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From AMOEBA to BRAIN:

How Nature process Information∗

Ignacio Ozcariz†

RQuantech - Geneva (Switzterland)

Criptosasun - Madrid (Spain)

‡

(Dated: September 10, 2020)

ABSTRACT

The paper presents a new Information Processing paradigm based on the dynamics encountered

in life from organisms as diﬀerent as Amoebas and the Mammalian brain.

Our thesis contemplates that life supports information processing via metabolic dynamics in self-

organized enzymatic networks which have the capacity to represent functional catalytic patterns

that can be instantiated by speciﬁc input stimuli. Furthermore, the information patterns can be

transferred from the functional dynamics of the metabolic networks to the biochemical enzymatic

activity information encoded by DNA.

The metabolic dynamics are governed by fractional dynamics that evolve in topological fractal

spaces with multiscale time parameters generating complex attractors.

The complete dynamic information process is driven by the shorterm process of metabolic dy-

namics and the longterm process of DNA expression via epigenetic mechanisms.

“Once the development was ended, the founts of growth

and regeneration of the axons and dendrites dried up

irrevocably. In the adult centres, the nerve paths are

something ﬁxed, ended, and immutable. Everything may

die, nothing may be regenerated. It is for the science

of the future to change, if possible, this harsh decree.”

Santiago Ram´on y Cajal

CONTENTS

I. Introduction. 2

II. Origin and evolution of the nervous system. 2

A. Introduction 2

B. Amoeba’s information processing system 2

1. Amoeba’s Immune System 2

2. Amoeba’s development of the immune

system 2

3. From immune system to nervous system 3

C. Nervous system evolution 3

D. Mammalians Nervous system 4

III. The tetrapartite synapses 5

A. Introduction 5

B. Extracellular Matrix 5

C. Glial Networks 6

D. Neural Networks 6

E. The holo-network 8

IV. Fractional Dynamics 8

A. A history dependent and non-local dynamics 8

∗New Information Processing framework

†Also at Universidad Politecnica Madrid. Doctorate Student;

im.ozcariz@alumnos.upm.es

‡i.ozcariz@rquantech.com

1. Multiplicative calculus (Geometric and

Bi-geometric) 9

2. Fractional Calculus 10

B. Network fractional dynamics 12

1. Algebraic considerations 12

2. Dynamical model 12

C. Shorterm vs. longterm information

processing 15

V. Practical Device implementation 15

A. Light and sound correlated as an input

stimuli to the network 15

B. Potential eﬀects in Epilepsy and other

neurological diseases 16

C. Potential eﬀects on COVID-19 patients 17

VI. CONCLUSIONS AND FURTHER WORK 17

A. Summary 17

B. Disclaimer 17

Acknowledgments 17

A. Biological papers 17

B. Mathematical Papers 18

References 18

2

I. INTRODUCTION.

The aim of the paper is to represent the organism as

a Holo-Network formed by the integrated assemblage of

neural, glial, extracellular molecular networks and the

immune system that until now has always been consid-

ered in isolated compartments.

The extracellular matrix (ECM) is produced and dy-

namically modulated by several cell types that include

neurons and glial cells. The ECM plays a role in the

communication and control of the dynamics of neuroglial,

intra-neural, and intra-glial networks. Also the immune

system is playing a central role in the molecular dance

established by these networks and the full interplay be-

tween all of them achieves the objective of the homeosta-

sis state of the organism with its environment.

II. ORIGIN AND EVOLUTION OF THE

NERVOUS SYSTEM.

A. Introduction

To understand how the most complex nervous systems

works we must ﬁrst understand the species evolution of

these overly complex functions.

The nervous system is contingent on the development

of the diﬀerent eukaryotes families. We will develop in

the present paper the thesis that the specialization, ﬁrst

of the immune system of the primordial eukaryotes in its

information processing needs, and second the integration

of some of the networks supporting these communication

functions within the organisms, drove the development

of the vertebrate nervous system.

An eﬃcient and integrated, in dynamic terms,

perception-action system linking information receptors

(external and internal) to actuators, was the vital func-

tion on which evolution acted to enhance the capabilities

of the nervous system. Also, the dynamics acted over

the epigenetics machinery for the cell and exerted evo-

lutionary pressures to achieve the best ﬁtted organism

according to the characteristics of the environment.

To have achieve a complete Mammalian nervous sys-

tem (central and autonomous) required almost two bil-

lion years of evolution. During three-quarters of this

time, roughly 540 million years ago, animal life erupted,

diversifying into a kaleidoscope of forms in what is now

known as the Cambrian explosion, information process-

ing was then based on metabolic dynamics supported by

the communication channels in the internal structural

matrix of the individual cells. We will see these Amoeba

systems in the subsection B of this section.

Subsection (II C) of this section represents the part of

the evolution devoted to the mechanisms that multicel-

lular arrangements makes to maintain signal processing

coherent between the sets of cells. Also, we will look over

the nervous system of the mollusc as a ﬁrst step to see

the evolution of the nervous system.

Subsection (II D) will present the full complexity of

the Mammalians nervous system. To study these next

steps in the evolutionary history of the animal kingdom,

we shall consider the fertile ground of embryology. In

these grounds we can see how the nervous systems emerge

from a cluster of cells, the morula. We will quickly run

through the fundamental discoveries of homeotic genes,

thanks to experiments on the fruit ﬂy, Drosophila, during

the 1980s in which homeotic genes were discovered. They

are responsible to arrange the development of the embryo

and impose its ﬁnal shape. We will ﬁnally get to the

current complexity of the diﬀerent families of specialized

nervous cells (Neural and Glial) and the Extracellular

matrix in which these cells are structured that sits in the

core of the Mammalians nervous system.

B. Amoeba’s information processing system

1. Amoeba’s Immune System

Amoeba is one of the most abundant organisms in

ecosystems such as pond water. Under a microscope,

it is possible to see them wandering around on the slide,

but it is also remarkable to see them feeding on micro-

organisms, like a culture of macrophages. Although It is

still up for debate that the amoeba could be the earliest

form of macrophage, and by an unknown evolutionary

pathway, give raise to the modern macro phage. One

of the clear lessons from the study of the Amoeba’s be-

haviour is the complexity of the tasks that it accom-

plishes. From the feeding of Paramecium to the discrim-

ination of food from other Amoebas it is amazing the

complex dances that the simple organism develops.

One of the theories regarding the onset of innate im-

munity in eukaryotes is that it started in microorganism

such as an amoeba to accomplish the function of discrim-

ination between food and other Amoebas. The rationale

supporting this theory is that amoebas that do not make

this distinction would vanish incredibly earlier from the

ecosystem. Therefore, some kind of communication sys-

tem in the surface of the amoebas gets information from

the outside that will makes the molecules be able to dis-

tinguish between food, that is safe to be eaten, and an-

other amoeba, or even another part of the same amoeba.

The characteristics of this communication mechanism is

not yet known, but one of its basic functions is to dis-

criminate between what is self and what is not, being

extremely speciﬁc as it is the core function of the im-

mune system.

2. Amoeba’s development of the immune system

The movement of amoebas is seemingly at random,

but the more evolutionary macrophages, act with an aim

if they are exposed to a chemo-attractant. In that case

all of themwould head in the same direction. Therefore,

3

under the inﬂuence of these environment signals amoe-

bas behave like macrophages. A potential interplay with

the ﬁrst multicellular organisms would have happened

and amoebas may have had a parasitic action. Inverte-

brates and vertebrates present phagocytic cells that have

much in common with amoebas and ﬁll the role of the

”police” in the blood vessels and tissues, searching for

aliens. One theory is that these phagocytic cells, the

ancestors of macrophages, come from a population that

retained an ancestral, unicellular morphology, as they are

today reﬂected in the amoebas. One important thing to

bear in mind, is that complex means of host defence were

present in the genome by the time eukaryotes developed

into plants and animals.

