# Cortical Hubs Form a Module for Multisensory Integration on Top of the Hierarchy of Cortical Networks

**Abstract**

Sensory stimuli entering the nervous system follow particular paths of processing, typically separated (segregated) from the paths of other modal information. However, sensory perception, awareness and cognition emerge from the combination of information (integration). The corticocortical networks of cats and macaque monkeys display three prominent characteristics: (i) modular organisation (facilitating the segregation), (ii) abundant alternative processing paths and (iii) the presence of highly connected hubs. Here, we study in detail the organisation and potential function of the cortical hubs by graph analysis and information theoretical methods. We find that the cortical hubs form a spatially delocalised, but topologically central module with the capacity to integrate multisensory information in a collaborative manner. With this, we resolve the underlying anatomical substrate that supports the simultaneous capacity of the cortex to segregate and to integrate multisensory information.

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 1

NEUROINFORMATICS

ORIGINAL RESEARCH ARTICLE

published: 19 March 2010

doi: 10.3389/neuro.11.001.2010

2007; Zhou et al., 2007; Hagmann et al., 2008). While the organisa-

tion of cortical areas into clusters permits the segregated processing

of information of different modality, the large number of connec-

tions involves that sensory information is highly accessible to all

cortical areas, regardless of its modal origin. A detailed analysis of

the corticocortical communication substrate has revealed the central

role of the cortical hubs, by facilitating the communication between

the different sensory modalities (

Zamora-López et al., 2009).

Whether the cortical hubs act as passive transmitters of informa-

tion, or they perform a more active function is a relevant question

that we try to answer in the present paper. We start by summarising

principles of complex network analysis and information theory in

Section “Materials and Methods”. Section “Topological Capacity

of Integration” contains a thorough application of graph theo-

retical measures which reveal that the cortical hubs form an addi-

tional module, expressed as a higher hierarchical level. In Section

“Functional Capacity of Integration”, we challenge the intuitively

assigned integrative properties of this central module by means of

dynamical and information theoretical measures. On the one hand,

we ﬁ nd that only simultaneous lesion of particular hubs leads to

a dynamical segregation of the sensory modules (visual, auditory,

somatosensory-motor and frontolimbic). On the other hand, the

same hubs form a dynamical cluster after simultaneous excitation of

primary sensory areas, a clear sign of their integrative capacities.

MATERIALS AND METHODS

GRAPH ANALYSIS

We ﬁ rst introduce basic concepts of graph theory. A network is

an abstract manner to represent different aspects of a real system,

providing it with a form (topology) which can be mathematically

INTRODUCTION

The mammalian nervous system is responsible for collecting and

processing of information, and for providing adaptive responses

which permit the organism to survive in a permanently changing

environment. Sensory neurones encode environmental information

into electrical signals which propagate in a “bottom-up” manner

through different processing stages (

Kandel et al., 2000; Bear et al.,

2006

). Each level provides responses of increasing complexity and at

different time scales, e.g. reﬂ ex arcs, emotional responses and more

elaborate cognitive responses. Information of the same modality (e.g.

visual, auditory, somatosensory, etc.) traverses the body together,

typically separated from the processing paths of other modalities.

This permits that particular regions of the cortex specialise in detect-

ing different features of the sensory stimuli, e.g. orientation, velocity

and colour of the visual input; or frequency and pitch of the audi-

tory stimuli. However, in order to generate a coherent perception of

the reality, the brain needs to combine (integrate) this multisensory

information at some place (

Robertson, 2003) and during some time

(

Fahle, 1993; Singer and Gray, 1995; Engel and Singer, 2001). For that,

the paths of information need to converge.

It has been argued that the functional capacity of the NS to bal-

ance between segregation (specialisation) and integration might be

facilitated by its structural organisation (

Sporns and Tononi, 2001).

Analysis of the connectivity between regions of the cerebral cortex

in macaque monkeys and cats has revealed the following character-

istics: (i) clustered organisation of the cortical areas (

Scannell and

Young, 1993; Scannell et al., 1995; Hilgetag et al., 2000; Hilgetag

and Kaiser, 2004

) (see Figure 2), (ii) a large density of connections,

and (iii) a broad degree distribution containing highly connected

areas which are referred as hubs (

Zemanová et al., 2006; Sporns et al.,

Cortical hubs form a module for multisensory integration on

top of the hierarchy of cortical networks

Gorka Zamora-López

1

*, Changsong Zhou

2,3

and Jürgen Kurths

4,5

1

Interdisciplinary Center for Dynamics of Complex Systems, University of Potsdam, Potsdam, Germany

2

Department of Physics, Centre for Nonlinear Studies, Hong Kong Baptist University, Hong Kong, China

3

The Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems, Hong Kong Baptist University, Hong Kong, China

4

Transdisciplinary concepts and methods, Potsdam Institute for Climate Impact Research, Potsdam, Germany

5

Institute of Physics, Humboldt University, Berlin, Germany

Sensory stimuli entering the nervous system follow particular paths of processing, typically

separated (segregated) from the paths of other modal information. However, sensory perception,

awareness and cognition emerge from the combination of information (integration). The

corticocortical networks of cats and macaque monkeys display three prominent characteristics:

(i) modular organisation (facilitating the segregation), (ii) abundant alternative processing paths

and (iii) the presence of highly connected hubs. Here, we study in detail the organisation and

potential function of the cortical hubs by graph analysis and information theoretical methods.

We ﬁ nd that the cortical hubs form a spatially delocalised, but topologically central module

with the capacity to integrate multisensory information in a collaborative manner. With this,

we resolve the underlying anatomical substrate that supports the simultaneous capacity of the

cortex to segregate and to integrate multisensory information.

Keywords: corticocortical networks, cortical hubs, multisensory integration, segregation, integration

Edited by:

Claus C. Hilgetag, Jacobs University

Bremen, Germany

Reviewed by:

Steven Bressler, Florida Atlantic

University, USA

David Meunier, University of

Cambridge, UK

*Correspondence:

Gorka Zamora-López, Interdisciplinary

Center for Dynamics of Complex

Systems, University of Potsdam,

Komplex II – Golm (Haus 28) ,

Karl-Liebknecht-Str. 24, D-14476

Potsdam, Germany.

e-mail: gorka@agnld.uni-potsdam.de

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 2

Zamora-López et al. Cortical module for integration

tractable. A network G (N, L), composed of N nodes interconnected

by L links, is described by an adjacency matrix A with entries A

ij

= 1

when there is a link pointing from node i to node j, and A

ij

= 0

otherwise. The density of G is the fraction between the number

of links L and the total number of links possible:

ρ=

−

L

NN()1

. In

order to characterise the topological scales of networks, there exist

many statistical descriptors, all measurable from the information

encoded in the adjacency matrix. The output degree

ki A

oj

N

ij

()=∑

=1

of a node i is the number of efferent connections that it projects to

other nodes, and its input degree

ki A

ij

N

ji

()=∑

=1

, is the number of

the afferent connections it receives. The degree distribution p(k)

is the probability that a randomly chosen node has degree k. One

of the key discoveries that triggered a renewed interest in graph

theory is that the distribution p(k) of many empirical networks

approximately follows a power-law p(k) ∼ k

−γ

(Newman, 2003),

where γ is the degree exponent. In such scale-free(-like) networks

the majority of nodes possess a small number of neighbours, and

few nodes (the hubs) are highly connected.

