TRACING ONE NEURON ACTIVITY FROM THE INSIDE OF ITS STRUCTURE

8/12/2019 TRACING ONE NEURON ACTIVITY FROM THE INSIDE OF ITS STRUCTURE

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International Journal on New Computer Architectures and Their Applications (IJNCAA) 2(1): 8- 22

The Society of Digital Information and Wireless Communications, 2012 (ISSN: 2220-9085)

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Tracing one neuron activity from the inside of its structure

Luciana Morogan

Military Technical AcademyBucharest, Romania

[email protected]

ABSTRACT

As new areas of neural computing are

trying to make at least one step beyondthe definition of digital computing, the

neural networks field, was developed

around the idea of creating models ofreal neural systems. The key point is

based on learning rather than

programming. We designed the modelwe present in this paper, based on the

the needs introduced by new trends in

neural computing. We created a model

for information processing insideneurons. It is constructed, from a neural

inside point of view, as a feedback

system that controls the flow of

information passing through the neuron.Processes of internal neural learning take

place.

We translated them as processes oflearning at molecular level. Our work

is disseminated into the construction of

an algorithm of learning, that weintroduce in the end of this paper.

Keywords-Neural networks; hybridintelligent systems;nature inspired computing technique;

evolutionary computing.

I. INTRODUCTIONIn classical neural networks (see [1], [2],

[3], [4], or

[5]) the electrical information exchangedbetween neurons is usually viewed as

physical information. For our model,

this was not enough because of our needin dealing with the internal processes

taking place into one neuron device

from a neural-like network. In order to

materialize this kind of internalinformation, we introduced the

molecular information as objects or

object compounds (we alreadydetailed this aspect in [6] and [7]). They

are viewed as both computational and

information carrying elements. Inthis manner, one neuron device

represents itself a complex system for

information processing making use of

object molecules that offer support forcomputational processes, [6]. A very

important role, played by this objects,

can bealso underlined. Specific internal

structure, [6], of each neuron can bemodeled as neuron nucleus containing

its genetic material (the role of genetic

control is presented in the second sectionof this article). This can be represented

in our model as different objects that are

grouped together in order to expresssome specific properties that can

materialize the neuron genetics. The way

this kind of information can berepresented and used will be furtherdetailed in the forth section of this paper,

while in the third section we introduce

the platform on which our model is

based on. We also need to introduce, inthe second section, some genetic

background in order to explain our

choices in one neuron potentialconstruction. The way information


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can be controlled in modeling the neuronbehavior and an algorithm of the

learning process at the molecular level

are also presented in the final section.A network is created by placing a finite

set of neuron devices in the nodes of a

finite directed graph. A neural-like

system structure is represented by anensemble composed of network units. In

addition, we create a parallel distributed

communication network of networks of

neurons. There are possible two types ofinteractions translated as neural

communication: local (between neurons)

and global interactions (betweennetworks, or neurons of different

networks). The mean by which the

communication can be realized is theexistence of a finite set of directed

links between neurons in this graph. The

directed links are called synapses. For

this model, it is necessary also thecommunication with the surrounding

environment. We also refer with the

same term - synapse - to the directed

communication channel from one neuronto the environment and vice-versa we.

The system may evolve or involve

depending on the synaptic creation ordeletion from one network during the

computations. The design of the binding

affinities between neurons was alreadydescribed in [8], [9] and [10], as not any

neuron can bind to any other neuron.

Connections in a network are dependent

on sufficient quantities of the

corresponding substratesinside the neurons and the

compatibilities between the typesof the transmitters1 in the presynaptic

neurons and those of receptors in the

posts-synaptic neurons. Synapseweights, representing degrees of the

binding affinities, are assigned

to each synapse. The value of thebinding affinity degree is considered to

be zero for neurons with no synapse

between them and for the so-calledsynapses between neurons and

the environment.1The transmitters and receptors are

specific protein-like objects inthe pre and postsynaptic neurons

respectively, in accordance with the real

biological transmitters and receptors

proteins ([11], [12], [13]).The main idea (we already introduced in

[14]) is focused on potential changes (or

mutations), that may occur into oneneuron, influencing the expression of its

genetic material. Those changes, as in

real biological cases, will influencethe binding degrees (between neurons

and between neurons and the

environment) inducing this way both

modified synaptic communicationsand/or modifications into the

network structure (see [9] and [10]).

