A COMPUTATIONAL STUDY OF THE DIFFUSE

NEIGHBOURHOODS IN BIOLOGICAL AND ARTIFICIAL

NEURAL NETWORKS

P. Fernández López, C. P. Suárez Araujo

Institute for Cybernetics, University of Las Palmas de Gran Canaria, Spain

P. García Báez

Dept. of Statistics, Operations Research and Computation, University of La Laguna, Spain

Keywords: Nitric oxide, Artificial neural networks, Cellular signalling, Volume transmission, Diffuse neighbourhood.

Abstract: This paper presents a computational study on a fundamental aspect concerning with the dynamic of nitric

oxide (NO) both in the biological and artificial neural networks, the Diffuse Neighbourhood (DNB). We

apply the compartmental model of NO diffusion as formal tool, using a computational neuroscience point of

view. The main objective is the analysis of DNB by the observation of the AI-NOD and CDNB variables,

defined in this work. We present a study of influences and dependences with respect to associated features

to the NO synthesis-diffusion process, and to the environment where it spreads (non-isotropy and non-

homogeneity). It is structured into three sets of experiences which cover the quoted aspects: influence of the

NO synthesis process, isolated and multiple processes, influence of distance to the element where NO is

synthesized, influence of features of the diffusion environment. The developments have been performed in

mono and bi-dimensional environments, with endothelial cell features. The importance of this study is

providing the needed formalism to quantify the information representation capacity that a type of NO

diffusion-based signalling presents and their implications in many other underlying neural mechanisms as

neural recruitment, synchronization of computations between neurons and in the brain activity in general.

1 INTRODUCTION

The understanding of brain structure and function

and its computational style is one of the biggest

challenges both in Neuroscience and Neural

Computation. To reach this aim it is essential to

know underlying mechanisms of the brain activity.

The activity of the brain has as principal

responsible the BNN, the cellular communication

and learning. Neural recruitment, or synchronization

of computations between neurons, the existence of

an information indexing schema at the Biological

Neural Network (BNN), or the LTP expression, are

aspects that can depend in a direct way on an

underlying neural signalling schema. We understand

that such aspects will be able to have a considerable

role in the information representation capacity, and

so, in the BNN and ANN computation potential.

Among all set of cellular signals that affect

globally the brain activity, the volume transmission

(VT) is located. Its underlying mechanism is the

diffusion of neuroactive substances and diffusible

signals, like Nitric Oxide (NO). NO is one of the

liposoluble molecules generated by cells from the

own tissue which allow a volumetric transmission. A

key property of NO is its extreme diffusibility in

both aqueous and lipid environments, which allows

a fast three-dimensional spread of the signal

irrespective of the presence of membranes (Suárez

Araujo, 2000). Because of this, it freely diffuses

through membranes affecting all neighbouring cells

(Hawkins, et al., 1993), (Schuman, et al., 1994),

(Zhuo, et al., 1998) and (Garthwaite, et al., 1995).

The presence of a molecule in the brain such as

NO, opens new perspectives in the study of the brain

functioning. NO can help as an element of control

for several systems. It can act as a retrograde

neurotransmitter; it can be involved in learning and

490

Fernández López P., P. Suárez Araujo C. and García Báez P. (2009).

A COMPUTATIONAL STUDY OF THE DIFFUSE NEIGHBOURHOODS IN BIOLOGICAL AND ARTIFICIAL NEURAL NETWORKS.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 490-495

DOI: 10.5220/0002326004900495

 SciTePress

memory, and in the LTP process. It is capable to

produce hybrid neuromodulation, diffusive hybrid

neuromodulation (DHN) (Suárez Araujo, 2000). NO

has also opened a new dimension in our concept of

neural communication, overlaying the classical

synaptic neurotransmission, where information is

passed between neuronal elements at discrete loci

(synapses).