Distance and centrality

The distance d

ij

between two nodes i and j is the length of the short-

est path between them, i.e. the minimal number of links crossed

to travel from i to j. If there is a link i → j, then d

ij

= 1. If there is

no other choice than going through an intermediate node k such

that i → k → j, then d

ij

= 2, and so on. When there exists no path

from i to j then d

ij

= ∞. The average pathlength l is the average

distance between all pairs of nodes. The shortest path between two

nodes is usually not unique and there are several alternative shortest

paths. In order to characterise the inﬂ uence of individual nodes

on the ﬂ ow and the spread of information through a network, the

betweenness centrality C

B

(i), is deﬁ ned as the fraction of all shortest

paths passing through i (

Anthonisse, 1971; Freeman, 1977):

Ci

i

B

st

st

siti

N

()

()

,

=

≠≠

∑

σ

σ

(1)

where σ

st

(i) is the number of shortest paths starting in s, running

through i and ﬁ nishing in t, and σ

st

is the number of all shortest

paths from s to t.

Matching index

The topological similarity of two nodes can be characterised as the

number of common neighbours they share. In the extreme case,

two nodes are topologically identical if both have the same set of

connections. The neighbourhood of node i is deﬁ ned as the set of

nodes it connects with, Γ(i) = {j : A

ij

= 1}. In graphs without mul-

tiple links the size of the neighbourhood |Γ(i)| equals the degree

of i. The matching index of two nodes i and j is thus the overlap

of their neighbourhoods: MI(i,j) = |Γ(i) ∩ Γ(j)|. Deﬁ ned in this

manner MI(i,j) depends on the degrees of i and j, and the values for

different pairs are not comparable. Imagine two nodes with degrees

k(i) = k(j) = 3 which are connected to the same neighbours. As

Γ(i) = Γ(j) their matching is MI(i,j) = 3. Imagine other two nodes

with degrees k(i′) = 3 and k(j′) = 4. Maximally, they could share

three neighbours and have MI(i′,j′) = 3 as well, despite i and j

are topologically equivalent but i′ and j′ are not. In order to com-

pare the values for different pairs the measure can be normalised by

the number of distinct neighbours of the two nodes, i.e. the union

of the two neighbourhoods |Γ(

i) ∪ Γ(j)| as illustrated in Figure 1.

The normalised matching index can be computed as:

MI i j

ij

ij

AA

ki k j A A

in jm

nm

N

in

(, )

|() ()|

|() ()|

() ( )

,

=

∩

∪

=

+−

=

∑

ΓΓ

ΓΓ

1

jjm

nm

N

, =

∑

1

(2)

Now, MI(i,j) = 1 only if i and j are connected exactly to the

same nodes, Γ(i) = Γ(j), and MI(i,j) = 0 if they have no common

neighbours.

Reference surrogate networks

Graph theoretical measures help understand the topological organ-

isation of networks. Equally relevant is to uncover the features

which are characteristic to the underlying system and the funda-

mental properties of its development. In this sense, the question

is not whether a graph measure takes a speciﬁ c numerical value,

but whether this value distinguishes the empirical network G

emp

from others of similar characteristics. For that, the formulation

of appropriate null-models is required. A typical such null case is

to generate surrogate networks with the same size N, number of

links L and degree distribution p(k) as in G

emp

. The link switching

method (

Katz and Powell, 1957; Holland and Leinhardt, 1977; Rao

et al., 1996; Kannan et al., 1999; Roberts, 2000

) consists of the fol-

lowing iterative process: starting from G

emp

, at each iteration two

links are chosen at random (i

1

→ j

1

) and (i

2

→ j

2

). The links are

rewired as (i

1

→ j

2

) and (i

2

→ j

1

) provided that the new links do not

already exist and do not introduce self-loops, i.e. i → i. Repeating

the process sufﬁ cient times the resulting surrogate network con-

serves the initial degree distribution but any higher order structure

is destroyed.

DATA

The classical textbook illustration of the cerebral cortex as a sur-

face (grey matter) which can be subdivided into functional or

cytoarchitectonic regions is only a limited picture. Additionally,

long-range ﬁ bres link the cortical areas via the white matter forming

FIGURE 1 | Schematic representation of the normalised matching index,

computed as in Eq. 2. For proper comparison between pairs, the measure is

normalised by the number of different neighbours of v and v′ .

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 3

Zamora-López et al. Cortical module for integration

a complex network which is neither regular nor completely random.

This intricated structure enhances the richness and complexity of

information processing capabilities of the cerebral cortex. In this

paper we focus on the analysis of the cortical connectivity of the

cat because it is, up to date, the most complete and reliable dataset

of this kind.

Corticocortical connectivity of the cat

The dataset of the corticocortical connections within the cortex of

cats was created after an extensive collation of literature reporting

anatomical tract-tracing experiments (

Scannell and Young, 1993;

Scannell et al., 1995

). It consists of a parcellation into 53 cortical

areas and 826 ﬁ bres of axons between them as summarised in

Figure 2. The connections are weighted according to the axonal

density of the projections. After application of data mining methods

(

Scannell and Young, 1993; Hilgetag and Kaiser, 2004), the network

was found to be organised into four distinguishable clusters which

closely follow functional subdivisions: visual (V), auditory (A),

somatosensory-motor (SM) and frontolimbic (FL).

Surrogate data

In order to perform signiﬁ cance tests of the graph measures, an

ensemble of 1000 surrogate networks has been created following

the link switching method (see Section “Graph Analysis”). All the

resulting networks have the same size N = 53, the same number of

links L = 826 and the same degree distribution as the corticocortical

FIGURE 2 | Weighted adjacency matrix W of the corticocortical connectivity

of the cat comprising of L = 826 directed connections between N = 53 cortical

areas (

Scannell and Young, 1993; Scannell et al., 1995). For visualisation

purposes, the non-existing connections (0) have been replaced by dots. The

network has clustered organisation, reﬂ ecting four functional subdivisions: visual

(V), auditory (A), somatosensory-motor (SM) and frontolimbic (FL).