Analyzing the neuron

activity rate, we can control the flow ofinformation and the effect of the events.

A result resides in the fact that

modifications into the synapticcommunications will also determine the

neuron to adapt to its inputs and

modeling this way its behavior (firsttime we mentioned this idea in

[8]). The neuron ability to adapt to its

inputs leads to a

process of learning at the molecular level

and defines the neuron as a feedbackcontrol system. Implications involved

by such a dynamic neuron design overthe information processing are

introduced.

II. VIEW FROM INSIDE ONE

NEURON: REPRESENTATION


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AND THE ROLE OF GENETICCONTROL

In order to understand the processes we

model into ourneuron functional designwe felt the need to introduce some

biological explanations about what

means and what is the meaning of

genetic control (see [15] or [16]).Neurons are controlled by DNA strands

forming chromosomes in the nucleus of

the neuronal cell. All neurons have the

same chromosomes and the same DNA.What makes the difference then? The

answer is found in the fact that

differentiation is realized at the level ofDNA portions that are being expressed

by different genes depending on proteins

with which they are associated anddelimited by. One may say that the

proteins are those that governs the

neuron structure and functionality. Two

classes of genes are already known:structural genes that produce both

functional proteins and the messenger

RNA (mRNA), and regulatory genes that

control precisely the expression of firstclass of genes, the structural genes. The

process of mRNA production (leaving

the neuron nucleus in forming proteinsor enzymes in the cell by a process

called translation) is called transcription.

mRNA production from a DNA templateis catalyzed by an enzyme. The process

of

transcription begins with the attachment

of this enzyme by a regulatory region in

the DNA called promoter. At anotherlocation into the DNA is found a so

called super-gene that codes theregulatory genes. For short, during the

processes of neuronal differentiation and

neuronal development, one super-regulatory gene can govern other genes

which in their turn can activate sub-families of regulatory genes.

In biology, there are processes that lead

to internal neuronal changes in case ofindirectly activation of the

transfer channels of the postsynaptic

neuron (see [15] or [16]). To explain the

biological phenomena, we must saythat the transmitters action into the

postsynaptic neurons are not dependent

on their chemical nature, but the

properties of the receptors of the bindingneurons. There are the receptors that

determine if the synapse is excitatory or

inhibitory2. The receptors are alsodetermining the way transfer channels

are activated. In case of direct gating

they are quickly producing synapticactions. In case of indirect gating the

receptors are producing slow actions that

usually serve for neuron behavior

modeling by modifying the affinitydegree of the receptors. From our point

of view, we chose for our model

precisely this second case as the

biological secondary messenger(we refer to [16] for the explanation of

this term) has a profound effect over the

alteration of genes expression. In thismanner, we chose to model bellow

learning at a molecular level. This is

defined as the process thatsimulates the neuron device ability to

adapt to its inputs by

altering the expression of its genes.

The neuron input, represented bytransmitters-like objects, is modeled

bellow as being capable of determininginternal changes both into the neuron

structure and influencing the neuron

device behavior. To the neuron device agenetic internal timing is associated. In

normal conditions, this internal timing


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leads to a gene activation. In abnormalconditions it leads to gene malfunction.

The robustness comes at two stages:

non-functional processes are changeableby feed-back mechanisms and non-

functional neurons are killed.

III. BACKGROUND FOR MODELINGTHE NEURON BEHAVIOR

The following definitions in this section

were already introduced by the authorsin a series of already published papers

(see [8], [9], or [10]). In this section, we

chose to remind them as they representthe background for modeling the neuron

behavior.

One neuron in the network represent asystem for information processing. It

offers objects (biological protein

molecules equivalents) as a support for

computational processes. We introducedthe tree - like architecture of such a

neuron in [6] as the construction (n) =

(0,j0,1,j1,,r,jr) where the finite

number of r compartments are structuredas an hierarchical tree-like arrangement.

Their role is to delimit protected regions

as finite

2An excitatory synapse is a synapse in

which the spike in the presynaptic

cell increases the probability of a spikeoccurring in the postsynaptic cell.

spaces and to represent supports for

processes involving the objects that are

embedded inside ([6] is recommendedfor more details).0;j0- found at the j0=0 level of the tree root - is the inner

finite space of the neuron (containing allthe other regions). The inner hierarchical

arrangement of the neuron is represented

by the construction

1,j1.. r,jr , such as for j1 jrnatural numbers, not necessarily disjoint,

with jk r for k {1 ,.., r}, wedefine the proper depth level of the k - thcompartment.We may have a maximum of r inner

depth levels. The initial neuronal

architecture of n and the objects found in

each of its compartment regions aredescribed by the initial configuration.