This defence system, shared by plants and animals,

is the Toll receptor signalling pathway and is based on

NFκB (nuclear factor kappa-light-chain-enhancer of acti-

vated B cells) activation of gene function. This pathway

has been demonstrated in vertebrates, invertebrates and,

but not conclusively, in plants.

Experiments carried out with mice demonstrated the

following: Mice strains that cannot respond to bacte-

rial lipopolysaccharide (LPS) due to a defective Toll

gene cannot mount an adaptive immune response against

gram-negative bacteria, which carry LPS on their surface.

This is a proof that the loss of innate immunity had a

discernible eﬀect on the adaptive immune response.

More experiments have demonstrated the above role

of genes involved in immunity to diseases in ﬂies and

that these genes have homologues that also operate in

humans. Toll genes in ﬂies and TLR genes in humans.

Homologues of these genes have also been found in other

rather diﬀerent organisms like sharks, nematodes, and

plants.

In summary the innate immune system provides early

defence against pathogen attack, and communicates to

the adaptive immune system that a pathogen invasion

is happening. As we have seen a very ancient signalling

pathway, the Toll pathway, is acting to support this dual

function. This pathway is hundreds of million years older

than the adaptive immune system and is present in prac-

tically all the superior organisms today. The other com-

ponent of the innate immunity, the phagocytic cells, may

have their ancestors in amoeba-like eukaryotes. One of

the key points of the above discussion is that immune

system bears a signalling (information processing) mech-

anism at its deepest roots.

3. From immune system to nervous system

One of the new paradigms that have arisen in the ﬁeld

of physiology is the interplay between the immune system

and the nervous system. [1] The idea that neurotrans-

mitters could serve as immunomodulators emerged with

the discovery in complex organisms, that their release

and diﬀusion from nervous tissue could lead to signalling

through lymphocyte cell-surface receptors and the mod-

ulation of the immune function. [2]

The main idea is that the immune system and the

nervous system, that handle thousands of neuro-immune

transmitters can sense and respond to multiple environ-

mental conditions and aggressions. Cross talk between

these two systems has been reported in all the organisms

and all states, and the neuro-immune interactions can

operate as important immunoregulatory hubs.

In the previous section we have introduced the role of

the immune system in amoebas that now we will see, can

also support nervous system-like functions. In a pub-

lished paper [3] it was presented the shocking behaviour

of the true slime mold Physarum. When these amoebas

were exposed to unfavourable conditions as three con-

secutive pulses at constant intervals, they reduced their

locomotive speed in response to each episode. When

the plasmodia was subsequently subjected to favourable

conditions, they spontaneously reduced their locomotive

speed at the time when the next unfavourable episode

would have occurred. This implies the anticipation of

impending environmental change.

Moreover, it has been also demonstrated that,

Physarum Polycephalum, displays other abilities that

could be initially tagged as intelligent. It can solve mazes

[4] and geometrical puzzles [5], control robots [6], and in

a recent paper there is presented evidence of conditioned

behaviour in the organism. [7]

In our current nature’s information processing

paradigm this behaviour would only apply to organisms

that have a nervous system. So, information processing

by unicellular eukaryotes could be thought as a precursor

of nervous-dependent functions in multicellular organ-

isms. All the functions seen in the previous paragraph, no

matter the way in which they are performed, have some

information processing abilities that until know have re-

mained unexplained.

We introduce here the hypothesis that these capabil-

ities arose from the previous signalling pathways devel-

oped by the early immune system that was adapted to

new information processing functionalities like the ones

presented.

C. Nervous system evolution

To develop in more detail the thesis introduced at the

end of the previous section we must now have a look

at the evolution of the nervous system from the simple

multicellular organisms to Mammalians. The idea is to

tackle the complexity of mammalian models, which are

comprised of highly interconnected multiple networks,

studying ﬁrst simpler organisms.

One of the most promising discoveries regarding early

development of electrical conductance in multicellular ar-

rangements was made by Dr. Lars Peter Nielsen [8] at

the start of this century in 2009.

Dr.Nielsen introduced the concept of cable-bacteria

that more recently has been complemented with

4

nanowire-bacteria. These bacteria are arranged in a

manner that they create a cylinder of conducting wires

that encases a chain of cells. The wires allow the mi-

crobes to transfer electrons involved in redox reactions

that take place at spatially separated loci (centimetres

in cable-bacteria and micrometres in nano-bacteria).For

the cable-bacteria, the electrons are gained by oxidizing

hydrogen sulphide and are then transferred to oxygen-

rich sediment, where the electrons are linked to wa-

ter molecules. In the nanowire-bacteria the electrons

are shuttled between the oxidation side of organic com-

pounds along the so-called protein nanowires to the

electron-accepting side. There is still a lot of debate

about how the bacterial nanowires conduct electrons. Dr.

Derek Lovely [9] introduced the idea that chains of pro-

teins called pilins, which consist of ring-shaped amino

acids, are the key mechanism that support the electron

transport creating the electric current.

Once we have seen that Nature developed the mecha-

nisms to generate electric currents, we can make the hy-

pothesis that similar processes were utilized by eukary-

otes to be used in specialized cells to fulﬁl the signal

transfer in multicellular arrangements.

The specialized cells that we commonly associate to

the functions performed by the nervous systems are the

neurons. We will present further on in this paper that

neurons are only a part of the complex arrangement that

is necessary to carry on all the functionality of the ner-

vous system.

In any case and starting by the current paradigm, Dr.

Michael Bate [10] introduced the concept that “neurons

are born and diﬀerentiate in ways that are not condi-

tioned by their future functions as elements of neural

circuits” He also stated that “To understand how func-

tions ... can emerge from these beginnings, it is worth

remembering that fundamental attributes of the nervous

system such as the circuitry underlying locomotion or

escape behaviour are probably also present as a rather

stereotyped and evolutionary conserved set of cells and

connections. It is at least possible to envisage that there

is a fundamental framework of circuitry just as there is

a scaﬀolding of initial pathways”.

The nervous system would then be composed of a mix

of, evolutionary proven, signal pathways mechanisms,

many of which were already developed in early life or-

ganisms.

The classical works to understand the basics of the ner-

vous system has been performed in the squid giant axon.

In this “simple” system main studies have been devoted

to the models supporting functionality of the neuronal

network of the squid. One of the approaches that we can

take to get a deeper glimpse into the nervous system of

these organisms regarding the biomolecules implied in the

signalling paths is to get the response against anaesthet-

ics agents that as we know now are a powerful inﬂuencers

in the functionalities of the system.

One of these early works was performed by (Shrivastav

et al., 1976) [11] They exposed a giant squid axon to the

volatile anaesthetic halothane and recorded membrane

depolarization at low anaesthetic concentrations. They

observed a similar depolarizing eﬀect with the volatile

anaesthetic trichloroethylene, which also increased the

threshold potential for action potential ﬁring and reduced

the amplitude of resulting action potentials. The key

point of his experience was that the mechanism was act-

ing at the level of the synapse and generated the anaes-

thetic eﬀect on the neuron functionality.

More experiences were carried later by Dr. Ryden

Armstrong [12]. Even if they recognized that “there re-

mains signiﬁcant uncertainty as to how and where these

compounds act at the molecular and cellular levels.“, re-

garding the anaesthetics actions over nervous systems of

mollusc the results of their experiences were that “work

showed that volatile general anaesthetic compounds di-

rectly interact with ion channel proteins, a potassium

channel that hyperpolarized neurons, preventing neuro-

transmitter release. Subsequent work then showed that

general anaesthetics also directly target and suppress ion

movement through the excitatory acetylcholine receptor.

Thus, show that some general anaesthetics target both

presynaptic and postsynaptic sites on neurons. These

studies also highlight the potential target sites of anaes-

thetic actions that include classical and peptidergic neu-

rotransmitter synapses. Finally, the data from various

studies on Lymnaea demonstrate that chronic exposure

of cultured neurons to anaesthetic compounds might ren-

der neuronal growth and synaptic connectivity dysfunc-

tional.”