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 4

Zamora-López et al. Cortical module for integration

network of the cat. To assure that any further internal structure is

destroyed, each surrogate network is the product of 10 × L itera-

tions. In the following, this set will be referred as the rewired ensem-

ble {G

1n

}, and the original corticocortical network of the cat as G

cat

.

The ensemble average of graph measures applied on the surrogate

set {G

1n

} will be considered as the expected values.

INFORMATION THEORY AND INTEGRATION

Information theory has been very successful to describe transmis-

sion of information, encoding and channel capacity. At the root

of this success lies the original idea of Shannon to apply concepts

of statistical physics to represent the nature of communication.

Consider a system A with M possible states. That is, a measurement

made on A yields the values a

1

,a

2

,…,a

M

, with a probability p(a

i

).

The average amount of information gained from a measurement

that speciﬁ es one particular value a

i

is given by the entropy of the

system (

Shannon, 1948; Cover and Thomas, 1991):

HA pa pa

ii

i

M

() .=−

() ()

=

∑

log

1

(3)

The entropy can be interpreted as the amount of surprise one

should feel upon reading the result of a measurement (

Faser and

Swinney, 1986

). It vanishes when the system has only one accessible

state because the value a is always obtained, i.e. there is no surprise.

H(A) is maximum when all the states are equally likely, i.e. there

are no preferred states.

The statistical dependence between two systems x

1

and x

2

is quan-

tiﬁ ed by their mutual information:

M

Ix x Hx Hx Hxx

12 1 2 12

,,.

(

)

=

(

)

+

(

)

−

(

)

(4)

By deﬁ nition, the joint entropy is H(x

1

,x

2

) ≤ H(x

1

) + H(x

2

). The

equality is only fulﬁ lled if x

1

and x

2

are statistically independent,

hence MI(x

1

,x

2

) = 0, and otherwise MI(x

1

,x

2

) > 0.

Integration

In a series of papers Tononi and Sporns proposed a particular meas-

ure of integration (

Tononi and Sporns, 1994; Tononi et al., 1996,

1998

). Given a system X composed of N subsystems x

1

, integration

is deﬁ ned as:

I

XHxHX

i

N

i

() ()=

()

−

=

∑

1

(5)

where H(x

i

) is the entropy of one subsystem and H(X) = H(x

1

,x

2

,…,x

n

)

is the joint entropy of the system considered as a whole. I(X) = 0

only if all x

i

∈ X are statistically independent of each other, and posi-

tive otherwise. After this deﬁ nition, integration is the extension of

mutual information for more than two systems. In other words, I(X)

measures the internal level of statistical dependence among all the

subsystems x

i

∈ X.

Linear dynamical systems

The steady-state of a linear system whose N subsystems

x = (x

1

,x

2

,…,x

n

) are driven by a Gaussian noise ξ = (ξ

1

,ξ

2

,…,ξ

N

),

is described by

xg Ax

ijij

t

ji

=∑ +ξ

, where g is the coupling strength

and A

t

is the transpose of the adjacency matrix. Otherwise the

dynamics of x

i

would be characterised by its own outputs, not

by the inputs it receives. Written in matrix form:

xAx=+g

t

.

(6)

In practical terms the variable x

i

might be interpreted as the

activity level of the cortical area i (

Kötter and Sommer, 2000; Young

et al., 2000

), or as the mean ﬁ ring rate of the neurones in the area i

(

Graben et al., 2007). The entropy of such a multivariate Gaussian

system can be analytically calculated out of its covariance matrix

such that

HX e COVX

N

() ( )| ()|=

⎡

⎣

⎤

⎦

1

2

2log π

, where |·| stands for

the determinant (

Papoulis, 1991; Tononi and Sporns, 1994). The

entropy of an individual Gaussian process is

Hx e

ii

() ( ),=

1

2

2log πν

where ν

i

is the variance of x

i

, say, the i

th

diagonal element of the

COV(X) matrix. Replacing H(X) and H(x

i

) into Eq. 5 and apply-

ing basic algebra, we reduce the integration of such a multivariate

Gaussian system as:

IX

COV X

i

i

N

()

()

.=

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

=

∏

1

2

1

log

ν

||

(7)

This expression shows that I(X) of the linear system is prop-

erly normalised and is independent of system size N. The covari-

ance matrix can be analytically computed by solving the system

such that

x =⋅=⋅

−

1

1A

t

g

ξξQ

, and averaging over the states pro-

duced by successive values of ξ one ﬁ nds: COV(X) = 〈x · x

t

〉=

〈(Q · ξ)·(ξ

t

· Q

t

)〉 =Q · Q

t

.

Comparing different systems

To compare I(X) of different systems, the matrix A

t

needs to be

adequately normalised because application of the same coupling

strength g to different networks might set them into different

dynamical states. Hence, they might not be comparable. The lin-

ear System (6) has several poles depending on g. The smallest pole

corresponds to

g

1

1

=

λ

max

where λ

max

is the largest eigenvalue of the

transposed adjacency matrix A

t

. The solutions only have physical

meaning for g < g

1

, otherwise the stationarity condition does not

hold. In Figure 3 the poles corresponding to the corticocortical

network of the cat are shown. Notice that at the poles, both entropy

and integration diverge. To make the comparison of the dynamics

of different networks possible, we normalise the adjacency matrices

as

ˆ

.AA

t

A

t

max

==g

1

λ

In this manner, all systems have the smallest

pole at g = 1.

Finally, a proper coupling strength g needs to be chosen. For

that, we have estimated the covariance matrices of the cat cortical

network under different coupling strengths (Figure 4). They are

similar to the correlation patterns arising from more complex

models (

Zemanová et al., 2006; Honey et al., 2007; Zhou et al.,

2006, 2007

). This similarity indicates the validity of the simple

linear System (6) for the exploratory purposes here intended.

All networks considered in Section “Functional Capacity of

Integration” are normalised by their ﬁ rst pole and a coupling

strength of g = 0.5 is applied. Unless otherwise stated, the noise

level is set to ξ

i

= 1.0.

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 5

Zamora-López et al. Cortical module for integration

A

B

C

FIGURE 3 | Parametric study of the linear System (6) using the cortical

network of the cat. (A) Integrability range. When the determinant |1 − gÂ| = 0

the system has a pole. Negative values lead to non-physical solutions. (B)

Entropy and (C) Integration diverge around the poles.

