-detectors (Anholt,
1994) also, since they link metabotropic receptors as well as G-proteins to
cAMP signaling pathways and also to Ca
2+
-activated calmodulin, which
are essential for many brain functions. Therefore, AC can be conceived of
as a coincidence detector for intracellular processes that occur in a longer
time window than obtains for the NMDA receptor. While the latter is ade-
quate for perceptual processes, AC could be better suited to the timing of
mnemonic and emotional processing.
Both Ca
2+
and G-protein pathways lead to the activation of AC, cAMP
release and then to kinase (second order catalyst) activation. Therefore, the
activity of the kinase family of enzymes is a part of all stps activated by
86 4 Neural Computational Mechanisms Supporting Cognitive Processes
transmitter-receptor binding at the neuronal membrane. A diagram of some
stps activated by glutamate receptors, including two kinases (PKA and
CamKII), and their possible role supporting conscious processing is shown
in Figure 4.2.
The kinase enzyme family has a major function in cell metabolism: the
catalysis of phosphate transfer from ATP to a protein substrate. Through
the action of kinases, other proteins becomes activated and can perform a
variety of biological functions. Therefore, kinases are second-order cata-
lysts, the catalysts that control the other catalysts.
Fig. 4.2. An example of d([s
1
,...,s
c
],s
t
): providing the energy for quantum compu-
ting
A recent genome mapping (Kostich et al, 2002) revealed that the kinase
family of enzymes corresponds to 510 sequences in the human genome,
approximately 2% of the total genome. The kinases determine several op-
erating characteristics of neurons by controlling the proteins that control a
variety of functions, as cytoskeleton spine density, post entry Ca
2+
signal-
ing, activation of transcriptional factors (CREB), and control of glucose
metabolism. They also control the life (cell growth and division) and death
(apoptosis) of neurons. The common aspect of all these functions is the
3.8 Summary 87
transfer a group of atoms (the phosphoryl group) between different mole-
cules. This mechanism, that is proper to the kinases, has been compared to
a switch that causes biochemical pathways to work slower or faster.
The role of some members of the kinase family in the brain, in learning
and memory processes, has been well studied by several researchers (see
Sweatt, 1999, for a review). Mainly protein kinase A (PKA), protein ki-
nase C (PKC), calmodulin-dependent protein kinase II (CaMKII), and mi-
togen-activated protein kinase (MAPK) have been implicated in such pro-
cesses.
It is clear from Fig. 4.2 that the processing of Ł(G | H, S
i
) by the cell S
i
results from a complex set of concurrent molecular transactions involving
many different d(s
o
, s
t
)s at the same time. These huge parallel processes
become very complex, especially when the results obtained in an assembly
of d(s
o
, s
t
)s doesn't match the results obtained by other assemblies of other
d(s
o
, s
t
)s. However, the possibility of successful non-linear coupling in bi-
ochemical computation is limited by the local character of the allosteric
mechanism. In other words, the allosteric mechanism, that is the universal
coupling mechanism for proteins, depends on the spatial proximity of ef-
fectors, receptors and substrates.
Cerebral processing is distributed over a huge number of neurons S
i,n
,
supporting complex distributed language Ł(G(H, S
b
))
= {{Ł(G
(H, S
i,n
))}
n =
1 to k
}
i = 1 to l
. In this condition, brain processing will very likely generate
many instances of conflict, many of them not solvable through the forego-
ing biochemical forms of interaction. Binocular rivalry in the visual sys-
tem, as well as many other modalities of mismatching between distributed
parallel units, are the rule in cerebral computational activity. Consequent-
ly, the integration process, in addition to biochemical coupling, should in-
volve non-local interactions between assemblies that recognize different
features of the stimulus and/or represent different goals to be achieved
through voluntary action. It thus ought to involve quantum computations.
In this context, CD stps will help to promote superposition and entangle-
ment.
4.3 Current Physical Implementations of Quantum Computers
Although most experimental results obtained with quantum computing are
limited to a few particles encoding a small (e.g. 2 to 4) number of qubits,
the properties of quantum computing are perfectly suitable for the model-
ing of brain function. The level of integration of millions of parallel pro-
cessing units in the brain cannot be explained through a serial coupling of
88 4 Neural Computational Mechanisms Supporting Cognitive Processes
operations, which would lead to an extremely long processing chain, or by
convergence to a single region, which would be overloaded. The brain
surely has some convergence regions, but this architecture is limited to re-
gional integration and cannot explain how the results of millions of proces-
sors are bound together in the performance of a complex cognitive task, or
in the generation of conscious states.
If brain function resembles the operations of a QC, the most relevant
property to be highlighted is entanglement. This property allows any num-
ber of particles, belonging to millions of cells, to instantaneously bind their
internal quantum informational states, generating a unified waveform that
would correspond to a moment (or to a temporal unit) of consciousness.
Also, quantum cryptography may be required to explain how one single
sensory (e.g. visual, auditory, etc.) quantum- computational line of pro-
cessing would not be corrupted by simultaneous computations carried on
in each of these sensory modalities (this subject will be discussed in Chap.
5).
Nuclear Magnetic Resonance (NMR) is being successfully used to built
QCs (Warren, 1997). Sequences of radio frequency pulses (rfp) manipulat-
ing spin orientations constitute quantum logic gates and perform unitary
transformations on the QC‘s state. In the first stage rfps create state super-
position. In the sequence, rfps modify spin orientations to write the desired
instruction. Finally, rfps change state amplitudes to enhance the probability
of the desired answer. NMRQCs have been built based on
1
H and
13
C nu-
clei in chloroform, on
3
Ce atom (Ahn et al., 2000), using two
1
H nuclei in
cytosine, as well as on alanine and trichloroethylene. Since quantum com-
puting may be implemented over organic molecules, in particular on amino
acids, it will be assumed here that state superposition may be achieved on
specific proteins.
Another technique being proposed to build QCs employs ions in micro-
traps (Cirac and Zoller, 2000). In this approach, the QC is composed of a
set of N ions confined in independent harmonic potential wells, which are
separated by a constant distance d, large enough to prevent Coulomb re-
pulsion from exciting the vibrational states of the ions, and so allows the
ions to be individually addressed (Fig. 4.3).
Quantum information is encoded in the internal states of particles (as
the spins of electrons or protons, and electronic states of ions), but the
transfer and retrieval of such information requires the manipulation of ex-
ternal states (values related to movement). Therefore, quantum- informa-
tional devices are two-level systems (Nagerl et al., 2000). Experimental
manipulation of such systems is based on the achievement of a strong cou-
pling between external and internal states. In ion-trap computers, internal
states of one ion are coupled to the vibrational degrees of freedom of the
3.8 Summary 89
ions in the trap (Nagerl et al., 2000) in such a way that the emission or ab-
sorption of phonons (quasi-particles) by the ions, caused by a change in
movement, is accompanied by changes in internal states.
In the ion-trap quantum computer, the coupling between the vibrational
and the internal states of the ions is made by an oscillator. Laser pulses
have been used as oscillators, allowing the transferal and retrieval of in-
formation to the internal states of ions. Rfps forming a quadripole control
the movement of a population of ions in a magnetic trap. Laser beams are
used to individually access the ions and change/measure their internal
states (Kielpinski et al., 2002).
Fig. 4.3. Ion-Trap QC (adapted from Kielpinski et al., 2002)
In this setting, the CNOT logical gate has been implemented with a de-
gree of realibility around 75% (Schmidt-Kaler et al., 2003). A target ion
has its internal state changed to the state labeled as 1 in binary code. The
controlled ion is then changed (from 1 to 0 or from 0 to 1) only when the
state of the target ion is 1. As theoretically predicted, the experiments have
shown that once the string of ions in the trap gets entangled, the change in
the internal state of the target ion naturally produces the expected change
in the controlled ion(s).
90 4 Neural Computational Mechanisms Supporting Cognitive Processes
4.4 The Dendritic Spine as Quantum Computing Device
There are several kinds of chemical elements and molecules that together
with their constituent particles could be used by the brain to generate quan-
tum computational systems. Among the experimentally proven alternatives
reviewed in the last section, we have chosen trapped Ca
2+
ions as a plausi-
ble neurobiological model, by considering the already known functions of
such cations, which are frequently trapped in the dendritic spine or in intra-
cellular compartments (for a review of Ca
2+
functions see Alkon et al.,
1998; Ghosh and Greenberg, 1995). These Ca
2+
movements are central to
all kinds of cognitive and emotional processes in the brain. Ca
2+
ions
trapped in cellular compartments would quite arguably satisfy the require-
ments for a quantum computing device, while retaining information from
the biochemical processes they participate in.
Spines are morphological specializations of mammalian neurons that re-
ceive the majority of synaptic excitatory inputs. Their role has recently
been studied by using new techniques, such as two-photon microscopy and
the generation of transgenic mice. A consensus has emerged among neuro-
scientists that ―spines are biochemical units that compartmentalize calci-
um...the specific function of spines could be calcium-derived signal trans-
duction, restricted to individual synaptic inputs, which could implement
input-specific learning rules‖ (Holthoff et al., 2002).
By imaging spines from layer V pyramidal neurons of the rat primary
visual cortex, the above authors discovered that the position of the spine in
the dendrite is related to its Ca
2+
dynamics and to its putative visual pro-
cessing rules.
Spines are suited to the accumulation of Ca
2+
since they have a "head"
that is larger than the "neck" that connects it to the dendrite (see Sabatini et
al., 2001, 2002). The STPs internal to the spine are compartmentalized by
several scaffold, anchoring and adaptor proteins (see Fig. 4.3).
Ca
2+
enters into the spine through AMPA and NMDA membrane recep-
tors, and also through voltage-dependent calcium channels (VDCCs). Such
membrane receptors are ion channels controlled by the binding of a ligand
(mainly the neurotransmitter Glu). VDCCs, in turn, are ion channels con-
trolled by the voltage (action potential) of the membrane. The three kinds
of Ca
2+
channels work cooperatively. AMPA and VDCCs can work inde-
pendently of the other channels, while the NMDA channel is dependent on
the others, because of its complex physiology. Being an activity-dependent
mechanism, NMDA opening obeys a more restrictive set of conditions. On
the other hand, it works as the faster controlling mechanism of massive
Ca
2+
entry, and therefore has a central role in the processing of afferent in-
formation.
3.8 Summary 91
A central factor determining Ca
2+
dynamics in the spine is the presence
of Ca
2+
pumps, which cause its differential distribution among the spine's
internal compartments. Holthoff et al. (2002) also found that the Ca
2+
pump‘s strength is positively correlated with its distance from the soma of
the neuron. Ca
2+
ions trapped in the dendritic spine are moved from one in-
ternal compartment to another, or through the neck to the endoplasmatic
reticulum, by the action of Ca
2+
pumps.
Fig. 4.3. The dendritic spine: the dendritic spine has a very complex structure
supporting advanced functions
92 4 Neural Computational Mechanisms Supporting Cognitive Processes
The energy prompting Ca
2+
movement in the spine is made available
through the reduction of ATP by one member of the kinase family, the
calmodulin-dependent protein kinase (CaMKII). This kinase is capable of
autophosphorylation, or self-catalysis. It can also be activated by binding
to calmodulin (CaM), a protein that is activated by Ca
2+
and available in a
large number of configurations, which are selected by the entering Ca
2+
(see discussion in Pereira Jr., 2003).
The cognitive profile of CaMKII is completely adequate to the new, pu-
tative functions we assign to it. CaMKII has been proposed as a candidate
for the molecular basis of memory: "The analysis of CaMKII autophos-
phorylation and dephosphorylation indicates that this kinase could serve as
a molecular switch that is capable of long-term memory storage" (Lisman
et al., 2002). Also, Vianna et al. (2000) suggest that ―memory formation of
spatial habituation depends on the functional integrity of NMDA and
AMPA/kainate receptors and CaMKII activity in the CA1 region of the
hippocampus [and that] the detection of spatial novelty is accompanied by
the activation of at least three different hippocampal protein kinase signal-
ing cascades‖.
The CaMKII cycle in the dendritic spine has recently been elucidated
(Fox, 2003). Its movement in the spine is controlled by Ca
2+
-induced ex-
citatory activity. In turn, CaMKII provides the energy for Ca
2+
movements
in cellular compartments. These movements, we hypothesize, would be re-
lated to the encoding of information in internal electronic states of the ions.
Considering also the existence of surrounding fields generated by the
movements of other ions (Na+, K+), the dendritic spine has an impressive
similarity with current ion-trap quantum computers, which use the same
Ca
2+
caged in magnetic traps, moved by magnetic quadripoles and cod-
ed/decoded by laser beams.
In our framework, entanglement of Ca
2+
internal states in spines would
be generated by the coordination of actions of membrane channels and cel-
lular agents that manipulate this cation. For instance, the NMDA channel
CD would transform a temporal coincidence of Glu pulses generated by
perceptual mechanisms into a quantum coherent state by erasing the tem-
poral order and altering the Ca
2+
vibrational state to generate superposition
(see the conditions for the generation of entanglement in Bouwmeester and
Zeilinger, 2000).
The possibility of the dendritic spine working as a Ca
2+
trap QC depends
on the demonstration of its compatibility with the requirements for a phys-
ical system to perform quantum computation. Since the spine has
mesoscopic mechanisms that manipulate individual atoms, we cannot rule
out the possibility that such mechanisms could prepare the ions in super-
posed and entangled states, and then perform transformations correspond-
3.8 Summary 93
ing to logical gates by means of metabolic operations on the trapped Ca
2+
s.
The conjoint activation of AMPA, NMDA, and VDCC would control dif-
ferent quantum computational operations in spines. Neurobiological data
has shown (see Krystal et al., 1999) that the balance of Ca
2+
channels is
necessary for the generation of normal consciousness, while disturbances
(e.g., AMPA and VDCCs activation without NMDA) generate perceptual
distortions and hallucinations (see Pereira Jr. and Johnson, 2003). These
phenomena would be explained by the specific role of each channel in spi-
nal QC, as proposed in more detail in the following chapter.
A popular objection to neural quantum computation derives from the
warm and noisy conditions of the brain, as compared to the experimental
settings where QCs have been realized (see discussion in Pereira Jr.,
2003). However, it should be noted that Ca
2+
ions are cooled in artificial
ion-trap QCs to improve the accessibility of initial and output states as
classical binary states (an important requirement for digital computers,
corresponding to Nielsen and Chuang's last requirement), not for the gen-
eration and manipulation of superposition/entanglement. Isolation is im-
portant to assure the unitarity (also referred as the "reversibility") of the
operation in the time window necessary for the artificial device to encode
and decode binary data reliably. Such experimental constraints have not
been proved to forbid natural quantum computing from occurring in short-
er time windows or with less reliability than artificial QCs. Isolation and
low temperatures are not therefore absolute requirements for the existence
of QCs, but only conditions for the engineering of artificial QCs as digital
computers.
94 5 The Brain and Quantum Computation
5 The Brain and Quantum Computation
Quantum computing is becoming a reality both from the theoretical point
of view as well as from the realm of practical applications. New techniques
are being discussed and implemented, making quantum operations feasible
at room temperature. Moreover, new architectures are being proposed for
future quantum machines. Nuclear Magnetic Resonance and Ion Traps (IT)
are among the technologies most used in quantum computation experi-
ments.
The brain is the most sophisticated processing machine developed by
nature thus far. Quantum information and quantum computation have been
considered important issues in the effort to understand neural function and
the phenomenon of consciousness.
Dendritic spines (DS) are specialized synaptic structures with low en-
dogenous Ca
2+
buffer capacity, allowing large and extremely rapid Ca
2+
changes under physiological conditions. Ca
2+
diffusion to the dendrite
across the spine neck is negligible, and the spine head functions as a sepa-
rate compartment on long time scales, allowing localized Ca
2+
build-up
during trains of synaptic stimuli. Also, DS are very plastic structures, in-
volved in both rapid learning, or imprinting, as well as in the slower learn-
ing due to environmental changes.
Here DS are assumed to be IT devices that may be used to build Quan-
tum Charge-Coupled Computers with an architecture similar to that pro-
posed by Kielpinski et al (2002). We use the Deutsch-Josza algorithm (see
a review in Nielsen and Chuang, 2000) to propose a model of a Quantum
Cortical Pattern Recognition Device (QCPRD). Since the Deutsch-Josza
algorithm is also probabilistically computable in a constant number of
steps, we implemented a probabilistic QCPRD version and tested it as a
Visual Pattern Recognition Device (VPRD).
We propose that quantum processing controls the allosteric (R,T) states
of proteins which are involved in coincidence detector stps (d([s
1
,...,s
c
],s
t
)).
In this condition, superposition, entanglement and quantum processing are
assumed to be dependent on phosphorylation of specific kinases, triggered
by the entry of calcium through the NMDA channels and mR-Glu in the
5.1 The Dendritic Spine as an Ion Trap Quantum Computing Device 95
support of higher brain functions such as consciousness and working
memory.
5.1 The Dendritic Spine as an Ion Trap Quantum
Computing Device
An ITQC involves:
1. confining ions in a single narrow trap;
2. employing ionic electronic and motional states as qubit logic levels; and
3. transferring quantum information between ions through their mutual
Coulomb interactions.
Ion trapping (Fig. 5.1) is realized by using a combination of radio fre-
quency and quasi-static electric fields generated by a set of electrodes
(Kielpinski et al, 2002). By varying the voltage on these electrodes, ions
are confined in a particular region or transported along the local trap axis.
Applying radio frequency voltage to outer layers creates a quadripole field
that confines the ions transverse to the local trap axis (see Fig. 4.3). Laser
pulses are used to implement quantum gates (see Kielpinski et al, 2002).
Photodetectors are used to read the results of the quantum computation.
Fig. 5.1. The dendritic spine as a Quantum Computing Device: nature has en-
dowed the brain with a huge computational capacity.
96 5 The Brain and Quantum Computation
We propose here that Ca
2+
ions are trapped in molecular cavities of
some special molecules composing the Post-Synaptic Density (PSD) and
that energy released by phosphorylation of other PSD proteins implement
different kinds of quantum gates. At the end of the computation, Ca
2+
ions
are moved into the spine´s endoplasmatic reticulum (SER) or removed to-
ward the extracellular space, depending of the obtained results.
Electric
Motion
Fig. 5.2. The DS qubits: building the neural qubits
The electronic qubits for Ca
2+
are
1
:
|0> = s
1
and |1> =
3
p
0
(5.1)
and the energy E required to promote the state transition (Harris, 1996)
S
1
3
p
0
(5.2)
is equal to 25 .21507 eV such that the frequency of the activating photon
is calculated from (Harris, 1996):
E = h
(5.3)
where h is Planck‘s constant. Its value is 50 nm (UV). The motion qubits
are
|0> = ground state; and |1> = first excited state
(5.4)
with ground state energy of 3.2 eV calculated for a cavity of around 0.7A.
This value was chosen because Ca
2+
radii is smaller than the atom having
1
http://physics.nist.gov/cgi-bin/AtData/levels_form
5.2 The Deutch-Josza Algorithm 97
the same number of electrons in the outside shell, which corresponds to
Argonium.
Remark 5.1. It is interesting to note here that the frequency of energy
absorption for Tyrosine transition electronic state is around 250 to 269 nm.
Tyrosine is one of the aminoacids composing a family of kinases, called
Tyrosine Kinases.
5.2 The Deutch-Josza Algorithm
The DS physiology was discussed in Chap 4. In order to understand DS as
a quantum IT device it is sufficient to recall the following from the physi-
ology of glutamate (Glu) and its receptors (Fig. 5.3):
1. Glu binding to AMPA receptors allows the entry of Ca
2+
and promotes a
depolarization of the membrane potential;
2. Ca
2+
also enters the spine by means of the voltage sensitive Ca channel
(VSCC in Fig. 5.3);
3. The NMDA receptor functions as a coincidence detector (CD), since
depolarization removes the Mg
2+
attached to it, and a posterior Glu-
binding, within a temporal window , promotes the entry of Ca
2+
. Next,
calcium-bonded Calmodulin (CaM) takes energy from ATP sources and
delivers it to other biochemical processes;
4. Glu binding to metabotropic receptors (mR-Glu) promotes the activation
of many types of G-proteins which control other processes;
5. Glu receptors are anchored in the membrane by a set of proteins; and
6. Ca
2+
concentrations inside the cell are controlled by systems moving it
to other organelles (e.g., SRE in Fig. 5.3).
Conjecture 5.1. The NMDA coincidence detector paradigm is proposed
to create entangled quantum states in the brain, as follows:
1. A first Glu released by a pre-synaptic cell is received by the AMPA
channel, promoting EM depolarization and Mg
2+
release from the
NMDA channel (Fig. 5.3A).
2. EM depolarization opens the VSSC channels; at this juncture Ca
2+
en-
ters and is trapped at the Post-Synaptic Density (PSD) (Figs. 5.1 and
5.3A).
3. A second release of Glu (by another pre-synaptic cell), within the tem-
poral window , transports Ca
2+
to bind to CaM. CaM provides energy
to create state superposition of other proteins, or trapped Ca
2+
ions (Fig.
5.3B).
98 5 The Brain and Quantum Computation
4. A third Glu release (by a third pre-synaptic cell) over GluMR activates
G-proteins and then implements quantum gates with the entangled Ca
2+
(Fig. 5.3C and D); and
5. Ca
2+
moves into the SRE at the end of the quantum computations.
Fig. 5.3. The dendritic spine as a Quantum Computing Device: using
molecular machinery for quantum computation
The Deutsch-Jozsa algorithm (DJA) was the first explicit example of a
computational task performed exponentially faster using quantum effects
instead of classical means
(Nielsen and Chuang, 2000).
Given a one-bit function, one has either two constant functions:
(0) = 0 and (1) = 0, or (0) = 1 and (1) = 1;
or two ―balanced‖ functions:
(0) = 0 and (1) = 1, or (0) = 1 and (1)= 0.
5.2 The Deutch-Josza Algorithm 99
A one-qubit QC (Fig. 1B) can decide whether is constant or balanced in
just one step.
DJA is proposed as a mechanism to provide a one-step answer to the
following question: Is constant or balanced?
Algorithm 5.1: DJA works as following (Fig. 5.3):
1. Starting with the standard state |0> |0> at the input (I) and output (O)
registers;
2. a NOT operation is applied to I;
3. Next, the Hadamard transformation (H in fig. 5.3) is applied to both reg-
isters.
Thus:
NOT H
|0> |0> |0> |1>
2
1|0|
2
1|0|
(5.5)
In the sequence, the unitary transformation U
f
is applied to both registers
U
f:
2
1|0|
|1
2
1
2
1|0|
2
1|0|
)(
Bx
xf
x
(5.6)
5. In this condition, O remains in the state (|0> - |1> / 2). If is constant I
is (|0> - |1>/2) and if is ―balanced‖ it is (|0> + |1>/2). If H is ap-
plied again to I, it becomes (|0>) if is constant and (|1>) if is ―bal-
anced‖.
6. Finally, these O states are reliably distinguished by a measurement in
the standard basis, thus distinguishing balanced from constant functions af-
ter just one query.
Conjecture 5.2: DS process the DJA as follows (Fig. 5.3).
1. The AMPA channel is associated with the O register, whereas the VSSC
is assigned to the I register;
2. The initial EM depolarization promoted by Ca
2+
entry through the
AMPA channel removes the Mg
2+
from the NMDA channel and opens
VSCCs;
3. Next, NMDA channels are activated, and CaM is used to perform the
Hadamard transformation upon the Ca
2+
trapped in the spine head;
4. Next, mR-Glu receptor is activated, and a G-protein is used to imple-
ment U
f
, as well as
100 5 The Brain and Quantum Computation
5. to perform another Hadamard transformation, and
6. the result is obtained by moving (or not) the Ca
2+
into SRE.
Remark 5.2: A point worth remarking is that DS is amenable to many
different and sophisticated experimental manipulations, which may be
used to check the above assumptions.
Now, let‘s consider another function : B
n
that is either constant
if the 2
n
values are either 1 or 0, or balanced if exactly half (i.e. 2
n-1
) of the
values are 0 and half are 1.
Algorithm 5.2: DJA is extended to answer if : B
n
balanced, if
1. one starts with a row of n I qubits and one O qubit and to applies the
same step procedures above . At the end, the n Is are in state
2. a NOT operation is applied to I;
3. Next, the Hadamard transformation (H
n
) is applied to all n qubits of the
I register and to the O register;
4. In the sequence, the unitary transformation U
f
is applied to both regis-
ters, resulting in
|
> =
n
x
xf
n
x
2
)(
|1
2
1
(5.7)
5. If is constant then
will be just an equal superposition for all the
|x>‘s with an overall plus or minus sign, whereas if is a balanced func-
tion then
will be an equally weighted superposition with exactly half of
the |x>‘s having the minus signs;
6. Recalling that H has its own inverse (HH=1) and that H applied to each
qubit |0> of |
> results in an equal superposition of all |x>‘s. Therefore, if
is constant then the resulting state is
x = |0>|0>...|0>, and if it is balanced, then |x>‘s is x |0>|0>...|0>.
Remark 5.3. The reading of each of the n qubits completes the meas-
urement. DJA requires O(n) steps to distinguish balanced from constant
functions, whereas classical algorithm demand O(2
n
) steps for the same
task. However, a probabilistic algorithm is able to solve the same task in k
steps with a probability of (1-) for correct answer and less 1/2
k
.