Formally, this is represented by

(0,j0: o0,1,j1: o1,,r,jr: or) .

For each k,jk, with k {1,..r}, okmay be written under the form of thestring objects 1

m1 2

m2.p

mp,

where p N is finite. Each imi

represents the quantity mi(miN) of

aifound in the region k,jkof the neuronn. If mi= * , then we face an arbitrary

finite number of copies of ai.

The model of synaptic formation was

already introduced in [9] and [10]. We

underline a restriction coming frombiology field, and representing the main

idea, we focused on, in modeling thesynaptic formation: not any neuron can

bind to any other neuron. We consider

N, the finite set of neurons in a networkcomponence, network contained

into the system structure. The following

finite sets are defined: (1) OCniis the set

of all clusters of objects

compounds/complexes for the neuron ni

and OCnjthe set of all clusters of objectscompounds/complexes for the neuron

nj (OCni OCnj 0). (2) OCntiOCnithe set of all neurotransmitters for ni and

OCrj OCnj the set of all receptors fornj . (3) For manipulating more easier the


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sets in (2), we refer to the set ofneurotransmitters for ni by

OCntand the set receptors for nj by OCr.

A finite set L of labels is considered.There is a labeling function of each

object compound/complex defined by l :

OCn L, where

n is ether ni, or nj . Tr= l(OCnt) 2Listhe set of labels of all neurotransmitters

for ni and R = l(OCr) 2Ltheset of labels of all receptors for nj.

We consider T =

{i.\i N , =1/k , k N *fixed}

The set of discrete times. For all i OCnfound in the regionk,jk of n and mi its multiplicity, thequantity found into the substrate (in the

region k,jkof neuron n) of one organiccompound/organic complex ai at thecomputational time

t, t T is the function Cn:k,jk: T OCn N {*}defined by

Cn:k,jk(t, i) =

{mi , if mi represents the number of

copies of ai into the substrate

*, if there is an arbitrary finite number

of copies of ai into the substrate .}

A few properties of quantities of organic

compounds/

organic complexes into the substrate are

presented below:1) The quantity of substrate, at the

computational time

t , t T found into k,jk is

Cn:k,jk (t) = Cn:k,jk (t, 1) + Cn:k,jk (t,2) + Cn:k,jk(t, p) =

()

for all mi N. If there is i {1 , ., p}such as

Cn:k,jk(t, i) =*, then Cn:k,jk(t,) = * .

2) If at the moment t we Cn:k,jk(t, i) =

miand at a later time t'a new quantity mi

of aiwas produced, supposing that in the

discrete time interval [t, t'] no object ai

was used (one may say consumed), then

Cn:k,jk(t', i)=Cn:k,jk(t, i) + m'i= mi+m'i

3) If at the moment t we Cn:k,jk(t, i) =

miand at a later time t'a new quantity mi

of ai was consumed, supposing that in

the discrete time interval [t, t'] no object

ai was used (one may say consumed),

then

Cn:k,jk(t', i)=Cn:k,jk(t, i) - m'i= mi-m'i

We make the observation that

mi-- m'i 0 (mi m'i)

because it can not be consumed more

than it exists.

4) ) If at the moment t we Cn:k,jk(t, i)= miand at a later time t

'we have

Cn:k,jk(t, i) = m'isupposing that in the

discrete time interval [t, t'] no object ai

was produced or consumed, thena) if mi< m'iwe say that there is a rise in

the quantity of ai;

b) if mi > m'i we say that there is a

decrease in the quantity of ai;

c) if mi = m'i we say that no changes

occurred in the quantity of ai.


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The binding affinity depends on a

sufficient quantity of substrate and the

compatibility between the type oftransmitter and receptor type at time t of

synaptic formation (t T).The sufficient quantity of substrate is afair ratio admitted r between the number

of neurotransmitters released by ni

into the synapse and the number of

receptors of the receiver neuron nj .