An intrinsic feature of the NO diffusion is the

formation of not-wired neighbourhoods, diffuse

neighbourhoods (DNB), which supports the

emerging of complex structures. The formation of

these structures has been a subject studied by other

authors (Krekelberg, 1996, 1997) and (Krekelberg

and Taylor, 1998). These studies have been specific

studies concerning with the cortical map formation

and their relationship with neighbourhoods in the

Kohonen SOM. They have not considered the DNB

as a possible underlying communication schema in

BNN and ANN. Our studies consider this capacity.

Our work introduces and studies fundamental

concepts in the formation of DNB, product of the

dynamics of NO diffusion. We understand DNB as

an auto-contained element in the behavior of NO

and the study of its dynamics as essential,

independent from the learning model or the neural

architecture where it is embodied in. This will

provide us a better understanding of the

computational skills that NO has; computational

skills based on the neural recruitment mechanisms,

emergence of complex structures and increase of

information capacity and processing. This justifies

the need of a generalist and in-depth development

like the one our work presents. We provide concepts

located in the Theoretical Framework of the study of

NO (Suárez Araujo, 2000). It will be able to manage

studies of causal aspects of the NO dynamics, and

comparative studies with concepts and classical

architectures.

In this paper, we focus our effort on the analysis

of DNB dynamics and its possible influence in

mechanisms and processes at the neural circuit

and/or higher level. An important aim is to infer

from the analysis a possible implication of VT in the

increase of the information representation capacity

in both BNN and ANN, in their architectures and in

the functional complexity of its main computation

element, the neuron.

We present a computational analysis of DNB,

based on our compartmental model of diffusion of

NO (Suárez Araujo, et al., 2006). This work requires

of a set of own concepts of the diffusive phenomena,

which we have defined and that are: Directionality

of NO dynamics (DNO), Average Influence (AI),

Diffusion Centre of the DNB (CDN) and DNB Limit

(DNBL).

The importance of this study is providing the

needed formalism to quantify the information

representation capacity that a type of NO diffusion-

based signalling can present.

2 METHOD

Diffusion is the main axis in the study of the NO

dynamics (Wood and Garthwaite, 1994), as well as

the main responsible of the NO influence to different

brain zones from a functional and structural point of

view. This influence is materialized, essentially, by

means of the Diffuse Neighbourhoods (DNB). This

concept allows to analyze how NO influences move

by means of diffusion as well as what is their

dependence with non-isotropy and non-

homogeneity. The establishment and analysis of

DNB, which we will perform using the

compartmental model of NO diffusion, precise to

formalize intrinsic aspects to the diffusion

phenomena and to the NO dynamics. On one hand,

we have the directionality measure of the NO

dynamics, which provokes different spatial-temporal

influences in the diffusion environment. This takes

us towards the concept of Average NO Influence,

key variable in the DNB definition. On the other

hand, its dynamic, adaptive and no local character

justifies the need of variables which formalize that

dynamism and its effect as diffusion centre of the

DNB and DNB limit.

2.1 Compartmental Model of NO

Diffusion and Concepts of Diffusion

The compartmental model of NO diffusion (Suárez

Araujo, et al., 2006) is a discrete computational

model that allows us to study the dynamic of NO,

generation, diffusion, self-regulation and

recombination, in biological and artificial

environments. Its main feature is its simplicity, it

can be considered as a general formal tool with

biological plausibility. It gathers real features of the

diffusion environment such as the no homogeneity

and the no-isotropy and possible morphology of the

NO synthesis.

It represents an important tool for designing and

interpreting biological experiments on NO behaviour

and its effect on brain structure and function.

The model is based in compartmental systems

(Anderson, 1983) and it is defined by a system of

A COMPUTATIONAL STUDY OF THE DIFFUSE NEIGHBOURHOODS IN BIOLOGICAL AND ARTIFICIAL

NEURAL NETWORKS

491

first order differential equations, like eq. (1), where

we can consider specific cyclic contour conditions.

i,i−1

i−1

−C

)+D

i,i+1

i+1

−C

)−

(1)

Where D

i,i-1

and D

i,i+1

are the coefficients of

diffusion between the compartments i and i-1 and

between i and i+1, respectively. λ

is the self-

regulation parameter of NO. It is being considered,

for this case, a self-regulation of NO dynamics

proportional to the quantity of concentration, and F

is the function of generation of NO.