A

BC

FIGURE 4 | Covariance matrix of the cat cortical network as a linear system. The adjacency matrix has been previously normalised by 1/λ

max

and the noise level

set to ξ

i

= 1.0. Coupling strengths are: (A) g = 0.52, (B) g = 0.84 and (C) g = 0.92.

RESULTS

TOPOLOGICAL CAPACITY OF INTEGRATION

In order to characterise the connectional organisation of the nervous

system and to understand its functional implications, the complex

network approach has been applied in the recent years, particu-

larly at the level of the cerebral cortex. This analysis has revealed

several organisation properties, e.g. the clustering of cortical areas

according to their sensory modality (visual, auditory, somatosen-

sory-motor and frontolimbic). Recently, it has been reported that

communication paths between cortical areas in different sensory

modules are not random, but mediated by the hubs of the network

(

Zamora-López et al., 2009). In this section we present a more

detailed graph analysis aiming to characterise the potential func-

tion of the cortical hubs.

Inter-modal communication

The betweenness centrality C

B

(ν) quantiﬁ es the relevance of a node

v within the communication paths in a network. As represented

in Figure 5A, we observe that within each of the sensory systems,

few cortical areas possess a large betweenness. With C

B

(ν) > 500 we

ﬁ nd: visual areas 20a, 7 and AES; auditory area EPp; somatosensory-

motor areas 6m and 5Al; and frontolimbic areas Ia, Ig, CGp, 35

and 36. On the contrary, only the visual primary cortex (area 17)

and the hippocampus have C

B

(ν) = 0. In general, we observe that

cortical regions known to perform highly specialised sensory func-

tion have few connections and very low centrality, e.g. primary and

secondary visual or auditory areas, and early somatosensory-motor

areas. These areas typically contain ordered mappings of the sen-

sory stimuli such as retinotopic or tonotopic maps, see Appendix

of

Scannell et al. (1995).

The centrality of a node usually correlates with its degree,

hence, it is trivial to find out that precisely the hubs have larger

centrality. Drawing any further conclusion requires performing

a proper significance test. For comparison, the average C

B

(ν)

of the nodes in all the 1000 rewired networks of the surrogate

ensemble {G

1n

} has been computed. The ascending line in Figure

5B shows the expected dependence of the betweenness centrality

on the degree of the nodes. As a node receives k

i

(ν) inputs and

projects k

o

(ν) outputs, the number of shortest paths passing

through v is linearly proportional to k

i

(ν)k

o

(ν) in the surrogate

networks. The most prominent observation is that, while C

B

of

the low degree areas follow the expected centrality, the centrality

Subsets of elements

The entropy of a subset of systems S ⊆ X can be obtained by

ﬁ rst computing COV(X) as indicated above, and then extracting

the covariance submatrix COV(S) out of COV(X) by consider-

ing only the elements x

i

∈ S. The entropy of the subset is then

HS e COVS

N

S

() ( ) | ()|=

⎡

⎣

⎤

⎦

1

2

2log π

, and its integration I(S) is:

IS H x HS

COV S

j

j

j

N

xS

S

j

() ()

()

.=

()

−=

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

=

∈

∏

∑

1

2

1

log

ν

||

(8)

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 6

Zamora-López et al. Cortical module for integration

φ() ,k

L

NN

k

kk

′

′

′′

=

−

()

1

(9)

where

N

k′

is the number of nodes with k(v) ≥ k′ and

L

k′

is the

number of links between them. Notice that φ(k) is an increasing

function of k. As

φ()

()

0

1

=

−

L

NN

is the density of the network, after

the nodes with low degrees are removed the remaining reduced

network contains more links per node. Thus, a plain measure of

φ(k) is not very informative because hubs have a higher intrinsic

chance of being connected to each other. Again, a conclusive inter-

pretation requires the comparison to random networks with the

same degree distribution. The question is then whether φ(k) of

the real network grows faster or slower with k than the expected

k-density φ

1n

(k) out of the surrogate networks {G

1n

}. If φ(k) grows

faster than φ

1n

(k), it means that the hubs are more connected than

expected and form a dense module (a rich-club). On the contrary,

if φ(k) grows slower than φ

1n

(k), the hubs are more independent

of each other than expected.

In Figure 6A the k-density φ

cat

(k) of G

cat

is presented together

with the ensemble average φ

1n

(k). For low degrees, φ

cat

(k) follows

very close the expectation, but for degrees k(v) > 15, φ

cat

(k) starts to

grow faster showing that the hubs of the network form a rich-club.

The largest difference occurs for k = 23, comprising of 11 cortical

hubs from all the four sensory systems (Figure 6B). Compared to

the internal density of the four modules of the network, we ﬁ nd

that the hubs form an even denser module (Table 1).

Topological similarity of cortical hubs

A central assumption in systems neuroscience is that the func-

tion of brain regions are speciﬁ ed by their afferents and efferents

(

Passingham et al., 2002). Under this assumption, it is to be expected

that cortical areas of similar function, i.e. specialised in the process-

ing of same modal information, should display a similar pattern of

FIGURE 6 | Rich-club organisation. (A) k-density of the corticocortical

network of the cat φ

cat

, compared to the expectation out of the surrogate

ensemble {G

1n

}. The largest difference occurs at k = 23 (vertically dashed line)

giving rise to (B) a rich-club composed of 11 areas.

A

B

FIGURE 5 | Centrality of cortical areas. (A) Betweenness of cortical areas

shows that at each sensory system few areas are very central. (B) Comparison

between C

B

of cortical areas and the expected centrality due to their degree

(brown line). As a consequence of the modular and hierarchical organisation of

the network, low degree areas closely follow the expected centrality but hubs

are signiﬁ cantly more central than expected. Communication paths between

sensory systems are centralised through the hubs.

of the hubs is largely significant. This is an evident consequence

of the modular organisation of the network and the particular

role of the cortical hubs for the inter-modal communication.

Communication paths running between low-degree areas of dif-

ferent modules are usually mediated through the hubs (

Zamora-

López et al., 2009

).

This signiﬁ cance test permits us to uncover the most likely candi-

dates to be a hub of the network, not only in terms of their number

of links, but considering their contribution for the corticocortical

communications. The hubs found here are potential candidates

to perform high level integration because they have access to the

information of different modalities. However, with the current

results we can only afﬁ rm with certainty that the hubs are useful

for the transmission of information from one modality to another.

Concluding whether they perform any further function or not, it

requires a more careful analysis.