Without restraining generality, we

consider the sufficient quantity of

substrate at time t (t T) of synapticformation as a boolean function of the

assessment ratio evaluation. For trans Trwith m its multiplicity (where for

OCntwith l() = trans and trans Trwehave Cni:0,0(t, ) = m) and rec R with

n its multiplicity (where for b OCr with

l(b) = rec and rec R we have

Cni:0,0(t, ) = n), we defineq

t: Tr R {0,1},

qt(trans, rec) =

{ 0 , if m/n r

1 , if m/n = r }The compatibility function is a

subjective function Ct:

Tr R {0,1},such as for any trans

Tr and any rec R,

Ct(trans; rec) = {

0 , if trans and rec are not compatible

1 , otherwise } .

The binding affinity function is the onethat models the connection affinities

between neurons by mapping an affinity

degree to each possible connection. We

consider Wt N the set of all affinity

degrees (at time t) ,

wt={ w

tijN ,i,j{1,2,,|N| , ni, nj

N }

For any ni , nj N , trans Tr (the

transmitters type of neuron ni), rec R(the receptors type in neuron nj ), C

t

(trans, rec) = with {0,1} , andq

t(trans, rec) = y with y {0,1}, we

design the binding affinity function as

a function Atf:

(N N) TrR) Ct(TrR) q

t(TrR)

W where

Atf

((ni, nj); (trans, rec),, y) ={ 0 , if x= Ct(trans, rec) =0 y{0,1}0 , if (x= C

t (trans, rec) =1) (y= qt

(trans, rec) =0)

wtij, if (x= C

t(trans, rec) =1) (y= qt

(trans, rec) = 1)( wtij 0) }Theorem 1 (The binding affinity

theorem): For Pi Pbnia finite set ofbiochemical processes of neuron

ni by which it produces a multiset of

neurotransmitters of type trans (trans Tr), at the computational time t (t T),

and, in the same time, for Pj Pbnja finite set ofbiochemical processes of neuron nj by

which it produces a multiset of receptors

of type rec (rec R), we say thatthere is a binding affinity between ni and

njwith the binding affinity degree wtij (

wtij 0) if and only if there is

w

t

ijWt*

such as A

t

f((ni, nj), (trans,rec), 1, 1) wtij .

Definition 1: For any ni,njN, trans Tr(the transmitters type of neuron ni) and

rec R (the receptors type in neuron nj)at time t, if there is a binding affinity


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between ni and nj with the binding

degree (wtijW

t) and w

tij 0

then we say that there is a connection

formed from ni to nj. This connection iscalled the synapse between the two

neurons and it is denoted by synij. Each

synapse has a synapse weight wtij= w;w

N*. If instead of ni we have e then thesynapse synej represents the directedlink from the environment to the neuron

njand if instead of njwe have e then the

synapse synie represents the directed

link from the neuron ni to theenvironment. The synapses weighs are

considered to be zero (wtie= wtie = 0).Definition 2: We say that there is no

connection from neuron ni to neuron nj ifand only if w

tij= 0.

Theorem 2 (One way synaptic direction

theorem): For Pi Pbni a finite set ofbiochemical processes of neuron

ni by which it produces a multiset ofneurotransmitters of type trans and

receptors of type rec', at the

computational time t (t T), and in thesame time for Pj Pbnj a finite set ofbiochemical processes of neuron nj bywhich it produces a multiset of receptors

of type rec andneurotransmitters of type

trans', if there is wtijin W

twith

wtij0 such as At

f((ni, nj), (trans,rec), 1,

1) = wtij then there is no w

tijin W

twith

wtij0 such as At

f ((ni, nj), (trans',rec'),

1,1) = wtij . (If there is w

tijin W

tsuch as

Atf((ni, nj), (trans',rec'), 1,1) = w

tij then

wtij= 0 .

IV. A WAY OF INFORMATIONCONTROL IN MODELING

THE NEURON BEHAVIOR

The neuron genome is represented as amemory register of the result of the

previous information processing and the

effect of events. The temporary memoryof all informations of both the cellular

and surrounding environment and the

partially recording of the results of the

previous computations are translated asDNA sequences. From the information

processing point of view, for each

neuron in N, of great importance are

considered:1- the neuron architecture and the

corresponding initial neuron

configuration. For each compartment,initial objects are introduced and

processes handling objects

are defined;2- the initial quantities found into the

substrates of each compartment;

3- the neuron genetics represented

bellow byneuron genome in terms of totality of

all genes encoders of genetic

information that is contained into DNA.