From these experiences we can get the point that

synapses, and signal transmission in broad terms, more

than cells are the key actors of the ﬁlm that nature per-

forms in the information processing representation.

D. Mammalians Nervous system

Back in 1983 Dr.Walter Gehring [13] studying

drosophila, fruit ﬂy, discovered the existence of homeobox

genes responsible for a homeotic transformation where

legs grow from the head instead of the expected anten-

nae. It was the demonstration that the development of

multicellular organisms is based on a program of diﬀer-

ential expression of genetic information.

One of the subsets of homeobox genes are the Hox

genes. They are the genes that determine the identity

of embryonic regions along the body axis. The precursor

cells of the nervous system cells arise within an epithelial

ﬁeld of cells made competent through the expression of

one or more control genes of the bHLH type (proneural

genes) mediated by the Notch-Delta interaction.

In vertebrates, the embryogenic process known as neu-

rulation are the stages that go from the neural plate to

neural tube and to the neural crest. All the cell types

during this neurulation process arise from pluripotent

embryonic stem cells (ESCs). The adult neural stem

cells (NSCs) persist in two main areas: the ventricular-

5

subventricular zone, where NSCs give rise to olfactory

neurons, and the hippocampus, where new neurons in-

volved in cognitive processes are generated. In both re-

gions, the stem cells that give rise to neurons are special-

ized populations of astrocytes that maintain close inter-

actions with the brain vasculature and can be activated

by behavioural and pharmacological stimuli. [14]

The important point in the previous paragraph is that

the pluripotent adult cells of the nervous system derives

from an astrocyte population.

If now we come back to the immune origins of the ner-

vous system presented in (II B 3) we will see the current

interrelations between the two systems in the vertebrate

kingdom.

In a paper [2] Dr. Franco presented “Initially, the idea

that neurotransmitters could serve as immunomodulators

emerged with the discovery that their release and diﬀu-

sion from nervous tissue could lead to signalling through

lymphocyte cell-surface receptors and the modulation of

immune function. It is now evident that neurotransmit-

ters can also be released from leukocytes and act as au-

tocrine or paracrine modulators”. Dr. Franco ﬁnished his

paper signalling that “Current and future developments

in understanding the cross-talk between the immune and

nervous systems will probably identify new avenues for

treating immune-mediated diseases using agonists or an-

tagonists of neurotransmitter receptors.”

Last year Dra. Cristina Godinho-Silva, Dra. Filipa

Cardoso, and Dr. Henrique Veiga-Fernandes in their pa-

per [1] introduce the concept “Neuro–Immune Cell Units:

A New Paradigm in Physiology”. Their thesis is “The in-

terplay between the immune and nervous systems has

been acknowledged in the past, but only more recent

studies have started to unravel the cellular and molec-

ular players of such interactions. Mounting evidence in-

dicates that environmental signals are sensed by discrete

neuro–immune cell units (NICUs), which represent de-

ﬁned anatomical locations in which immune and neu-

ronal cells co-localize and functionally interact to steer

tissue physiology and protection. These units have now

been described in multiple tissues throughout the body,

including lymphoid organs, adipose tissue, and mucosal

barriers. As such, NICUs are emerging as important or-

chestrators of multiple physiological processes, including

haematopoiesis, organogenesis, inﬂammation, tissue re-

pair, and thermogenesis.”

This year it has been reported in [15] the regulation be-

tween Neuro-Immune Circuits mainly related to gut tis-

sues and Organ Homeostasis. Main result of the report

is “Preclinical studies targeting neuro-immune interac-

tions upon stimulation of the vagus nerve, application

of acetylcholine agonist, and b2 adrenoreceptor agonists

have emerged the potential successful treatment in in-

ﬂammatory diseases. Of note, the site-speciﬁc control of

immune functions by the nervous system via neurotrans-

mitters/neuropeptides suggest that the nervous system

can exert a rapid and local control of immune cells”.

We ﬁnalize this introduction to the Mammalian ner-

vous system emphasizing two points;

1. the origin of the multiple cell types of the system

from a unique type of “germ” that in the adult

phase are the Astrocytes.

2. the full relation at the signal processing level be-

tween the immune system and the nervous systems

III. THE TETRAPARTITE SYNAPSES

A. Introduction

We could have headed this section also by the title

“The nervous system: the known unknown”, but we have

preferred to focus on the theme that in the next section

will be developed as the core of the paper. The dynamics

that underscore not the individual networks that inter-

twined in the nervous system nor the full hyper-network

which nodes are the complex synapses.

We consider that synapses are the places in which

presynaptic neurons, postsynaptic neurons, astrocytes

and all the molecules in the extracellular matrix make

their mutual interrelations. All the actors are inﬂuenced

by the results of the processing in the synapse. So, the

former paradigm that the nervous system was based on

the responses of the postsynaptic neurons to the signals

generated by the presynaptic neurons, has now changed

to a global dynamic process in which Neurons, Glial cells

and Extracellular matrix composites, evolution.

We will present in the next sections several references

for the subsystems implied that would aid to support the

change of paradigm

B. Extracellular Matrix

From the onset of the century the role of the Extracel-

lular Matrix (ECM) on the nervous system has started

to be recognized. It was not until 1971 that was accepted

the existence of the ECM initially based on the predomi-

nance of hyaluronan and chondroitin sulphate proteogly-

cans (CSPG). Taking in account that ECM suppose 20%

of the volume of the adult brain it is shocking that its

role was so diminished.

In 2008 Drs. Zimmermann in their paper [16] estab-

lished that ECM is rich in hyaluronan, CSPG (aggrecan,

versican, neurocan, brevican, phosphacan), link proteins

and tenascins (Tn-R, Tn-C) and its role is to regulate

the cellular migration and axonal growth. Thus, ECM

participates actively in the development and maturation

of the nervous system. ECM swift assembly and remod-

elling was associated with axonal guidance functions in

the periphery and with the structural stabilization of

myelinated ﬁbre tracts and synaptic contacts in the mat-

urating central nervous system.

More recently has been reported the organization of

the CSPG into either diﬀuse or condensed ECM. Diﬀuse

6

ECM is distributed throughout the brain and ﬁlls peri-

synaptic spaces, whereas condensed ECM selectively sur-

rounds parvalbumin-expressing inhibitory neurons (PV

cells) in mesh-like structures called perineuronal nets

(PNNs).

In [17]) is reported that “ECM not only forms physical

barriers that modulate neural plasticity and axon regen-

eration, but also forms molecular brakes that actively

controls maturation of PV cells and synapse plasticity in

which sulphation patterns of CS chains play a key role.

Structural remodelling of the brain ECM modulates neu-

ral function during development and pathogenesis.”

The main conclusion is ECM components and the

molecules they interact with, will provide new insight

into the molecular networks that regulate neural plastic-

ity

C. Glial Networks

One of the strangest things in neurophysiology studies

has been for most of the time of the past century the

secondary role attributed to the Glial system. Starting

with the pejorative name gave to most of 50% of the to-

tal number of cells of the nervous system, glia, from the

Latin and Greek for glue, and continuing for the asso-

ciate role as merely “support” cells of the big kings, the

neurons.

It was at the end of XIX century that the Spanish

Nobel price Ramon y Cajal brought to light the potential

role of these cells in two seminal articles, “Something

about the physiological signiﬁcance of neuroglia” (1897)

and “A contribution to the understanding of neuroglia in

the human brain” (1913). [18].

The posterior oblivion of these cells in the opinion of

the author was due to the electrical characteristics of the

neuronal system in comparison with the biochemical op-

eration of the glial system. The measure of the activity of

neurons in the century of electricity was more straightfor-

ward than the comprehension of the biomolecules implied

in the functionality of glia cells.

Uniquely at the end of the century and with a big

impulse from the Spanish Institute Ramon y Cajal and

Dr. Alfonso Araque, Dra. Gertudris Perea and Dra.

Marta Navarrete the role of glia has started to become

mainstream.