5.3 The Quantum Cortical Recognition Device 101
5.3 The Quantum Cortical Recognition Device
The cortex is composed of a ordered set of layers (Fig. 5.4.a) numbered 1
to 6 from outside to inside. Their output neurons are called pyramidal
cells. Input information from thalamus arrives at layer 4 to the stellate
cells, which in turn conveys it to the pyramidal cells at layers 2 and 3. The
output from these layers are distributed over other cortical areas, whereas
the pyramidal cells from layer 6 send its axons to the thalamus, colliculus
and brainstem, besides returning collateral branches to layer 2 and 3, and
to layer 1. In this layer, the axons form the so-called parallel fibers also
spreads to other cortical areas.
Many interneurons operate in the cortex, controlling the traffic of in-
formation between the stellate and pyramidal cells and between pyramidal
cells. Many of these interneurons are GABA inhibitory neurons.
Many modulating circuits (RAS, 5-HR, DA and NA in Fig. 5.4.a) con-
trol the excitability of the pyramidal cells. These control circuits are in
charge of setting the electrical code of the pyramidal cells as either a tonic
or phasic encoding (Rocha, 1997), by depolarizing the membrane.
5.3.1 The Model
Definition 5.1: The electrical encoding by the pyramidal cell is:
1. in hyperpolarized states, when the membrane electrical potential EM
approaching the K equillibrium potential E
k
is supported by a phasic en-
coding (see Rocha, 1997 and Rocha et all, 2001) of short words formed
by bursts of action potentials. The phasic encoding is mainly determined
by an oscillating Ca
2+
current. Also, hyperpolarized states favors the
Mg
2+
binding to the NMDA channel; and
2. in depolarized states, when the membrane electrical potential EM is
growing less negative, supported by a tonic encoding of long words
formed by a continuous spiking. The tonic encoding is mainly deter-
mined by Na
+
, Ca
2+
and K
+
ions. Also, depolarized states favor the Mg
2+
uncoupling from the NMDA channel.
Thus if a neuron n
i
O(G | H, S
b
) is able to express two languages
Ł(G| H, n
i, 1
), Ł(G| H, n
i, 2
)
there must exist d(s
o
, s
c
) Ł(C, | G, H, S
b
) to specify which of these lan-
guages are to be used in a given processing.
102 5 The Brain and Quantum Computation
Model 5.1: Let the the DJA processing device CPRD in Fig. 5.4.b be
formed by:
1. a retina R as an array of r = p x p bits;
2. a device B (cortical stelate cells) of m states of 1 bit;
3. a device Q (cortical pyramidal cells) of m states of 1qubit;
4. a device D (cortical stelate cell) of 1 qubit; and
5. a measuring device M of 1 bit.
Fig. 5.4.a. Cortical layers and neurons (A) and connections (B)
A pattern recognition device is constructed by:
1. M
1
mapping neighbor n
1
(n
1
=1 in Fig. 5.4) bits of R to a bit of Q;
2. M
2
mapping neighbor n
2
bits of R to a bit of B;
3. one to one mapping M
3
from B to Q;
5.3 The Quantum Cortical Recognition Device 103
4. one to all mapping M
4
from D to Q,
5. one to all mapping M
4
from Q to A;
6. a function (x), such that either (0) = (1) and it is said constant or (0)
(1) and (x) is said to be balanced.
Fig. 5.4.b. The Cortex and the QCPRD: building a powerful pattern recognition
device using quantum computing
It is assumed herein that:
1. stellate cells B in Fig. 5.4 are strongly inhibited by OFF stellate neigh-
bors and excited by ON cells, such that
2. they clearly recognize and amplify edges (for contrast) in a given direc-
tion. Also:
3. pyramidal cells Q in Fig receive information from:
Edge Detector Stellate Cells B;
ON stellate cells activated by the retinal cells; and
D stellate cells: on cells with large receptive fields.
104 5 The Brain and Quantum Computation
Fig. 5.5. The cellular structure of QCPRD: enhancing contrast to detect bor-
ders.
Algorithm 5.3: CPRD is assumed here a QCPRD if:
1. the D bit is set 1 at the moment t
1
if all m bits of R mapping to it are set
1 at time t
o
(t
o
<t
1
) otherwise it is set 0;
2. the input register I (or VSCC channel) of Q is set by R and the output
register (or AMPA channel) is set by D at time t
1
;
3. If 0 < t
1
- t
o
< then m bits of Q are set in superposition by Hadamard
gates H triggered by the NMDA channel, and
4. a bit of B is set to 1 at the moment t
1
if the n neighbor bits of R mapping
to it are set 1, otherwise it is set 0;
5. the function (x) at the i
th
bit of R is set at time (t
1
< t
2
) by the mR-Glu
system as constant
if the i
th
bit of B is 1;
6. otherwise, the function is set as balanced, and the m qubits of R are
transformed at the time t
3
by the unitary matrix
U
: | x, d > | x, d > (x)
7. after this, the IP
3
channel applies another H to I at time t
4
;
5.3 The Quantum Cortical Recognition Device 105
8. M measures states of I in O(n) steps at time t
5
, such that its bit is set 1 if
(R) is constant, otherwise it is set 0, and
9. finally, the stimulus pattern P at the retina R is recognized if (R) is
constant. This pattern is defined by the maps M
1
and M
2
, and it is a di-
agonal in Fig. 5.4. The complexity of P is directly related to the number
of bits used by B and Q.
Algorithm 5.3. A probabilistic version (PCPRD) of a CPRD is defined
if Q is regarded as a classical device and if M performs probabilistic meas-
urements. The QCPRD structure is modeled herein upon the visual cortical
column (Verkhratskt, 2002, Kasai, 2003 and Miller, 2003), especially con-
cerning the behavior of the device D and its relations with Q
(Kasai 2003).
Remark 5.4: The evolution from PCPRD to QCPRD may account for
the increase on cortical processing in nature.
5.3.2 Learning
The learning of the maps M
1
, M
2
is implemented by beginning with large
neighborhoods n
1
, n
2
and pruning n
2
those connections at B and Q not
used for P identification. If n
1
, n
2
are initially very large, any P presented
to R will be associated to a constant (R) and assumed as a known P. Now:
Proposal 5.2: The following are basic CPRD learning rules:
Rule 1: A rapid learning (imprinting) occurs if most (if not all) not used
connections of M
1
, M
2
are drastically (n’
1
, n’
2
1) pruned. In this condi-
tion, P is fixed and easily recognized in any other future occasion. Imprint-
ing is efficient if the variability of P (v(P) in the closed interval [0,1]) is
low, otherwise the number q of CPRD specialized to recognize {P
i
}
i=1 to r
increases as r1 and v(P) 1.
Rule 2:A slow learning occurs if n’
1
,n’
2
<< r bits, the learning ve-
locity being inversely proportional to . Slow learning is preferable if v(P)
is large, since it allows the selection of the most significant s bits of P to be
used in its recognition. However, the s 0 as v(P) 1. Again, different
CPRD may be created to recognize sets similar {P
i
}
i = 1 to r
.
Spine pruning is demonstrated in both imprinting and slow learning in
the zebra finch (Helmchen, 2002; Lieshoff and Bischo 2003). Sexual im-
printing occurs mainly at the lateral neo-hyperstriatum (LNH) and slow
learning at archineostriatum caudale (ANC). Both types of learning re-
quires an initial increase of the number of spines in both (hormonally in-
duced) LNH and ANC (environmentally induced). Spine density decreases
in LNH within 2 days after exposition of to the female, whereas but re-
106 5 The Brain and Quantum Computation
quires around 3 weeks to stabilize in ANC. The density increase promoted
by changing the animal from isolation to a social condition occurs in 3
days.
Fig. 5.6. Examples of PRCD recognition
5.3.3. Recognizing Faces
Algorithms 4 and 5 were implemented in the system Sensor that is being
developed in our laboratory since 1997 (Rocha, 1997,
www.eina.com.br/sensor) using the formalism of Fuzzy Formal Grammars
to simulate natural vision systems. The efficacy of those algorithms in
learning to identify defined images was tested using the Enscer Figure Da-
ta Base (www.enscer.com.br). This was done according to the following
steps:
5.4 Fuzzy Logic and Conflict in O(G|H,Sb) 107
1. select a searching element: this is the element to be used to select the
figures in the database. It is selected from one or more of the figures of
this database. In the case of the example of Fig. 5.6, the chosen element
was Laura, a female character;
2. select the element features to be used in the search: these are the com-
ponents of the query to be used to search data base figures. In the pre-
sent example, the query features were the face and hair contours;
3. use the query elements for PCPRD training: in the case of imprinting
simulation it is sufficient to use just one example, whereas in the case of
slow learning it is necessary to use examples from more than one train-
ing figure. These training examples are used to define the pattern of the
query representative points, that will be used as the template to disclose
similar elements in the queried images;
4. test the efficacy of the system using another set of figures: it is necessary
to have a selected set of figures containing and not containing the query
element, in order to test the efficacy of the system in correctly identify-
ing the figures containing the query element, without missing any one or
mistakenly taking other elements as the query one.
PCPRD attained an efficiency of 90% in cases similar to that of Juca
and Laura‘s identification (Fig. 5.6) and imprinting simulations, when
dendritic spine pruning preserved neighborhoods of 10% or less near the
query representative points and the probabilistic identification was set to
admit an error of 10%. Imprinting worked fine with query elements that
are very prototypical as Laura. In the case of more variable query ele-
ments, slow learning was a better approach to characterize specific points
of a small neighborhood variance, which are better key elements for a suc-
cessful query. Such a characterization depends on successive pruning pro-
moted by a given family of training examples. The PCPRD algorithm in-
creased Sensor efficiency when compared to the previous contour
identification algorithm in use, based on comparison of those points at
which the contour experienced a significant change of direction (Rocha,
1997).
5.4 Fuzzy Logic and Conflict in O(G|H,S
b
)
The ambiguity of L(G | H) together with the distributed organization of
O(G | H, S
b
) with respect to process, are certainly important sources for
conflict among agents n
i
O(G | H, S
b
).
Definition 5.5. Given
108 5 The Brain and Quantum Computation
d(s
o
, s
t
| H, S
b
) and d(~s
o
, s
t
‘
| H, S
b
), (s
o
, ~s
o
) = 0
, s
t
, s
t
‘
V
t
(5.7)
conflict occurs if
(s
t
, s
t
‘) 1
(5.8)
implying that if confidence in or truth of is defined as
(d(s
o
, s
t
| H, S
b
)) = (s
t
, V
t
)
, (d(~s
o
, s
t
‘
| H, S
b
))= (s
t
, V
t
‘)
(5.9)
then
(d(s
o
, s
t
| H, S
b
))+ (d(~s
o
, s
t
‘
| H, S
b
)) > 1
(5.10)
Proposal 5.3. DIPS conflict may arise from many causes, but mainly
because distinct agents n
i
, n
j
O(G | H, S
b
) may shares pieces of infor-
mation d(s
i
, s
j
) d(s
o
, s
y)
or d(~s
o
, s
t
‘
) collected at different moments t
i
, t
j
or for different sources n
r
, n
s
; they may use different tools or (V
n
, P)s to
process d(s
o
, s
t
| H, s
i
) and d(~s
o
, s
t
‘
| H, s
j
)), etc. That is because two or
more agents n
i
, n
j
O(G | H, S
b
) provide conflicting information when en-
rolled in solving a given task.
Theorem 5.1. The increase of the mean ambiguity <(Ł(G | H, S
b
))> of
Ł(G| H, S
b
) enhances conflict in O(G| H, S
b
).
Proof: The mean ambiguity (d(s
o
, s
t
) | H, S
b
) of d(s
o
, s
t
) is
(d(s
o
, s
t
)|H, S
b
) =
((d(s
o
, s
t
)) log (d(s
o
, s
t
)) - (1-(d(s
o
, s
t
))) log(1- d(s
o
, s
t
))
(d(s
o
, s
t
))> =
5,0)),|,((
1
0
1
bt
m
i
i
SHssd
m
where m is the number of d(s
o
, s
t
) accepted as belonging to Ł(G| H, S
b
) be-
cause (d(s
o
, s
t
)) 1. Therefore
<(Ł(G | H, S
b
))> m bits
The increase of m augments the possibility that given two agents n
i
, n
j
O(G | H, S
b
) then
(d(s
o
, s
t
| H, n
i
)) 1, (d(~s
o
, s
t‘
| H, n
j
)) 1,
and
5.4 Fuzzy Logic and Conflict in O(G|H,Sb) 109
(d(s
o
, s
t
| H, n
i
)) + (d(~s
o
,s
t
‘
| H, n
j
)) > 1
Conflict may be resolved by means of:
1. Default Logic: because matching priority supports derivation chains of
the type
(s
i
, s
j
) s
i
s
j
unless d(s
i
, s
k
) s
i
s
k
(5.11)
That is, given d(s
i
, s
j
) <d(s
i
, s
k
), s
j
rewrites s
i
unless s
k
is available to
rewrite s
i
. Here, the symbol means to support.
2. Threshold Logic: because it may assume that a sentence s
i
is al-
lowed to be rewritten only if its number of available copies q(s
i
),q(s
j
) are
greater than a given minimum that is
q(s
i
), q(s
j
) > θ
(5.12)
3. Temporal logic: temporal reasoning may be supported by fuzzy lan-
guages, because (d(s
o
,s
t
) are subjected to temporal restrictions , such
that
(d(s
o
, s
t
| H, n
k
, ) 1 only if t <
(5.13)
and t is proportional to the number k = 2 of derivation steps of
d(s
o
, s
1
, ... , s
k
, s
t
).
As a matter of fact, the above logics may be considered as special cases in
a broader family of logics, called
4. Fuzzy Logic: because it may assume that a sentence s
i
is rewrilten as
(d(s
o
,)) =
Q
n
i 1
((d(s
o
, s
t
| H, n
i
) )
(5.14)
where Q is a logical operator (e.g., the majority of the rules, x of n rules,
unless, etc.);
Remark 5.5. Many types of human reasoning are supported by the
above logic (Rocha, 1992, Rocha, 1997). Many types of conflict are solved
by means of consensus inspired in a logic of this type if
“most of the relevant pieces of information are true... then”
110 5 The Brain and Quantum Computation
that is
(d(s
o
, s
t
)) =
Q
n
i 1
((d(s
o
, s
t
| H, n
i
) (d(s
o
, s
t
| H, n
i
)) ) 1
(5.15)
However, not all types of conflict may be solve by the above techniques
once the control of ((d(s
i
, s
j
) | H, S
b
)) is restricted by the power of the
control language Ł(C, | G, H, S
b
) as stated by Theorems 2.5 and 2.6.
5.5 Recurrent Architecture Generates Entanglement
Supporting Consciousness
A ubiquitous strategy used by large assemblies is signal reinforcement
through recurrent processing. Computational loops correspond to circular
sentences supported by Ł(G| H, S
b
).
Definition 5.7. A CD sentence d([s
1
... s
j
... s
c
], s
t
), s
o =
[s
1
, ..., s
j
,..., s
c
]
(Definition 4.1) is circular if there exists d(s
1
, s
j
) such that (d(s
1
, s
j
))1.
Forming loops is primarily a way to increase the computational weight of
an assembly. Also, as a positive by-product, competition ends by increas-
ing the coherence of the whole-brain activity, since the loops activate CDs
and second-order catalysts. Therefore loops become critical factors for at-
tention/consciousness. By increasing coherence they prepare the system
for quantum computing, a resource that is able to overcome mismatching
and destructive competition.
In the visual system, a sensory signal arriving at proximal dendrite sites
of the pyramidal cortical neuron activates AMPA channels, depolarizes the
membrane and displaces Mg
2+
from the NMDA channel. The degree of
this depolarization is assumed to be greater for tonic spiking state associat-
ed to wakefulness than for rhythmic burst firing observed during slow
wave sleep. This AMPA activation encodes sensory information into a
spike volley at the axon, which is distributed to other high order cortical
neurons.
Later reentrant information from the last neurons is distributed over the
distal sites of the pyramidal neuron dendrite and activates NMDA chan-
nels. If both the sensory information and the reentrant signal are coherently
associated, then Ca
2+
entrance is prompted and activates a stp, binding to
CaM and then to CaMKII and other kinases, increasing the coherence of
neuronal activity among all neurons that participate in the whole circuit
(Fig. 5.7). Such loops contribute to activate the coincidence-detectors and
5.5 Recurrent Architecture Generates Entanglement Supporting Conscious-
ness 111
second-order catalysts, thus increasing the spatio-temporal coherence of
brain activity, which is important for selective attention (LaBerge, 2001)
and also to the putative generation of quantum entanglement among dis-
tributed agents, supporting complex cognitive functions and consciousness
(Rocha et al., 2001).
Fig. 5.7. Reentrant loops in the brain: consciousness is entirely dependent on
reentrant information
One of the main characteristics of neuronal assemblies is the emergence
of synchronous oscillations, in the theta and gamma frequencies (see a dis-
cussion in Pereira Jr. and Rocha, 2000), which are believed to increase the
coherence among the cells belonging to the assembly, and to distinguish
the assembly from its environment. By doing so, the assembly also pre-
pares itself for quantum computing. The existence of convergence regions
112 5 The Brain and Quantum Computation
together with mechanisms of temporal synchronization of neuronal oscilla-
tions are important, but limited forms of integration. We have argued (Ro-
cha et al., 2001) that possible and frequent conflict between specialized
agents generates mismatches that could impair cognitive processing and
context-related generation of adequate behavior. The putative and plausi-
ble existence of quantum computing in the brain would be a natural and
scientifically-based alternative to explain how brain function could be in-
tegrated to such a higher level, and how a unified state of consciousness
could be generated from a distributed parallel system.
Fig. 5.8. Quantum processing in reentrant loops
Therefore, we propose that cognitive processing in the brain uses com-
putational mechanisms that are very close to the strategies of parallel pro-
5.5 Recurrent Architecture Generates Entanglement Supporting Conscious-
ness 113
cessing and non-local integration of information used in quantum compu-
ting. Entanglement in the whole brain is created and maintained by recur-
rent loops and temporal synchronization provided by the brain´s reentrant
architecture, e.g. over sensory and associative areas (Fig. 5.8).
The coherence-generating mechanisms involved in such processes – as
the formation of loops, and temporal synchronization among several areas
– are automatically impressed on Ca
2+
dynamics. Considering that the ac-
tion of electromagnetic forces upon a ion that is vibrating in the proper
frequencies can change its internal state, then the action of such forces up-
on a population of ions can generate a inter-correlated change in the inter-
nal state of the whole population. This operation would mean that the ions
can spontaneously become entangled for a brief period of time, in spite of
the fact that the brain operates at physically high temperatures.
Another alternative is that quantum processing may be achieved by con-
trolling the allosteric (R,T) states of proteins involved in coincidence de-
tector stps (d([s
1
,...,s
c
],s
t
)). In this condition, superposition, entanglement
and quantum processing is assumed to be dependent on electronic states of
molecules controlled by the entry of calcium through the NMDA channels
and mR-Glu.
Of course, a population of Ca ions or Ca-interacting molecules that be-
comes entangled should decohere (reduce the superposed state by interac-
tion with the environment) very fast. However, this tendency should not be
used as an argument against quantum computing in the brain for three rea-
sons.
First, because the brain possesses compartments where such populations
can be functionally segregated for a period of time that would be sufficient
to generate a conscious state.
Second, the most important reason is that the waking brain is continu-
ously generating new correlations between Ca
2+
ions and/or biological
macromolecules, which can give continuity to conscious experience even
when the correlations that generated the previous conscious state have van-
ished (of course, we know that some of these correlations are preserved
through continued activity in some intra-cellular stps—corresponding to
the phenomenon of memory).
Third, because decoherence is a necessary condition (for the conscious-
ness-supporting quantum system) to influence behavior, as consciousness
is assumed to do. By back-action on the biochemical systems where Ca
2+
also participates (and has a central role), the interferences that occur during
the entangled phase can have an influence upon which proteins are activat-
ed and how they are activated. Therefore, by considering this role of deco-
herence we can account for the apparent fact that consciousness is not an
114 5 The Brain and Quantum Computation
epiphenomenona but has a causal role, i.e., we can voluntarily determine a
part of our behavior.
5.6 Superdense Codes
Amplitude interference of quantum states happens when amplitudes from
different sources come together, since they may add in some places and
subtract or cancel in others. Quantum computing takes advantage of inter-
ference to change state amplitudes for both writing and reading instruc-
tions (Gilmore, 1995). Quantum superdense coding (QSC) is the property
that "in order to switch from any one of the four Bell states to all other four
it is sufficient to manipulate only one of the two qubits while in the classi-
cal case one has to manipulate both" (Zeilinger, 1998).
Fig, 5.9. Superdense codes: We do not mix information from different sensory
channels.
5.7 Quantum Computing and Working Memory 115
A problem that QSC solves is to enable two protagonists, Alice and Bob,
who share no secret information initially, to transmit a secret message x
under the nose of an adversary Eve, who is free to eavesdrop on all their
communications. This area of study has been called quantum cryptography
(Fig. 5.9).
If Alice and Bob are limited to classical communication, they cannot de-
tect eavesdropping. Now, if Alice and Bob‘s public communication is sup-
plemented by a quantum channel, any eavesdropping disturbs quantum
transmission in a way likely to be detected by Alice and Bob (Bennett and
Shor, 1999). But channel noise also causes code corruption in quantum
transmission. If the quantum channel is too noisy the best strategy (Bennett
and DiVicenzo, 2000) is for Alice not to send the input qubit through the
channel at all, but instead prepare a number of pure EPR pairs, and share
them through the noisy channel with Bob, resulting in noisy EPR pairs.
Then, using their ability to communicate classically, Alice and Bob distil a
smaller number of good EPR pairs, and additional classical communica-
tion, to teleport the input qubit safely to Bob.
While generating an unitary state of consciousness, the brain should also
separate the pre-processing of different 'qualia' up to the point when it all
comes together. In the framework presented here, reliability of quantum
processing in specialized brain areas would be obtained by using cripto-
graphic strategies to overcome undesired interference through noisy chan-
nels and provide coherent binding of quantum informational patterns. The
strategy would be to keep a quantum communication channel together with
classical axon-dendrite connections between such specialized regions.
Combining both kinds of long-range communication in the brain gives
us a picture of brain function that is largely compatible with a popular
model of information processing in contemporary cognitive neuroscience,
the Working Memory (WM) model.
5.7 Quantum Computing and Working Memory
Neuroscientists have suggested that the prefrontal cortex might integrate
spatial (where) and object (what) information, since cells responding to
these two kinds of information intermixed in the prefrontal cortex engage
in general-purpose temporary storage across many processing domains.
Such lateral prefrontal function was named Working Memory, and as-
sumed to be a conscious task. It is a short-term memory that is activated to
support thinking processes, e.g. while performing a mental arithmetic task
as the mental addition of 434 + 87. One common strategy is to add the
116 5 The Brain and Quantum Computation
units, tens and hundreds digits; while adding the tens it is necessary to
keep other pieces of information in memory. This holding-in-memory
while performing another task is the modus operandi of WM (Jonides,
1996).
Executive functions, in fact, seem to be spread across multiple regions
of the frontal cortex, located on the outside surface of the lateral frontal
cortex (LFC). However, the anterior cingulate cortex (ACC), located on
the medial surface of the frontal cortex, is also activated in functional im-
aging studies of WM, and it is also considered an important executive area.
This region receives important inputs from sensory systems and is con-
nected to LFC. Both ACC and LFC are part of what is proposed as the
frontal lobe attentional network, which involves also areas in the parietal
cortex. This cognitive system is involved in selective attention, mental re-
source allocation, decision-making process, voluntary movement control,
and/or resolving conflict between competing stimuli (LeDoux, 2002).
WM includes an "executive processor" and a "buffer". The ―buffer‖ is
supposed to be widely distributed over the brain, according to the contents
of the information to be kept in WM. Finally, both LFC and ACC have
important connections with the ventral frontal cortex (VFC), which in turn
is one of the main entries of the neorcotical system in the emotional limbic
system (LeDoux, 2002). In this way, emotions are automatically associated
to WM contents.
There are limits of how much we can attend at one time. Some limita-
tions are related to the similarity of the attended information. It is more
difficult do attend to sources of information when both are presented to the
same modality than when they are presented to different modalities. In a
similar way a task of transforming a stimulus into a similar code, or deal-
ing with similar semantic contents, is more difficult to attend to than when
contents or codes are different (Posner, 1995). This may be explained by
the required modality segregation by means of dense codes during quan-
tum conflict solving.
Beyond this, there is a more general limitation on how much one can at-
tend at one time. This general limitation can be demonstrated most clearly
when all the specific sources of interference were removed. Perhaps be-
cause of these limitations, much of perceptual input goes unattended while
some aspects become the focus of attention. Attending, in this sense, is
jointly determined by environmental events and current goals and con-
cerns. When appropriately balanced, these two kinds of inputs will lead to
the selection of information relevant to the achievement of goals and lends
coherence to behavior (Posner, 1995). This more general WM limitation
may be explained by the limitations of maintaining long-lasting entangled
circuits required to support the consciousness of WM contents.
5.7 Quantum Computing and Working Memory 117
The task
The entangled working memory
Fig. 5.10. Working Memory: a quantum computational space
The existence of quantum communication channels between such areas,
besides the classical connections, implies new properties to the WM func-
tion (Fig. 5.10). The processing of information in WM circuits cannot be
considered to be purely serial, but includes parallel processing by several
agents that are bound together through non-local integration. One of the
consequences is that the role of the "executive processor" is relative, i.e., it
is not the only agent able to apply the processing rules and integrate partial
results.