We will refer to it by G;genetic code represented as

* the set G = (g1,..,gk); k N of genesin the genome componence along with

their appropriate expressions (later

defined in this paper);* the controllers of genes expression as a

set of objects representing the class of

genes controllers. We denote this class

by C = (c1,.., cq); q N;genetic information represented by all

information about the cellular andexternal environment.An internal timing that sets up the

neuron activity rate is considered. An

important observation must beunderlined: this internal timing must not

be confound with the computational


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timing representing the time unit forboth internal processing and the neural

network manner of external electrical

exchanges of spikes between neurons.One may say that the computational

timing may be the real time. The

reason is found again in biology where

molecular timescale is measured inpicoseconds (10

-12s). On such timescales

chemical bonds are forged or broken

developing this way the physical process

that we call it life.The internal timing is characterized by

an intracellular feedback-loop. It is

measured between consecutiveactivations of groups of genes. On

activation, the expression of a group of

genes encode proteins after which theywill be turned off until the next

activation. The computational timing

we defined as the set

T = {i.\i N, =1/k , k N *fixed}

meanwhile the internal timing TnR,, TnR

T, that sets up the neuron activity rateasTnR= { Tk\kN , Tk= ti, iN tiT}

where Tk represents the moment of a

gene (group of genes) activation.Corresponding to the computational time

ti, i N, in parallel, we may deal with anactivation of a gene (group of genes) at

time Tk(Tknotation = ti) and then

the gene (group of genes) is turned off at

a later moment in time ti+p; 0 < p < j, p,

j N. It is possible that not onlyone computation may take place until the

next activation either of the same gene(group of genes) or a different

one (group) at Tk+1 (Tk+1notation = tj ).

The internal timing is measured betweenconsecutive activation of the same

gene (group of genes). If in the discretetime interval

[Tk ,Tk+1] the neuron device computes

starting from ti to tj (ti, ti+1,.. tj ), thenthe time interval will represent the

internal neuron timing characterizing the

gene (group of genes) that were

activated at Tk. This can be illustrated inFigure 1. We generalize the assumptions

presented in the above statements by

presenting two successive activations of

all genes of genome G. We underline thefact that if the arrangement of the genes

into the genome is (g1; : : : gk),

bellow they will be arranged followingthe moment of theirexpression. We

denote by Tig the time of the activation

of

Figure 1. Internal timing setting up theneuron activity rate (measured

between consecutive activation of the

same gene/group of genes) is not the

computational timing. In parallel,corresponding to the computational time

ti the activation of a gene (group of

genes) may take place at the internal

time Tk. At time ti+p; 0 < p < j the gene(group of genes) is (are) turned off. Not

only one computation may take place

between consecutive activations of thesame gene (group of genes) or a

different one (group).

It is considered Tk+1such as Tk+1= tj. Inthe discrete time interval [Tk; Tk+1] the


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neuron device may compute startingfrom ti; ti+1.. tj . the gene g and Tg

i +1

the time of the next activation of the

same gene g.

For k1 + k2 ..+ kr = k and for each gjl

with j {1,r} and l {1,.,kj}

representing genes that are activated

in the same time Tij, we have:

at one activation of all genes:at time Ti1 , the genes considered to be

activated are represented by (g11,

g12,, g1k1 ). Their corresponding

activation times, representing in fact

the same Ti1

, are represented by

( Ti1

g11, Ti2

g22,..,Ti2

g2k2)

at time Tir, the genes considered to be

activated are represented by (gr1 , gr2

,, grkr) Their corresponding

activation times, representing in factthe same T

ir, are represented by

( Tir

gr1, Tir

gr2,..,Tir

grkr)

at time Ti1+1

, the genes considered tobe activated are represented by (g11 ,

g22 ,, g1k1) Their corresponding


i1+1 , are represented by

( Ti1+1

g11, Ti1+1

g12,..,Ti1+1

g1k1)

at the next activation of all genes:at time T

i2+1 , the genes considered to

be activated are represented by (g21 ,g22 ,, g2k2) Their corresponding


i1+1 , are represented by

( Ti2+1

g21, Ti2+1

g22,..,Ti2+1

g2k2)

at time Tir+1 , the genes considered to

be activated are represented by (gr1 ,gr2 ,, grkr) Their corresponding


ir+1 , are represented by

( Tir+1

gr1, Tir+1

gr2,..,Tir+1

grkr)

Two observations are arising from these

considerations:

1) For i1, i2 ,., ir such as the orderrelation that is considered between the

internal times of genes activations is

represented by Ti1

< Ti2

< .< Tir,

is not necessarily that the next genesactivation internal times preserve the

same order relation (is not necessarily

that T

i1+1

< T

i2+1

< ..< T

ir+1

).2) In case of a gene suffering some

changes because of mutations

(mutation case will be discussed later

in this paper), for i {i1,..ir} the

gene gjlactivated at time Ti

gjlis not necessarily the same gene gjlactivated at T

i+1gji .