The computational analysis of DNB, using this

model requires of a set of own concepts of the

diffusion phenomena, which will be defined and that

are: Directionality of NO dynamics (DNO), Average

Influence (AI), Diffusion Centre of the DNB (CDN)

and DNB Limit (DNBL).

NO Directionality, (NOD)

NOD allows to assign an i state to every

compartment, associated to the movement that NO

has along each dimension. Basing on this, it can be

measured the permanence time of compartment i, in

such state, and to calculate the Average NO

Influence that a source i compartment can be

causing on a destination compartment j.

NOD is formally defined for an i compartment

by equation 2. Its value is function of the NO

concentration dynamics associated to the adjacent

compartments, according to the propagation schema

defined for each dimension.

()

22,1

112,1,1

,1,1

112,1,1

22,1

−

−−−

−−−−−

−+

+++++

+++

−

+++

−+

++−

iii

iiiiii

iiiii

iiiiii

iii

CDD

(2)

Basing on the NOD value in each dimension, an

i compartment can be located on 3, 9 or 27 states, for

environments with one, two or three dimensions,

respectively, Table 1.

Basing on time proportion, over the total time of

the diffusion process in which compartments are

located in one state, it can be calculated the Average

Influence that an i compartment is performing into

another j compartment.

Table 1: Possible states of a compartment in mono-

dimensional environments and its NOD values.

State G

Condition

←

< 0

→

> 0

↔

= 0

Average Influence (AI)

Average Influence between compartments by the

NO dynamics (AI) is a magnitude that quantifies the

influence k compartment is performing into r

compartment.

()

⎟

⎠

⎞

⎜

⎝

⎛

∏

∈ (...))),,,(,,,(

,,max),,(

TrqpTrpkTi

tEitrkI

(3)

Where i is every compartment that is located on

T(k, p, r, T(p, q, r, T(…))), path, definition-recursive,

which goes from k compartment to r compartment,

passing by intermediate compartment p and using

path T(p, q, r, T(…)). The function 0 <

(i, E

,t) < 1

defines the time proportion, over the total process

time t, that i compartment is located on the E

state

in favor of the direction followed that path T defines.

This magnitude is computed for a determined

instant, being able to vary throughout time.

Diffuse Neighbourhood (DNB)

Diffuse Neighbourhood (DNB) of an i compartment,

(4), is constituted by a set of compartments which

fulfill certain criteria with relation to the AI i

compartment is performing in them.

(

)

(

)

{

}

tjiIQjtV

,,:)(

(4)

Where Q(I(i, j, t)) is the Q criteria over the AI

that i compartment performs in the j one, and j

represents every compartment that fulfills Q. Q can

be the exceed of a threshold value

by the AI. This

way, j

∈

(t)

⇔

Q(I(i, j, t)) is fulfilled, where Q is

defined according to the logical expression I(i, j, t) >

. Another criterion could be that AI is located

between two values,

. So Q criterion would be

defined by the logical expression

< I(i, j, t) <

The way in which Q criterion is defined causes

different types of neighbourhoods, and the DNB of a

compartment can change throughout time. This way,

the DNB is dynamic and adaptive that generates

complex structures. These aspects of the DNB can

be featured by tracing the diffusion centres of every

instance of the DNB throughout time. This diffusion

centre of the DNB (CDNB) determines a position,

which corresponds to the averaged position of all the

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492

influences that compartments belonging to the

neighbourhood perform between them

∑∑

∈∈

VjVj

tjiItjiItV ),,(),,(),( rC

(5)

Where r

corresponds to the position j

compartment has in the diffusion media.