118 6 Memetics and Cognitive Mathematics
6 Memetics and Cognitive Mathematics
The purpose of this chapter is to introduce the logic and basic assumptions
of memetics, presenting its definition and presenting memetics as a poten-
tial answer to the problem of brain evolution and the human capacity of
dealing with complex mathematical processes.
6.1 Memes
Humans are capable of imitation and so can copy from one another ideas,
habits, skilled behavior, inventions, song and stories. These are all memes,
a term which first appear in Richard Dawkins‘ book The Selfish Gene
(Dawkins, 1976). In that book, Dawkins dealt with the problem of biologi-
cal (or Darwinian) evolution as differential survival of replicating entities
(Dawkins, 1976). By replicating entities Dawkins meant, obviously, genes.
Then, in the final part of his book, Dawkins asked the question ‗are there
any other replicators on our planet?’ to which he answered ‗yes.’ He was
referring himself to cultural transmission and fancied another replicator – a
unit of imitation (Blackmore, 1997; 1999). Dawkins first though of
‗mimeme’, which had a suitable Greek root (Dawkins‘ words) but he want-
ed a monosyllable word which would sound like ‗gene‘ and hence the ab-
breviation of mimeme, or meme. A revolutionary new concept (actually, a
truly Kuhnian paradigm shift) was born. Like genes, memes are replica-
tors, competing to get into as many brains as possible.
One of the most important memes created by humans is the concept of
numbers. Numbers are cultural inventions only comparable in importance
to agriculture or to the wheel (Ifrah, 1985). Counting, however, is a pro-
cess observable in a great number of non-human species. Millions of years
before the first written numeral was engraved in bones or painted in cave
walls by our Neolithic ancestors, animals belonging to several species
were already registering numbers and entering them into simple mental
computations (Dehaene, 1997). We, modern humans differ from the other
species by being able to deal with numbers in a highly sophisticated man-
6.1 Memes 119
ner, rather than the simple block-counting process characteristic of lower
species, which is typically limited to 3 or 4 units.
The Oxford English Dictionary offers the following definition:
Meme: An element of a culture that may be considered to be passed on
by non-genetic means, esp. imitation.
Memes can be thought as information patterns, held in an individual‘s
memory and capable of being copied to another individual‘s memory. The
new science of memetics is a theoretical and empirical science that studies
the replication, spread and evolution of memes. As the individual who
transmitted the meme continues to carry it, the process of meme transmis-
sion can be interpreted as a replication, which makes the meme a truly rep-
licator in the same sense as a gene. Like the evolution of traits by natural
selection of those genes that confer differential reproductivity, the cultural
evolution also occurs by selection of memes with differential reproductivi-
ty, that is, those memes with the highest copying rates will take over those
with lower copying rates.
Dawkins listed three characteristics for any successful replicators:
1. copying-fidelity: the more faithful the copy, the more will remain of the
initial pattern after several round of copying;
2. fecundity: the faster the rate of copying, the more the replicator will
spread; and
3. longevity: the longer any instance of the replicating patterns survives,
the more copies can be made of it.
Just think of the example provided by Blackmore (1999), regarding the
song ‗Happy Birthday to You,‘ and you have a tremendously successful
replicator, already copied (with high fidelity) thousand of millions of times
(high fecundity) all over the world for several decades (longevity). In these
characteristics, memes are clearly similar to genes and the new science of
memetics imitates, to a certain extent, the example of genetics (a
metamemetics phenomenon?).
Memetic and genetic evolution can interact in rich and complex ways, a
phenomenon described as ‗meme-gene-coevolution‘ (Blackmore, 1999).
The study of cultural evolution is a branch of theoretical population genet-
ics and applies population genetics models to investigate the evolution and
dynamics of cultural traits equivalent to memes (Kendal and Laland,
2000). Research in gene-culture coevolution employs the same methods to
explore the inter-relations of genes and cultural traits. Meme evolution
may occur either exclusively at the cultural level, or through meme-gene
interaction, both mechanisms having important consequences.
120 6 Memetics and Cognitive Mathematics
6.2 The Formal Meme
Let there be given two organisms:
O(G(H, S
i
)), (d(s
i
, s
j
) | H, S
i
b
))
and O(G(H, S
j
)), (d(s
t
, s
o
| H, S
j
b
))
(6.1)
sharing the same grammar G, but having different knowledge K (see Defi-
nition 3.9)
K
i
= {d(s
o
, s
t
) | d(s
m
, s
n
) = d(s
o
, s
t
) ∩ d(s
o
, s
t
) }
K
j
= {d(s
o
, s
t
) | d(s
r
, s
s
) = d(s
o
, s
t
) ∩ d(s
o
, s
t
) }
(6.2)
about H because
(d(s
t
, s
o
| H, S
i
b
)) (d(s
t
, s
o
| H, S
j
b
))
(6.3)
This implies from Eq. 3.8 that there exists
d(s
m
, s
n
) = d(s
o
, s
t
) ∩ d(s
o
, s
t
) , d(s
m
, s
n
) K
i
but d(s
m
, s
n
) K
j
(6.4)
Definition 6.1. O(G(H, S
j
b
)) reproduces (imitates) a sensory-motor pro-
cess d(s
o
, s
t
| S
i
b
), as executed by O(G(H, S
i
)), if it produces
d(s
o
, s
t
| S
b
i
),
which reaches a final state similar to that reached by S
i
; that is:
(s
t
| S
i
b
,
s
t
| S
j
b
) 1.
Theorem 6.1. O(G(H, S
j
)) learns about d(s
o
, s
t
| S
j
), as executed by
O(G(H, S
i
)), if it is able to express d(s
o
,s
t
| S
j
) supported by the same action
d(s
o
,s
t
), reproduced by both O(G(H, S
j
)) and O(G(H, S
i
)). This learning oc-
curs if d(s
o
, s
t
) belongs to both Ł(G(H, S
i
)) and Ł(G(H, S
j
)).
Proof: As a consequence from the Definitions 3.8 and 6.1 and the fact
that both O(G(H, S
j
)) and O(G(H, S
i
)) are able to execute d(s
o
, s
t
) because
it belongs to the languages expressed by both S
j
and S
j
.
6.3 Improving Meme Spread 121
Remark 6.1. Theorem 6.1 is about learning by imitation or by observa-
tion, a very important type of learning for some kinds of animal, including
man. For instance, Petrosini et al. (2003) reviewed a series of papers dis-
closing the main properties of the means whereby the rat learns by observ-
ing its peers in solving maze problems. It is worth remarking that these au-
thors also stressed the importance of the so-called ―mirror neuron‖
(Rizzolati and Arbib, 1998) in supporting this type of learning. Here, the
―mirror neuron‖ is assumed to be in charge of some of the derivations in
d(s
m
, s
n
), since G is distributed over the neurons of S
b
.
Definition 6.2. d(s
m
, s
n
) in Eq. 6.4 becomes a formal meme if d(s
o
, s
t
)
supporting it, is reproducible or imitable by another organism.
Theorem 6.2. The complexity (difficulty) in meme learning is directly
proportional to
| (d(s
t
, s
o
| H, S
i
b
)) - (d(s
t
, s
o
| H, S
j
b
)) |
Proof: As a consequence of Definition 3.8f and Theorem 6.1.
Remark 6.2. Theorem 6.1 is also central to the concept of meme intro-
duced by Dawkins (1976), since it set the basic conditions by which formal
memes may spread (copied) in a population of O(G(H, S))s. In this way,
meme dispersion is governed by the same evolutionary rules discussed in
Chaps. 2 and 3. It is worthwhile to remember that those rules govern gene
evolution, as well (Rocha and Massad, 2003b). Theorem 6.2 clearly de-
fines one of the main restrictions on meme learning.
Proposition 6.1. O(G(H, S
i
)) uses d(s
o
, s
t
| S
i
) associated to d(s
o
, s
t
| S
i
)
in order to induce O(G(H, S
j
)) in reproducing d(s
o
, s
t
| S
i
) and to recover
d(s
o
, s
t
| S
i
) as d(s
o
, s
t
| S
j
). In this way, O(G(H, S
i
)) signals d(s
o
, s
t
| S
j
) to
O(G(H, S
j
)) using d(s
o
, s
t
| S
i
).
Proof: Follows from Definitions 3.8f, 6.1, 6.2, and Theorem 6.1.
Remark 6.3. Meme learning has generally been characterized as a deci-
sion by the organism S
j
, resulting from observation of the behavior of a
knowledgeable organism S
i
. Proposition 6.2 shows that meme spread may
also be due to a decision by S
i
to promote the learning of a meme d(s
m
, s
n
).
In this way, the meme spreading may also be supported by instruction.
6.3 Improving Meme Spread
Let S and M be such that is possible to guarantee initially that there exists
S
i
, S
j
S and M
i
, M
j
M
(6.5)
122 6 Memetics and Cognitive Mathematics
and
d(s
o
, s
t
| S
i
, M
i
), (d(s
o
, s
t
| S
i
, M
i
))1,
d(s
o
, s
t
| S
j
, M
j
), (d(s
o
, s
t
| S
j
, M
j
))1
(6.6)
such that
d(s
m
, s
n
) = d(s
o
, s
t
| S
i
, M
i
) d(s
o
, s
t
| S
i
, M
i
),
(d(s
i
, s
j
) ) 0.5.
(6.7)
In this condition
Theorem 6.3. O(G©(H, S
i
)) is able to learn to signal about d(s
o
, s
t
) sup-
ported by d(s
o
, s
t
| S
i
, M
i
) using d(s
o
, s
t
| S
j
, M
j
).
Proof: From conditions 6.5 to 6.7 there always exists
d(s
m
, s
n
) = d(s
o
, s
t
| S
i
, M
i
) d(s
o
, s
t
| S
i
, M
i
),
(d(s
i
, s
j
)) 0.5.
Given that G is a self-controlled grammar, it is possible to evolve s
c
C
(see Definition 2.5 and Eq. 2.34) such that (d(s
i
, s
j
)) = (q(s
c
)). In this
condition it is possible to (d(s
i
, s
j
)) 1, such that d(s
o
, s
t
| S
j
, M
j
) be-
comes expressible as (d(s
o
, s
t
| S
j
, M
j
)) 1 as a consequence from
d(s
i
, s
j
)) 1.
In this way, O(G(H, S
i
)) may use d(s
o
, s
t
| S
j
, M
j
) to sign d(s
o
, s
t
) sup-
ported by d(s
o
, s
t
| S
i
, M
i
).
Remark 6.4. The language Ł(G(H, S
j
, M
j
)), which is composed of all
d(s
o
, s
t
|S
j
, M
j
) used to signal about d(s
o
, s
t
) composing Ł(G (H)), is the
type of human-like language that may evolve in G. If S
j
, M
j
signify the
phonetic motor system, then Ł(G© (H, S
j
, M
j
)) is an oral language; other-
wise it is a signed language.
Theorem 6.4. The expressiveness (Ł(G(H, S
j
, M
j
))) increases as the
cardinality of S
j
, M
j
is augmented by evolution.
Proof: as a consequence from Eqs. 2.22 to 2.5 and 6.5 to 7, Theorems
2.1, 2.2, 2.5, 6.1, and 6.3.
Remark 6.5. As recently proposed by Hauser et al. (2002), the evolu-
tion of human language is a consequence of the evolution of the brain for
more general purposes. Also, according to recent findings published by
Enard et al. (2002), the complexity of human language is mainly due to the
6.4 Memetic Channels and the Brain 123
increase of the complexity of oral-pharyngial motor control. The increase
of the (Ł(G(H, S
j
, M
j
))) in the case of human languages allowed another
mechanism for meme spreading, based on the rules defined by Proposition
6.1, that we used to call teaching.
6.4 Memetic Channels and the Brain
Communication among DIPS‘ agents is established by means of two main
strategies:
1. Mail addressing: both the sending and the receiving agents know them-
selves, that is to say they have the capacity to address messages specifi-
cally to each other. Imitation is mainly a mail-addressing memetic
channel, since it is based in individual "contacts."
2. Broadcasting: agents deliver messages that are not specifically ad-
dressed to another defined agent, but to those interested in the subject.
Instruction is the first mechanism to implement a broadcasting memetic
channel since the action of one organism is communicated to a group of
peers. Language, by supporting teaching, increases the capacity of a
broadcasting memetic channel. Writing widely broadens this capacity.
Since, unlike genes, memes do not come packaged with instructions for
their replication, our brains provide this function, strategically guided by a
fitness landscape that reflects both internal drives and a world-view that is
continually updated through meme assimilation (Gabora, 1997).
The evolution of the human brain began with the macro evolutionary
events that culminated with the first species of the Homo genus, about two
and a half million years ago. By about 160,000 years ago H.sapiens
(Conroy, 1997; White et al., 2003) had brains as large as ours and the other
sapiens sub-species, the Neanderthals had brains with a volumetric capaci-
ty larger than ours. They controlled fire, had cultures and probably had
some form of language as well. The evolution of the brain is achieved by
both increasing the number of its neurons (brain size increase) and by al-
lowing new specializations for these new neurons.
The increase in brain size, however, had a price (Blackmore, 1999).
Oversized brains are expensive to run and ours consumes 20% of the
body‘s energy for a mass corresponding to only 2% in weight. In addition,
brains are expensive to build. The amount of protein and fat necessary for
the development of the human brain forced the first members of the Homo
genus to increase their meat consumption, which entailed better hunting
124 6 Memetics and Cognitive Mathematics
strategies, which in turn fed back to increased brain size. Finally, big
brains are dangerous to produce. The increase in brain size, along with the
bipedal gait of the Homo species resulted in severe birth risks. Big brained
human babies have enormous difficulty passing through the birth canal. In
addition to higher maternal and fetal mortality, this results in the human
baby being born prematurely, as compared with other primates. On one
hand, this has the beneficial consequence that our brains have greater neu-
ronal plasticity, which increases their learning capacity. On the other hand,
the complicated twisting maneuvers the human fetus has to do in order to
pass through the birth canal means that the human female rarely is able to
deliver without assistance. This also contributes to socialization and addi-
tional selection for brain growth.
Our brains have changed in many ways other than just size. The modern
human prefrontal cortex, oversized when compared with other hominids, is
fed by neurons coming from practically all other parts of the brain. Its role
in the complex cognitive abilities of modern humans is still to be fully un-
derstood, but we already know that when damaged by accident or surgical
removal (a common practice some decades ago) the victim is severely lim-
ited in calculation performance (in addition to personality changes). How-
ever, this size increase may free the brain to invent new types of neurons,
by modifying gene expression, whenever an increasing social selective
pressure appears.
6.5 The Evolution of our Mathematical Competence
Let us imagine the African environment of about 150,000 years ago. It is
now widely accepted that our species evolved in a lake environment at
about this time (Fig. 6.1). The first humans began to organize themselves
into small groups of hunter-gatherers, with few contacts outside their own
clans. With the abundant supply of animal protein and fish fat, their brains
achieved the size and organization observed in current members of our
species. However, their numerosity was probably restricted in the same
ways as that seen in modern hunter-gatherer societies. As those primitive
human groups started out on their journey toward the northern parts of Af-
rica (and, mainly after the out-of-Africa migration waves towards the Cau-
casus and then Europe) they began to come in contact with other groups of
primitive humans, the Neanderthalers, who had been dwelling in Europe
for some 100,000 years at that time. The clash of cultures had well known
disastrous effects for the latter with our own sub-species prevailing and the
disappearance of our cousins some 30 to 50 thousand years ago.
6.5 The Evolution of our Mathematical Competence 125
Fig. 6.1. The primate hominides: it started around the lakes ..
It was not, however, until the first nomadic tribes of humans settled in
communities, following the agricultural revolution some 10 thousand years
ago, that trade between different groups and/or individuals pressed for the
development of a more sophisticated numerical system. Trade, and the ne-
cessity for book-keeping, was also a selective pressure for the evolution of
mathematics beyond simple counting. In our own country, it is possible to
find evidence of such a process.
The archeological site known as Pedra Furada, in the northern part of
Brazil, is one of the richest sites of primitive human groups (Guidon,
1998), aged between six and twelve thousand years (as in any other dating,
those figures are subject to ferocious argument within the archeological
community). Its archeological findings are currently divided into two cul-
tural traditions, named the "Nordeste" tradition and the "Agreste" tradition.
The Nordeste tradition had two phases, the Pedra Furada and the Pedra
Talhada. It is characterized by pictorial (humans, animals and vegetables)
and non-pictorial graphics (Fig. 6.2). The pictorial graphics represent ac-
tion, as a general rule, and are more prevalent than the non-pictorial
graphics. Human figures are represented with several cultural attributes
and are depicted in their daily life activities. Four main themes can be
identified: dancing, sexual activities, hunting, and activities around a tree.
One specific figure, a non-realistic picture dated from around nine thou-
sand years ago, appears repeatedly in other archeological sites of the
northeastern part of Brazil. The styles of the Nordeste tradition are named
Serra da Capivara (SC), which is the oldest, Serra Talhada (ST) and, most
recently, Serra Branca (SB). The ST, which is intermediate in time, is the
most complex, denoting an evolutionary process from SC to ST, but a re-
126 6 Memetics and Cognitive Mathematics
gression in style to SB. This process is believed to have happened around
six thousand years ago.
Fig. 6.2. The paintings of Pedra Furada: are they depictions of early arithme-
tic?
The Nordeste tradition disappeared, probably due to a combination of
climatic changes and the arrival of better warriors groups, which forced the
former inhabitants to abandon the area.
6.6 How Memes Spread 127
The Agreste tradition is characterized by human graphics and rare ani-
mals. Graphics representing action are rare and are restricted to hunting.
Pure graphics are more prevalent than in the Nordeste tradition. Frequent-
ly, the Agreste graphics were done inside the Nordeste panels, but can be
distinguished from these by their lower technical quality. The Agreste and
Nordeste pictures coexist in the area by about 10 thousand years but the
Agrest can be disappeared some 4-3 thousand years ago.
6.6 How Memes Spread
Several attempts have been made to provide the new science of memetics
with a mathematical framework for modeling the spread of memes. The
Journal of Memetics (http://jom-emit.cfpm.org/all.html), an electronic
journal dedicated to memetics, presents a number of articles dealing with
the mathematics of memetics. The great majority (if not the totality) of
these words are essentially adaptations of population genetics. In the paper
by Edmonds (1998), for instance, a classification of memetics models is
presented, with an interesting discussion of the possibilities of modeling
memetics. Kendal and Laland (2000) discuss the phenomenon of meme-
gene coevolution. The authors argue that whether cultural evolution occurs
purely at the level of the meme, or through meme-gene interaction, is a
question for which a body of formal theoretical work already exists that
can be readily employed to model empirical data and test theoretical hy-
potheses. These works exemplify cultural evolution and gene-culture co-
evolutionary theory, the branch of theoretical population genetics referred
to above (Boyd and Richerson, 1985; Cavalli-Sforza and Feldman, 1981;
Feldman and Laland, 1996). The authors reject the argument that meaning-
ful differences exist between memetics and population genetics methods.
One of the goals of present is to point out the similarities between memet-
ics, cultural evolution, and gene-culture co-evolutionary theory, and to il-
lustrate the potential utility of genetic models to memetics.
The authors conclude that cultural evolution and gene-culture co-
evolutionary modeling paradigms can be effectively employed to enhance
the quantitative study of memetics. Simple and complex cultural phenom-
ena such as behavior patterns, belief systems, and institutions can be ana-
lyzed by characteristics of associations between easily definable and quan-
tifiable memes. The quantitative approach can be used to describe meme
diffusion dynamics, and make sense of patterns of variation in memes. The
methods can also clarify why and how human attributes evolved in con-
128 6 Memetics and Cognitive Mathematics
junction with memes, how they continue to evolve, and what is the basis of
any stability or maintenance of the trait
In another interesting model, Gatherer (2001) shows simple computer
simulations of the interaction of genetic factors and memetic taboos in
human homosexuality. These simulations clearly show that taboos can be
important factors in the incidence of homosexuality under conditions of
evolutionary equilibrium, as for example in states produced by heterozy-
gote advantage. However, frequency-dependent taboos, i.e. taboos that are
inversely proportional to the incidence of homosexuality, cannot produce
the oscillating effect on gene frequencies predicted by Lynch (1998). Ef-
fective oscillation is only produced by rapid withdrawal and re-imposition
of taboos in a non-frequency-dependent manner, and only under conditions
where the equilibrium incidence of homosexuality is maintained by heter-
ozygote advantage, or other positive selective mechanism. Withdrawal and
re-imposition of taboo under conditions where homosexuality is subject to
negative selection pressure, produce only feeble pulses, and actually assist
in the extinction of the trait from the population. Additionally, it is shown
that frequency-dependent taboos assist in a more rapid achievement of
equilibrium levels, without oscillation, under conditions of heterozygote
advantage. An attempt is made to relate the simulations to past and con-
temporary social conditions, concluding that it is impossible to decide
which model best applies without accurate determination of realistic values
for the parameters in the models. Some suggestions for empirical work of
this sort are made.
Fig. 6.3. Logistic model for innovation spread
6.6 How Memes Spread 129
The above discussion is hence related to the population genetics ap-
proach to memetics. Let us see now an ecological approach—that is, how
to model meme spread by dynamical systems. As a matter of fact, the first
approach to the spread of ideas by dynamical systems was that due to Rog-
ers (1995), whose classical book, the Diffusion of Innovations, is a land-
mark of modeling the spread of ideas and concepts. Actually, when Rogers
wrote the first edition of his book forty years ago (Rogers, 1962), the very
concept of meme has not been proposed yet. In its fourth edition (Rogers,
1995), however, this book still misses the idea of a meme. Notwithstand-
ing this fact, the spread of innovations is not an alternative to memetics
dynamics. The so-called logistic model of innovation spread (Fig. 6.3) is
one way of simulating this dynamics.
In this model, it is assumed that, from a total population n, a fraction a
adopted a given novelty. Therefore, there remains a fraction n – a individ-
uals ´´susceptible` to the innovation. The model also assumes a contact
rate between ‗infected‘ and ‗susceptible‘ individuals.
The rate of growing of adopters is therefore given by:
which can easily be solved as:
This is one of the forms of the logistic equation. When the rate of contact
varies with time, the equation is:
The reader familiar with the mathematical theory of epidemic spread
will easily recognize the above model as an epidemiological model in a
new disguise. It is indeed a model of spread which could be made more
))()((
)(
tanta
dt
tda
1
0
0
exp
1
1)(
t
a
a
ta
1
0
0
0
)(exp
1
1)(
s
dss
a
a
ta
130 6 Memetics and Cognitive Mathematics
sophisticated in order to take into account other facts related to transmis-
sion, like the reproduction rate of ideas spreading.
An interesting example of the spread of a new meme is represented by
the case of hybrid corn in Iowa farms in the late 20‘s and early 30‘s, de-
scribed by Rogers (1995). At the time, farmers chose the best seeds from a
given year‘s yield for the following year‘s planting. The hybrid corn seeds,
in contrast, besides giving a greater yield, were handicapped by the need to
buy new seeds every year. The example of hybrid corn is described in de-
tail by Rogers. In the following analysis, we revisit the spread of hybrid
corn in Iowa, applying an original model.
Fig. 6.4. Modeling hybrid corn technology diffusion: Number of farms per year
adopting hybrids
The model assumes two types of farmers, called ―susceptible‖ to the in-
novation, denoted S
1
and S
2
, one of them more opinionated than the other.
Both susceptible types of farmers are subject to broadcast advertising, and
adopt the innovation at a rate of farmers per time unit in this way. Once
having adopted the innovation, susceptible farmers pass to a new state,
called ―infected‖ by the innovation, denoted I
1
and I
2
, depending on the
previous states, whether S
1
or S
2
, respectively. In addition to the broadcast
advertising, farmers could adopt the new meme by a kind of ―infectious‖
contact with the farmers who had already adopted the innovation. This oc-
curred at rates
1
and
2
potentially infectious contacts per time unit, de-
pending on the contact states of S
1
or S
2
, respectively. Once the meme was
6.6 How Memes Spread 131
adopted, the farmers were removed from the infectious state to a new, ―re-
sistant‖ state, denoted R
1
and R
2
, from the states I
1
and I
2
, respectively,
with rates
1
and
2
. The more opinionated farmers, I
2
, influence the more
susceptible farmers S
1
, through a new contact rate r, but are, in turn, sus-
ceptible to innovation by contact with farmers from the class I
1
. The mod-
el assumes also that the broadcast advertising rate decreases with time
according to a logistic function, (t(exp[-t]) and that the direct
contact rates
i
, i = 1, 2, increased according to another logistic function,
i
(t(exp[
i
t]). The model‘s dynamics are described by the follow-
ing set of ordinary differential equations:
)(
)(
)()()()()()(
)()2.0)(()()()()()()()(
)()()()()(
)2.0)(()()()()()()()(
22
2
11
1
221222
2
112211111
1
1222
2
2211111
1
tI
dt
dR
tI
dt
dR
tItItSttSt
dt
dI
tItItItrStItSttSt
dt
dI
tItSttSt
dt
dS
tItItrStItSttSt
dt
dS
The results of the numerical simulation of the model can be seen in Fig-
ure 6.4, which demonstrate the good fitting capacity of this model with re-
spect to real data from the Iowa farmers.