Considering the statements above, we

can now define the neuron activity

rate.Definition 3: For the neuron n (n N)

the neuron activity rate is the mapping

nRate : TnRN N ..N

(the cartesian product of N .. N is

considered of k times) that, for allgenes (g1, .., gk) composing the

neuron genome, is defined by

nRate (Ti+1, i+1) = (Tg1i+1

-- Tg2i, Tg2

i+1+

Tg2i,Tgk

i+1-- Tgk

i)

We introduce next a few properties of

one neuron activity rate.Property 1: Genes that are activated at

the same step are the genes from the

same group. There are some values in

the array defining the neuron activityrate that may be equal for genes that

are activated at the same step. The

next two properties are presenting


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cases when some changes may occurinto the neuro functionality (cases of

mutation, of at least one of the genes

involved into that processes,influencing the neuron activity rate).

In this manner, there are considered

the iteration steps nRate(Ti+1, i + 1) =

Tg1i+1

- Tg2i , Tg2

i+1- Tg2

i ,Tgk

i+1-

Tgki)

and

nRate(Ti+1, i + 2) =

Tg1i+2

- Tg1i+1

, Tg2i+2

- Tg2i+1

,Tgk

i+2- Tgk

i+1)

Property 2: If there are some valuesTgj

i+1- Tgj

i= Tgl

i+1- Tgl

i , then it is not

necessarily that at the next step

of the iteration Tgji+2

- Tgji+1

= Tgli+2

- Tgli+1

Two meanings are arising from this

property. Genes from the same group

of genes are activated at the same

time. This is the case considered inProperty 2: genes gj and gl are

activated at the same step of the

iteration. From Property 1 all values in

the neuron activity rate array ofdifferent genes from the same group, at

any iteration step, should be the same.

If any differences occurs, then we facethe case in which mutations occurred to

at least one of the genes in the same

group, if Tgji+2

- Tgji+1

Tgli+2

+ Tgli+1

Wecan obtain the influences over the

neuron activity rate can be seen by

analyzing Tgji+2

- Tgji+1

Tgli+2

- Tgli+1

Property 3: If it is considered to beaffected the expression of one gene,

let us say it would be gj , then we aredealing with some changes into the

neuron activity:

mutations occurred, and in thiscase gj can be analyzed.

Tgji+1

- Tgji Tgl

i+2+ Tgl

i+1

Property 4: (This property can be seen as

a conclusion drawn from the previousproperties.) There can be detected

some changes into the neuron

functionality by analyzing the

differences that may occur into theneuron activity rate values.

In the design of the system, at the level

of individual neuron, a finite set of

symbol states is assignedis composed by a finite set of functional

states and the non-functional state that

corresponds to the neuron death.The neuron activity rate function

provides the neuron states by the aid

of a mapping f such as at eachiteration step

f : N N , where the

cartesian product of k times represents

the array containing the returnedvalues of the neuron activity rate

function. The functional states are

mapping levels of the neuron

functionality, passing through variousdegrees of functionality.

In order to model the neuron behavior,

we needed a way of controlling theinformation. In this manner a solution

we found was to assign the neuron

with a new ability, the onethat makes it to adapt to its inputs by the

alteration of its genetic material. This

process we defined as the process of

learning at a molecular level. The neuron

device will be viewed as a feedbackcontrol system.

We suppose Tktime of expression of thegene g, encoding protein-like object

a. After expression, g will be

deactivated. For each controller objectc from the set of all considered

controllers C, with c OCn, there is a


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gene g G such as we define theprocess of genetic expression of one

gene as the gene expression function

in relation to the internal timing TkasExp(Tk) : G OCn CD,

Exp(Tk)(g , c) = avar

,

where c is the controller of the

expression of gene g, a is the producedobject and var its multiplicity

(representing the quantity of aproduced at the expression of gene

g). Three remarks arise from the

definition of the genes expression

process:Remark 1: At a specific moment in time

the expression

of a gene under the influence of acontroller encodes only one type of

object. (At a specific moment in time

one gene controlled by a controllercan encode only one object.)