One of the extraordinary ideas of Cajal in his paper

of 1913 was that “The gray matter neuroglia would con-

stitute a vast endocrine gland intertwined with neurons

and nerve plexus, intended perhaps to produce hormones

associated with the brain activity”. Current research has

demonstrated that the so-called gliotransmitters as par-

allel to the neuronal neurotransmitters regulate widely

neuron functionality.

The role of the Astrocytes releasing gliotransmitters

and controlling transmission and plasticity at the synap-

tic level led to a new concept in synaptic physiology, the

Tripartite Synapse, in which astrocytes are integral ele-

ments of the synapses and actively exchange information

with the neuronal elements (Araque et al.; [19] Halassa

et al.,[20] ;Perea et al. [21]).

More recently, March 2020) a Science paper written by

thirty authors [22] goes further in the relation between

neurons and glia establishing a powerful link between

these two systems at the level of the bodies of the cells via

a purinergic junction. Even if they initially stated that

“Microglia perform dynamic surveillance of their micro-

environment using motile microglial processes that con-

stantly interact with neurons. However, the molecular

mechanisms of bidirectional microglia–neuron communi-

cation are unclear” the results of their research are “

All of these results unequivocally indicate that microglia

continuously monitor neuronal status through somatic

junctions, rapidly responding to neuronal changes and

initiating neuroprotective actions”.

Other aspect of the glia functionality that we want to

take in consideration is that the most reported imaging

techniques of the brain are based on the activity of the

glial cells. Two of these techniques to monitor brain ac-

tivity, functional Magnetic Resonance Imaging (fMRI)

and Diﬀusion-Tensor Imaging (DTI) are not based on

the electrical activity of neurons, nor the oxygen content

supplied by blood capillaries (fMRI), or by anisotropic

water diﬀusion (DTI). Oxygen consumption measured in

a fMRI voxel is the result of the metabolism of diﬀer-

ent cell types in which, glial cells (astrocytes, microglia,

endothelial, neutrophils, pericytes, NG2 glia, 0ligoden-

drocytes) are the main source with neurons and vascular

as secondary. In the case of DTI is even more focused

the role of the glia because DTI signal are based on the

myelin-axon unit and support-providing myelin sheaths

are the role of the oligodendrocytes.

So, the neuron centred brain is still today the main

paradigm in the signal processing function but as we

have seen glia cells have functions that clearly make this

paradigm complicated to support.

The functionality of the so called “tri-partite” synapse

in which Astrocytes regulate extracellular ion and trans-

mitter homeostasis [23] as well as the role of the neuron-

glia junctions is out of question and by the modulation of

neuro and glia transmitters [24], and peptide hormones,

fully inﬂuences the dynamics of neurons. Also, the role

of the oligodendrocytes metabolically supports neuronal

axons, as the base of neural circuits is clearly a powerful

inﬂuence in the dynamics of these circuits.

D. Neural Networks

Hundreds of thousands of papers have been devoted to

the functionalities of Neural Networks from a full range

of perspectives going from computational approaches to

cell biochemistry.

Taking into account that the purpose of the paper is

dealing with the dynamics of the diﬀerent systems im-

plied in the information processing tasks we will focus

7

brieﬂy in two aspects of the neural networks; i) the dy-

namics of the system isolated ii) the potential inﬂuence

of external factors of the dynamic system.

The historical model for the dynamics of the neuron

isolated has been the Hodgkin–Huxley and its multiple

derivations. One more developed model was introduced

by Eugene M. Izhikevich in 2003 [25]. The model con-

templated the analyses of the dynamics using bifurcation

methodologies and with only four parameters he was able

to reproduce the spiking and bursting behaviour of some

types of isolated neurons. In ﬁgures 1 and 2 we present a

Matlab c

simulation of the model using a network of one

thousand randomly coupled spiking neurons. As we can

see, we obtain some common features in the spike model

of the neuron of the neuron 927. Figure (2).

FIG. 1. 1000 Network Firing

FIG. 2. Neuron 927 Spiking Pattern

Regarding models of complete neural circuits, one that

contemplates high order dynamics is the Freeman model

(KIV model). The model was derived from the olfactory

system and presents a dynamic with chaotic evolution.

To achieve this chaotic result the model is based on dif-

ferential equations of second order.

The full model is presented in [26] and the summary of

the model states, “We postulate that the meaning of the

information exists in the interrelations of the enormous

number of neurons carrying each a fragment of informa-

tion, which constitutes the knowledge brains create from

information. This knowledge must be stored and later

mobilized in the serial processes for categorizing sensory

information and then deciding on courses of action in re-

sponse to the input. Mobilized knowledge has the form of

bursts of oscillatory electroencephalographic (EEG) and

electrocorticographic (ECoG) potentials carrying spatial

patterns of amplitude modulation (AM), which emerge

from chaotic oscillations of background activity through

episodic bifurcations (state transitions) by which the cor-

tex jumps from chaotic disorder into phase-locked limit

cycle activity and back again.”

From this model we will see in the next section, that

the interrelation with the rest of the network systems im-

plied, will get a more elaborated information processing

framework.

Going to the inﬂuence of external factors in the net-

work dynamics and with the aim to only present one of

them, we focus on the eﬀects of alcohol on the nervous

system.

The complete actions of ethanol in the nervous sys-

tem cannot be covered in this very brief mention and

of course the eﬀects of ethanol includes actions on glia,

and extracellular matrix, but we will focus only on the

actions of ethanol with the molecular signalling path-

ways involved. “Among the central actions stimulated

by alcohol, its inhibitory gabaminergic eﬀect, in con-

junction with the inhibition of certain excitatory glu-

tamatergic receptors, endocannabinoids, cerebellar cal-

cium channels and hippocampal proteins which are es-

sential for memory formation, results in sedation, loss

of inhibitions, relaxation, loss of cognitive functions, at-

tention deﬁcit, impaired sleep-wake regulation (blackout

eﬀect), and the ﬁnal state of psycho-motor depression.

On the other hand, the excitatory action of alcohol on

mu receptors of the opioid system and subsequent acti-

vation of the limbic system by dopamine and of 5-HT1B

receptors by serotonin result in the eﬀect of well-being

and mood elevation. In addition, the down regulation

of dopamine and GABA receptors explains the increase

in alcohol consumption and subsequent development of

chemical dependency”[27].

The key point that we want to emphasize here is that

the eﬀect of a single molecule could alter the entire dy-

namics of the complete neural network.

8

FIG. 3. Organism as a Holo-Network

E. The holo-network

In the previous subsections we have made an approach

to the diﬀerent networks that must be taken into account

to fully grab the inner functionalities of the nervous sys-

tem.

Here we will make a summary of the interrelations of

these networks and the ones implicated in the immune

system with the aim to delineate a common ground for

the dynamics of an entire organism. Of course, the dy-

namics must take also into account the environment in

which this organism evolves.

The objective is to demonstrate that Homeostasis of

the complete system Organism plus Environment is the

driver for the dynamic.

Starting with the tri-partite synapse contemplated in

the glia section we can now enlarge this concept to the

tetra party synapse as had been presented in (Dityatev

and Rusakov [28] ; Verkhratsky and Nedergaard [29];

Smith et al. [30] Dzyubenko et al.[31]). These papers

support the concept of the fully integration and inter-

relation of the Extracellular matrix components and the

other two networks taken into account that both, neurons

and astrocytes, synthesized components that become key

agents at the ECM level and have a critical role in the

functionality of synapses in the nervous system.

So, the actors at the synaptic level are.

(i) the presynaptic terminal

(ii) the postsynaptic element

(iii) the peri-synaptic processes of the astrocytes

(iv) the processes of neighbouring microglial cells that

periodically contact the synaptic structure

(v) the extracellular matrix (ECM), which is present

in the synaptic cleft and also extends extrasynaptically.

With this complex structure in mind we can make the

hypothesis that tetra-partite synapses are the nodes of a

kind of hyper-network (Marc) that will be the substrate

for the dynamics of the nervous system. Remembering

now the considerations done in subsection(II D) regard-

ing the relations between the nervous system and the

immune system we can fully expand the hyper-network

presented to a Holo-Network that will also include these

interactions.