Collective organisation of cortical hubs

A relevant question is now whether the cortical hubs are func-

tionally independent of each other, i.e. each hub has a specialised

function, or they perform some collaborative function. A graph

measure to characterise the relation between the hubs of a net-

work is the rich-club phenomenon. The k-density φ(k), is deﬁ ned

as the internal density of links between the nodes with degree

larger than k′:

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Zamora-López et al. Cortical module for integration

projections. In the case of the cortical hubs, it has been shown in

the previous section that they form a tightly connected module.

Whether this module could be regarded as a functional module, at

least from a topological point of view, is the goal of the following

analysis. The matching index MI(v,v′) is a graph measure to estimate

the topological similarity of two nodes, by counting the number of

common neighbours of v and v′ (Section “Graph Analysis”). In order

to compare the values obtained for different pairs, the measure is

normalised such that MI(v,v′) = 1 only if all the neighbours of node

v are also all the neighbours of v′. See the example in Figure 1.

We have computed the matching index for all pairs of cortical

areas and the result is shown in matrix form, Figure 7A. Visual

inspection reveals the modular organisation of the network. This

is reﬂ ected by the fact that MI(v,v′) is typically larger if both v

and v′ belong to the same anatomical module, than if they belong

to different modules. To highlight this difference, in Figure 7B

the distribution of the matching values is shown: when the areas

belong to the same module (internal matching), or to different

modules (external matching). The external matching has a broad

skewed distribution but peaking near MI = 0.15. The internal

matching displays a more constrained distribution with maximum

at approximately MI = 0.55. In Table 1 the average matching of the

network is compared to the average internal matching for each of

the anatomical modules V, A, SM and FL. The internal averages are

FIGURE 7 | Topological similarity of cortical areas. (A) Pairwise matching

index MI(v,v ′) for all areas summarised in matrix form. Self-matching MI(v,v) is

ignored for visualisation. (B) Distribution of the MI values in (A) if the areas v

and v′ are in the same anatomical module V, A, SM or FL (dashed line), and if

they belong to different modules (solid line). (C) Recomputed distribution of MI

if the areas belong to different modules, but cortical hubs are discarded (solid

line). And distribution of MI(v,v′) only if v and v′ are hubs in the Rich-Club

(dotted line).

Table 1 | Comparison between the anatomical modules and the Rich-Club. Both the internal density of links and the average matching of the areas in each of

the functional modules V, A, SM and FL are larger than the whole network averages. The same happens for the areas in the Rich-Club, with values comparable to,

or larger than those for the anatomical modules.

always larger than the global average despite the broad deviations,

conﬁ rms the expected functional cohesiveness of the modules; not

only in terms of their internal density of connections, but also in

terms of their common connectivity.

As pointed out, the distribution of external matching is skewed

and contains some larger values up to MI ≈ 0.6. We ﬁ nd that most of

these larger values are contributed precisely by the links between the

cortical hubs which lie in different modules. We have recomputed

the distribution of external matching, but ignoring the matching

between the cortical hubs (solid line in Figure 7C). The distribution

decays now faster than in Figure 7B. Finally, the distribution of the

internal matching for the 11 hubs forming the rich-club is displayed

(dotted line of Figure 7C). It appears clearly separated from that of

the distribution of external matching and peaking near MI = 0.55.

Its average is 0.52 ± 0.10, comparable to, or larger than, the internal

matching of the anatomical modules, Table 1. These observations

support the idea that the cortical hubs form a functional module

on their own, as the anatomical modules do.

Hierarchical organisation and integration capacity

The two structural properties of the cortical hubs here presented, (i)

hubs are densely connected with each other and (ii) they are func-

tionally interrelated in terms of their inputs and outputs, extend the

current understanding of cortical networks by uncovering that the

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 8

Zamora-López et al. Cortical module for integration

multisensory hubs form yet another module which lies at a higher

level in the hierarchical organisation. In the complex networks liter-

ature one ﬁ nds two types of hierarchical topologies. The model after

Arenas et al. (2006) considers hierarchies as the agglomeration of

modules, say, small modules join to form larger modules, Figure 8A.

Another type of hierarchy after

Ravasz and Barabási (2003) can be

regarded as a tree-like fractal structure which produces modular

networks with scale-free degree distribution. At each level, there

is a central community connecting to all the modules at the same

level, and to all modules in the hierarchies below. Such centralised

patterns are repeated through different scales, Figure 8B.

The organisation that we uncover here is none of these two, but

it might be regarded as a combination of them. Notice that in the

model by

Arenas et al. (2006), the small communities are randomly

linked to each other such that their union forms a larger commu-

nity. In the present case, the inter-community links are not random,

but centralised. Therefore, the highest hierarchical level is formed

by a partial overlap of the underlying modules. See Figure 8C for

a schematic representation.

The functional implications of the topological ﬁ ndings

described in this section, necessarily arise from intuitive inter-

pretation of the intrinsic relationship between structure and func-

tion in neural systems. To provide a more solid ground to these

intuitive interpretations, in the following section we challenge

them by means of dynamical and information theoretical meas-

ures. We focus in a very simple dynamical model which has the

beneﬁ t of being analytically solvable, although its validity for our

purposes is conﬁ rmed by comparison to the dynamical output

of more complex models, see Section “Information Theory and

Integration”.

FUNCTIONAL CAPACITY OF INTEGRATION

The structural organisation described in the previous section

supports the idea that the cortical hubs might be responsible for

combining the multisensory information hence facilitate the emer-

gence of a global (integrated) perception. In this sense, we aim for

a deﬁ nition of integration which characterises the capacity of one

or more nodes to receive information of different character and com-

bine it to produce new useful information. Certainly, this deﬁ nition

involves crucial theoretical problems, e.g. what the character of

information is, or what are the rules under which information is

combined. Nevertheless, within a networked system, the nodes with

a capacity to integrate information should obey certain measurable

conditions. We propose the following:

1) Accessibility to information: A node can perform an integra-

tive function only if it has general access to the information

contained within the system.

2) Sharing of information: Two or more nodes can perform

integrative function in a collaborative manner only if they are

sufﬁ ciently connected with each other.

3) Segregation after selective damage: If a node has an integra-

tive function, its removal should lead to a decrease of the inte-

grative capabilities of the whole system.

From the structural point of view, the hubs listed in Figure

6B obey these three conditions. They are the most central areas

and they are densely connected to each other. Besides, robust-

ness studies (

Kaiser et al., 2007) have shown that intentional

lesion of the highly connected cortical areas largely affect the

communication within the network. In the following, we intro-

duce a framework to characterise the integrative function of the

hubs by means of dynamical systems and information theory.