Remark 2: At different activations a

controller can influence the expression

of different genes from a group.(After the deactivation of a gene that

encoded one type of object, the samecontroller can activate, at the next

iteration, either the same gene, either

another gene in the sequence of genesfrom the same group.)

Remark 3: In the same time, there are

controllers that activate differentgenes, each one encoding different

objects. (There is a set of genes that canbe activated in the

same time, each one of them being underthe influence of different controllers

and encoding different types of

objects.)

We consider T1 TnR the moment of agene (group of genes) activation.

Bellow, expressed as moments of T1,we present the properties of the genes

expressions.

Property 5: At time T1 2 TnR, differentcontrollers of different genes from the

group of genes activated in the same

time will lead to different results of

their gene expression functions.Formally, for all ci cj and for

all gigj (i j) such as ci controls the

expression of gi and cj controls the

expression of gj , we haveExp(T1)(gi, ci) Exp(T1)(gj, cj).

Property 6: A gene controlled by

different controllersat different moments in time will lead to

different results of the gene expression

function. Formally, for all ci cj (i j) controllers of the same gene g and

for all TkTnR, k 1 such as T1 < Tk,we have

Exp(T1)(g , ci) 6= Exp(Tk)(g, cj).

Property 7: Different genes controlled by

the samemcontroller at different

moments in time will lead to differentresults of their gene expression

functions. Formally, for all c that cancontrol different genes gi 6= gj and

for all TkTnR, k 1 such as T1 < Tk,we have

Exp(T1)(gi , c) Exp(Tk)(gj , c).

Property 8: A gene controlled by thesame controller at different

consecutive times leads to the same

result of the gene expression function.Formally, for all c that controllers

a gene g, we have Exp(T1)(g , c) =

Exp(T2)(g , c).

Property 9: For all c and for all g, the

expression of gene g being controlled

by c, if at time T1we have


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Exp(T1) (g , c) = var1

and at time T2(such as T2- T1is a positive value not

necessarily T2= T1+ 1) we have

Exp(T2)(g , c) = bvar2

such as we obtaintwo different object types a and b (a

b) with var1 var2, then we

say that the gene g was mutated

(underwent a genetic mutation).A. Algorithm of learning at molecular

level We introduce in this subsection

an algorithm of learning at molecular

level. At neuron level, the process oflearning

finds its definition into the neuron ability

to adapt to its inputs. This adaptationprocess is done by the adoption

of the products objects resulted from the

alteration of the gene(s) expression.The conditions of this to happen are

also introduced. Before presenting the

algorithm, we introduce bellow the

necessary algorithm pre-conditions:the region in the neuron n architecture

in which the objects are to be encoded

by processes of genes expressions

(corresponding to the neuron nucleus)must be chosen; a discrete time

interval is considered. The steps in

this algorithm correspond to theinternal times of activation of the

same group of genes (we consider

working with genes from the samegroup of genes); the gene considered

to suffer a mutation (at step m) is

g, g 2 G and its controller is considered

to be c, c C.

Algorithm of learning at molecular levelStep 0. (Step of internal time T0 of the

group of genes activation. Itrepresents a previous step in which the

controller c of gene g was produced):

The controller c is produced. It will beable to control the expression of gene

g at a next activation of the group ofgenes from which gbelongs to.

Step 1. (Step of internal time T1 of the

group of genes activation, T0 < T1):The gene g controlled by c will

encode object a in var1 number of

copies. The gene g expression

function isExp(T1)(g , c) =

var1.

..........

Step j. (Step of internal time Tj of the

group of genesactivation such as T1< Tj Tm-1, for m

N,m > 1):

The gene g controlled by c will encodethe same object ain var1 number of

copies (for all steps from Step 1 to

Step j). The results of this step areconsidered to be: the neuron activity

rate:

nRate(Tj, j) =Tg1

j Tg1

j-1,. Tg

j- Tg

j-1, Tgk

j-

Tgkj-1

)

the product of the gene g expressionprocess:

Exp(Tj)(g , c) = avar1

the quantity found into the substrate in

region p,jpof neuron n:

Cn:p,jp(Tj) = Cn:p,jp(Tj,) +

() var1

()

..........

Step m. (Step of internal time Tm of the

group of genes activation, Tm-1

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TRACING ONE NEURON ACTIVITY FROM THE INSIDE OF ITS STRUCTURE