We are using the preﬁx “holo”, with the meaning of

whole, entire, capturing the idea that the organism has

an underlying global network that processes all the in-

formation coming from external and internal sources in

order to incorporate them into an integrated framework.

The clear aim of this information process is to main-

tain the life of the organism and we will take life deﬁ-

nition as the homeostatic state of the ensemble organ-

ism/environment.

IV. FRACTIONAL DYNAMICS

A. A history dependent and non-local dynamics

Newtonian calculus is based on the “Method of Flux-

ions”, a book of Sir Isaac Newton completed in 1671 but

published posthumously in 1736. This late publishing

was the cause of a bitter dispute with his contemporary

Gottfried Leibniz regarding the priority of the calculus

invention.

9

In fact, Leibniz had a deeper understanding of the con-

cept, as we will see later, due to his potential broad scope

that he envisaged.

In the second half of the 17th century, several mathe-

maticians were occupied in the inﬁnitesimal analysis in-

troduced earlier by Barrow and the genius of Newton and

Leibniz was to systematize the concept of derivative.

Newton deﬁned the derivative Das the local behaviour

of a function around one-point x0taking into account one

inﬁnitesimal δx, that would be as little as needed.

Df (x0) = f(x0+dx)

dx (1)

One of the key points in this deﬁnition is the locality

of the derivative taking into account that Dis related to

the point x0.

Also, it is important to note that this deﬁnition has

in its root a kind of average around the point x0, so

if the function has a huge variation in this point, like

exponential functions, the approach could be misleading.

1. Multiplicative calculus (Geometric and Bi-geometric)

Robert Katz and Michael Grossman introduced in

[32] one idea that even Galileo had discussed brieﬂy;

to use ratios of the function instead of diﬀerences for

the derivative deﬁnition. From this base multiplicative

calculus also known as “star” calculus was born.

a. Deﬁnition of the * derivative.

The * derivative of the function fis given by:

f?(x) = lim

h→0(f(x+h)

f(x))1

h(2)

Doing some calculations (see: [32]) we can obtain the

relation between the classic (prime) and * derivatives:

f?(x) = exp(f0(x)

f(x))= exp(ln ◦f)(0)(x)(3)

considering that (ln ◦f)(x) = ln(f(x)). If f(x) is a posi-

tive function and f(n)exists,

f?(n)(x) = exp(ln ◦f)(n)(x)(4)

In this expression is included the case for n= 0 and so,

f(x) = exp(ln ◦f)(x)(5)

b. Geometric interpretation of the * derivative.

If we remember the geometric interpretation of the

classic derivative it is the slope of the line that is tan-

gent to the curve at a particular point.

y0(x) = lim

∆x→0

∆y

∆x≈∆y

∆x(6)

then

∆y≈y0(x)∆x(7)

This last relation is the change rate of the function for a

change rate of the variable. Remembering the deﬁnition

of the * derivative in (2) and considering ∆x=x2−

x1the variation of the independent variable and ∆?y=

y2

y1≈y(x2)

y(x1the star variation of the dependent variable.

From y?(x) = lim∆x→0(∆?y)1

∆x≈∆?y)1

∆xwe get ﬁnally

∆?y≈(y?(x))∆x≈expy0(x)

y(x)∆xThen we can establish

that,

y2≈y1expy0(x)

y(x)∆x(8)

Summarizing we have:

Classical derivative

y(x+ ∆x)≈y(x) + y0(x)∆x(9)

Star derivative

y(x+ ∆x)≈y(x) exp y0(x)

y(x)∆x(10)

c. Linear approximation versus exponential.

approximation. A function y(x) that is diﬀerentiable at

a point x=x0has a linear approximation Lnear this

point deﬁned as follows

L{y(x)}=y(x0) + y0(x)×(x−x0) (11)

Another approximation to a function could be using ex-

ponential functions as in , y(x) = a×bx.These functions

have constant star derivative and y(x+1) = y?(x)×y(x).

The following two properties are true for all positive dif-

ferential functions [33]

1. If we scale a function with a constant factor, the

function and the scale one has the same star deriva-

tive that is the constant factor. This is the idea

behind the multiplicative rate of change. So, if

Sy(x) = c×y(x), then Sy?(x) = y?(x) = c.

2. Any positive diﬀerentiable function at a point x0

has an exponential approximation Enear x0de-

ﬁned by;

E{y(x)}=y(x0)×y?(x0)(x−x0)(12)

We can show the diﬀerences between the two approxima-

tions with next example:

y(x) = 1

x2y0(x) = −2

x3y?(x) = exp−2

x

y(1) = 1 y0(1) = −2y?(1) = exp−2

Linear approximation at x= 1 is

L{y(x)}=y(1) + y0(1) ×(x−1) = 3 −2×x(13)

Exponential approximation at x= 1 is

E{y(x)}=y(1) ×y?(1)(x−1) = exp2×(1−x)(14)

We can see the results of the approximations in the Fig-

ure (4) below

10

FIG. 4. Linear vs. Exponential approximation

So, for the kind of curves as the exponential, the expo-

nential approximation has a better match than the linear.

d. Bi-geometric Calculus.

One further variation of the deﬁnition of derivative

gives raise to the bigeometric calculus. In this case the

bigeometric derivative is deﬁned as being:

y??(x) = lim

x→x0

[y(x)

y(x0)]1

ln x−ln x0(15)

And the relation with the classic derivative is:

y??(x) = exp x×y0

y(16)

If we remember from the economy manuals that elasticity

is the ratio between ∆y

∆xand y

xis the relation of quantities

and prices, we have with the bigeometric derivative the

perfect tool to deal with the functions that arise in this

domain as well as others like biology in which similar

ratios, (quantities/catalysers) have a main role. (Scale

law functions).

2. Fractional Calculus

a. Deﬁnitions. Fractional Integrals and Derivatives.

In the previous section we have seen that classic deriva-

tive gives the linear approximation of smooth functions,

and multiplicative or star derivatives, allow an exponen-

tial approximation. In both cases the derivative is linked

to a point of the curve and so has a purely local charac-

teristic.

Now going further on the power laws, we will introduce

the concept of Fractional Calculus that we can deﬁne as

the ﬁeld with objects that are characterized by power

law non-locality, power law longterm memory or fractal

properties by using integration and diﬀerentiation of non-

integer orders.

Returning to the quarrels between Newton and Leib-

niz, it was this last one that invented the notation dny

dnx.

Perhaps it was naive play with symbols that prompted

L’Hospital in 1695 to ask Leibniz, “What if n be 1/2?”

Leibniz [34] replied: “You can see by that, sir, that one

can express by an inﬁnite series a quantity such as D1

2xy

or D1:2xy . Although inﬁnite series and geometry are

distant relations, inﬁnite series admits only the use of

exponents which are positive and negative integers, and

does not, as yet, know the use of fractional exponents.”

Later, in the same letter, Leibniz continues prophet-

ically: “Thus it follows that D1

2xwill be equal to

x√dx :x. This is an apparent paradox from which, one

day, useful consequences will be drawn.”

Continuing with the history Euler (1730) mentioned

interpolating between integral orders of a derivative.

Laplace (1812) deﬁned a fractional derivative by means

of an integral, and it was Lacroix (1819) the ﬁrst to write

a derivative of fractional order.

For example if we use potential functions as y=xm,

Lacroix expressed it as,

dny

dxn=m!

(m−n)!xm−n=Γ(m+ 1)

Γ(m−n+ 1)xm−n(17)

Replacing m= 1 and n=1

2we obtain the fractional

derivative of order 0.5 of the function y=x

d1

2y

dx 1

2

=Γ(2)

Γ(3

2)x1

2=2

√π√x(18)

In which we can see the result advanced by Leibniz.

To summarize a long history, we will use now on the

concept of fractional derivative and integral elaborated

by Liouville and reframed by Riemann:

We start with the integral presentation.