Additionally, we perform a probabilistic analysis of the compo-

sition of the dynamical core, rather than a deterministic one.

The reason is that even if the corticocortical networks of the

cat is the most complete and reliable dataset of its kind up to

date, it is not free of experimental errors. For example, some of

the real connections might still be absent in the data. We aim

to discriminate those hubs which, grouped together, possess

a larger potential to integrate multisensory information from

those groups which might have lesser capacities. For that, we

FIGURE 8 | Hierarchical organisation of complex networks. (A) Hierarchies

as agglomeration of modules (

Arenas et al., 2006). (B) Centralised and fractal

hierarchical model (

Ravasz and Barabási, 2003). (C) Illustrative representation

of the modular and hierarchical structure found in the corticocortical

connectivity of the cat. The highest hierarchical level is formed by a densely

interconnected overlap of the modules.

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Zamora-López et al. Cortical module for integration

arbitrarily choose all the areas with output degree k

o

(ν) ≥ 20 as

potential members of the integrator module giving rise to a set

of N

S

= 19 areas:

S

hubs

= {20a, 7, AES, EPp, 6l, 6m, 5Am, 5Al, 5Bm, 5Bl, SSSAi,

SSAo, PFCL, Ia, Ig, CGa, CGp, 35, 36}.

The statistical analysis consists in measuring the integrative

capacities of all the 524,097 combinations of sizes N

S

= 1 to N

S

= 19

out of the 19 hubs in S

hubs

.

Integration capacity after sensory stimulation

Consider the linear System (6) with Â being the transposed and

normalised adjacency matrix of the cat G

eat

. All areas are driven

by a small Gaussian noise level ξ

i

= 1.0 and coupled by g = 0.5.

This case might be regarded as the activity of the network in the

resting-state because all x

i

are driven by noise of small intensity

and there is no sensory input. Now, we intend to illustrate the

joint capacity of a group of areas to integrate information of

different character. Even if it is unclear how to deﬁ ne the character

of information, in the case of cortical networks it is known that

sensory information enters the cortex through speciﬁ c regions

termed as primary sensory areas: primary visual cortex (area

17), primary auditory cortex (area AI) and primary somato-

sensory cortex (areas 1, 2 and 3b). According to

Scannell et al.

(1995) the cortical areas 1, 2 and 3b are subregions of the primary

somatosensory area, named by some authors as SI. Hence, we

simultaneously excite all the primary sensory areas {17, AI, 1, 2,

and 3b} by assigning them a larger noise level ξ

j

= 10.0) and we

measure the integration I(S) of all the subsets S of hubs out of

S

hubs

. Because of the excited condition, we denote the integration

of the subsets as I

e

(S).

The results depicted in Figure 9A show that I

e

(S) can largely dif-

fer. For example, among all the subsets of size N

s

= 10, the integra-

tion of some of them is very small, I

e

(S) ∼ 0.1, while the integration

of others becomes much larger, I

e

(S) ∼ 0.5. These differences permit

us to identify those cortical hubs which, grouped together, become

more statistically dependent among them as a consequence of the

multisensory stimulation. Considering only those subsets whose

I

e

(S) lies within the largest 10% (red crosses in Figure 9A) a co-

participation matrix C is constructed such that C

ij

is the number

of times (given in frequency) that two cortical hubs participate

together in one of the maximal sets, Figure 9B. It is observed that

areas {7, AES; EPp; 6m; Ia, Ig, CGp, 35, 36} participate together in

over 75% of all the maximal sets. Visual area 20a and the soma-

tosensory-motor area 6l participate only in 50% of the occasions

with those areas in the core. The remaining areas, {5Am, 5Al, 5Bm,

5Bl, SSSAi, SSSAo and PFCL}, can be discarded as members of the

dynamical core.

FIGURE 9 | Functional segregation and integration. (A) Local

integration I(S) of cortical hubs after stimulation of the primary sensory areas.

(B) Co-participation matrix of cortical hubs within the subsets leading to large

I

e

(S) (red dots). (C) Modular integration

I

P

4

of the sensory modules V, A, SM and

FL after simultaneous lesion of cortical hubs. N

S

is the number of hubs removed.

(D) Co-participation matrix of the hubs within the subsets S which lead to a

larger decrease in the dynamical dependence

I

P

4

()

()

S

of the sensory modules

(marked by red dots).

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 10

Zamora-López et al. Cortical module for integration

Dynamical segregation after multiple lesions

Within a networked system the removal of critical nodes should

lead to a decrease in its integrative capacities. In the following, we

study the impact of targeted lesions of the corticocortical network

of the cat, G

cat

. For all the possible subsets S composed of hubs in

S

hubs

, we perform a lesion to the network by simultaneously remov-

ing the nodes x

i

∈ S and characterise the consequent functional

segregation of the network G

S

= G

cat

− S as the change in statistical

dependence between the four modules (V, A, SM and FL). Lesion

of areas critical for the integration capacities of the system should

lead to a dynamical segregation of the modules, i.e. a decrease in

their statistical dependence.

Recall that integration I(X) as deﬁ ned in Eq. 7 is an extension

of the mutual information for more than two systems. It repre-

sents the limit case in which the statistical dependence among

all the elements x

i

in the system X is quantiﬁ ed. To cover differ-

ent scales of organisation we propose to characterise the statisti-

cal dependence between groups of elements. Imagine a partition

P = {S

1

,S

2

,…,S

n

} into n groups (modules) of the elements x

i

such

that X = S

1

∪ S

2

∪…∪ S

n

. Then, we deﬁ ne the modular integration

of the partition P as:

I

XHSHX

j

j

n

P

() ().=

()

−

=

∑

1

(10)

Note that when n = N, then I

P

(X) = I(X).

Considering the partition P

4

= {V, A, SM, FL} and the cortico-

cortical network of the cat, then

I

P

G

4

()

= 0.29

2

cat

. The modular inte-

gration of each lesioned network G

S

is computed for the partition

P

4

. Notice that (a) the nodes are also removed from the partition

and (b) every G

S

is adequately normalised by its largest eigenvalue

such that the measured observables are comparable across reali-

sations (see Section “Information Theory and Integration”). The

results in Figure 9C permit us again to discriminate between sub-

sets of hubs whose simultaneous removal lead to a large segrega-

tion of the network, while removal of other subsets has barely no

effect. For example, among all the possible lesions of size N

S

= 10,

some trigger a large segregation of the modules,

I

PS

G

4

005

()

∼ .

while other lesions do even increase their dynamical dependence:

I

I

PS P

GG

44

035

()

∼>

()

.

cat

.