We also calculated the basic reproduction ratio of the innovation, the
threshold number of ―infected‖ farms, below which the innovation would
not spread to the other farms. This is a parallel to the basic reproduction ra-
tio, R
0
, (Anderson and May, 1991) of infections and it is considered the
key parameter related to infectious dynamics. For the proposed model, the
threshold condition is given by = 0; that is, provided the broadcast adver-
tising is positive, the innovation will spread. When this is not the case, that
is, when the spread is dependent only on the contact between farmers who
adopted the new meme and those who are still susceptible, the threshold
condition is given by
21
2
1
1
0
r
R
132 6 Memetics and Cognitive Mathematics
As the rates ,
1
, and
2
are time-dependent, this parameter is variable
with time, starting with zero (when is different from zero) and growing
to more than 8000 at the peak of the ―epidemic‖ when calculated with the
values used in the simulation of the model.
6.7 How the Number Meme Spread
According to Rogers (1995), the characteristics of innovations, as per-
ceived by individuals, which help to explain their different rates of adop-
tion are:
1. Relative advantage, the degree to which an innovation is perceived as
better than the idea it supersedes;
2. Compatibility, the degree to which an innovation is perceived as being
consistent with the existing values, past experiences, and needs of po-
tential adopters;
3. Complexity, the degree to which an innovation is perceived as difficult
to understand and use;
4. Triability, the degree to which an innovation may be experimented with
on a limited basis; and
5. Observability, the degree to which the results of an innovation are visi-
ble to others.
Therefore, innovations that are perceived by individuals as having
greater relative advantage, compatibility, triability, observability, and less
complexity will be adopted more rapidly than other innovations.
Now, returning to the main theme of this chapter—the spread of the
number meme—we may imagine a scenario from about 10,000 years ago,
when trading was starting to blossom and the ‗natural‘ capacity of numera-
cy, that is, counting in blocks up to four or five, was not enough anymore.
Those individuals with intelligence slightly above the average of the time
could invent counting procedures and techniques superior to KFN. Those
new techniques that were perceived by others as having greater relative
advantage, compatibility, triability, observability, and less complexity cer-
tainly spread rapidly, representing, therefore, a paradigm shift in the cur-
rent culture of the time.
It is easy to conclude that a growing arithmetic capacity in a local group
of individuals represented a greater relative advantage at a competitive
stage of the primitive trading system. The new way of dealing with num-
bers should be easily tried, the results readily observable and, if the system
6.8 Future Perspectives in the Memetics of Cognitive Mathematics 133
evolved in a gradual way, the complexity should be built up on each new
step, in order not to be a restrictive factor.
It may, therefore, be concluded that the evolution of mathematics, as
known today, started with the spread of a new meme (or rather a set of
new memes) in a manner rather like the way described by Rogers (1995),
that is, for each step, early adopters learned by imitation, followed by a
take-off phase of quick spread, reaching eventually an equilibrium with the
late adopters. The theory of memetics, along with the theory of innovation
diffusion, explains in a rather plausible way such evolutionary events
which changed human culture forever.
Finally, it is worth mentioning the curious fact that the evolution of
number systems very similar to each other, occurred in an independent
manner in places far away in time and space, such as Egypt and Guatema-
la! This indeed demonstrates the power of new mathematical ideas in
providing their inventors and adopters with a great competitive advantage.
6.8 Future Perspectives in the Memetics of Cognitive
Mathematics
So far, we have addressed the hypothesis of the evolution of our mathe-
matical capacity based on plausible assumptions of meme-gene coevolu-
tion of brain size and complexity. It is possible, however, that the evolu-
tion of mathematical ability is more dependent on a more active form of
meme transmission, namely, teaching. If this is the case then the develop-
ment of mathematics beyond the KFN level occurred later in our evolu-
tionary history. Teaching is a more sophisticated form of meme transmis-
sion than simple imitation. In addition, teaching is a kind of social
interaction, which occurs in more complex communities. It may have
evolved as a particular case of altruism. Teaching is a form of giving
someone else precious information that could result in differential repro-
ductivity in the receiving individual. Therefore, teaching probably evolved
by the same mechanisms currently accepted for the evolution of altruism:
kin selection or reciprocation. In the former, the individual who teaches
some useful trick to another individual increases his/her inclusive fitness if
the receiver shares a substantial number of genes with him/her. In the latter
mechanism, the altruistic teacher gives his/her precious information with
the expectation that in later interactions the current receiver will share oth-
er kinds of information with him/her. Both mechanisms make sense from
the gene and meme points of view. A third mechanism could also be in-
134 6 Memetics and Cognitive Mathematics
volved in the transmission of a mathematical memeplex: selfish teaching.
In this form of ‗altruism,‘ the teacher shares information with the receiver
based on the belief that this will increase his/her own survival or the sur-
vival of the group. The meme-gene coevolution aspect of this kind of
meme transmission, however, should occur under the setting of group se-
lection. Group selection, although still controversial among population ge-
neticists, may indeed occur in cultural evolution, as also discussed by
Blackmore (1999).
The important point we would like to emphasize is that mathematical
memeplexes can be transmitted, either by imitation or by teaching, and that
both mechanisms can be inferred in the evolution of our cognitive mathe-
matical abilities, and through the same meme-gene coevolutionary mecha-
nisms.
Another area that could be subject to future investigation related to the
meme-gene coevolution of mathematical memeplexes would be the devel-
opment of mathematical tools to deal with the dynamics of meme trans-
mission and the evolutionary aspects, much in the flavor of the neo-
Darwinian synthesis. Population genetics is a mathematically rich area of
evolutionary biology and perhaps one could propose a new specialty that
could be called population memetics in which the adaptive value of meme
transmission could be quantified and predictions on meme dynamics could
be made. But this also is matter for future works.
Mathematics is a tremendously rich collection of ideas, concepts, tools,
etc., that can be characterized as a memeplex. It is always evolving and, in
addition to its obvious role in changing the world, and our world vision,
has certainly helped in the evolution of our cognitive capacity. Since the
first hominids surpassed the imitation threshold, individual mathematical
memes more complex than our innate subtizing (in addition to all other
memeplexes that characterize human culture), have been transmitted by
imitation and later on, by teaching, creating an autocatalytic virtuous circle
that culminated in the human brain.
6.8 Future Perspectives in the Memetics of Cognitive Mathematics 135
136 7 Modeling of Arithmetic Reasoning
7 Modeling of Arithmetic Reasoning
In this chapter we extend the scope of an earlier model (Rocha and Mas-
sad, 2002 and 2003a), wherein innate cerebral circuits were proposed to
deal with numbers, and where some of these circuits were specialized in
dealing with distinct types of number. On this view, the arithmetical pro-
cess is distributed among several brain areas, and different strategies are
used to solve the same arithmetical calculations. It is also assumed that
some important properties of the counting system may be disclosed by
studying the relation between calculation time and the size of numbers.
7.1 Counting
Counting is a process dependent on a distributed representation of number
in the brain (e.g., Butterworth, 1999; Dehaene, 1997; Dehaene, 2002). The
efficacy of the processing is mainly dependent on the optimization of the
control of the eyes and hands in attending the objects to be counted (e.g.,
Butterworth, 1999; Massad and Rocha, 2002, Rocha and Massad, 2003a).
Counting is also proposed to be the underlying process for arithmetic cal-
culation since it has been demonstrated that the (response) time required to
produce the result of an arithmetic calculation is dependent on the size of
the relevant numbers (e.g., Ashcraft, 1992; Dehaene, 1997; Fayol, 1996;
Hinrichs et al. 1991; McCloskey, 1991; McCloskey et al., 1991; Siegler,
1996). It has also been demonstrated that different strategies may be used
by the same person to solve the very same calculation (e.g., Siegler, 1996).
Despite these advances, no formal model other than the triple-code model
proposed by Dehaene and colleagues (Dehaene et al. 1998, Cochon et al.
1999) has been proposed so far to describe the actual experimental data on
brain arithmetic. The triple-code model assumes that numbers are repre-
sented in the human brain in three distinct formats: as Arabic numerals, as
sequences of words, and as analog representations of the corresponding
numerical quantities. Also, arithmetic calculation is mostly dependent on
verbal encoding.
7.2 A Model for Number Sense 137
Because of the foregoing considerations, a new class of fuzzy numbers,
called K Fuzzy Numbers (KFN), was proposed by Rocha and Massad
(2002) to implement arithmetical knowledge in a Distributed Intelligent
Processing System (DIPS). The proposed system is designed to simulate
brain function and experimental data about arithmetic capability in both
man and animals. KFN‘s main property is the dependence of the size of
the base of its membership function
di
() (Pedrycz and Gomide, 1998)
on the value of the number d
i
encoding .
Another interesting KFN property allows for many different solutions to
be tried on the very same arithmetic problem. Indeed, this seems to be the
strategy employed by the brain in the case of arithmetic processing. Many
studies of the means whereby children in different countries solve standard
arithmetic problems have revealed that youngsters use multiple strategies
(Butterworth, 1999; Dehaene, 1991, Dehaene, 1997; McCloskey et al.,
1991; Siegler, 1996):
1. total manipulation: the child counts separately each set to be processed
by pointing, marking, etc. each of their elements, and then counts by the
same process each element of the union, or the complement, etc. of the-
se sets to get the final result;
2. simplified manipulation: the result is obtained by counting each element
of the union, or the complement, etc. of the sets to be processed;
3. optimized manipulation: the result seems to be obtained by performing
the minimum counting, which varies according to the type of calculation
to be performed, and
4. mental calculation: the result is quickly processed by specialized cir-
cuits.
Also, the type of function explaining the effect of number size, as re-
ported in the literature, correlates with the kind of manipulation used (e.g.,
McCloskey et al., 1991). Finally, training changes the frequency of use of
the different types of strategy, such that adults tend to rely more on mental
calculation than do children (Siegler, 1996).
7.2 A Model for Number Sense
Counting, as for example a collection of animals on a farm (Fig. 7.1), is a
very complex process requiring the participation of sets of neurons located
in different areas of the brain, with each set in charge of solving one par-
ticular segment of the entire process. These sets are specialized to collect
sensory data in order to recognize the object to be counted, to perform cal-
138 7 Modeling of Arithmetic Reasoning
culations, to keep track of the counted objects, and to speak about the re-
sults. To count is, therefore, a task for a DIPS.
Fig. 7.1. The KFN counting circuit: an ancestral neural circuit for counting.
7.2.1 Identifying numerosities
Let the counting DIPS labeled KFN (Fig. 7.1) consist of two sets of
agents:
S: collection of sensory agents s which measure (m) variables v in
U as m =
(v), in a given subspace f U; and
R: a collection of recognition agents specialized for identifying objects
o
i
U, by means of a set
i
of relations between the above measures m
i
of
these o
i
. The image I (o
i
) used by R to identify o
i
is the minimum set of the
relations uniquely associated to o
i
and no other o
j
in U; or I (o
i
)
i
.
7.2 A Model for Number Sense 139
Many objects o
i
may be simultaneously identified by R because of the
redundancy inherent to any DIPS. The maximum number
i
of I (o
i
) simul-
taneously identified by R is dependent on and limited by the agent redun-
dancy r, which defines the subspace F = {f
i
}
i = 1 to r
, and is called the senso-
ry field of S. This kind of block quantification is named subtizing, and is
considered a special kind of counting (Butterworth, 1999, Dehaene, 1997,
Fink et al. 2001; Piazza, et al. 2002).
The identification of the cardinality of any set O
i
of objects o
j
in U,
called here cardinality quantification (CQ), requires also the following set
of agents:
1. C: a collection of control agents in charge of moving F over U, in order
to cover the subspace V containing the objects o
i
to be quantified, or
V {F
t
}
t = 1 to u
(7.1)
where F
t
is the subspace of U sensed by S at the step t, and u is the number
of steps required to cover V;
2. G: a collection of agents receiving information about the quantities of
o
i
identified by each R,
3. A: a collection of agents in charge of accumulating () the quantities
of o
i
identified at each covering step t (e.g., Meck and Church, 1983)
such that
t+1
=
t
– (1 - t),
(7.2)
4. Q: a collection of agents classifying (
i
) the quantities accumulated by
A such that
i
= d
i
if < < , otherwise
i
=
(7.3)
where
i
become the actual inputs to the quantifier q
i
Q; , > 0, and d
i
is a label in the dictionary
D = [d
1
, ... , d
n
, ]
(7.4)
obeying the following rewriting rules , used to order D:
d
i + 1
= d
i
d
1
, d
1
= d
1
and d
i - 1
= d
i
d
1
, d
1
= d
1
(7.5)
Let the classifying capability of Q to be dependent on the topographical
location g of q
g
Q recognizing as
g
(Fig. 7.1), that is
g
= d
g
if -
g
< < +
g
, otherwise
g
= ,
(7.6)
140 7 Modeling of Arithmetic Reasoning
g
,
g
= (d
g
)
(7.7)
If the possibility for
di
() of to be recognized by d
i
is defined as
di
() 1 if d
i
or
di
() 0, if -
g
or +
g
(7.8)
then it may be proposed that the output n
g
of the quantifier q
g
is
n
g
=
dg
()
(7.9)
KFN works as follows. The controlling agents (c C) move the eyes to
the locations where the objects to be counted reside. At each of these loca-
tions, the sensory neurons (s S) collect information to be used by agents
(r R) in identifying the objects to be counted. Data concerning the num-
ber of objects recognized by R at each location is then delivered to the gat-
ing agents (g G). Each g G may receive information from different
sets of R, and from the same or different sensory systems. In this way, the-
se agents free counting from sensory boundaries. Another task of the gat-
ing agents allows for block counting. Whenever more than one object may
be jointly identified in a focused location (as, for example, in subtizing–
Butterworth, 1999; Dehaene, 1997; Fink et al. 2001; Jensen et al. 1950; Pi-
azza et al. 2002), their quantities may be the signal recorded by g G.
When all objects in the focused location are recognized, and their quanti-
ties stored in g G, the controlling agents then move the eyes to the next
location to be explored and signal the neurons g G to send their data to
the accumulators a A. The output of these accumulators is classified by
the neurons q Q. The result of the counting is provided by those q
g
Q
for which n
g
is a maximum.
The KFN output n
g
of q
g
signals the confidence of being encoded by
q
g
as d
g
. Fuzzy Logic and Fuzzy Number theories were introduced by Za-
deh (1965) as the most adequate mathematical formalism for dealing with
the types of approximate reasoning and calculation performed by humans.
Fuzzy numbers are numbers of the type around n used to do most of our
daily life quantification (Pedrycz and Gomide, 1998). KFN is a particular
type of Fuzzy Number, and was specially tailored to describe the proper-
ties of counting in animals (Massad and Rocha, 2002, Rocha and Massad,
2003a). The above KFN circuit describes the main properties of those
quantifying neurons, recorded at both the frontal and parietal lobes
(Dehaene, 2002, Nieder et al. 2002, Nieder and Miller, 2003, Sawamura et
al. 2002). Also, the parietal KFN circuit is proposed to correspond to the
set of neurons forming the mental number line in humans (Butterworth,
1999; Dehaene, 1999; Fink et al. 2001; Gobel et al. 2001, Nuerk et al.
2001, Zorzi et al. 2002).
7.2 A Model for Number Sense 141
7.2.2 Quantification trajectory control
One of the most complex tasks in counting consists in the control of the
counting pathway, because it is necessary to avoid counting the same ele-
ment twice (or more), and to avoid forgetting an object in the counting col-
lection. This type of control is part of the problem known as the ―travelling
salesman problem,‖ a very well known one for AI scientists. It seems that
animals (Brannon and Terrace, 1998; Carpenter et al. 1999) and children
of school age learn to easily solve this task even for collections having a
large number of objects. Children learn not to miss or miscount an object
in the counting collection by labeling each counted element with a defined
finger. Each identified object is associated to one (overt or covert) motor
action (e.g., finger extension) by the children before they focus their eyes
on the next element to be counted (Fig. 7.2). In this way, they easily dis-
cover when the same place is being visited twice, or when a possible un-
marked object is found at the end of the counting trajectory.
Fig. 7.2. Ordinal quantities: Making sure everyone is counted; no one is missing.
This kind of correspondence allows for the construction of another
counting system—the ordinal KFN numbers—because each time the con-
troller sends a signal to move the eyes to the next object, it may also send
142 7 Modeling of Arithmetic Reasoning
another pulse to an accumulator whose output is filtered by another class
of neurons, called here ordinal neurons (Oi in Fig. 7.2), which in turn may
be associated to any motor action. This may be the type of neurons record-
ed by Carpenter et al. (1999) in the monkey motor control cortex, which
signals the order of a stimulus in a given sequence.
Fig. 7.3. Creating Base Fuzzy Numbers
7.3 The Crisp Numbers
The KFN circuit may evolve by increasing the complexity of A and Q
(Fig. 7.3), that is, by:
1. changing from the linear accumulation (Fig. 7.1) defined by Eq. 7.2 to a
non-linear function (Fig. 7.3):
t + 1
= Z
1
*
t
*
t
L
*
t + 1
=
t
- Z
1
* Z
2
*
t
*
t
L
* + Z
3
(7.10)
with t as the observation step, Z
1
, Z
2
and Z
3
as constants;
2. creating a hierarchy of subsets of accumulators (A
1
, A
2
,...) to monitor
the discontinuities on the lower level accumulators, such that whenever
the accumulator A
i
is reset, a pulse is accumulated by A
i + 1
;
3. creating a hierarchy of sequential quantifiers (q
1
, q
2
,...) associated to that
of accumulators (see Fig. 7.3), such that the accumulator A
i
feeds the
quantifier q
i
.
7.3 The Crisp Numbers 143
This KFN evolution creates a new kind of number system, called a
Fuzzy Base Number System (FBN), whose base size is determined by the
periodicity of the accumulating function. For instance, if the accumulator
is reset after counting 10 objects, then the number base is 10, as in the dec-
imal system.
KFN evolution may continue by adjusting the filtering or classifying
function to allow SQ
g
Q to perform a crisp classification of (Fig. 7.3),
such that
g
= d
g
if d
g
g
< < d
g
g
, otherwise
g
= 0
(7.11)
dg
() 1 if d
g
g
< < d
g
g
otherwise
dg
() = 0
(7.12)
because
g
= k (d
g
) 0
(7.13)
In such a condition, SQ
g
attends to A only if = d
g
.
It is now possible to create a special family of Crisp Base Numbers
(CBN) of KFN (Fig. 7.4), such that the output i
g
of the g quantifier i
g
Q
becomes
i
g
= d
g
if d
g
g
< < d
g
g
, otherwise i
g
= 0
(7.14)
and the output is made reentrant over the same set A of accumulators ini-
tially activating it, that is, if in Eq. 7.11 is made
i
g
, g = 1, j
(7.15)
In these conditions, the I output crisply codifies the cardinality of ab-
stract sets bearing (or not) a relation to collections of objects in the real
world. If the CBN base is set equal to 10, then we have the classical (or
crisp) decimal number system.
The evolution of the KFN circuit provides animals with two counting
systems, using fuzzy (Q) and crisp (I) quantifiers. Both systems may have
different functions in sophisticated number systems like those developed
by man. On the one hand, the Qs may easily control motor neurons in Bro-
ca‘s or the hand‘s areas to speak or write about the quantities identified by
them (Fig. 7.4), because each Q may map to specific pre-motor and/or mo-
tor neurons. On the other hand, verbal sensory temporal neurons or visual
parietal-occipital cells may directly map to I neurons, which in turn may
change the adequate amount to the accumulators A
i
(Fig. 7.4) to provide
the adequate semantics for spoken and written numbers.
144 7 Modeling of Arithmetic Reasoning
CBN
Speaking, writing, reading numbers
Fig. 7.4. Crisp numbers: KFN evolution creates CBN number circuits
7.3 The Crisp Numbers 145
There is a lot of redundancy in the complete model proposed here to
cope with the processes of identifying quantities by animals and of count-
ing by humans. This is a very important DIPS property. The same
knowledge has to be available to different neighborhoods and to different
agents specialized to solve the same task. Neurons that signal quantities
are found both at parietal and frontal sites (Dehaene, 2002, Nieder et al.
2002, Sawamura et al. 2002) and they may be associated to the quantity
identification (parietal neurons) and the use of this information in other
neighborhoods (frontal neurons). Also, other frontal neurons may be in
charge of computing ordinal numbers associated to sequences of motor ac-
tions (Carpenter et al, 1999), whereas parietal neurons assume the task of
computing cardinal numbers associated to set numerosity. Ordinal and car-
dinal counting will agree in one to one counting when only one object is
identified in the focused location, but they will differ in the case of block
counting when more than one object are jointly identified (subtizing) in the
focused location. The partial redundancy provided by KFN and CBN cir-
cuits is also in accordance with both experimental findings and theoretical
propositions about the existence of approximate and crisp quantification by
humans (e.g., Barth et al, 2003; Dehaene et al. 1999; Gallistel and Gelman,
2000; Nuerk et al. 2001; Polk et al. 2001).
Another interesting feature of the model is that a CBN may be a KFN
evolution due to changes in the expression of:
1. genes which control the ionic channels in the membrane of the accumu-
lator neuron; depending on the number and/or the quality of the Na, K,
Ca and Cl channels, neurons may use monotonically increasing or peri-
odic codes to encode the same kind of information (Rocha, 1992, Ro-
cha, 1997). Thus, the ionic properties required by Eqs. 7.2 and 7.10 may
be provided by changing ionic channel gene expression;
2. genes that control the number and quality of quantifying neurons; in-
crease in cerebral size is one of the most salient features of animal evo-
lution, and changes in the dynamics from Q to I quantifiers may easily
be explained by ionic channel gene encoding; and
3. the complexity of the frontal circuits involved in the motor control of
the counting trajectory; the growth and sophistication of the frontal pro-
cessing capacity is one of the well documented differences between man
and the other primates.
In consideration of the foregoing items, we propose that the genetic in-
formation for the development of CBN circuits was already available to
the early hominids, and that increasing complexity in the lives of human
societies may have pressed man to invent crisp arithmetic in at least four
146 7 Modeling of Arithmetic Reasoning
different cultures, viz., the Sumerian, Egyptian, Greek and Mayan cultures
(Butterworth, 1999, Dehaene, 1997; Devlin, 2001, and Ifrah, 1985).
Fig. 7.5. The real numbers: quantifying distances.
7.4 The Real Numbers
The necessity for quantification extends beyond that of ordering actions
and evaluating cardinalities. It also comprises quantification of distances.
If the ability to quantify amounts of food and numbers of predators in-
creases the odds of survival, so the ability to estimate distances in the dif-
ferent sensory-motor spaces is also a basic necessity for any animal, all of
which need to quantify both cardinalities and continuous variables (Gallis-
tel and Gelman, 2000). The KFN and CBN circuits are specially tailored
for such a task. It suffices to substitute sensory input to the gate agents g
G with the output of the sensory-motor areas which determine the amount
of movement in any of the different sensory-motor spaces (Fig. 7.5). For
example, the different human action spaces—hands (very closed), arms
(peri-personal), eyes and head (near), trunk and head (reachable) and legs
(distant) spaces—all of these may be used by both KFN and CBN circuits
to approximately or crisply evaluate distances and to support the genesis of
7.5 Doing Arithmetic 147
the notion of real numbers (recently introduced in human culture) as well
as the metric system, where it is used to measure distances.
7.5 Doing Arithmetic
KFN and CBN circuits may be designed for doing both fuzzy (Pedrycz
and Gomide, 1998) and crisp arithmetic. On the one hand, man uses fuzzy
arithmetic to deal with approximate daily calculations of the following
type: spending around $X to buy more or fewer Y items; or to fill around
Z liters of gas to travel more or less W kilometers, etc. On the other hand,
crisp mathematics is required to maintain a checking account, to deal with
complex commercial transactions, and to do scientific calculations.
Addition is neatly processed by both KFN and CBN circuits, whether as
formal or simulated operations (Fig. 7.6). In the case of simulated opera-
tions, numbers representing quantities (chickens in Fig. 7.6) may first be
decoded as a set of elements that are then used to mentally simulate the
process of counting described in section 2.1. Each time a chicken is men-
tally imagined (or focused on), the corresponding value is loaded into G.
Each time another chicken is mentally focused on, the value of G is accu-
mulated by A, etc. In the case of formal calculations, spoken or written
numbers are associated to the corresponding agents i I, which in turn
load directly and adequately to the accumulators a A, such that the result
is obtained by the classifications provided either by the neurons q Q or i
belonging to another set I. Hybrid calculations may be processed loading
one number directly into a A using the corresponding i I and decoding
the other as a set of elements, whose elements are sequentially accumulat-
ed in the same a A through the gate g G. If the operands are ordered
first, then this hybrid calculation may be optimized by loading directly the
highest operand and decoding the others as a set of elements. Thus, both
the KFN and CBN may allow man to make use of distinct adding strate-
gies (Fayol, 1996; Gelmann and Gallistel, 1991; and Siegler, 1996). It is
easily verified that simulated calculations may render addition time de-
pendent on the size of operands, whereas formal calculations make it con-
stant. Since different sets of agents may enroll to solve the same task pro-
posed to a DIPS, then it may be that man activates different KFN and CBN
circuits to a parallel processing of the same addition operation, the result
being furnished on a ―first come, first served‖ basis if speed is required, or
by majority if precision is the goal. Also, the types of errors pointed by
McCloskey, Harly and Sokol (1991) will be those expected for the kind of
calculations proposed above, if mistakes are made in operand association
148 7 Modeling of Arithmetic Reasoning
Fig. 7.6. Doing arithmetic: adding and multiplying.