Diffuse Neighbourhood Limit, (DNBL)

We define Diffuse Neighbourhood Limit (DNBL) of

an i compartment as the max(|r

-r

|), where r

is the

position vector of the i compartment, r

is the

position vector associated to the j compartment, and

such compartment corresponds to any of the

compartments belonging to the DNB of the i

compartment i,j

∈

(t).

3 RESULTS AND DISCUSSION

We have performed a computational study of NO

dynamics, using the compartmental model of NO

diffusion by the observation of the AI-NOD and

CDNB variables.

We have focused our efforts in the analysis of

one of the great potentialities of NO as diffusive

signalling, the emergency of DNB and so, of

complex structures. We have established influences

and dependences with respect to associated features

to the NO synthesis-diffusion process, and to the

environment where it spreads (non-isotropy and no-

homogeneity) (Syková, 2001). We have structured

the study into three sets of experiences which cover

the quoted aspects: influence of the NO synthesis

process, isolated and multiple processes, influence

of distance to the element where NO is synthesized,

influence of features of the diffusion environment,

isotropy and homogeneity, presenting our obtained

results in this section.

The first two studies have been performed in a

mono-dimensional environment, figure 1, with

endothelial cell features. The values of the diffusion

and auto-regulation constants are, respectively, D =

3,3·103

-1

and

= 1,3863 s

-1

and average life

=0.5s. (Malinski, et al., 1993)

Figure 1: Environment of 401 compartments. The

synthesis processes are in the compartment 201 for

synthesis alone, and at 101, 201 and 301 for multiple

syntheses.

Figure 2: I(201, 251, t), I(201, 301, t), I(201, 351, t) AI

Profiles that compartment 201 produces into

compartments 251, 301, 351.

Figure 2 shows the AI which exerts the

compartment 201 into compartments placed at

different distances from it. It is observed the AI

dependence with the distance to the NO-generator

compartment and time. The synthesis process at

compartment 201 starts at t = 0 s., takes 0.2 s. and

presents trapezoidal morphology (Suárez, et al.,

2006). We have observed that I(201, 251, t) gets its

maximum, I

max

(201,251,t)

≅ 45%, at t ≅ 2.75 s.

Initially, compartments produce influence into other

compartments, in a fast way, the more the less

distance. This is fundamental to propitiate neural

recruitment processes, computations

synchronizations, adaptations to changes of

environment. Thus, when synthesis is over, there

already exists an AI greater than 15% of I

max

(201,

251, t). After reaching its maximum, I(201, 251, t)

decays at a faster speed, getting at t = 5 s., negligible

values. I(201, 301, t) takes a 25% more than I(201,

251, t) in reaching the maximum, and I(201, 351, t)

a 30% more, being the max influences much lower

in these compartments. When increasing the distance

in 50µm, the max AI value falls in a 50%, and when

increasing it in 50 µm. more, I

max

decreases a 75%,

Figure 1. This analysis takes us to another concept

as “the importance of neighbourhood” is threshold

value of I which compartments have to reach to

consider them as belonging to a DNB. Thus, DNB

are dynamic and adaptive, being formed gradually in

time by means of the incorporation of compartments

and, at the same time, modifying their order at the

neighbourhood. Importance will be an indicator of

the generated DNB stability; the more stable the

DNB, the lower its importance. This changing

character of DNB handles working in non-stationary

environments, real environments. It can help the

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493

formation of cortical maps (Krekelberg, 1996) and

(Krekelberg and Taylor, 1998), hybrid complex

structures, DNB + wired neighbourhoods, as well as

the incorporation of volume learning into BNN and

ANN.

All these computation and/or information

representation potentialities will be more justified

with the emergence of complexity in DNB when we

increase complexity in NO generation processes.