Left side:

0Iα

tf(t) = 1

Γ(α)Zt

0

f(τ)(τ−t)(α−1)dτ (19)

Right side:

tIα

bf(t) = 1

Γ(α)Zb

t

f(τ)(τ−t)(α−1)dτ (20)

The derivative of Riemann Liouville is deﬁned by

RL

aDα

xf(x) = 1

Γ(n−α)(d

dx )nZx

a

f(τ)

(x−τ)α−n+1 dτ

αreal number, ninteger, (n−1) ≤α < n (21)

M. Caputo in 1967 made a little reformulation of the

derivative

C

aDα

xf(x) = 1

Γ(n−α)Zx

a

f(n)(τ)

(x−τ)α−n+1 dτ

αreal number, ninteger, (n−1) ≤α < n (22)

11

Properties of Fractional Derivatives.

1. Linearity Fractional diﬀerentiation is a linear

operation-

Dα(κf(x) + λg(x)) = κDαf(x) + λDαg(x) (23)

2. Scaling and Scale invariance

dαf(λx)

dxα=λαdαf(λx)

d(λx)α(24)

3. Sequential Composition

Dαf(x) = Dα1Dα2...Dαnf(x)

α=α1+α2+... +αn

αi<1

(25)

b. Geometric Interpretation of fractional integration.

In this section we will get the model of Dr. Igor Pod-

lubny [35] that introduced a very visual interpretation of

the fractional integration and subsequent derivation.

If we refer to equation (19)and consider the function

gt(τ) = 1

Γ(α+ 1){tα−(t−τ)α}(26)

this function has a scaling property. So, if we take t1=kt

and τ1=kτ , then

gt1(τ1) = gkt(kτ ) = kαgt(τ) (27)

Now we can write the integral in the form

0Iα

tf(t) = Zt

0

f(τ)dgt(τ) (28)

This expression can be considered as a Stieltjes integral

and we will try to understand it from a representation in

the axes τ, g andf , for diﬀerent values of t.

First we draw the curve that results of the interception

of the surfaces gt(τ) and f(τ) for (τ, g ) for 0 ≤τ≤tand

diﬀerent values of tfrom t= 10 and intervals of ∆t=−1.

Let us establish f(τ) = |tau + 0.5 sin(τ)

Figure (5), shows in blue this curve.

FIG. 5. Geometrical Representation of Fractional Integral

Now if we consider the projection of this curve over

the three planes, projection over the plane f(τ),(τ, t) is

simply f(t) and the area under the curve in this plane

will represent the integral

0I1

t(t) = Zt

o

f(τ)dτ (29)

So, the integer integral.

Considering now the projection over the plane

f(τ), gt(τ) , and the area under this curve in this plane

it is precisely the Stieltjes integral. The key point is

that diﬀerent values of tas parameter does not change

the classical integral and of course changes the fractional

one.

From this reﬂection we can point out a geometric in-

terpretation of fractional integration, as the addition of a

third dimension gt(τ) to the classical pair f(τ)), τ . This

third dimension will act as a non-lineal time scaling over

the ffunction.

We will see later more on that with the Mellin trans-

form that act as a kernel function deﬁned for this geo-

metrical explanation.

If we now consider the function fas a velocity or rate

of change of a variable, the ﬁrst integral of the velocity is

the space travelled and if we now consider the fractional

integral;

SO(t) = Zt

0

v(τ)dgt(τ) = 0Iα

tv(t) (30)

The ”space” travelled in this case SOis the result coming

from a kind of geometrical time.

From here we can also develop a geometrical interpre-

tation of the Rieman-Liouville derivative. Taking into

account that the velocity is the derivative of the space

v(t) = 0Dα

tSO(t) (31)

with 0Dα

t, the Rieman-Liouville derivative of order αand

12

0< α < 1 is deﬁned by

0Dα

tf(t) = 1

Γ(1 −α)

d

dt Zt

0

f(τ)dτ

(t−τ)α(32)

Then left-sided Rieman-Liouville fractional derivative of

the distance SOis equal to the individual speed vO(rate

of change) of that object.

vO(t) = d

dt 0Iα

tv(t) = 0D1−α

tv(t) (33)

B. Network fractional dynamics

1. Algebraic considerations

In this section we must acknowledge the seminal con-

tributions of I.C. Baianu to the algebraic approach to

biology in which we have based the next paragraphs as

the roots for our Holo-Network dynamics.

As we have presented in the section (III E) an organism

would have a dynamic system supported through tightly

coupled, biochemical subsystems, dependent upon spe-

ciﬁc, metabolic processes, information processing-linked,

that we named Holo-Network.

Information processing of the kind envisaged is more a

dynamic process than a digital computation as the cur-

rent approach presents. If we consider the dynamic under

an algebraic perspective we can make the comparison be-

tween the Boolean logic of digital computers and many-

valued logic and logic algebras that is the approach that

we propose here.

Robert Rosen and I. Baianu [36] presented algebraic

models for ”ultra-complex, in the sense of extreme com-

plexity, in order to avoid any possible confusion, and

clearly distinguish it from the complicated model systems

that are routinely called Complex Systems”, They intro-

duced ” the presence of a higher dimensional algebra and

mathematical structure in categories that are represent-

ing these systems and their transformations”

Capabilities of an ultra-complex system for both

metabolism, M, and repair, R, must be generated from

within the system’s own suﬃciently complex organiza-

tion; this implies also that metabolism leads naturally to

repair, or Metabolism-Replacement, or (M,R), system.

It is considered to be non-reducible to its components

and algorithmically non-computable, in the sense of not

being valuable as a function by a Turing machine. From

the point of view of ultra-complex dynamics evolution is

extremely complex, chaos-like.

In the paper [37] Baianu formulated the Non-

linear dynamics of non-random genetic and cell net-

works through categorical constructions enabled by the

Lukasiewicz–Moisil Logic algebras.

This is the algebraic approach that we will also ap-

ply to the Holo-Network, In fact this approach has only

certain broad similarities to the well-known method of

McCulloch and Pitts.There are major diﬀerences aris-

ing from the fact that we consider the sub-networks of

the Holo-Net acting in a multiple, quantum manner, and

strongly coupled via signaling pathways.

The use of a Lukasiewicz–Moisil (LM) Topos expands

the potential dynamics range to any sub-Network like

Neural,Glial, EMC, or Genetic-Epigenetic processes.

Non-commutative and non-distributive varieties of

many-valued LM-logic algebras, as applied by Baianu

([38]), open a broad ﬁeld for applications both to our

Holo-Networks and also to quantum systems, as we will

see later on.

2. Dynamical model

Now we will develop the dynamics of our model. Based

on the algebraic considerations introduced in the pre-

vious section and using the calculus tools presented in

the (IV A 1 and IV A 2) sections. Holo-networks in our

new framework are Categories of the Lukasiewicz–Moisil

Topos, which Functors will evolve according to the mul-

tiplicative fractional calculus.

The key point is that dynamics get the non-locality

and historical properties from fractional dynamics and

power law “additivity” via the multiplicative geometric

and bigeometric calculus.

a. Info-Mechanics [39]

The above framework could be nominated Info-

Mechanics as a new dynamic framework related to the

classical mechanics and quantum mechanics. To brieﬂy

remember the basics for these systems: Classical Me-

chanics:

1. Dynamic evolution in a spacetime that is 3D Eu-

clidean and 1D time independent.

2. Galileo relativity principia. All the nature laws are

the same for all the Inertial systems (Systems that

move straight and uniformly)

3. Newton determination principia. All the move-

ments are fully determined by the initial position

and velocity

Hamiltonian Mechanics:

1. Dynamics is described in the Phase-Space that has

the structure of a sympletic variety and the Poisson

Integral as the Invariant.

2. A new relativity principium (Einstein’s special rel-

ativity) could be introduced in the framework.

3. Hamiltonian Function. Each uniparametric group

of symplectic diﬀeomorphisms of the phase space

that keeps constant the Hamiltonian are associated

to a prime integral of the movement.