7.5 Doing Arithmetic 149
of i I; in operand set decoding; by a c C bad performance, or even by
misclassifications of q Q or i I.
Subtraction may be easily performed as the inverse operation of addi-
tion. In this case, the subtrahend is loaded in a A and then c C controls
a step-by-step increase of the load in a A until the classified output by q
Q or i I equals the minuend. But subtraction may also be calculated by
a decreasing counting, when the minuend is firstly loaded in a A and
then c C controls a step-by-step decrease of the load in a A using in-
hibitory neurons. The result is classified by q Q or i I after a number
of steps corresponding to the subtrahend. These are the so-called counting
up and counting down strategies used to simulate subtraction (Fayol, 1996;
Gelmann and Gallistel, 1991; and Siegler, 1996). These strategies may
render the subtraction time dependent on the size of the subtrahend (count-
ing down) or of the result (counting up). Since both strategies will always
be activated in a DIPS, the calculation time may be optimized, if the first
circuit to finish calculation provides the solution. But subtraction may also
be formally achieved by associating the operands to their corresponding i
I, which in turn will load a A through excitatory (minuend) and in-
hibitory (subtrahend) synapses.
Multiplication may be computed either as a repeated summation or as a
block counting (counting by multiples). In the case of both simulated and
formal calculations, one of the operands is loaded in g G and repeatedly
added to a A under the control of c C. The difference between simu-
lated and formal calculations is defined by the way g G and c C are
loaded: directly from i I in the case of formal computations, or by means
of a set decoding in the case of simulated operations. Hybrid operations
will be also allowed. It may be hypothesized that the calculation time will
be greater in the case of simulation even if formal processing will also re-
quire repeated calculations. This is because the neuronal complexity re-
quired for loading g and c for simulation is greater than that needed for
formal calculations. Also, the multiplication time may be expected to be
inversely related to the size of the block used for repeated operations and
to be directly dependent on the number of repeated calculations.
Division may be calculated as the inverse operation in respect to multi-
plication or as a case of repeated subtraction. In the first case, the divisor is
loaded in g G and repeatedly added under the control of c C until the
output at q Q or i I equals the dividend. In the second case, the divi-
dend is loaded in a A; the divisor loaded in g G, and repeatedly sub-
tracted from a A until the output q Q or i I equals 0. In both cases,
150 7 Modeling of Arithmetic Reasoning
the result is the number of repeated pulses of c C computed by the
frontal ordinal circuit.
As in the case of multiplication, the repeated calculations for division
may be either simulated or formally computed. By the same reasoning dis-
cussed for multiplication, the time for simulation will be greater than that
for formal computations. In the case of simulation, the division time may
become directly dependent on the size of the divisor or inversely propor-
tional to the size of the dividend. But division may also be simulated as
sectioning of the number line fed by sensori-motor data and generating real
numbers (Zorzi et al, 2002). This may be a simulation that is independent
of the size of operands.
7.6 The Evolution of Arithmetic Knowledge
The model proposed here is based on the assumption that initially a set of
DIPS units (neurons) are able to discriminate the objects of interest in the
real world and to provide basic information about their quantities to other
specialized agents in charge of accumulating the results of sequential or-
dered observations. The accumulator output is then classified by another
set of units, called here quantifiers, into a small set of classes allowing a
primitive manipulation of quantities.
The quantity of sense was introduced here by associating a sensory iden-
tifier to two other different agents: the accumulator and the quantifier neu-
rons, such that the number of identified objects in a given sensory system
at each focused location is accumulated, followed by an estimation of their
quantity. However, in such a simple sensory-specific model, let us say,
five pictures have no relation to five sounds, even when both are associat-
ed to the same set of real world objects. This is because each kind of sen-
sory information is specifically channeled to a defined set of accumulators,
which in turns activates a specific set of quantifiers within a given sensory
modality. In this way, we may speak of different quantifying circuits, each
one sensory bounded.
The first important evolutionary step from the quantity of concept to that
of numerosity was to channel different sensory information to the same set
of gates, which in turn feed the same accumulators and quantifiers. Now,
five pictures and five sounds are associated to the same quantity and may
now refer, if necessary, to the same set of real world objects. This is be-
cause the same quantifying circuit is activated by information provided by
7.6 The Evolution of Arithmetic Knowledge 151
different sensory systems. This was what may have happened in evolution
with the emergence of the multi-sensorial (parietal) cortex.
The second important step in the development of the concept of number
required the creation of a specific set of quantifiers—the I quantifiers—
and the reentrance of the output from the I quantifier to the same gates G
feeding the accumulators A, providing information to the very same I
quantifiers. This loop freed quantification from any outside sensory input.
In other words, reentrance freed quantification from observation in the ex-
ternal environment. Therefore, this step created the concept of number as a
set of symbols referring to quantities, or the cardinality of abstract sets.
This generated number lines in the parietal cortex, as proposed by many
authors. (Butterworth, 2000; Dehaene, 1999; Fink et al. 2001; Gobel et al.
2001, Zorzi et al. 2002). It is worth stressing that reentrance is an im-
portant feature of brain processing, as mentioned by Edelman (1989 and
1997) and Rocha et al. (2001). This highly evolved model supports the
human number capability and perhaps a degree of number sense in the
other most evolved primates.
Both the accumulation and filtering functions used to generate any of
the above evolutionary steps of our model may be understood as supported
by ionic properties of the neural membrane. Different dynamic membrane
systems may generate monotonically increasing or periodic changes in the
membrane potential (Rocha 1992; Rocha 1997). Different genes governing
the expression of ionic channels may be responsible for such different en-
codings and changes in the expression of these genes may account for
KFN evolution. These ionic genes are known to have appeared early in
brain phylogeny. This implies that at least the concept of numerosity may
be at hand for many species. The distinct spatial configurations of the
DIPS agents required by the different steps of our model may also be easi-
ly understood from our knowledge of brain phylogeny, and governed by
genes responsible for axonal growth and addressing (e.g., Edelman, 1997).
Again, these genes are known to be ancient.
The number sense is now a well-known animal capability (Butterworth,
1999; Dehaene, 1999; Devlin, 2001) and humans invented numbers at least
in four different cultures at four different times (Ifrah, 1985). In this con-
text, we may propose that an Animal Mathematical World is a product of
the interaction between genes and their habitats, and by the same token the
Human Mathematical World is determined by the interplay between the
brain (genes) and culture (habitat).
152 7 Modeling of Arithmetic Reasoning
7.7 Representation of Other Abstract Entities
Mathematics is much more than the ‗science of numbers.‘ If mathematics
is considered as the science of patterns, we can demonstrate that other ab-
stract mathematical constructs may be modeled in a similar way to the
manner in which the number concept was built up, step-by-step, towards
increased generalization. Patterns are stable relations among features of
concrete or abstract objects recognized by a set R of specialized agents.
The same general procedure used above to construct both KFN and
CBN numbers may be applied to construct other abstract mathematical
concepts, starting from the identification of other specific patterns in the
environment. In the same way that identifiers provided accumulators with
information about quantity, these operators may furnish other types of in-
formation about defined sensory patterns (e.g., line relations) to other spe-
cific processing agents (e.g., angle evaluation), so that similar kinds of
(geometric) concept generalization are achieved through multi-sensory pat-
tern channeling, with information reentrance through other specialized
processing circuits. As a matter of fact, there is no a priori difference be-
tween visual pattern recognition and classical geometric processing. What
is the difference between the recognition of the pattern of an elderly male
human face by some specialized cell at the temporal lobe and the formal
construction of an object in any equivalent order-dimensional geometric
space? Certainly the increased brain capacity for geometric analysis of vis-
ual information in some animals is also a characteristic with high adapta-
tional value.
How is it possible to think about something that does not exist? This is a
high level of abstraction (level 3 of abstraction according to Devlin, 2001)
that may be supported by simulation performed by DIPS agents using ab-
stract objects of the type described above for integers. Simulation only re-
quires other specialized agents capable of recruiting abstract level 3 agents.
DIPS simulation is the keystone of imagination and it is supposed to ap-
pear in evolution with the enhancement of the frontal lobe, especially in
humans. In fact, imagination is nothing but mental simulation controlled
by a top-down process, where most frontal neurons recruit more posterior
agents in an orderly process. In such a way, DIPS simulation may allow
man to create a sophisticated number theory or other abstract mathematical
object.
7.7 Representation of Other Abstract Entities 153
154 8 Brain Maps of Arithmetic Processes in Children and Adults
8 Brain Maps of Arithmetic Processes in Children
and Adults
Despite the increasing number of experimental mappings of a widely dis-
tributed neural circuit subserving human arithmetic cognition, the theoreti-
cal reasoning concerning these data remains mostly metaphorical, and
guided by a connectionist approach. Herein, the DIPS theory developed in
previous chapters is used to develop a new technique for EEG mapping of
the brain activity associated with cognition, and to study arithmetic cogni-
tion in elementary school aged children and in adults. Performance in solv-
ing arithmetic calculations was studied in adults and children while their
EEG activity was recorded. Experimental data showed a) a clear-cut dis-
tinction between genders, males being faster than females in providing
equally correct answers; b) quick learning, characterized by a decrease in
calculation time, which is dependent on the order of problem presentation;
and c) a number size dependence that differed between children and adults.
Factor analysis showed three distinct patterns of neuronal recruitment for
arithmetic calculations in all experimental groups, which varied according
to the type of calculation, age, and gender.
8.1 The Study of the Mathematical Brain
Experiments carried out in humans demonstrated the involvement of the
inferior parietal cortex as well as multiple regions of the prefrontal cortex
in the course of numerical performance (e.g., Butterworth, 1999, Cabeza
and Nyberg, 1997, 2000; Carpenter et al., 1999; Dehaene, 1997; Dehane et
al. 1999; Göbel et al. 2001; Iguchi and Hashimoto, 2000; Jahanshahi et al.
2000; Kong et al. 1999; Menon et al. 2000; Nieder et al. 2002; Pesenti et
al. 2000; Ratinck et al. 2001; Sawamura et al. 2002; Skrandies et al., 1999;
Zorzi et al. 2002). It is now accepted that the inferior parietal region is im-
portant for the translation of numerical symbols into quantities, and the
representation of relative magnitudes of numbers. The prefrontal cortex is
proposed to be responsible for the processes involved in sequential order-
ing of successive operations, in control over their execution, error correc-
8.1 The Study of the Mathematical Brain 155
tion, inhibition of verbal responses, etc. But other central and temporal
cortical areas are also involved in many arithmetic calculations. It seems,
therefore, that human arithmetic capabilities result from complex cerebral
processing involving different types of neurons widely distributed
throughout the brain, each of them in charge of solving a particular subtask
of the whole problem.
Despite the increasing number of authors (e.g., Cowell et al. 2000;
Dehaene et al. 1999; Jahanshahi et al. 2000; Zago et al. 2001) experimen-
tally mapping a widely spread neural circuit supporting human arithmetic
cognition, the theoretical reasoning about these data is not well developed.
Although neurons at distinct areas in the brain are assumed to take charge
of different duties in the solution of the experimental task, the results are
always discussed by hypothesizing some association between the different
areas without questioning any distinct behavior at the level of the neurons
at each of these areas. Also, the role played by each brain area in the dif-
ferent arithmetic tasks is, in general, discussed metaphorically rather than
formally. In the DIPS approach, however, the task of counting is assumed
to be performed by a large number of specialized agents distributed all
over the brain, such that mathematical skills becomes dependent on both
the different types of specialization by different agents as well as on how
these agents engage themselves in solving given mathematical problems.
Another interesting DIPS property is that many different means are tried
to solve the very same arithmetic problem. Indeed this seems to be the
strategy used by the brain in the case of arithmetic processing. Many stud-
ies of how children in different countries solve standard arithmetic prob-
lems has revealed that the subjects use multiple different strategies such as
(Butterworth, 1999; Dehaene, 1991; Dehane, 1997; McCloskey et al.,
1991; Siegler, 1996):
1. total manipulation: the child counts each set to be processed separately
by pointing, marking, etc. each of their elements, and then counts by the
same process each element of the union, or the complement, etc. of the-
se sets to get the final result;
2. simplified manipulation: the result is obtained by counting each ele-
ment of the union, or the complement, etc. of the sets to be processed;
3. optimized manipulation: the result seems to be obtained by performing
the minimum counting, which varies according to the type of calculation
to be performed, and
4. mental calculation: the results are quickly processed by specialized cir-
cuits. Also, the type of function explaining the number size effect re-
ported in the literature correlates with the kind of manipulation used
(e.g., McCloskey et al, 1991).
156 8 Brain Maps of Arithmetic Processes in Children and Adults
Finally, training changes the frequency of use of the different types of
strategy, such that adults tend to rely more on mental calculation than chil-
dren (Siegler, 1996). Rocha et al., (2003c,d) showed that children and
adults use different strategies for solving any kind of arithmetic calcula-
tion, because they found different types of correlation between the calcula-
tion time and the size of the different operands. Also, they showed that the
size effect dependence was more complex for adults in comparison with
children, and they concluded that learning enriches arithmetic knowledge
by increasing the number of strategies available for the same calculations.
Here, the distributed properties of arithmetic cognition were studied in
three experimental groups while having their EEGs recorded in the course
of solving arithmetic calculations (Rocha et al., 2003c). The performance
in solving the arithmetic problems was measured by the time interval re-
quired for the calculation, since the error rate was low in all groups. The
EEG analysis provided different brain mappings (Rocha et al., 2003c) that
revealed the distributed properties of the mathematical brain.
8.2 The Technique
The DIPS approach to cerebral physiology also results into new technolo-
gies for the analysis of EEG activity related to cognitive tasks (Foz et al.
2001; Rocha and Rocha, 2002; Rocha et al., 2003c).
8.2.1 Theory
Definition 8.1: Let n
i
, n
k
) measure, in the closed interval (0,1), the pos-
sibility of any two neurons n
i
, n
k
to jointly involve themselves in solving a
task t, such that
n
k
kjj
nn
n
n
1
),(
1
)(
(8.1)
)(
j
n
-0.5 = ξ
(8.2)
N is said to be a strongly (un)connected system N
s
if 0.5 for most of
a
j
N. Strongly (un)connected systems are of no interest where cognition
is concerned, because either their agents have difficulties in enrolling to-
gether to solve a task, or the relations shared by their agents tend to be ste-
reotypical rather than versatile. Also, their agents are more likely to share
strong, rather than plastic, commitments.
8.2 The Technique 157
N is said to be a loosely connected system N
L
if 0 for most of a
j
N
even if a
j
, a
k
) 1 or 0 for some, but not all a
k
N
L
. A DIPS is a loosely
connected system, because each of the agents retains the maximum capa-
bility to enroll with different groups of other agents in the effort to solve
different tasks, since
m
n
j
) .5.
Definition 8.2. The following brain entropies are defined from Eqs. 2.2
to 2.25:
1. the commitment h(n
j,k
) of n
j
, n
k
in jointly solving a given task is calcu-
lated as:
h(n
j, k
)
= - n
j
, n
k
) log
2
n
j
, n
k
) - ~n
j
, n
k
) log
2
~n
j
, n
k
)
(8.3)
~a
j
, a
k
) = 1 - a
j
, a
k
)
(8.4)
2. the enrollment capability h
m
(n
j
)
of n
j
is calculated as
h
m
(n
j
)
= -
m
n
j
) log
2
m
n
j
) - ~
m
n
j
) log
2
~
m
n
j
)
(8.5)
~
m
n
j
)
= 1 -
m
n
j
)
(8.6)
that is, h
m
(n
j
) is a function of the mean probability
m
n
j
) of n
j
to com-
municate with the other agents n
k
N, and
3. the actual commitment h(n
j
)
of n
j
to solve the task is
)()()(
,
1
kjj
n
k
m
nhnhnh
(8.7)
Proposal 8.1. The linear correlation coefficient r
i,k
for the EEG activi-
ties recorded at site d
i
, d
k
is assumed to be an indirect measure of the pos-
sibility (n
i
, n
k
) of the enrollment of the neurons n
i
, n
k
at these locations in
solving a given task. In this way, the (mean) EEG activity recorded, and
associated to a given event e of such task t, is used to calculated the corre-
sponding h
e
(n
j
) for each recording site d
i
. The set H
e
of all h
e
(n
j
) values for
all similar events e are used to generate different brain mappings (fig. 8.2)
according to the statistics applied to analyze H
e
.
Remark 8.1. Foz et al. (2001) used this technique to study the plasticity
of neural circuits for language in brain-damaged children, where the clas-
158 8 Brain Maps of Arithmetic Processes in Children and Adults
sical Broca or Wernicke areas were destroyed during their fetal life. These
children experienced severe delays in language development, but by the
age of 11 years their brain mappings, associated with oral charades and
texts, revealed an extensive use of the right hemisphere areas to compen-
sate for the lesion of their homologous left sites.
8.2.2 The Methods
Data were obtained from the three experimental groups by Rocha, Rocha
and Massad (2003 c,d). These groups were equal in regard to gender bal-
ance, in mean age differences, and in cognitive development.
1. Group CHI2: 20 children — mean age: 7 yrs, 7 mo; enrolled in 2
nd
and
3
rd
semesters of elementary school; mastering addition and subtraction,
as well as reading and writing simple phrases;
2. Group CHI4: 24 children — mean age 8 yrs, 3 mo; enrolled in 3
rd
and
4
th
semesters of elementary school; mastering addition, subtraction, and
multiplication; in their initial training for division; also, reading simple
texts and writing simple phrases;
3. Group AD: adults: 20 adults — mean age 28 yrs; enrolled in graduate
courses in the field of exact sciences in a university near São Paulo, and
attending a special training program in Biotechnology.
The children were selected from a group whose parents agreed with the
explained experimental protocol and who signed a special permission
form. Each group was formed by children from two equivalent classrooms
in respect of their cognitive profile, as evaluated by the principal of a mid-
dle class school in the city of Guarulhos. Adults volunteered after having
the experimental protocol explained to them and they also signed the per-
mission form, whose terms were approved by the university‘s Ethics
Committee.
The experimental protocol consisted in solving 30 different problems
(Fig. 8.1) for each arithmetic calculation while the volunteers‘ EEGs were
recorded (Fig. 8.2). Each question was visually presented on a computer
screen, and the volunteer had to choose the correct response among a set of
displayed numbers. The questions were presented in two different visual
formats. In one of them (VA format) the quantities were represented by
numbers and elements of a given class of objects (toys, fruits, etc.), where-
as in the other format (VB format) quantities were only represented by
numbers. Two different series of questions were prepared, each concerning
a given display format, and containing 15 problems each. The questions in
each series were randomly selected when they were initially programmed.
8.2 The Technique 159
Fig. 8.1. The experimental protocol: Testing arithmetic knowledge.
Each question in the series was numbered to allow the study of any de-
pendence of performance concerning the order of question presentation.
Whereas adults solved both series, the groups CHI2 and CHI4 manipulat-
ed only the questions in the VA format. This was mainly because initial
tests revealed that children could become tired if submitted to long series
of calculations. CHI4 children were tested in two different epochs: at the
end of the 3
rd
and 4
th
semesters, in an attempt to quantify possible learning
via reduction in errors and in response time. Adults solved the VA series
for all four types of arithmetic calculations before the VB series were pre-
sented. This was done in order to allow comparisons between children and
adults to be studied independently from the type of visual presentation, and
to assure that the VA format could not be interpreted as infantile by the
adults. All mistakes and correct responses were clearly signaled to the vol-
unteers after they clicked their choice of answer to the question posed.
The set of numbers available for response selection were displayed in lines
of increasing but not consecutive quantities. These sequences were ran-
domly selected when programming the experimental protocol, such that all
volunteers were presented with the same response set for each question
posed. All sessions were videotaped for further inspection of the volunteer
160 8 Brain Maps of Arithmetic Processes in Children and Adults
performance whenever necessary. Adults were encouraged to comment on
the experiment at the end of the session.
Fig. 8.2. Game playing while 20-electrode EEG is registered
Each subject solved the tests while his/her EEG was registered with 20
electrodes placed according to the 10/20 system; impedance smaller than
10 K ohm; low band passing filter 50Hz; sampling rate of 256 Hz and 10
bit resolution (Fig. 8.3).
Two networked personal computer were used: one for the EEG record-
ing and the other for game playing. Timing of test events (e
1
, e
2
, ...), like
the beginning of the visual information display, decision making, etc.,
were written as corresponding marks (m
1
, m
2
, ...) in the file of the recorded
EEGThe EEG was visually inspected for artifacts before processing, and
the events associated with a bad EEG were discarded. The mean time for
solving the text was calculated for each experimental group. The recorded
EEG was averaged for epochs of duration equal to this mean time, referred
to the beginning of the visual display, in order to generate the Game Event
Related Activity (GERA) file for each volunteer (Fig. 8.4).
8.2 The Technique 161
Fig. 8.3. EEG Recording
Fig. 8.4. Game Event Related Activity
162 8 Brain Maps of Arithmetic Processes in Children and Adults
These GERAs were used to calculate different brain mappings accord-
ing to the following rationale (Foz et al. 2001; Rocha et al. 2003c). The
linear correlation coefficients r
i, j
for the averaged activity at each record-
ing site d
i
, referred to the averaged activity for each of the other 19 deriva-
tions d
j
, were calculated for each GERA (Fig. 8.5).
Fig. 8.5. Regression Analysis
This r
i, j
is considered a measure of the possibility (n
i
, n
j
) that neurons
n
i
at d
i
are exchanging information with those neurons n
j
at d
j
, and this
number was used to calculate h(n
j
) for each d
i
(Fig. 8.6). These data were
used to calculate the following brain mappings: a) MCC: plot of the nor-
malized mean h(n
j
), and b) FM: three factors were extracted by using Prin-
cipal Components Analysis, and these factors were rotated using the vari-
max normalized method. If these extracted factors explained more than
50% percent of the total h(n
j
) variability, then the analysis was considered
acceptable, and the color plots of the calculated correlation coefficient for
each electrode and each factor were produced.
The reaction (or calculation) time was measured by the computer as the
time interval from the beginning of the visual display up to the moment the
volunteer clicked a number as the selected answer. Whenever an error oc-
curred, the volunteer was signaled for error and the computer clock reset.
The computer also kept track of the number of errors and correct responses
of each volunteer. Non-parametric static was used to evaluate differences
8.3 Agent Commitment Experimentally Measured 163
between groups and gender, and multiple regression analysis was em-
ployed to study possible correlations between calculation time, number
size, order in the calculation series and gender.
Fig. 8.6. EEG Correlation Analysis
8.3 Agent Commitment Experimentally Measured
The mean, maximum and minimum values of the actual commitment
(h(d
j
)) differed among groups and type of calculations (See Table 8.1). The
differences between the mean values obtained for the adults and children
were statistically different for all type of calculations, as well as for male
and females. No statistical difference was observed between CHI2 and
CHI4 groups.
164 8 Brain Maps of Arithmetic Processes in Children and Adults
Table 8.1. Commitment (h(d
i
)) vs. Calculation
ADU
MALE
FEMALE
MEDIA
MAX
MIN
MEDIA
MAX
MIN
AU
2,12
9,01
0,07
2,50
8,26
0,11
*
SU
2,08
7,04
0,13
2,42
7,62
0,27
*
MU
2,08
6,76
0,00
2,59
8,21
0,35
*
DI
2,31
6,66
0,24
2,67
7,62
0,48
*
CHI4
MALE
FEMALE
MEDIA
MAX
MIN
MEDIA
MAX
MIN
AU
3,18
11,24
0,12
3,29
9,34
0,00
*
SU
2,60
6,98
0,25
3,16
8,51
0,00
*
MU
2,47
6,38
0,04
3,41
9,48
0,00
*
DI
2,56
7,65
0,11
3,08
7,52
0,00
*
CHI2
MALE
FEMALE
MEDIA
MAX
MIN
MEDIA
MAX
MIN
AU
2,65
6,97
0,16
2,85
9,34
0,11
SU
2,60
6,79
0,13
3,15
8,51
0,27
*
AU: Addition; SU: Subtraction; MU: Multiplication; DI: Division. ADU:
adults; CHI4: children enrolled in 4
th
and 5
th
semesters of elementary school;
CHI2: children enrolled in 2
nd
and 3
rd
semesters of elementary school. Mean, Max
and Min: mean, maximum and minimum h(d
i
) values, respectively. Statistical dif-
ferences between male and female means are marked with a *.
8.3 Agent Commitment Experimentally Measured 165
The commitment h(d
j
) of any agent d
i
in solving a given arithmetic cal-
culation, was calculated according to Eqs. 8.1 to 7, taking into considera-
tion that the correlation coefficient r
i, j
between the EEG activity recorded
at the sites d
i
, d
j
, measures the possibility n
i
, n
k
) that the neurons n
i
, n
j
in
these sites enroll themselves in the solution of the proposed task. The h(d
j
)
max/min values in Table 8.1 clearly show the adequacy of this formal ap-
proach, since the actual values h(d
j
) calculated for each arithmetic calcula-
tion and each experimental group obeyed the theoretical conditions re-
quired to guarantee that agent commitment has always a positive value.
Also, the mean h(d
j
) was statistically smaller for male than females in all
groups, which may be an explanation of the fact that the males were
quicker than females in solving arithmetic questions (Rocha et al., 2003c).