Selecting only those subsets whose lesion leads to a larger

segregation of the modules, i.e.

I

P

G

4

(

)

S

lies among 10% of

the minimal modular integration for each size N

S

(red dots

in Figure 9A), a co-participation matrix C is constructed,

Figure 9D. The entries C

ij

are the number of times (given in

frequency) that two areas participate together in one of the

minimal subsets. A core of cortical areas is found which par-

ticipate together in over 70% of these cases: {7, AES; EPp; Ia,

Ig, CGp, 35, 36}. Somatosensory-motor areas 6m, 5Al and 5Bl

join them in over 50% of the cases.

In summary, both the multiple lesion and the multisensory

excitation analysis performed in this section lead to the identi-

ﬁ cation of the same cortical hubs as responsible for the integra-

tion of multisensory information in the corticocortical network

of the cat. Moreover, this set largely coincides with the top hier-

archical level found by the graph analysis in Section “Topological

Capacity of Integration”, corroborating the integrative function

assigned to the hubs by intuitive interpretation of their topologi-

cal characteristics.

SUMMARY AND DISCUSSION

In this paper we have analysed the modular and hierarchical organi-

sation of the corticocortical network of the cat and its relationship

to the intrinsic necessities of the brain to simultaneously segregate

and integrate multisensory information. From the topological point

of view, we have extended the current understanding of cortical

organisation with the ﬁ nding that the cortical hubs form a central

module on top of the cortical hierarchy; which is expressed as the

partial overlap of the four anatomical modules (visual, auditory,

somatosensory-motor and frontolimbic). By means of dynamical

and information theoretical measures, we have corroborated its

capacity to integrate multisensory information, i.e. after simultane-

ous excitation of visual, auditory and somatosensory primary areas,

a particular set of hubs becomes statistically dependent forming a

dynamical cluster. Additionally, the simultaneous lesion of these

hubs leads to a largest decrease in the integrative capacities of the

network. Both structural and functional results indicate that visual

areas 7 and AES, auditory area EPp and frontolimbic areas Ia, Ig,

CGp, 35 and 36 are the most likely candidates to form the top

hierarchical module. The participation of somatosensory-motor

areas is less clear, although area 6m is the strongest candidate of

them. Visual area 20a and somatosensory-motor areas 5Al and 5Bl

are also potential candidates.

The modular and hierarchical organisation here detected agrees

with the behaviour observed in dynamical simulations of cortical

networks. The resting state dynamics are typically governed by the

formation of dynamical clusters which closely relate to the anatomi-

cal modules, but the inﬂ uence of the hierarchical organisation is

also expressed. In

Zemanová et al. (2006) and Zhou et al. (2006,

2007)

it was shown that the correlation between the dynamical

clusters is mediated by the cortical hubs. In

Honey et al. (2007) the

centrality of the hubs was found to oscillate in time. Simulation

of excitable dynamics on hierarchical networks (

Müller-Linow

et al., 2008

) has shown that the dynamical behaviour of the corti-

cal network of the cat may be dominated either by the modular

structure or by the hubs, depending on the time scales.

SEGREGATION, INTEGRATION AND LOCALISATION

The separation of modal information paths is a relevant charac-

teristic of organisation in the nervous system that permits simul-

taneous (parallel) processing of sensory input and detection of

its features. Cortical regions containing neurones specialised in

similar function, e.g. in processing information of the same sen-

sory modality, lie geographically close to each other (Figure 10A).

However, a coherent perception and the emergence of mental

states such as awareness and consciousness require that infor-

mation is integrated at different levels: the binding of sensory

features into entities, the combination of entities with memo-

ries (personal experiences) into events, etc. While experimental

techniques have led to a deep understanding about the basis of

sensory perception, the nature of integration and the localisation

of brain regions involved in it, is still under the subject of debate.

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 11

Zamora-López et al. Cortical module for integration

As stated by Fuster (2003), simple extrapolation of the principles

of sensory organisation do not lead to the identiﬁ cation of the

substrate for cognition.

Several models have proposed that high-level functions are rep-

resented by distributed, interactive and overlapping networks of

neurones, which transcend any of the traditional subdivisions of

the cortex by structural (cytoarchitecture) or functional criteria

(

Damasio, 1989; Fuster, 2003, 2006; Tononi, 2004). During the

recent years increasing experimental evidence has conﬁ rmed this

hypothesis and the networked perspective has gained the favour

against the assumption of a single brain region fully responsible

for integration (

Stam and Reijneveld, 2007; Bullmore and Sporns,

2009; Knight, 2009

). The anatomical networked connectivity may

serve as the basis in which localised and distributed functional

networks rapidly emerge and dissolve governed by coordination

dynamics according to the sensory stimulation and the ongoing

activity (

Bressler and Kelso, 2001).

As a further evidence, our results resolve the anatomical organi-

sation substrate that supports the capacity of the cerebral cortex

to simultaneously segregate and integrate information. In the

light of this organisation, it could be envisioned that multisensory

integration emerges from the collaborative function of the cortical

hubs. While early sensory cortical regions perform specialised

processing of the sensory input, the hubs of the network may

work together to combine the multisensory information. A relevant

organisation difference is that the cortical hubs form a module

which is densely connected by axonal paths through the white mat-

ter, but is geographically delocalised (Figure 10B).

LIMITATIONS AND OUTLOOK

The current paper focuses in the corticocortical connectivity

of cats because it is, up to date, the most complete and reliable

dataset of its kind. Hence, it is the most suitable for a detailed

and statistically consistent analysis. The main limitation is that

it comprises of interconnection between cortical areas in only

one cerebral hemisphere. Because of the known inter-hemisfere

differences in many mammals, particularly in humans, it will

be very valuable in the future to acquire the connectivity within

and between both hemispheres in animal and human models.

Based on current literature in which the cortical networks of the

macaque and cat models display similar features, we expect that

the general organisation principles here exposed to be valid in a

wide range of mammals.

An interesting challenge is now to explain the emergence of

this modular and hierarchical organisation in terms of evolution

and development, in particular how the delocalised cluster of

hubs could have evolved if, apparently, areas of similar function

tend to be grouped close to each other. Very likely, the balancing

between short wiring requirements (leading to minimisation of

energy costs) and short processing paths allowing for robustness

and fast responses (

Kaiser and Hilgetag, 2006) plays a major role.

It would also be of relevance to ﬁ nd out whether similar hierarchi-

cal patterns are repeated across smaller scales within the cortex,

i.e. the interconnections between cortical columns and micro-

columns. This would imply an underlying fractal-like complex

architecture which can emerge from simple rules of assembly

during development.