Info Mechanics

1. Dynamic evolution over a Lukasiewicz–Moisil

Topos.

13

2. Nature is structured in Categories which inner mor-

phisms are based on holo-networks.

3. Functors between Categories evolution according to

multiplicative fractional calculus.

We introduce the hypothesis that the diﬀerent models of

mechanics that apply to Nature are related to the spatial-

temporal scale in which we make the analysis.

With this approach Quantum Mechanics (a Hamil-

tonian quantized mechanics) and its broad application

Quantum Field Theory (QFT) are applied to Nature

in the microworld., from Plank distance to micrometre.

Info-Mechanics will appear as the necessary framework

for the mesoscale, to planet wide scale, and ﬁnally Gen-

eral Relativity (GR) as the frame for the Universe scale.

The inconsistencies between QFT and GR could now

be contemplated as a scale factor in a wide theory Info

related. Considering Mellin transform as the convolution

model for the RL Integral with kernel tswith respect

to the Haar measure dt

t(multiplicative invariant),

some nice relations have been developed by Dr. Liam

Fitzpatrick, showing that Mellin space serves the Info

role in the context of the AdS/CFT correspondence. [40].

b. Toy Model

In this section we will develop one simplest model

based on the previous statements.

The model takes in account the evolution of the Func-

tors between a simple Category on Info, a virus, and the

three subcategories, Neural Nets, Glial Nets, and Im-

mune Nets, of a Superior Organism Category (Holo-net).

For the dynamic equations we will use the Chua sys-

tem that initially contemplates an electronic circuit that

achieves chaotic dynamics (Figure 6). The system was

implemented in Matlab c

, Simulink c

. This circuit has

a memristor component that according with [41] intro-

duces the current according to the parameters shown in

the circuit,

IM(t) = W(φ(t))V1(t) (34)

where W(φ(t)) is the incremental memductance deﬁned

as

W(φ(t)) = dq(φ)

dφ (35)

FIG. 6. Chua’s circuit with Memristor

The dynamics of this circuit with a passive memristor

(ﬂux-controlled memristor and negative conductance) are

given by the ODE (Ordinary Diﬀerential Equations) sys-

tem

dV1(t)

dt =1

C1

[V2(t)−V1(t)

R+GV1(t)−W(φ)V1(t)]

dV2(t)

dt =1

C2

[V1(t)−V2(t)

R+IL(t)]

dIL(t)

dt =1

L[−V2(t)−RLIL(t)]

dφ(t)

dt =V1(t)

(36)

If we translate these variables to the Info space:

V1=ImmuneNet .Info

V2=GliaN et.I nf o

IL=NeuralN et.I nf o

φ=V irus.Inf o

That reﬂects the action of the virus (Memristor com-

ponent in the Chua system) over the three sub-networks,

Immune system, Neural system and Glial System.

If we now transform the variables in dimensionless pa-

rameters according to [42]

x=V1y=V2z=ILw=φ,

C2= 1 R= 1

α=1

C1

β=1

Lγ=RL

L=G

(37)

The system (36) is transformed to

dx(t)

dt =α(y(t)−x(t) + x(t)−W(w)x(t)

dy(t)

dt =x(t)−y(t) + z(t)

dz(t)

dt =−βy(t)−γz(t)

dw(t)

dt =x(t)

(38)

where piecewise-linear function W(w) is deﬁned by

W(w) = a for |w|<1

=b for |w|>1(39)

Now, to obtain our system dynamics, we will change the

ﬁrst derivative by fractional ones.

0Dq1

tx(t) = α(y(t)−x(t) + x(t)−W(w)x(t)

0Dq2

ty(t) = x(t)−y(t) + z(t)

0Dq3

tz(t) = −βy(t)−γz(t)

0Dq4

tw(t) = x(t)

(40)

To solve numerically the system of fractional deriva-

tives equations we will use a method described by Petras

14

[43]. It is a time domain method that leads to equations

described by

x(tk)=(α(y(tk−1)−x(tk−1) + x(tk−1)

−W(w(tk−1)x(tk−1))) hq1−

k

X

j=ν

c(q1)

jx(tk−j)

y(tk)=(x(tk)−y(tk−1) + z(tk−1)) hq2

−

k

X

j=ν

c(q2)

jy(tk−j)

z(tk)=(−βy(tk)−γz(tk−1)) hq3−

k

X

j=ν

c(q3)

jz(tk−j)

w(tk) = x(tk)hq4−

k

X

j=ν

c(q4)

jw(tk−j)

(41)

Where

W(w(tk−1)) = a for |w(tk−1)|<1

=b for |w(tk−1)|>1(42)

The total time for simulation that we will use is 200

seconds and his the interval of calculation that is 0.005

seconds. So, total number of points per curve for each

calculation is forty thousand.

The coeﬃcients c(qi)

jj= 0.1....) are binomials (−1)j(qi

j

that we will calculate according to the expression [44]

c(q)

0= 1, c(q)

j= (1 −1 + q

j)c(q)

j−1(43)

Value of the parameters for calculation are:

α= 10, β = 13 γ0.1= 1.5a= 0.3b= 0.8

q1= 0.97 q2= 0.97 q3= 0.97 q4= 0.97

x(0) = 0.8y(0) = 0.05 z(0) = 0.007 w(0) = 0.6

The results are presented in the Figures (7, 8, 9, 10).

FIG. 7. Holo-Network Axes (x, y, z)

FIG. 8. Holo-Network Axes (w, x, y)

FIG. 9. Holo-Network Axes (w, x, z)

FIG. 10. Holo-Network Axes (w, y, z)

15

In the complementing materials( [45], [46], and [47])

you will ﬁnd three videos that contain the graphics pre-

sented related to ”Virus.Info” variable interactions, with

a variation of the αparameter.

C. Shorterm vs. longterm information processing

In the previous sections we have focused on the math-

ematical approach to the Info space, here we will present

brieﬂy how the short and long time scales of the space

are processed.

If we start considering organism’s biomolecular pro-

cesses, they would be the substrate that could allow to

implement one of the instantiations of the Info Functor

previously described.

Anther clear instantiation of the Info Functor is the

structure of DNA/RNA as we have used in our Toy model

with the Virus role.

As we have seen in the dynamics graphics (Virus.Info),

ﬁgures (8, 9, and 10) introduces two separate regions on

the phase space. The permanence of the system in one of

the regions is abruptly broken with a transfer to the other

region that takes place after a usual long time, compared

with the time of the transfer.

We can speculate that dynamics that happen within

the regions are short term processes based on the

biomolecular mechanisms while the transfer region corre-

sponds to DNA/RNA processes. These last plastic pro-

cesses could be genetic or epigenetic.

In our toy example Virus.Info, DNA/RNA instantia-

tion gets the system from its initial state to one of the re-

gions. From this moment metabolic processes in the three

networks (Holo-Net) lead the dynamics of the system un-

til a new DNA/RNA process happens, usually replication

of the Virus.Info, but could also be DNA/RNA processes

on the Holo-Net. From this moment a totally diﬀerent

dynamic of the system takes place. It is not highly spec-

ulative to associate these dynamical changes to the dif-

ferent states that organisms suﬀer under virus stress.

We will not go any further in this part of the analysis

because a more detailed mathematical model beyond the

toy model presented is necessary.

Only as a matter of quantiﬁcation of the diﬀerent time

scales for the diﬀerent Nets.Info, we see that for the

biomolecular/metabolic processes it takes times from mi-

croseconds to seconds, while plastic processes take place

during hours (circadian) or several days.

V. PRACTICAL DEVICE IMPLEMENTATION

A. Light and sound correlated as an input stimuli

to the network

The device that we have developed proﬁting from the

concepts presented in the previous sections, is mainly a

Light and sound generators that focus the visual and ear

channels of the patient in the two signals that we can

advance are highly correlated.

The current prototype is shown in the picture of the

Figure (11 )

FIG. 11. Device Sound & Light (Prototype)

showing the light matrix and the earphones for the

sound.