Finally, the mean h(d
j
) was smaller for adults in comparison to children,
but adults were faster than children in doing the same calculations.
8.3.1 Males are Faster than Females in Arithmetic Calculations
One of the main results of the present experiments was a clear gender dif-
ference for all groups. In general, males were quicker than females in
arithmetic calculations (Fig. 8.7), but there was no consistent gender dif-
ference concerning errors.
Adult males were quicker than adult females in all kinds of arithmetic
calculations, although the difference tended to be smaller in the case of
multiplication. These differences were statistically significant at the level
of p < 5%, as evaluated by the Man-Whitney U test. Also, the index I =
(T
m
– T
f
)/SD is that proposed by Halpern (1992) to quantify gender differ-
ences was calculated for each arithmetic task. (SD is the standard deviation
calculated for the entire group ADU, and T
m
, T
f
the mean male and female
times, respectively.) This index was positive and greater for addition and
division in comparison with subtraction and multiplication. It reached the
highest value of 38% in the case of addition and the minimum value of 9%
in the case of multiplication.
CHI4 males were quicker than their female classmates in the case of ad-
dition, subtraction and multiplication, but not in the case of division.
Again, the differences were statistically significant at the level of p < 5%
as evaluated by means of the Man-Whitney U test. The value of I steadily
increased from 5%, in the case of addition, to 12% in the case of subtrac-
tion, and to 19% for multiplication. CHI2 females were slower (p < 5%)
than their male classmates in the case of addition and subtraction, but not
in the case of multiplication. The value of I was 5% in the case of addition
and only 1% for subtraction.
166 8 Brain Maps of Arithmetic Processes in Children and Adults
Fig. 8.7. Calculation time: Males are faster than females in arithmetic calcula-
tion. Top: ADU; middle: CHI4 and bottom: CHI2
Another general result was the difference in the calculation time be-
tween the experimental groups. Adults were quicker than children in all
kinds of calculations, and CHI2 were slower than CHI4 children in adding
and subtracting. All these differences were established at a level of statisti-
cal significance of p < 5%, as evaluated by means of the Man-Whitney U
test.
8.3 Agent Commitment Experimentally Measured 167
The calculation time varied narrowly between 3–5 seconds in the group
ADU when all kinds of calculations were considered. There was no statis-
tical difference in these different calculation times for this group. CHI4
children spent between 8–12 seconds to get the results in the case of addi-
tion and subtraction and between 12-16 seconds in the case of multiplica-
tion. No statistical difference was observed for addition and subtraction,
but female multiplication times were different from their addition and sub-
traction times at the level of p < 6%. The addition and subtraction times in
the group CHI2 varied between 13–15 seconds, with no statistical differ-
ence concerning the type of calculation. CHI2 children were statistically
slower than CHI4 children in their calculations.
Adults did not err on any kind of calculation. Errors were around 3% for
addition, subtraction and multiplication in the group CHI4, and did not sta-
tistically vary with respect to the kind of calculation. However, the error
rate reached a peak at around 70%, and a mean of 48% in the case of divi-
sion, since this group was at the beginning of their division learning peri-
od. Errors were around 12% for both addition and subtraction in group
CHI2. Statistically, these children erred more than CHI4 children.
The most robust result from the multiple regression analysis was the in-
verse correlation between the calculation time and the order of question
presentation (see correlation coefficients labeled L in Fig. 8.8). The ADU
calculation time decreased as the order of question presentation increased
for all kinds of operations; this correlation ranged from 40% to 60%. CHI4
children were quicker at the end than at the beginning of the addition and
subtraction series, but not in the case of multiplication. The correlation for
this group ranged from 21% to 36%. CHI4 children were quicker at the
end than at the beginning of the addition and subtraction series, too. But
their correlation coefficients ranged from 11% to 30%. CHI4 children were
also quicker at adding and subtracting, but not at multiplication, in the se-
cond test as compared to the first one. The correlation coefficients in this
case varied between 13% and 18% (see correlation labeled Epoch in Fig.
8.8).
The calculation time was also gender related for all kinds of calculations
in the case of the ADU group; for addition and subtraction in the case of
group CHI4; and for addition in the case of CHI2 children. The correlation
was around 20%, and showed male calculation time was shorter than fe-
male processing time.
―Sex differences in mathematics achievement are well documented‖, as
Halpern put it in her 1992 book: Sex differences in cognitive abilities. The
importance of this subject is clearly acknowledged (Beal, 1999) in the spe-
cial issue of the journal Contemporary Educational Psychology devoted to
the Math-Fact Retrieval Hypothesis, proposed by Royer and Tronsky
168 8 Brain Maps of Arithmetic Processes in Children and Adults
(1999) to explain that males outperformed females in the SAT-Math exam
because they are quicker in math-fact retrieval. Finally, Gallager et al.
(2000) proposed that ―strategy flexibility is a source of gender differences
in mathematical ability assessed by SAT-M and GRE-Q problem solving.‖
Fig. 8.8. Number size effect: Calculation time depends on size of the operands.
Top: ADU; middle: CHI4 and bottom: CHI2
The gender differences seem, from the above, to start to appear at the
very beginning of academic training in arithmetic (Group CHI2) and to
persist into adulthood (Group ADU). It must be remarked that the adult
8.3 Agent Commitment Experimentally Measured 169
group was composed of graduate students in the exact sciences. This inval-
idates any strong claim that gender differences in math are the result of a
biased education, and seems to point to a phylogenetic explanation.
A common adult commentary at the end of the experiment was that they
tried to find a good strategy to quickly solve the tests. This is in clear
agreement with the findings of Gallager et al. (2000), and it is supported
here by another clear result: dependence of calculation time on the order of
question presentation. The strongest correlation observed in the present
experiments was the inverse relation between calculation time and the or-
der of the question in the test series, for all type of calculations and for all
experimental groups. This clearly shows that volunteers experienced some
sort of learning while solving the tests. Interesting also is the fact that re-
gression analysis shows that the rate of this learning is greater for males
than females. It must be remembered that the correlation coefficient for the
calculation time/gender relation was around 20% when all types of calcu-
lations and all groups are considered. Learning effects were also demon-
strated by Dehaene et al. (1999) and Spelke and Tsvikin (2001), who
trained bilingual adults in solving arithmetic calculations in one language
and then tested them in both languages.
Some female adult commentaries seem to corroborate the hypothesis of
strategy flexibility and its language dependence. Some female adults
comment that for them is difficult to use, e.g., the commutative property of
multiplication a * b = b * a, which may speed up calculation by using the
largest multiplicand as the block to be repeatedly summed over to answer
the question. The present results clearly show that learning in test solving
is not exclusively language dependent, since the present tests were all vis-
ually encoded and decoded. Taken together with Dehaene et al. (1999) and
Spelke and Tsvikin (2001), these data point to an important property of the
neural math circuits, viz., their plasticity, a condition necessary to explain
how ―number discovery,‖ imposed by the complexity of social transac-
tions, may be stabilized and incorporated into different human cultures.
The learning effect observed in the CHI4 children is further strong evi-
dence for such a hypothesis. The calculation time and error rate decreased
in this group when the test performances from two different epochs (six
months apart) were compared, but this group had not mastered division
yet.
But choice and optimization of strategies requires variability of tools to
solve the same problem. This property is basic to DIPS and to the model
proposed here. Number representation, counting, and calculations are sup-
ported by different neural circuits involving neurons located at different
brain regions and specialized for different kinds of tasks: cardinal and or-
170 8 Brain Maps of Arithmetic Processes in Children and Adults
dering representation, fuzzy and crisp numbers, addition, subtraction, mul-
tiplication, and division.
8.3.2 Size Number Effect
The use of different calculation strategies is confirmed by the regression
analysis of the calculation time dependence on number size. As already
showed and proposed by others (Fayol, 1996; Fink et al. 2001; Gallistel
and Gelman, 1991; Gelman and Meck, 1983; Groen and Parkman, 1972;
Jensen et al. 1950; Kaufmann et al. 1949; Mix, 1999; Siegler, 1996), the
present results indicate that both children and adult use different strategies
to solve each of the arithmetic calculations.
The size number effect was first observed in the group CHI2. Addition
time increased as both the minima of the operands (minimum counting
simulation) increased (S2 in Fig. 8.6) and the square root of their sum
augmented (R(N1 + N2) in Fig. 8.6). Subtraction time increased as both
the minima between the subtrahend (counting up simulation) and the result
(counting down simulation) increased (M(N2, R) in Fig. 8.6) and the
square root of the operands augmented. Similar results in adding and sub-
tracting were also observed in the case of CHI4 children. Furthermore, the
multiplication time in this latter group increased as the minima of the mul-
tiplicands increased (S2 in Fig. 8.6) and also as the square root of their
sum augmented.
The adult addition time increased as both the minima of the operands
increased (S2 in Fig. 8.6) and as the square root of their sum augmented
(R(N1 + N2) in Fig. 8.6), too. Their subtraction time also increased as both
the minimum between the subtrahend and the result (M(N2,R) in Fig. 8.6)
and the square root of the operands (as observed in the CHI4 group) aug-
mented. However, their subtraction time decreased as the square root of
the sum of the operands increased. The product time decreased as both the
maximum multiplicand (S2 in Fig. 8.6) and the square root of the sum of
the multiplicands increased. Finally, division time increased as the minima
between the divisor (division as inverse multiplication strategy) and the re-
sult (repeated subtraction simulation) increased (M(N2,R), as in Fig. 8.6)
and the as square root of the sum of the operands decreased.
The results with the CHI2 and CHI4 groups demonstrate that children
used full and optimized simulations for solving addition and subtraction
problems. Their calculation times were dependent on the square root of the
sum of the operands, and also obeyed the optimization strategies of mini-
mum counting for addition, or counting up or down in subtraction (Siegler,
1996). But the size effect dependence never resulted in high correlation
8.4 The Distributed Mathematical Brain 171
coefficients; they remained at around 20%. This means that much of the
calculation time variation may be due to the use of other strategies, such as
those discussed here as formal calculations and which are supposed to be
faster than simulation solutions. The CHI4 children also used different
strategies to solve the multiplication tests, since the calculation times de-
pended on both the square root of the sum of the multiplicands and on the
size of the smallest of these operands. The 40% correlation in the later case
may be an indication that block counting (or counting by multiples) was a
predominant strategy over full simulation and formal calculation. The
analysis of the videos clearly shows changes of strategies. Sometimes the
same children used finger counting, while at other times they finger-
pointed figures on the computer screen. In some instances the same chil-
dren used both hands for finger counting, while on other occasions he/she
used one-hand simulation. Most children used repeated movements of fin-
ger blocks to simulate multiplication. Similar results were obtained with
the twelve students of the second private school, although some correla-
tions did not attain statistical significance at the level p < .5, possibly be-
cause of the small number of volunteers compared with CHI2 or CHI4.
Some of the results with those children in public schools were similar also
to those obtained with the CHI4, but again the statistical significance of the
correlation was more variable due to the fact that the children in this group
were enrolled in many different school semesters.
Adults used different strategies for solving each kind of arithmetic cal-
culation, too. Since the size effect dependence was more complex for
adults in comparison with the children, it may be proposed that learning
enriches arithmetic knowledge by increasing the number of available strat-
egies for the same calculations. Adults persisted in using the same child-
hood strategies, since similar calculation-time/number-size relations were
disclosed when both groups were compared. Nonetheless, some unex-
pected inverse dependencies of calculation time with the square root of the
operand sum were observed for multiplication and division. Also, calcula-
tion was faster if the maximum operand increased. All of these point to the
use of other types of strategies to solve these types of calculations. At least
one volunteer reported ruler simulation, which is based upon visual opera-
tions with the number line (Zorzi et al. 2002).
8.4 The Distributed Mathematical Brain
The MCC and FMs associated with each arithmetic task corroborated the
male/female and adult/children differences in performance described in the
172 8 Brain Maps of Arithmetic Processes in Children and Adults
preceding sections, and disclosed some interesting characteristics of the
brain activity associated with the arithmetic calculations.
The adult MCCs were very similar for males and females (Fig. 8.9).
High values of h(d
i
) for the frontal and central electrodes where obtained
bilaterally in the case of addition and subtraction, and mostly over the left
hemisphere in the case of multiplication and summation. Different from
the MCCs, the factor mappings FM1, FM2, and FM3 greatly differed be-
tween sexes. The adult MCCs were very similar for both male and females
and for all types of calculation (Fig. 8.9). They show that anterior and bi-
lateral areas were strongly committed to solving any type of arithmetic
calculation. The three FMs show, however, that these anterior areas en-
rolled other neurons (distributed over the entire brain), in different ways
when male and females were considered.
Fig. 8.9. Adult brain mappings. MCC: h(d
j
) Mean mappings; FM1, FM2 and
FM3: Factor Mappings generated by Principal Components Analysis. AU: Addi-
tion; SU: Subtraction; MU: Multiplication; DI: Division. All mappings are nor-
malized into the closed interval [0,1].
In the case of addition and subtraction (Fig. 8.9), we propose a set of:
1. left frontal neurons (N
f
:): formed by cells recorded at FP1, F3 and FZ;
8.4 The Distributed Mathematical Brain 173
2. bilateral central-parietal (N
cp
) cells: formed by neurons recorded at C3,
CZ, C4, P3, PZ and P4, and
3. neurons (N
v
): distributed more laterally in both hemisphere, which are
involved in solving calculations in adults.
On the one hand, the female N
f
is more anterior than the male N
f
and the
latter is more lateral than the former. On the other hand, N
cp
may be as-
sumed to be more similar for both genders, although sharing an association
with N
f
that is strong in the case of man and almost absent in the female.
Male MF3 displays a strong correlation among areas in the right hemi-
sphere, whereas MF2 points to associations between sites in the left hemi-
sphere and the posterior cortex. Female MF3 display patterns similar to
male MF3 and includes also some of the areas appearing in man‘s MF2.
Also, the N
v
neurons are more correlated in males than in females. Fi-
nally, it may be assumed that N
f
and N
cp
are sets of neurons involved with
task solving, whereas N
v
is in charge of visual computations tasks.
Frontal and central-parietal, or temporal and parietal components, have
being described in many fMRI studies on addition and subtraction
(Cochon et al. 1999; Cowell et al. 2000; Dehaene et al. 1999; Jahanshahi et
al. 2000; Menon et al. 2000; Rickard et al. 2000; Stahian et al. 1999;
Stanescu-Cosson et al. 2000; and Zago et al. 2001) and ERPs (Iguchi and
Hashimoto, 2000; Kong et al. 1999; Skrandies et al., 1999). All these au-
thors reported widespread areas involved in arithmetic calculations, and
stressed both left frontal and parietal areas as common and important com-
ponents of the arithmetic brain. Some of these authors have suggested that
the frontal component of such circuitry (our N
f
) is much involved with the
complexity of calculation, besides other duties (Jahanshahi et al. 2000;
Kong et al. 1999; Menon et al. 2000; Stanescu-Cosson et al. 2000; Zago et
al. 2001). The N
cp
component is generally assumed to have a bilateral dis-
tribution and a specific role for arithmetic computations, and its activity is
described as mostly dependent on the type of calculation and on number
size (Cochon et al. 1999; Cowell et al. 2000; Dehaene et al. 1999; Menon
et al. 2000; Rickard et al. 2000; Stanescu-Cosson et al. 2000). Some au-
thors have also referred to other visual and verbal components associated
with arithmetic calculations that involve other neural circuits (e.g., Cowel
et al. 2000; Dehaene et al. 1999; Zago et al. 2000). Among all the papers
listed above, only Skandries et al., (1999) have reported that females con-
sistently have larger global field power in EEG than males, and they also
displayed different scalp field topography for various components. These
authors also stressed that early visual processing ERP components were
gender sensitive. The gender differences in calculation time for both addi-
tion and subtraction (Rocha et al., 2003c, d) may be explained by the use
174 8 Brain Maps of Arithmetic Processes in Children and Adults
of different strategies by men and women, reflected in a better coordina-
tion between N
f
and N
cp
(as shown by FM1) in the case of men than the as-
sociation disclosed by MF1 and MF2 in the case of women. Also, these
differences appear to be supported by a different N
v
enrollment in both
male and female task solving.
Now, let us propose the N
cp
as the set of neurons in charge of imple-
menting the set of accumulators A and the KFN and CBN circuits (see
Chap. 7), whereas N
f
is the set of neurons implementing both the ordinal
numbering control of the counting pathway (Rocha, Rocha and Massad,
2003c, d) and the gate control of the KFN and CBN circuits exercised by
the C neurons, one of the components of N
f
. If this is the case, solutions to
addition and subtraction problems may be arrived at by using the KFN and
CBN circuits, and by means of both simulation and formal calculation, as
proposed by Rocha et al., (2003c, d). In the case of formal calculation, the
visually displayed operands are recognized by visual neurons (R neurons;
see Fig. 7.6) that project directly to the corresponding I neurons. In this
way, e.g., 4 + 3 or 7 – 3 involves the decoding of the numerals 4, 7, and 3
by some specific neurons in the visual associative cortex and their seman-
tic evaluation by means of the I neurons. The type of calculation (+ or -) is
visually recognized and semantically decoded by other neurons of the N
f
set, since the C neurons are in charge of the different calculational simula-
tions proposed by different authors in the literature (Butterworth, 1999;
Dehaene, 1991, Dehaene, 1997; McCloskey et al., 1991; Siegler, 1996) . In
the case of simulated operations, numbers representing quantities may first
be decoded as set of elements that are then used to mentally simulate the
process of counting (up or down) of the sets representing the operands.
Hybrid calculations may be processed by loading one of the numbers di-
rectly into a A, using the corresponding i I and decoding the other
number as a set of elements, whose elements are sequentially accumulated
into the same a A through the adequate gate g G. If the operands are
first ordered, then this hybrid calculation may be optimized by loading di-
rectly the highest operand and decoding the other as a set of elements.
Thus, both the KFN and CBN may allow both males and females to use
distinct adding/subtracting strategies (e.g., Fayol, 1996; Gelmann and Gal-
listel, 1991; and Siegler, 1996), which in turn shapes their N
f
and N
cp
and
results in their distinct MFs. It is easily verified that simulated calculations
may render addition time dependent on the size of operands, whereas for-
mal calculation makes it constant. Since different sets of agents may enroll
to solve the same task proposed to a DIPS, then it may be proposed here
that men/women may use different strategies of recruiting the KFN and
CBN for their calculations, which could explain the gender differences ob-
8.4 The Distributed Mathematical Brain 175
served for the calculation response time (Rocha et al., 2003c,d) and for the
MFs.
Data on multiplication may be easily understood according to the pro-
posed model, too. Both male and female FM2 show that N
cp
neurons are
engaged in this task, being left (male FM1) or right (female FM1) con-
trolled by the N
f
circuit. Many fMRI studies have also disclosed both
frontal and parietal components associated with multiplication (Cochon et
al. 1999; Rickard et al. 2000; Skrandies et al., 1999; Zago et al. 2001).
FM3 (associated with N
v
) shows a strong correlation among right hemi-
sphere areas in the case of men, whereas it is almost absent in women.
Again, the neuronal enrollment is different between sexes. Counting by
block (or multiple counting) may explain product solving (Rocha et al.,
2003c,d). Either formal block counting is left-controlled in the case of
men, or block counting is visually (right) simulated in women. This could
explain males being faster than females, although the difference is the
smallest observed for all calculations. This small difference may be ac-
counted for by the heavy dependence of multiplication tasks on MF2 being
equally associated with N
cp
neurons for both genders. It must be remem-
bered that also Skrandies et al. (1999) pointed out that scalp field distribu-
tions were affected by gender, indicating the activation of different neu-
ronal assemblies during visual information processing of males and
females.
Division is the operation that most differed concerning the factor map-
pings. In the case of men, MF1 showed a strong correlation for FP2, F8,
FZ, and CZ, whereas MF2 displayed an association between F3, C3, P3,
PZ, P4, O2, and OZ. It also resulted in a bilateral pattern of MF3. To the
best of our knowledge, only Skrandies (1999) have also included division
among the calculations proposed to their volunteers, but these authors did
not describe their results as taking into consideration the distinct types of
calculation. Perhaps adults make more use (as reflected by MF3) of ver-
bally encoded rules of thumb (e.g., all even number are divided by even
numbers, etc.) to orient calculation. On the one hand, perhaps males used
this type of information to better control the N
cp
neurons distributed bilat-
erally (see male division MF2 in Fig. 8.9), or to perform other visual for-
mal operations (male division MF1 in Fig. 8.9) such as line number sec-
tioning (Zorzi et al., 2002) as reported by at least one volunteer. On the
other hand, females may have used the same type of information to orient
different counting up/down strategies to solve division, since their cerebral
patterns for this operation have some similarity to their subtraction MFs if
one considers that most of the subtraction MF2 is present in MF1 division,
and that there is some resemblance between subtraction MF3 and division
MF2. This may be an indication that females made more use of the multi-
176 8 Brain Maps of Arithmetic Processes in Children and Adults
ple counting up/down strategies discussed by Rocha et al. (2003c) and may
explain why division calculation time differences between men and wom-
en was the second most significant.
8.5 Building the Distributed Mathematical Brain
The CHI4 and ADU brain mappings for addition and subtraction were
more similar than those for multiplication and division. But children (see
Fig. 8.10) were less trained in these later calculations than on addition and
summation. Their addition and subtraction calculation times are statistical-
ly smaller than the corresponding multiplication and division times, as well
(Rocha et al., 2003c). It is also interesting to note that children used a lot
of finger pointing and marking while solving each arithmetic question,
whereas adults used mental simulations instead of these overt manipula-
tions for the same purpose. This could be the main source of the
adult/children brain mapping differences and explains the fact that children
are slower and err more than adults in doing any kind of arithmetic calcu-
lation.
CHI4 and ADU male addition mappings share more similarities than
those for females. The boys‘ MCC includes the adult cerebral areas plus a
moderate activation over P4, T5, and OZ; their MF1 disclose a correlation
pattern very similar to the adult one, and their MF2 may be assumed to be
a combination of the adult MF2 and MF3. By contrast, female mappings
differed between adults and children, if MF3 is not considered. High h(d
j
)
values were obtained for the CHI4 girls over C3, FP1, F4, FZ, C4, PZ, and
P4 and these areas seemed to be well correlated, as disclosed by MF1. Al-
so, F3, C3, and O1 are the only high correlated areas in children‘s MF2.
The similarities between children and adults are even greater in the case of
subtraction, although the gender differences appear to be greater between
boys and girls than between men and women. These differences between
boys and girls, and the similarities between children and adults, may be
understood if children are still developing N
f
and N
cp
sets, and boys are
more advanced than girls in building them. This could explain boys being
faster than girls in arithmetic calculation.
There are more similarities between girls‘ multiplication and division
mappings than between the corresponding women‘s or boys‘ mappings.
Also, these arithmetic mapping differences were clear between men and
boys. Taken together, the MFs disclose patterns of greater coordination
among enrolled areas in the case of boys as compared to girls, and in the
case of adults as compared to children. Also, the MCCs seem to provide
8.5 Building the Distributed Mathematical Brain 177
evidence for neuronal recruitment in solving multiplication and division
that is greater for girls than for boys and adults. All of these may be under-
stood as multiplication and division in children‘s N
f
and N
cp
buildup is
more primitive than their corresponding modeling of addition and multi-
plication, a fact that parallels the differences in the amount of their training
on these different arithmetic operations.
Fig. 8.10. Children brain mapping: MCC: h(d
j
) Mean mapping; FM1, FM2 and
FM3: Factor Mappings generated by Principal Components Analysis. AU: Addi-
tion; SU: Subtraction; MU: Multiplication; DI: Division. CHI4: Children enrolled
in 4
th
and 5
th
semesters of elementary school; CHI2: Children enrolled in 2
nd
and
3
rd
semesters of elementary school. All mappings are normalized into the closed
interval [0,1], and color encoded according to the rule: 1 is red and 0 is black.
178 8 Brain Maps of Arithmetic Processes in Children and Adults
Finally, although there are some similarities among the CHI2 and CHI4
MCCs, the differences between their MFs are well defined. But the CHI2
calculation times were also very different from those obtained for the
CHI4 group. Again MF patterns seem to correlate with the amount of
training experience by and the degree of proficiency attained by children.
8.6 Conclusion
As a general conclusion, it could be said that the main difference between
adult and child brain activity patterns, as disclosed by MCs and MFs, is a
larger neuronal enrollment among children in comparison to adults, due,
perhaps, to a more generalized use by children of simulated, rather than
formal counting and calculations. Another difference consists in better
adult N
f
and N
cp
circuit development. The search in the literature for pa-
pers dealing with arithmetic, learning, and brain activity selected only the
work of Menon et al. (2000), an fMRI study about optimization of arith-
metic processing in perfect performers. These authors showed that activa-
tion of the left angular gyrus was training dependent.
Also, the gender differences reported here, for both adults and children,
are very novel data. Although it is a well established fact that boys outper-
formed girls in the SAT-Math exam, and it has been proposed that test-
solving speed and strategy flexibility may be explanations for such a find-
ing (Royer and Tronsky, 1999, Gallager et all, 2000), the present work
seems to be one of the first attempts to provide an ample analysis of the
possible brain activity differences between genders concerning arithmetic
cognition. To our knowledge, only Skandries, Reik and Kunze (1999) have
reported that females consistently have larger global field power in EEG
than males, and they also displayed different scalp field topography of var-
ious components. These authors also stressed that early visual processing
ERP components were gender sensitive. The present results indicate that
marked differences exist between sexes in brain processes supporting
arithmetic calculation, and that these differences are present from the very
beginning of academic training. Also, differences were found for male and
female adults that experienced a more similar and advanced mathematical
training due to the fact that they are successful postgraduate students in the
field of technology. All these facts seem to point toward an important phy-
logenetic component of these gender differences that may in part be aug-
mented by a gender influenced culture, which also has phylogenetic roots.