We have studied the dynamics in an environment

where there coexist several simultaneous synthesis

processes. We analyzed the AI in compartment 251,

when synthesis process exists in two symmetrical

positions to 251 at both sides. We have observed a

dynamics where compartments with minimum

influence exist. I(201, 251, t) as well as I(301, 251,

t) are lower than 3,5%, Figure 3. Thus, a value of

importance higher than this 3,5% implies this

compartment does not belong to both

neighbourhoods, even though it is at short distance

from the NO generation. It can be understood that

compartment 251 plays an isolator role of the

several complex structures that are formed. This

feature can indicate the existence of isolation zones

in the biological level, which can cause

computational segmentation and information

indexing.

Figure 3: I(201, 251, t), I(301, 251, t), I(101, 251, t) AI

Profiles that compartments 201 and 301 exert into

compartment 251, and compartment 101 into compartment

251, when there exist two, and three, simultaneous

synthesis processes, respectively.

It has been performed the study of the IM-NOD

profiles throughout time, when 3 simultaneous NO

synthesis processes exist, separated by 100

m. one

another (compartments 101, 201 and 301).

Analyzing the IM-MOD compartment 101 exerts

into compartment 251, it is observed the additive

character that multiple and aligned generation

causes, Figure 3. To thwart this effect, it becomes

necessary the variability of the average NO lifetime

in the neural tissue. This is one of the biological

phenomena which can justify the environment non-

homogeneity.

Finally, it is shown the dependence that DNB

dynamics and CDNB have to environment non-

isotropy and non-homogeneity. We have worked in

a bi-dimensional environment with very low values

of the diffusion constant in two zones, which makes

the NO dynamics to be almost null. The diffusion

constant is in the range 3,3·10

± 0,2·10

-1

the rest of the environment. There are induced two

NO synthesis processes in two compartments i, k. It

is observed a formation of non-symmetrical and

non-local DNB

and DNB

at times t = 0,6 s. and t=

1,3 s., Figures 4a and 4b. Thus it is noticed a

changing trajectory in CDNB and its possible

dependence in the way DNB does. For t = 0,6 s., in

both neighbourhoods, CDNB matches with the

compartment where synthesis was caused; however

for t = 1,3 s. in V

the position of CDNB has

changed, moving in agreement to that

neighbourhood shape.

4 CONCLUSIONS

We present a work developed from a computational

neuroscience point of view which provides a step

forward in the understanding of the VT and their

implications in the biological and artificial neural

networks.

We have performed a computational analysis of

one of the great potentialities of NO as diffusive

signalling, the DNB. We have used the

compartmental model of NO diffusion,

showing its

high capacity to study the dynamics of NO.

We have proposed and defined concepts

associated to the diffusion phenomena which present

significant capabilities to characterize the NO

dynamics. These concepts are Directionality of NO

dynamics, Average Influence, Importance of the

Neighborhood, Diffusion Centre of the DNB and

DNB Limit.

We have established that the generation and

dynamical behavior of the DNB depend on

associated characteristics to the NO synthesis-

diffusion processes, and to the environment where it

spreads (non-isotropy and non-homogeneity). The

complexity in DNB emerges when the complexity in

NO generation processes is increased. In this paper,

it is also showed the existence of isolation zones in

the biological level, which can cause computational

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494

Figure 4: Snapshot of the formation of two

neighbourhoods, DNB

, DNB

and position of CDNB

(indicated by a circle) in different times. Non-isotropic

and non-homogeneity environment of 16x16

compartaments. Black zone corresponds to null NO

dynamics. a) t = 0,6 s., b) t = 1,3 s.

segmentation and information indexing, and the

possibility to generate non local and non symmetric

DNB. With this study, it is possible to explain some

important environment features like the non-

homogeneity. Finally, all these results allow us to

detect the implications of VT, by means of DNB, in

the increasing of information representation

capacity, in the neural recruitment, in the

synchronization of computations between neurons,

in the neural modulation, in both scenarios,

biological and artificial. These implications will also

allow confirming the possible role of the NO on

several neural circuits as the sleep-wake cycle

control.

We will go onto these analysis about behavior of

DNB, developing complementary studies as

complex systems using bifurcation theory and

analysis.

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