For the ﬁrst implementation of the device we are work-

ing in an Info related open loop, with the meaning that

the light and sound signals generated are independent of

the state in which that the holo-net system is taken.

The evaluation in this initial approach is more focused

on the demonstration of the reaction of the Holo-Net

system, more than the optimization of the actions and

potential improvements on the health (dynamics) of the

patient. As we will see in the next section, there are

health conditions, such as Epilepsy, in which an immedi-

ate change in the dynamics of the system entered after

the Epileptic seizure is more than enough to prevent ma-

jor damages.

Of course, a subsequent close-loop device will be devel-

16

oped after we obtain the results of the tests with the cur-

rent prototype. Unfortunately for this second device the

current status of the technique will not allow us to have

an on-line feedback from Immune-Net.Info and even will

be diﬃcult (Imaging techniques) from Glia-Net.Info, but

we hope that a full set of interfaces with these systems

will be developed in the near future. Even the interfaces

with Neural-Net.Info are in the ﬁrst stages but the news

coming from this front are very exciting (see: Neuralink

presentation [48]).

The characteristics of the sound and light waveforms

as well as the times and protocols for its implementation

are at the time being under industrial secrecy and we are

not able to fully disclose them in the present paper.

An example of a treatment with one of the devices is

shown in the picture of the ﬁgure (12.

FIG. 12. Treatment with the Device Sound & Light

B. Potential eﬀects in Epilepsy and other

neurological diseases

As we have advanced in the previous section the initial

tests of the device are being made with patients suﬀering

from Epilepsy.

Epilepsy is a grave central nervous system illness that

is normally related to other complex neurological disor-

ders, resulting frequently in the development of recur-

rent seizures. These seizures can potentially result in the

death of the patient as they usually occur without previ-

ous warning.

The WHO (World Health Organization) estimates that

epilepsy aﬀects more than 50 million people worldwide.

Taking into account that the total world population is

about 7 billion there are 7 people out of 1000 suﬀering

epilepsy in the world. Epilepsy has an enormous inﬂuence

on the quality of life of the patient with collateral eﬀects

such as body lesions related to seizures and in the case of

treatment with antiepileptic drugs, depression, anxiety

and other adverse neurological disorders induced by the

treatments.

An epileptic disorder can be classiﬁed as partial, gen-

eralized, or unclassiﬁed nature depending on the diverse

types of seizures suﬀered. The most damaging is the sta-

tus epilepticus (SE) in which the seizure is sustained for

prolonged time. SE has been deﬁned as a prolonged gen-

eralized tonic-clonic seizure persisting for more than 5

minutes (or 10 minutes for focal seizures with or without

impairment of consciousness), or more than one seizure

within a period of 5 minutes without recovery of con-

sciousness in between.[49].

Incidence of SE, according to diﬀerent estimates, varies

from 6.8 to 41 cases per 100.000. Taking into account

that 60% of the total cases are refractory to drug treat-

ment and death is common between the SE patients there

is so far a dim future for these patients.

We have focussed our tests on the use of the device

immediately after the ﬁrst symptoms of the seizure are

present.

For the current experiments of light stimuli methods

in the Holo-Net.Info system we introduce the work made

with mice by Kristie M. Garza, Lu Zhang, Ben Borron,

Levi B. Wood, and Annabelle C. Singer. [50]. The results

are summarized; “Many neurodegenerative and neuro-

logical diseases are rooted in dysfunction of the neuroim-

mune system; therefore, manipulating this system has

strong therapeutic potential. Prior work has shown that

exposing mice to ﬂickering lights at 40 Hz drives gamma

frequency (40 Hz) neural activity and recruits microglia,

the primary immune cells of the brain, revealing a novel

method to manipulate the neuroimmune system. How-

ever, the biochemical signalling mechanisms between 40

Hz neural activity and immune recruitment remain un-

known.”

Other remarkably interesting experiment was made in

this case with zebra ﬁsh by Carmen Diaz Verdugo, Sverre

Myren-Svelstad and several collaborators. They studied

[51] Glia-neuron interactions that underlie state transi-

tions to generalized seizures. The brieﬂy conclusion of

the work was “The transition from a preictal state to a

generalized seizure is an abrupt process accompanied by

strong alteration of the functional connectivity between

17

glial and neural networks”.

Even if the subjects of the experiments were mice and

zebraﬁsh and not humans the results for our thesis are

appealing and induce a high positive potential for the

results of the device.

Other tests related to other neurodegenerative dis-

eases, like Alzheimer disease, will be carried on in the

near future even as in this case the positive eﬀects of the

treatment could take more time to be acknowledged.

C. Potential eﬀects on COVID-19 patients

As a supplementary material due to the current dis-

mal situation caused by the COVID-19 we have been

committed to start without any delay a battery of tests

with special waveforms of sound and light in the device

to test in COVID-19 newly conﬁrmed patients.

One of the most characteristic symptoms of the disease

is the acknowledgement that patients cannot smell the

food nor taste that they are eating. Normally these early

symptoms are the ﬁrst indicators of the virus SARS-Cov-

2 in action in the human body.

Therefore, this action links, as in our toy model, the ac-

tion, information processing of a virus with the networks

of the nervous system. Several papers of the thousands

edited during the last months regarding SARS-Cov-2,

take into account the potential short term and long term

problems caused by the disease in the nervous system but

for the moment only general treatments aimed at known

therapy paths are being tested.

One of the points that we want to raise in this no-

ticeably short justiﬁcation for the ongoing tests of the

device in COVID-19 patients is the cytokine release syn-

drome (CRS)–induced ARDS and secondary hemophago-

cytic lymphohistiocytosis (sHLH). This cytokine “storm”

was observed in patients with SARS-CoV and MERS-

CoV in 2008 as well as in leukaemia patients receiving

engineered T-cell therapy. The suppression of the storm

was the base for the ﬁrst treatments in March and April

this year 2020, based on therapeutics suppressing CRS,

such as tocilizumab, that entered clinical trials with some

positive eﬀects but with a severe and grave secondary

ones.

Our current approach will try to change the dynamics

of the virus via the eﬀects that we will induce in the ner-

vous system (neural, glial, and potentially aﬀecting the

extracellular matrix), with the Sound and Light device

therapy.

The ﬁrst essays will be done in a way patients newly

diagnosed with COVID-19 will be subject, with their to-

tal and positive consent, to a session of 15-30 minutes

interaction with the device before their sleeping period.

The next morning, they will be checked again for the

virus absence or presence ideally quantitatively and qual-

itatively.

VI. CONCLUSIONS AND FURTHER WORK

A. Summary

Further work must be done with the aim to:

1. Develop more complex models of Holo-Nets and

evaluate the ﬁtting with natural occurring cases.

2. Evaluate potential implementations of more ad-

vanced devices according with the results obtained

from complex models.

3. Study advanced artiﬁcial implementations for the

dynamic models going beyond electronic circuits.

4. Make deeper studies in the relation Quantum Me-

chanics and Info Mechanics, that could help to es-

tablish a bridge between quantum computation and

Holo-Network dynamics. At this point a potential

implementation of quantum algorithms over Holo-

Networks is the most appealing result envisaged.

B. Disclaimer

All rights reserved. All content (texts, illustrations,

photos, graphics, ﬁles, designs, arrangements etc.) not

referenced to third parties on this paper are protected by

copyright and other protective laws. The contents of this

paper are to be used only in accordance with Intellectual

rights International regulations.

Without the explicit written permission of the compa-

nies of the Author,RQuanTech (Geneva) and Criptosasun

(Madrid), it is prohibited to integrate in whole, or in part,

any of the protected contents published on this paper into

any materials or to use them by any other means.

ACKNOWLEDGMENTS

The Author wishes to acknowledge the support of the

teams of his companies , RQuanTech and Criptosasun

for their full support with the corrections and marvelous

ideas.

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18

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methods used in the paper. Diﬀerent articles related to

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