8.6 Conclusion 179
180 9 Arithmetic Learning Capability in Congenitally Injured Brains
9 Arithmetic Learning Capability in Congenitally
Injured Brains
The arithmetic learning capability of brain-damaged children with low and
normal IQ is studied and discussed, taking into account the model pro-
posed. The results clearly demonstrated that:
1. widely distributed bilateral parietal lesions reduces the children arithme-
tic learning capability;
2. left frontal lesions may dissociate the capability of handling and operat-
ing quantities from that of reporting these results;
3. intermittent delta rhythmic activity is associated to dyscalculia in nor-
mal IQ children, and
4. brain plasticity allows children to overcome most of their arithmetic
learning problems.
Also, the results corroborate the conclusions about the organization of the
arithmetic neural circuits discussed in Chap. 8.
9.1 Dyscalculia
The knowledge of numbers has a phylogenetic root (Rocha and Massad,
2002; 2003; Rocha et al., 2003c; Wynn, 1988, 2000) and may be disturbed
in a variety of forms, due to different causes. Sometimes children show
particular problems with arithmetic. If their intelligence is considered to be
normal this is usually called dyscalculia, otherwise it is named numeracy
deficit (Ansari and Karmiloff-Smith, 2002). Gerstmann‘s syndrome is of-
ten described in adults and is caused by acquired lesions, usually vascular
lesions or tumors involving the angular gyrus of the dominant parietal
lobe; however, it has also been described in children with learning disabili-
ties by the name of developmental Gerstmann’s syndrome (Mayer et al.
1999; Suresh and Sebastian, 2000).
In a recent review (Suresh and Sebastian, 2000), clinical epilepsy, ab-
normal EEG findings, and nonspecific MRI changes were associated with
developmental Gerstmann‘s syndrome. Possible contributing factors to dy-
9.2 Damaged Brains 181
scalculia may be as diverse as genetic predisposition, neurological abnor-
malities, and environmental deprivation (Shalev and Gros-Tsur, 2001).
The cause of numeracy deficit is not well discussed in the literature, spe-
cially in those cases where the intelligence deficit does not have a genetic
source but is associated with gross or slight brain damage (Ansari and
Karmiloff-Smith, 2002; Rocha et al. 2003d).
Although cognitive neuroscience has greatly advanced our knowledge
of numerical cognition, it has neglected the development of strong formal
models of brain computations supporting human and non-human arithme-
tic capability, as well as the study of number in cognitively impaired chil-
dren (reported e.g. in Ansari and Karmiloff-Smith, 2002; Isaacs et al.
2001; Rocha and Massad, 2002; 2003; Rocha et al.,, 2003c; Rocha et al.
2003d).
Here, data from Rocha et al. (2003d) are summarized regarding arithme-
tic skill development in a group of impaired children. This data is then cor-
related with the children‘s possible brain damage, and further used to sup-
port the propositions introduced in Chapters 7 and 8, concerning the
composition of cerebral arithmetic circuits.
9.2 Damaged Brains
The boy WS (fig. 9.1, 9.2) was born in 1987 from a mother experiencing
high blood pressure during her pregnancy. He experienced delay in lan-
guage development, uttering his first words at the age of one year and
eight months, and beginning to produce SOV (Subject-Verb-Object) at the
age of seven. His IQ is around 60. His MRI revealed a huge lesion – leu-
komalacy – of the left parietal lobe, probably due to a pronounced increase
in his mother‘s blood pressure that resulted in a week of hospitalization.
He began elementary school at the age of ten, and exhibited slow devel-
opment in his reading, writing, and arithmetic skills.
By the year 2001 he was able to sum and subtract numbers of up to
three digits, to multiply numbers of one digit, and to solve simple prob-
lems involving summation, subtraction and multiplication, although he
used quantity manipulation in many instances. His performance in mathe-
matics was, however, better than in reading and writing. One year later, he
improved his addition and subtraction skills, but continued to use manipu-
lation to deal with one-digit multiplication. In 2003 he was attending the
last year of elementary school. His 1991 MCCs showed a reduced enroll-
ment of neurons located under FZ, CZ, C3 and C4 for both summation and
subtraction. His factor mappings (FM1 and FM1) disclosed two patterns of
182 9 Arithmetic Learning Capability in Congenitally Injured Brains
h(n
i
) covariation, one among areas in the left hemisphere, and the other in-
volving areas on the right side of brain. Despite the huge left parietal le-
sion, subject 1 seems able to enroll neurons in the left hemisphere in his
calculations.
Fig. 9.1. WS, a case of left parietal leukomalacy. MCC: h(d
j
) Mean mappings;
FM1, FM2 and FM3: Factor Mappings generated by Principal Components
Analysis; AD: Addition; SU: Subtraction. For the MRI: RH: right hemisphere;
LH: left hemisphere.
The girl KTS (Fig. 9.3, 9.4) was born in 1986, also from a mother with
high blood pressure, and also hospitalized during one month, in the last
trimester of her pregnancy. She experienced a delay in motor development,
9.2 Damaged Brains 183
standing up by the first year of age and beginning to walk by the age of
one year and nine months.
Fig. 9.2. WS development in arithmetic capability. Portuguese is the language
employed in copies of the original classroom exercises.
The KTS MRI revealed a huge lesion of the right parietal lobe, and her
present IQ is 65. She began elementary school at the age of nine; her read-
ing, writing and arithmetic were considered adequate, although her lan-
guage performance was above that for mathematics. Her arithmetic capa-
bility by the year 2001 was well advanced, and included summation and
subtraction for numbers up to four digits, multiplication of numbers up to
two digits, and division by numbers of one digit. She is now attending a
special program for young adults in high school education. Her 1991
MCCs also showed reduced enrollment of neurons located under FZ, CZ,
C3, C4, PZ and P4 for summation, subtraction and multiplication.
184 9 Arithmetic Learning Capability in Congenitally Injured Brains
Her factor mappings revealed three patterns of h(n
i
) covariation, one
among posterior areas in both hemispheres, another among areas in the left
hemisphere, and the third one involving areas in the right side of brain.
Despite the huge right parietal lesion, she seems to be able to enroll neu-
rons in both the left and right hemispheres in her calculations.
Fig. 9.3. KTS, a case of right parietal leukomalacy.
9.2 Damaged Brains 185
Fig. 9.4. KTS development in arithmetic capability.
One conclusion from the above cases is that very early (uterine) unilat-
eral lesion of the parietal lobe does not impair, but may delay, the devel-
opment of arithmetic skills. Thus, it seems that the N
cp
component of the
arithmetic circuits modeled in chapter 7, and revealed by the brain map-
ping in chapter 8, is fault tolerant, perhaps because of a redundancy of
number representation in both hemispheres. The N
cp
is hypothesized to
correspond to the set of accumulators and quantifiers located bilaterally at
the parietal lobe, whose activity could be monitored by means of the parie-
tal (P3, PZ, and P4) and central (C3, Cz, and C4) electrodes. The lesion of
such a component in one hemisphere may be compensated by increasing
186 9 Arithmetic Learning Capability in Congenitally Injured Brains
the number of neurons with the same specialization in the other hemi-
sphere. Cases 3 and 4 seem to confirm this hypothesis.
Subject 3 (Fig. 9.5) is a boy, with an IQ of 54, born in 1990, premature
from a mother with high blood pressure. He experienced motor develop-
ment delay, standing up by 1 yr 2 mo, and walking by 2 yr, 3 mo. His MRI
revealed a huge bilateral lesion of the parietal occipital lobes. He is now
attending kindergarten, and is beginning to recognize some letters; he con-
tinues to experience difficulties in recognizing quantities above five.
Fig. 9.5. Two cases of bilateral leukomalacy with different arithmetic capabil-
ity development
On the other hand, subject 4 (Fig. 9.5) is a girl, IQ 61, born ‘89, with no
history of disturbances during the pregnancy. She experienced language
development delay; her first phrases were spoken by the age of 3 yrs. Her
MRI revealed a small bilateral lesion of the parietal-occipital lobes. She
began elementary school in 1999, and is considered to have a regular de-
velopment in her reading and writing capabilities, but shows a better ac-
quisition of arithmetic skills. She is now able to perform additions and sub-
tractions of numbers of up three digits, but is experiencing important
9.2 Damaged Brains 187
difficulties in learning multiplication. Her brain mappings indicated the en-
rollment of neurons at the parietal electrodes in both hemisphere in the
course of solving addition and subtraction problems. Thus, a huge bilateral
posterior lesion resulted in a greater arithmetic capability than unilateral or
small bilateral damage.
Fig. 9.6. A case of left parietal-frontal leukomalacy and arithmetic capability
development
The histories of frontal damaged children are different. The boy RC
(Fig. 9.6) was born in ‗95 and his present IQ is 57. There is a history of at-
tempted abortion here, resulting in a huge frontal-parietal lesion. Because
of this, he experienced a motor development delay, standing up the age of
188 9 Arithmetic Learning Capability in Congenitally Injured Brains
1 yr, and taking his first steps at 1 yr, 6 mo; he is now attending the second
year of elementary school. He began to cope with quantities above five in
2001; at the same time, he started logographic reading of a small set of
words referring to animals. He continued to improve his performance dur-
ing 2002, and began to add and subtract small numbers. His 91 MCC, as-
sociated with quantity and number recognition, showed low enrollment of
neurons in F7, T3, C3, CZ, which could be explained by the location of his
lesion. Despite this, he was able to recruit both left frontal (see FM1) and
parietal (see FM2) neurons to solve quantification tests.
Fig. 9.7. A case of left frontal schizencephaly. MCC: h(d
j
) Mean map-
pings; FM1, FM2 and FM3: Factor Mappings generated by Principal
Components Analysis; AD: Addition; SU: Subtraction; for the MRI: RH:
right hemisphere; LH: left hemisphere.
9.2 Damaged Brains 189
The girl of Fig. 9.7 (Subject 6) has an IQ of 58; she was born in ‗88 from
a hypertense mother who experienced one week of hospitalization during
the first trimester of pregnancy. This subject experienced motor develop-
ment delay, standing up by the age of 1 yr 3 mo, and taking her first steps
by the age of 1 yr 7 mo. She also experienced an important language de-
velopment delay. Her first words were spoken at the age of five. Her MRI
revealed a huge frontal lesion named schyzencephaly, a consequence of a
fetal isquemia during her mother‘s one week of hospitalization. She began
to attend elementary school in 2000, and exhibited good learning in read-
ing and writing since then. She is now able to read small texts and write
small notes. Her development of arithmetic skills was almost nill until
2002, when she invented a strategy of her own to cope with her difficulty
in selecting the correct answer for subtraction and addition problems,
which results were above seven.
Fig. 9.8. Arithmetic capability development of Subject 6
190 9 Arithmetic Learning Capability in Congenitally Injured Brains
Whenever she has to report a result of any arithmetic problem that is
bigger than seven, she represents the quantities and writes the correspond-
ing series of numbers in order to find the answer. It seems that her problem
lies in encoding the results of the quantification, and not in operating on
the quantities. Her 2000 MCC seems to point to a difficulty in enrolling
frontal neurons for both addition and subtraction, although her FM3s ex-
hibit patterns of some association of these frontal neurons with posterior
cells. However, FM1 and FM2 showed an important involvement of right
frontal neurons in both addition and subtraction.
Damage to the frontal component N
f
seems to delay, but not to seriously
compromise, the capability of the brain to quantify the cardinality of sets
by using the N
cp
circuits. It seems that frontal lesions result in a complex
deficit of handling ordinal numbers and encoding quantities into numerals.
These two frontal-damage children use a lot of simulation and manipula-
tion to solve their mathematical problems, what may be an indication that
they have to turn to frequent visual simulations controlled by the intact
right brain, instead of using a better left-frontal control of the N
cp
circuits.
This reduced left-frontal counting control may be the reason for using an
explicit process of numbering the elements of the sets added or subtracted,
in order to find the numerals representing the result of the calculation per-
formed by using N
cp
simulation.
This hypothesis seems to be confirmed by data from subjects with small
frontal lesions (as in Fig. 9.9). This girl was born in ‗87, with no history of
problems during the pregnancy, and no important neuropsicomotor delay.
Her MRI revealed a frontal-parietal gliosis and thickening of the posterior
corpus callosum. She began elementary school in 2000, and experienced
good learning in reading and writing. Her arithmetic learning is progress-
ing, and at the moment she has mastered addition and subtraction of two-
digit numbers and multiplication of one-digit numbers. However, she ex-
periences difficulties with the ordinal numbers above twenty. She has a
clear dissociation between quantification evaluation/manipulation and
number sequence ordering. Contrary to subject 6, she does not have any
problem in labeling the results of addition and subtraction. Her 2000 MCC
displayed clear patterns in use of frontal and parietal neurons, and of a cor-
relation of activity between them.
It is proposed in the literature that parietal number representation in-
volves both hemispheres (Cochon et al. 1999; Cowell et al. 2000; Dehaene
et al. 1999; Fink et al. 2001; Gobel et al.; 2001; Menon et al. 2000; Rick-
ard et al. 2000; Stanescu-Cosson et al. 2000; Zorzi et al., 2002). This could
explain why subject 3 is still very compromised in his arithmetic capabili-
ties, whereas subjects 1, 2 and 4 have achieved advanced calculation skills.
Both frontal and parietal lesions delayed the learning of arithmetic opera-
9.2 Damaged Brains 191
tions, but frontal damage continues to compromise the handling of ordinal
numbers in subjects 6 and 7.
Fig. 9.9. A case of frontal-parietal Gliosis and arithmetic capability develop-
men
192 9 Arithmetic Learning Capability in Congenitally Injured Brains
This is consistent with the description of ordering-sensitive neurons in
the monkey motor cortex (Carpenter et al. 1999). Finally, it has been sug-
gested that the frontal component of such circuitry is much involved with
the complexity of calculation, in addition to other duties (Jahanshahi et al.
2000; Kong et al. 1999; Menon et al. 2000; Stanescu-Cosson et al. 2000;
Zago et al. 2001).
9.3 Neural Plasticity
It is clear from the above data that arithmetic capability is preserved in
those children suffering from huge cerebral lesions acquired during their
fetal life. The unilateral destruction of either the frontal or parietal lobes
(subjects 1, 2, 4, 5 and 6) resulted in an important development delay in
both language and arithmetic skills, but did not preclude these individuals
from attaining sophisticated capabilities in reading, writing and calculat-
ing. The language development of these children (subjects 1, 2, and 6)
were described and discussed in a previous paper (Foz et al. 2001), and
neural plasticity was assumed in the explanation of the reorganization of
the language neural circuits, allowing these children to achieve high levels
of performance in speaking, reading and writing. Here, the same hypothe-
sis may be used to explain the reorganization of the arithmetic brain in the
fetal damaged brains. Suresh and Sebastian (2000) also reported that inten-
sive training improved the arithmetic capability of 60% of their subjects
exhibiting developmental Gertzmann Syndrome. It is interesting to consid-
er that the bilateral parietal lesion of subject 3 severely impaired his learn-
ing with regard to both numbers and letters. However, the bilateral parietal
lesion necessarily severely reduced his neural plasticity, too.
Neural plasticity requires both function reassignment of the neuron and
establishment of adequate neuronal connections (Chugani, 1999; Thomp-
son et al. 2000; Villablanca and Hovda, 2000). The latter may be achieved
in a densely connected, distributed processing system like the brain, and
the former may be understood if neuronal function is dependent on genetic
instructions shared by all cells. Rocha et al. (2003) proposed that the spe-
cialization of the different types of neurons (accumulators, classifiers, con-
trollers, etc.) involved in the arithmetic circuits may be explained by the
control over genes defining the types and quantity of ionic gates, as dis-
cussed in Chap. 7. For example, continuous or periodic accumulating func-
tions may be specified by favoring tonic or phasic spike encoding by the
piramidal neurons (Rocha, 1997). The expression of the genes required to
specialize the arithmetic neurons may be assumed to be phylogentically
9.4 Developmental Dyscalculia 193
specified in the case of fuzzy numbers (KFN circuit in Rocha and Massad,
2003a) and culturally guided in the case of crisp numbers (CBN circuit in
Rocha et al., 2003a). This assumption helps in understanding both brain
plasticity and the fact that man has independently created (or discovered)
the crisp numbers at least four times in his evolution — by the Summeri-
ans, Epyptians, Greeks, and Mayans (e.g., Ifrah, 1985). Whenever the en-
vironment demanded it, genes responded, creating new types of arithmetic
neurons. In a similar way, the complexity of commercial or scientific (e.g.,
astronomical) transactions pressed for the invention of crisp numbers, alt-
hough adequate teaching and challenging may guide brain plasticity. All
subjects 1-6 attend a special education school where teaching is oriented
by the recent knowledge provided by neuroscience about language and
arithmetic cerebral circuits (www.enscer.com.br). As a matter of fact, their
arithmetic teaching was both guided and inspired by most of the research
described here and in Rocha et al. (2003c,d).
9.4 Developmental Dyscalculia
Dyscalculia in school-aged children has received less attention in the liter-
ature than language and literacy deficits (Ansari and Karmilof-Smith,
2002; Suresh and Sebastian, 2000). Also, number knowledge disruption is
much more frequently studied in adults than in children (e.g., Butterworth,
1999; Dehaene, 1997; Fink et al. 2001). Subjects 8 (Fig. 9.10a,b) and 9
(Fig. 9.11) are examples of children having difficulties in learning arithme-
tic, reading, and writing.
On the one hand, we have subject 8, a boy with a normal IQ, born in
‗97, from a normal pregnancy, and with a normal childhood history. He
began elementary school in 2002, when he began to exhibit important dif-
ficulties in reading words, associating numbers with quantities, and per-
forming addition, although he was clearly capable of recognizing quanti-
ties when asked to compare sets with equal or different cardinalities. He
ended the school year being unable to show any real progress in reading or
arithmetic. His 2003 MRI disclosed a pattern of delayed myelinization. His
EEG was recorded twice in 2000 and revealed in both instances the pres-
ence of Intermitent Rhythmic Delta Activity. His FMs disclosed only two
patterns of correlation of cerebral activity for quantity and number associa-
tion (NU in Fig. 9.10a), and for addition of small numbers (AD in Fig.
9.10a), instead of the usual three Factor Mappings associated with our
tests, and also exhibited by in him in the case of set cardinality comparison
(EQ in fig. 9.10a).
194 9 Arithmetic Learning Capability in Congenitally Injured Brains
On the other hand, there is subject 9, also of normal IQ, a boy born in
1994 and adopted by the age of four. He had a history of child abuse prior
to this date. He began (and continues) to have psychological aid just after
his adoption to cope with some emotional problems. His 1999 MRI did not
showed any sign of brain damage (Fig. 10.11).
Fig. 9.10a. An abnormal MRI …
9.4 Developmental Dyscalculia 195
Fig. 9.10b. …associated with dyslexia and developmental dyscalculia. EQ:
Quantity comparison; NU: Quantity evaluation; AD: Addition. For the MRI and
classical EEG: RH: right hemisphere; LH: left hemisphere; A: anterior, C: cen-
tral; and P: Posterior. Plot of the percentage of classic frequencies (DELTA,
THETA, ALFA and BETA) obtained during the selected epoch in the EEG.
Subject 9 started elementary school in 2001 and clearly showed signs of
dyslexia and developmental dyscalculia. His main difficulty in arithmetic
lay in his incapacity to deal with ordinals. He has to use a rule to sequen-
tially order the numbers up to the desired figure. Given an arithmetic oper-
ation (e.g., addition of one digit numbers), he is able to obtain the solution,
196 9 Arithmetic Learning Capability in Congenitally Injured Brains
but he has to use the rule to identify the numeral associated with the quan-
tity in the solution. His EEG was also recorded twice in 2000 and revealed
in both instances the presence of Intermitent Rhythmic Delta Activity
(IRDA). During one of these recording session, when solving one addition
problem (9 + 4) he explicitly made a verbal request for help in visually
identifying the number 13, saying: Where is the number thirteen? His FMs
showed two patterns of cerebral activity association in the task of quanti-
ty/number association (NU in Fig. 9.11) and the usual three FMs in the
case of addition (AD in Fig. 9.11).
Fig. 9.11. A case of dyslexia and developmental dyscalculia with a normal
MRI
9.4 Developmental Dyscalculia 197
Developmental dyscalculia has been defined as a pathology of otherwise
normal-IQ children that is most often diagnosed when the children begin
elementary school (Ansari and Karmilof-Smith, 2002; Suresh and Sebas-
tian, 2000). Dyscalculia in low-IQ children is called numeracy deficit and
generally believed to be genetic in origin (e.g., Ansari and Karmilof-
Smith, 2002). Developmental dyscalculia is poorly understood, albeit an
important issue in teaching. Only recently, Isaacs et al. (2001) have stated
that, using voxel-based morphometry, they
have been able to demonstrate
that there is an area in the left
parietal lobe where children without a deficit
in calculation
ability have more grey matter than those who do have this
deficit.
To our knowledge, this is the first report establishing a structural
neural correlate of calculation ability in a group of neurologically
normal
individuals.
Dyslexia and developmental dyscalculia are the main deficits of subjects
8 and 9. The analysis of their EEG revealed the presence of Intermittent
Rhythmic Delta Activity (IRDA). In the case of subject 8, MRI revealed a
delayed myelinization, whereas in the case of subject 9, it was classified as
normal. Abnormal theta activity and IRDA have been associated with dys-
lexia (Harmony et al. 1996; Klimesch, 1999; Klimesch et al. 2001; Rippon
and Brusnwick, 2000). IRDA can be seen in association with a wide varie-
ty of pathological processes, varying from systemic toxic to metabolic dis-
turbances to focal intracranial lesions (Cornelis and Pritchard, 1999;
Neufeld et al. 1999; Sharbrough, 1999). IRDA is likely to be associated
with the development of widespread brain dysfunction (Sharbrough,
1999). IRDA in subject 8 is predominantly frontal and associated with
signs of delayed neural maturation. In the case of subject 9, IRDA is rec-
orded over all electrodes, but associated with a normal MRI. Also, subject
9 exhibits dissociation between ordinal and cardinal numbers, whereas def-
icits of subject 8 involve both types of numbers. This seems to confirm
once more the existence of the proposed components N
f
and N
cp
that are in
charge of controlling and performing counting, respectively.
Harmony et al. (1996) have proposed that delta power increases in tasks
that require attention to internal processing, or what they have called ―in-
ternal concentration‖. They also published results showing that the in-
crease in delta power was enhanced by task difficulty. Klimesch (1999)
proposed that EEG oscillations in the alpha and theta band reflect cogni-
tive and memory performance in particular if a double dissociation be-
tween the types of EEG response (tonic versus phasic) in two different
EEG frequency ranges (in the alpha and theta range) is taken into account.
He assumed that this dissociation is due to the fact that phasic band power
increases in the theta, but decreases in the alpha frequency range, and that
the extent of a phasic EEG response depends at least in part on the extent
198 9 Arithmetic Learning Capability in Congenitally Injured Brains
of tonic power, but in opposite ways for the theta and alpha frequency
range. The delta tonic power increases under conditions that are associated
with reduced cognitive processing capacity, and this augmentation of the
tonic power decreases the phasic response. He also proposed that a phasic
increase in a narrow frequency band of 2 Hz widths, lying below the indi-
vidually defined alpha frequency, reflects encoding processes of a working
memory system (Klimesch et al. 2001). Recently, Klimesch et al. (2001)
showed that in the low theta band, the phasic power recorded at frontal
sites did not differ when dyslexic and normal children were compared; but
that registered at occipital electrodes in the dyslexic was smaller than that
recorded in the normal children. These types of results clearly show that
the EEG can display signals associated with abnormal cognitive processes
in children, like dyslexia and dyscalculia, as described here for subjects 8
and 9. Also, dyslexia and dyscalculia may have an important genetic com-
ponent (Ansari and Karmiloff-Smith, 2002; Hynd et al., 1998; Lindsay,
1996). Perhaps in some types of developmental dyscalulia a disregulation
of gene reading may impair the construction of CBN circuits even while
preserving much of the KFN circuits.
9.5 Conclusion
A final point to make here concerns the widespread belief in our culture in
the dependence of arithmetic capability on language skills. For instance,
the triple-code model introduced by Dehaene and colleagues (Cochon et al.
1999; Dehane and Cohen, 1995; Dehaene et al. 1999; Spelke and Tsvikin,
2001), proposes that crisp arithmetic knowledge is verbally formatted,
whereas approximate calculation is quantity dependent. Although most of
the children studied here experienced important language acquisition de-
lays, no clear correlation is observed between their actual arithmetic skills
and language achievements. For instance, subject 2 never exhibited any
important language impairment, but her acquisition of arithmetic skills was
delayed, although at the moment they have attained a high level. Subject 1
began his first phrases at age seven, but he started to master addition be-
fore he acquired a minimal writing and reading capability. Currently, his
arithmetic capability is much more advanced than his oral, visual, or motor
language skills. Subject 6 uttered her first worlds by the age of five, but
developed very good oral language before she started to master arithmetic.