FIGURE 10 | Spatial location of the areas according to their modality: visual (yellow), auditory (red), somatosensory-motor (green) and frontolimbic (blue).

While areas of similar modality tend to lie close to each other (A), the hubs form a topological cluster which is spatially delocalised (B).

Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 12

Zamora-López et al. Cortical module for integration

Finally, we should remind that current non-invasive techniques

such as EEG, MEG and fMRI reveal only the presence of brain

activity. They permit to identify which brain regions are associated

with certain experimental condition. However, at the current stage

it is very difﬁ cult, if not impossible, to understand what is exactly

an activated region doing. Is it ﬁ ltering a signal? Is it integrating

information? Is an activation detected only because that particular

region contains memories which are being retrieved and passed to

other regions for processing? In our opinion, it would be highly

interesting to further develop concepts of information theory as

the modular and local capacity of integration here presented which

applied to the time series of regional activity might help understand

the particular function of individual brain regions within a given

experimental task.

ACKNOWLEDGMENTS

We thank Lucia Zemanová, Claus-C. Hilgetag and Werner Sommer

for valuable discussions. Gorka Zamora-López and Jürgen Kurths

are supported by the Deutsche Forschungsgemeinschaft, research

group FOR 868 (contract No. KU 837/23-1) and by the BioSim

network of excellence (contract No. LSHB-CT-2004-005137 and

No. 65533).

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Frontiers in Neuroinformatics www.frontiersin.org March 2010 | Volume 4 | Article 1 | 13

Zamora-López et al. Cortical module for integration

in complex brain networks. Phys. Rev.

Lett. 97, 238103.

Zhou, C. S., Zemanová, L., Zamora-López,

G., Hilgetag, C. C., and Kurths, J.

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Copyright © 2010 Zamora-López, Zhou and

Kurths. This is an open-access article subject

to an exclusive license agreement between

the authors and the Frontiers Research

Foundation, which permits unrestricted

use, distribution, and reproduction in any

medium, provided the original authors and

source are credited.

02 February 2010; published online: 19

March 2010.

Citation: Zamora-López G, Zhou C

and Kurths J (2010) Cortical hubs form

a module for multisensory integration

on top of the hierarchy of cortical net-

works. Front. Neuroinform. 4:1. doi:

10.3389/neuro.11.001.2010

Conflict of Interest Statement: The

authors declare that the research was con-

ducted in the absence of any commercial or

ﬁ nancial relationships that could be con-

strued as a potential conﬂ ict of interest.

Received: 06 April 2009; paper pend-

ing published: 10 June 2009; accepted:

- CitationsCitations114
- ReferencesReferences57

- "In particular, high-degree structural and functional hubs appear to be central in the PLS-derived network patterns. This result provides further evidence that, by virtue of their high connectivity, hub regions integrate information and facilitate communication among multiple brain regions (Zamora-Lopez et al. 2010; van den Heuvel and Sporns 2011; van den Heuvel et al. 2012; Mišicét al. 2014). As a result, the formation of global network patterns naturally revolves around hub regions, which serve to promote synchronization among distributed areas. "

[Show abstract] [Hide abstract]**ABSTRACT:**The dynamics of spontaneous fluctuations in neural activity are shaped by underlying patterns of anatomical connectivity. While numerous studies have demonstrated edge-wise correspondence between structural and functional connections, much less is known about how large-scale coherent functional network patterns emerge from the topology of structural networks. In the present study, we deploy a multivariate statistical technique, partial least squares, to investigate the association between spatially extended structural networks and functional networks. We find multiple statistically robust patterns, reflecting reliable combinations of structural and functional subnetworks that are optimally associated with one another. Importantly, these patterns generally do not show a one-to-one correspondence between structural and functional edges, but are instead distributed and heterogeneous, with many functional relationships arising from nonoverlapping sets of anatomical connections. We also find that structural connections between high-degree hubs are disproportionately represented, suggesting that these connections are particularly important in establishing coherent functional networks. Altogether, these results demonstrate that the network organization of the cerebral cortex supports the emergence of diverse functional network configurations that often diverge from the underlying anatomical substrate.- "Our measure satisfactorily vanishes in both extremal cases: dynamical independence and global synchrony. On the contrary, the 'neural complexity' measure proposed in [2] grows monotonically with coupling strength and becomes infinity when the network is globally synchronised (see Supplementary Material for details). Our measure derives from the one introduced in [39, 40] . "

[Show abstract] [Hide abstract]**ABSTRACT:**The major structural ingredients of the brain and neural connectomes have been identified in recent years. These are (i) the arrangement of the networks into modules and (ii) the presence of highly connected regions (hubs) forming so-called rich-clubs. It has been speculated that the combination of these features allows the brain to segregate and integrate information but a de- tailed investigation of their functional implications is missing. Here, we examine how these network properties shape the collective dynamics of the brain. We find that both ingredients are crucial for the brain to host complex dynamical behaviour. Comparing the connectomes of C. elegans, cats, macaques and humans to surrogate networks in which one of the two features is destroyed, the functional complexity of the perturbed networks is always decreased. Moreover, a comparison between simulated and empirically obtained resting-state functional connectivity indicates that the human brain, at rest, is in a dynamical state that reflects the largest complexity the anatomical connectome is able to host. In other words, the brain operates at the limit of the network resources it has at hand. Finally, we introduce a new model of hierarchical networks that successfully combines modular organisation with rich-club forming hubs. Our model hosts more complex dynamics than the hierarchical network models previously defined and widely used as benchmarks.- "On the other hand, it will lead to great damage of the whole networks in most cases due to a key node's failure. It is well known that many mechanisms such as spreading, cascading, and synchronizing are highly affected by a tiny fraction of key nodes [11][12][13][14][15][16]. In recent years, a series of cascading large blackouts have occurred in some countries, which caused huge social economy loss. "

[Show abstract] [Hide abstract]**ABSTRACT:**Evaluating the importance of nodes for complex networks is of great significance to the research of survivability and robusticity of networks. This paper proposes an effective ranking method based on degree value and the importance of lines. It can well identify the importance of bridge nodes with lower computational complexity. Firstly, the properties of nodes that are connected to a line are used to compute the importance of the line. Then, the contribution of nodes to the importance of lines is calculated. Finally, degree of nodes and the contribution of nodes to the importance of lines are considered to rank the importance of nodes. Five real networks are used as test data. The experimental results show that our method can effectively evaluate the importance of nodes for complex networks.

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