Even now, her reading and writing skills are much more developed than
those for calculations. Subject 8 and 9 did not experience any important
9.5 Conclusion 199
impairment of their oral language development, but they are dyslexic and
exhibit very defined arithmetic deficits.
The differences between our modeling of the arithmetic brain and that
proposed by Dehaene et al. (Cochon et al. 1999; Dehane and Cohen, 1995;
Dehaene et al. 1999; Spelke and Tsvikin, 2001) may be understood be-
cause we are currently studying the process of arithmetic knowledge ac-
quisition by the school-aged children, whereas they based their conclusion
on studies of adults, who have lost their arithmetic capabilities due to brain
lesions, or on groups of bilingual adults. We believe that the innate charac-
ter of the arithmetic brain strongly supports our view that arithmetic and
verbal capability are founded on independent neural circuits, although lan-
guage is the most important interface for humans to share information
about quantities and the manipulations they perform on these quantities.
Even Chomsky (Hauser et al., 2002) has recently assumed that human lan-
guage capability must be a consequence of a more general brain evolution.
200 10 Learning Arithmetic: Why So Difficult?
10 Learning Arithmetic: Why So Difficult?
The journey taken in the previous chapters on brain function let us begin to
understand how the brain is genetically organized to do both fuzzy (KFN)
and crisp (CBN) arithmetic, and how culture influenced humankind in cre-
ating a sophisticated theory of numbers.
10.1 The Nature of Arithmetic Knowledge
Nature selected animals with an increasing capacity for quantifying their
environment in order to better survive in a hostile world, where they have
to get food and avoid predators in order to maintain their bodies and to
find sexual partners for the procreation of new offspring. Man inherited
this knowledge (see Chap. 1) by inheriting the genetic instructions required
to build up specific (CBN and KFN) neural circuits composed of special-
ized neurons (see Chap. 7). These mechanisms of genetic inheritance were
understood and formalized in Chaps 2 and 3, where neurons were de-
scribed as formal entities specialized in processing subsets of a class of
fuzzy formal languages, supported by a Self-Controlled Grammar, which
in turn is a class of the R Grammars (Chap. 2). In this context, the brain is
proposed to be a Distributed Intelligent Processing System for such types
of grammars, and to take profit from evolutionary strategies in learning
how to use knowledge encoded by these languages in order to increase the
odds of surviving in a continuously changing world (Chap. 4). The compu-
tational capacity of the brain greatly increased when nature discovered
quantum computing and implemented it in some special cell components
(Chaps. 5 and 6). In the same way that genes allow information to flow
from individuals to individuals to continually reproduce and renew genetic
models defining the different species, memes are proposed that allow in-
formation to flow from individual to individual(s) of a community so as to
continually reproduce and renew culturally encoded models (Chap. 6).
Memes were demonstrated to be supported by specific neural circuits pro-
cessing a defined set of sentences from a fuzzy grammar (Chaps. 3 and 6).
In this way, in the many instances in which mankind discovered how to in-
10.1 The Nature of Arithmetic Knowledge 201
crease the capacity of genetically encoded neural quantification circuits,
memes were used to spread this knowledge among their fellows, who
shared the same social interests (see Chaps. 6 and 7). The process of pro-
moting such meme diffusion evolved from simple imitation to instruction
(Fig.10.1), when brain processing achieved a degree of complexity allow-
ing animals living together in a community, and to guide their fellows in
repeating the same actions they have discovered by themselves, to better
solve their needs (Chap. 7). The invention of language by mankind al-
lowed instruction to evolve into teaching (Fig 10.1).
Imitation
Instruction
Teaching
Fig. 10.1. The evolution of meme transmission: the same meme...different ways
of transmission
Learning arithmetic is nowadays a process of developing inherited CBN
circuits under the guidance of teaching. It develops as an ordered process
that must begin with the construction of many different circuits in distinct
cerebral areas (Chaps. 7 and 8), triggered by the questions posed in a set of
problems of increasing complexity. First, it is necessary to recognize, to
compare, and to order quantities and dimensions, and to identify order in
202 10 Learning Arithmetic: Why So Difficult?
sequences of events. Also, it is necessary to visually recognize the numer-
als. Whenever these capabilities are attained, then it is time to learn to as-
sociate numerals with cardinalities, orderings, and, finally, dimensions. To
write and/or to name (but not necessarily both) the numerals is a necessary
condition in communicating the results of any quantification or serial iden-
tification. The basics of addition and subtraction may be learned while
learning to recognize cardinalities, if the results of such operations may be
signalized with sets of corresponding cardinalities. Later, after the associa-
tion of numerals with cardinalities is learned, the capability to add and to
subtract can be enhanced by taking profit from formal algorithms support-
ed by the knowledge of number as a formal structured system, and by the
understanding of the formal notation used to implement such algorithms.
After all these steps, the understanding of multiplication and division as
special cases of counting by multiples, or as special cases of repeated addi-
tions and subtractions, can be easily attained. Such complex, ordered learn-
ing can be achieved at early ages, and as soon as it begins, gender differ-
ences are clearly established (Chap. 8) confirming the hypothesis of its
innate foundations and the distributed character of numerical processing in
the brain (Chaps. 8 and 9). This kind of learning may be delayed but not
impeded by cerebral lesions experienced early in the life (Chap. 9).
However, if the basic neural circuits for both fuzzy and crisp numbers
are genetically inherited by man, allowing us to invent many number codes
on different occasions and in spatially and temporally different cultures…
why is learning arithmetic considered a hard task in all cultures?
Several variables have been blamed as sources of stress in learning
arithmetic (e.g. Dal Vesco, 2002); among them are:
1. the process of evaluating children‘s performance and its relative value in
different cultures;
2. teaching methodologies;
3. teacher-pupil relationships;
4. peer relationships; and
5. difficulties in grasping mathematical concepts as presented by the
teacher.
In the present chapter, we discuss the possible discrepancies between
such factors and the brain physiology discussed in previous chapters as a
source of disturbance and stress in the processes of formal arithmetical
knowledge acquisition.
10.2 The Invention of Crisp Numbers 203
10.2 The Invention of Crisp Numbers
The invention of numbers by man was achieved whenever the complexity
of human relations increased and pushed the development of CBN circuits
to support the required arithmetic transactions.
It may be assumed that the human newborn, like any other animal, is in-
itially equipped with KFN circuits because natural selection made the ex-
pression of their genes predominate over those required to define the CBN
circuits. Perhaps this is because the complexity of the arithmetic transac-
tions for survival is easily (and maybe better) solved by fuzzy arithmetic,
supported by KFN circuits. Therefore, the human invention of numbers
will demand, first of all, the enhancement of the expression of the genes
encoding CBN agents, and then the development of adequate connections
among them. DIPS learning of a new task may require both the creation of
new agents and the establishment of defined commitments among them.
Let k
max
and c
max
be, respectively, the maximum number of KFN and
CBN circuits that a given brain may build. Let also k
a
and c
a
be, respec-
tively, the number of KFN and CBN circuits that a given brain has actually
built. Finally, let
k = k
a
/k
max
and c = c
a
/c
max
(10.1)
Now, let be defined in the closed interval [0,1] to measure the relative
gene expression for building KFN and CBN agents, such that
if 1 then the KFN gene expression prevails
(10.2)
otherwise
if 0 then the CBN gene expression prevails
(10.3)
In such conditions:
Conjecture 10.1. The newborn capability (CBN | G,H ) for inventing
crisp arithmetic may be calculated as:
(CBN | G,H) = 1 – ( *k ) / c , 0 < < 1
(10.4)
Conjecture 10.2. It is assumed here that in the case of animals other than
man:
k >> c and 1
(10.5)
whereas in the case of the human newborn
204 10 Learning Arithmetic: Why So Difficult?
k > c and 0
(10.6)
In this condition, in the case of animals
(CBN | G,H ) < 0
(10.7)
whereas in the case of humans
( CBN | G,H ) > 0
(10.8)
Remark 10.2. Conjectures 10.1 and 10.2 are proposed to explain the
large difference in arithmetic capability between human and animals, de-
spite the fact that the grammar G defining their brains shares the genes re-
quired by both KFN and CBN circuits. On the one hand, as and k/c in-
creases, the chances an animal creates a CBN circuit decreases and
ultimately becomes impossible. On the other hand, the difficulty a human
may have inventing (or learning) crisp arithmetic is mainly determined by
the actual value of the relation k/c. People suffering specific brain lesions
may have this value increase, and thus having (CBN | G,H) decrease, be-
cause of the loss of CBN agents. But since they did not lose the genes de-
fining these agents, they retain the capability for recreating such circuits,
as has been shown in Chap. 9. Genetic disturbances may either reduce or
increase k/c. But even in these conditions, learning will be possible if CBN
gene expression is not totally abolished.
Conjectures 10.1 and 10.2 also stress the influence of the environment H
upon the human capability for creating different crisp number systems and
for doing so on different occasions, as well as in cultures that are spatially
and temporally isolated.
10.3 Learning by Observing
A series of very interesting experiments was performed by Petrosini et al.
(2003) to study the properties of rats learning by observing other rats in
solving a given problem. They investigated how rats may learn to find a
submerged platform in a swimming pool (Fig. 10.2).
When a rat is put in a swimming pool (Fig. 10.2 left), it tries to escape:
1. swimming around the borders to jump out of the pool. After a few at-
tempts, it discovers that this strategy is not a solution for its problem,
and then
10.3 Learning by Observing 205
2. randomly swims across the pool and incidentally discovers the platform,
where it may find a safe place. After that, it improves its performance
by
3. quickly learning how to find the platform.
Now, if another rat is first allowed to observe the first rat discovering
how to find the safe place in the swimming pool (Fig. 10.2 right), and then
put in the pool, it quickly finds the platform, demonstrating that it has
learned to solve the problem by observing the behavior of the first rat. This
clearly shows the meme diffusion in rodents, promoted by imitation.
Learning to find a safe place.
I know what you are doing.
Fig. 10.2. Learning by observation: I know what you are doing!
The interesting results from these experiments arose when the second rat
was allowed to observe only one of the phases of the learning being expe-
rienced by the first rat (Fig. 10.3). In all cases, the partial observation re-
206 10 Learning Arithmetic: Why So Difficult?
sulted in blocking the normal learning of the second rat. It not only failed
to profit from observing the first rat, as in the normal condition, (Fig. 10.3
upper row), but the incomplete observation of the task solution resulted
(Fig 10.3 middle and low rows) in very poor learning!
Fig. 10.3. If you do not correctly show me how…I can’t learn! Poor teaching...
10.4 Arithmetic Meme Diffusion in School 207
10.4 Arithmetic Meme Diffusion in School
As discussed and modeled in Chap. 6, meme diffusion is dependent on
two main communication modalities.
In the first, mailing system, a meme active in one brain is replicated in
another brain, by means of a direct interaction between them. Imitation and
instruction are the two processes by which meme replication occurs. Here,
imitation implies intentional action of the receptor, but not necessarily of
the transmitter, whilst instruction implies intentional action of both actors.
A good example of instruction is the tutorial teaching system, the basis of
guild training in the Middle Ages that is still in use in some academic areas
such as medicine and graduate courses.
In the second, broadcasting system, memes are hardware-stored in the
culture shared by a group of actors, and are intentionally searched for by
these actors. Probably the first broadcasting hardware is the painted cave
wall, like Pedra Furada in Brazil (Chaps. 1 and 2), and the latest being the
Internet.
Teaching relies on both mechanisms, because it has the purpose of rep-
licating knowledge stored in the culture using the teacher as a mediator of
such replication. The meme in the culture is first revived in the teacher‘s
brain in order to begin replication in the pupil‘s brain. The teacher is re-
sponsible for selecting the memes to be replicated, and then helping the
student to handle them.
As discussed in Chaps. 7 through 9, there exist in the brain at least three
systems of number, supported by different neural circuits:
1. the ordinals: supported by ranking operations over serial events, per-
formed by some specific frontal circuits;
2. the integers: supported by quantification of set cardinality, performed by
defined frontal and parietal circuits, and
3. the reals (R): supported by quantification operations over the three con-
tinuous space dimensions: length, width and high.
Here must be added a fourth system:
4. currency numbers: supported by adding emotional valuation to integer
and/or real quantities of trading interest.
As discussed in Chap. 7, abstract concepts may be constructed by dis-
covering common properties shared by similar well-developed neural cir-
cuits. In this line of reasoning, higher numerical concepts are constructed
from the learning of common properties of the number systems listed
above. In this way, man created/discovered the notion of an abstract num-
208 10 Learning Arithmetic: Why So Difficult?
ber system and number theory by learning about the common properties of
at least four (a through d, above) different number neural circuits. Such a
theory is comprised of a collection of related memes and it is this theory
that is available in present human culture. In this context, crisp number
representations and properties (number theory) and their operations (group
theories), evolved from the more primitive knowledge represented in the
neural circuits depicted above. Note that the construction of the numerical
knowledge base in human culture occurred via an evolutionary process
similar to the evolutionary knowledge construction where the rat learns to
find a safe place in the water maze. The difference between the two pro-
cesses is in the time scale. Whereas culture involves knowledge evolution
from generation to generation, individual knowledge evolution occurs dur-
ing the individual life span.
In this line of reasoning, two learning strategies may be considered in
the acquisition of crisp number representation and their operations by
school-age children. On one hand, the evolutionary process of arithmetic
knowledge acquisition by children must recapitulate the initial steps of the
evolutionary pathway of cultural development of number and group theo-
ries under the teacher‘s guidance. This is a process similar to that of the rat
learning by observing how to do it. On the other hand, children must recre-
ate the main steps of universal arithmetic evolution under the influence of
an adequate environment. This is a process similar to that of the rat learn-
ing by doing it.
It is proposed here that learning arithmetic is so difficult because chil-
dren are not shown how to develop their innate circuits and how to evolve
the abstract arithmetic embodied in our culture from this innate
knowledge. A clear example (Bransford et al., 2003, pp. 92) of how teach-
ers make arithmetic learning difficult is illustrated in Fig.10.4, involving
the confusion they create between ordinal and cardinal numbers. One of
the most popular number exercises is to copy, write, or name the numbers
in sequences from 1 to ..., as a good exercise in learning the quantities rep-
resented by them. The teacher‘s rationale is that the children will be asso-
ciating the numerals (1, 2, 3, ... ) to the quantities to be represented by
them, because each time the child copies, writes, or names the numeral,
he/she will be adding one to the initial quantity in the series. However, this
type of exercise is interpreted by the child as one of identifying the ele-
ments of a series, which is to say, learning the ordinal numbers. In this
kind of exercise, the child uses each element numeral as a label of one of
the elements of the series, as for example, the first, the second ..., etc. To
mistake two different innate neural circuits not only imposes a burden on
the children in the initial learning of arithmetic, but it will also contribute
10.4 Arithmetic Meme Diffusion in School 209
to making any future learning more difficult, as in the case of not correctly
showing the rat how to solve the water-maze.
Fig. 10.4. If you do not correctly show me how to do it … I can’t learn! Mak-
ing wrong assumptions about the child’s reasoning.
Subjects 6, 7 and 9 described in Chap. 9 are examples of initial poor
teaching. Despite their cerebral lesions the children were able to develop
their ordinal and cardinal numbering circuits. But they were over-trained to
use the ordinals to communicate arithmetical operations with the cardinals.
Since their lesions rendered the ordinal circuit less efficient than the cardi-
nal circuit, they had great difficulty telling the teacher about the results of
their calculations. One of them (subject 6) invented a process of her own to
solve the problem (see Fig. 9.5). She wrote down the numerals serially un-
der each element of the sets involved in the calculation in order to discover
how to tell others the result computed by her cardinal neural circuit. Sub-
ject 9 asked many times for some help in finding the numerals in a series
in order to relate his calculations. Once, he asked one of the authors how to
210 10 Learning Arithmetic: Why So Difficult?
write the number thirteen, which was the correct result of the addition
problem (9 + 4) he had solved.
10.5 Evolving Arithmetic Knowledge in the School
It may be assumed that man started to develop our actual crisp numbers
and arithmetic gradually, as the complexity of human society began to in-
crease.
The first attempts, on the cave walls (see Fig. 6.2 and Fig. 10.5), and
with bones, were one-to-one mapping between the objects to be counted
and the elements of representation. The next steps involved shortening
such notations, for instance in the case of the Sumerian clay jar, recording
commercial transactions, which quantities were encoded by one-to-one
mapping between a pebble inside the jar and a corresponding symbol writ-
ten outside. This was the beginning of an abstract number notation. Arith-
metic calculation began to be computed by handling pebbles in a primitive
abacus (e.g., by the Greeks and Romans) or rope knots (e.g., the quipua of
the Incas), etc. Only very late in the history of mathematics, with the intro-
duction of Arabic numerals and the Indian zero, did modern algorithms
begin to be used to solve arithmetic calculations (Ifrah, 1985, Joseph,
1990).
Pre-school children have a good knowledge of number and calculation,
as shown in their mimicking the first one-to-one mapping representations
and calculations (Fig. 10.5). By the time children begin school, most have
built a considerable knowledge store relevant to arithmetic. They have ex-
periences of adding and subtracting numbers of items in their everyday
play, although they lack the symbolic representations for addition and sub-
traction that are taught in school (Fig. 10.5a) (Bransford et al., 2003).
Teaching our formal number notations and arithmetic calculations with
small numbers may be facilitated by mimicking the Sumerian double pro-
cedure, that is, pairing one-to-one mapping procedures with formal number
notation (Fig. 10.5b). Also, calculations may be easily learned if primitive
(Greek or Roman) abaci are allowed (Fig. 10.5c). By following this path,
children start to understand symbolic notation and become ready to learn
the algorithmic approach to arithmetic calculation (Fig. 10.5d). This evolu-
tionary pedagogy simulates the history of number knowledge acquisition,
such that neural circuits for abstract numbers begin to be naturally created
in the brain as the common properties of the different initial CBN circuits,
which are then developed and correlated with the symbolic notation.
10.5 Evolving Arithmetic Knowledge in the School 211
Under the current pedagogy, instead of taking this evolutionary ap-
proach to arithmetic learning, the young child is frequently asked to com-
pare set cardinalities or evaluate dimensions using linguistic variables be-
fore he/she is allowed to work with crisp quantification. For instance, the
child has to select the set having few, many ... elements, or he/she has to
say who is tall, short, etc. Children are asked to strengthen their KFNs in
order to be able to create their CBNs. But from conjectures 10.1 and 10.2,
this reduces (CBN | G, H), and makes arithmetic learning much more
difficult.
Fig. 10.5 a,b,c,d: Evolution of arithmetic knowledge: Ontogeny repeats
phylogeny?
212 10 Learning Arithmetic: Why So Difficult?
The contemporary view of learning is that people construct new
knowledge and understanding based on what they already know and be-
lieve. However, if students‘ initial ideas and beliefs are ignored, the under-
standing that they develop can be very different from what the teacher in-
tends. Thus, teachers must draw and work with the preexisting knowledge
that their students bring with them (e.g., Bransford et al., 2003) and not try
to model the child‘s brain according to their adult knowledge.
It is our proposition, here, that if teachers take into account the physiol-
ogy of the arithmetic neural circuits, as described and discussed in this
book, in planning their classroom activities, they surely will make arithme-
tic learning a more pleasant and easy journey.
10.5 Evolving Arithmetic Knowledge in the School 213
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226
Index
acceptance, 28,29,63,72,74
accumulator, 139, 141,144,146
adapted, 43, 44, 74
addition, 146, 171
adenylyl cyclase, 83,84
agnosia, 16
allosteric mechanisms, 82
ambiguity, 26,30,35
AMPA, 84,89,91
amplitude interference, 113
amplitudes, 80,113
attainability, 71,72
attention control, 62
attention deficit, 18
axonic transportation system, 49
Bell states, 80
bipedalism, 18,19
blackboard posting, 65
broadcasting, 122
Brocca‘s area, 16
buffer, 115
bursts, 225
Ca
2+
pumps, 90
calmodulin, 84,91,96
calmodulin-dependent protein
kinase, 86,91
cardinalities, 31,201
cardinality quantification, 138
cellular specialization, 22
central nervous system, 60
central-parietal (N
cp
) cells, 172
chimpanzees, 9,12,18
code string, 22
coincidence-detectors, 82,84,110
commitment, 156,162
communicating, 62
competition, 76,79
continuous spiking, 100
control string, 23
controlled NOT, 81
cooperation, 79
copying-fidelity, 118
counting, 54,117,118,131
crisp base numbers, 142
currency numbers, 206
decision, 71
decoherence, 112, 225
degree of acceptance, 29,63
degree of similarity, 29
dendritic spine, 89,91,94
Deutsch-Josza algorithm, 93
developmental dyscalculia, 18,196
developmental Gerstmann‘s
syndrome, 179
DIPS reasoning, 54,61
division, 148
dyscalculia, 22,192
dissociation, 16,189
distributed intelligent processing
system, 4,21,53
distributed intelligent processor, 53
division, 148,168,174
double dissociation, 216, 196
dyscalculia, 18
dyslexia, 18,194,196
EEG mapping, 153
electric code, 49
electrical gradient, 47
electro-encephalography, 17
10.5 Evolving Arithmetic Knowledge in the School 227
embriogenesis, 57,77
endoplasmatic reticulum, 90,95
enrollment capability, 156
Enscer figure database, 105
entanglement, 79,80,87
entropy, 33,83
enumerate, 7,15
evaluating, 62,72
evolution, 19,33,43
evolutionary learning, 70,75
executive processor, 115,116
expressiveness, 33,35,51
fecundity, 118
frontal-parietal gliosis, 189
fuzzy alphabet, 29
fuzzy base number system, 142
fuzzy formal language theory, 30
fuzzy grammar G, 31,35
fuzzy numbers, 136,192
game event related activity, 159
gene mutation, 38
genetic network, 25,39
Gertzmann syndrome, 16,191
glutamate, 24 ,96
glycyne, 83
goal definition, 62
Hadamard gate, 81
Hadamard transformation, 98,99
hominids, 18,123
hormones, 22,65
hyperactivity, 18
implementing, 62
imprinting, 93,104,106
inducer substrings, 24
inferior parietal, 9,13,153
innate numerical competence, 9
intelligence, 54
intermitent rhythmic delta activity,
192
internal states, 87,91
interneurons, 100
ion-trap computers, 87,91
K fuzzy numbers, 136
kinase, 84,85
laser pulses, 88,94
lateral prefrontal, 12,114
layers, 100
learning by observing, 61,203
learning capability, 67,179
left frontal neurons, 171
leukomalacy, 180
local response, 48,49
longevity, 118
looking time, 14
magnesium, 83
Magnetic Resonance Imaging, 17
Magneto-Encephalography, 17
mail addressing, 64,122
Man-Whitney U test, 164
matching, 28
mean ambiguity, 33,107
meme-gene-coevolution, 118
memes, 117
memetics, 117,126
memorizing, 62
memory span, 12
mental calculation, 136,154
mental lexicon, 17
mental number line, 139
microtraps, 87
model, 70
molecular neurobiology, 21
motor cortex, 12,191
multi-organ, 42,47,57
multiplication, 148
natural selection, 13,15
nervous system, 59
neural network models, 55
neuronal specialization, 54
neurons (N
v
), 172
neurotransmitters, 65
new language, 44,68
nicotinic acetylcholine receptor, 84
NMDA receptor, 83,96
228
nuclear magnetic resonance, 87,93
number, 13,15,20
number grammar, 17
number sense, 137,150
numbers, 117
numeracy deficit, 179,196
numerosity, 11,13,149
ordinal neurons, 141
organ, 42,56
oscillator, 88
parallel fibers, 100
parietal number representation, 189
Pedra Furada, 124,206
phasic, 100,196
phonons, 88
planning, 62,71
plasticity, 17,76,129,191
Positron Emission Tomography, 17
possibility, 31,32,52
post-synaptic density, 95,96
prefrontal cortex, 12,114,123,153
principal components analysis, 161
proteome, 22
pyramidal cells, 100,102
quadripole, 88,91,94
quantifiers, 142,149
quantum bit, 80
quantum charge-coupled computers,
93
quantum computation, 3,79,93
quantum computer, 81,88
quantum cortical pattern recognition
device, 93
quantum cryptography, 87
quantum gates, 81
quantum information, 87,93
quantum superdense coding, 113
radio frequency pulses, 87
recurrent processing, 109
recursive enumerable grammars, 36
replication, 118,206
repressor substrings, 24
retrograde signal, 50
rewriting, 28,138
schyzencephaly, 188
second-order catalysts, 83,109
selective attention, 110,115
self-controlled grammar,
36,55,121,199
Sensor, 105
sequential ordering, 153
signal transduction pathways, 24
somato-sensory cortex, 13
specialized agents, 54,79
spike, 49
spine pruning, 104
stellate, 100,102
subtizing, 9,.133,138
subtraction, 148,164
superfamilies, 61
superposition, 79,80,87,112
switch, 91
synchronous oscillations, 110
TATA box, 23
tonic, 100,197
transgenic mice, 89
travelling salesman problem, 140
triple-code model, 135,197
two-level systems, 87
two-photon microscopy, 89
type 0 grammar, 28
unitarity, 92
ventral frontal cortex, 115
vibrational states, 87
visual pattern recognition device,
100
voltage-dependent calcium
channels, 93
waveform, 87
Weber‘s law, 8
working memory, 114
zero, 15
230
