Functional Brain Networks - Javier M. Buldù

FUNCTIONAL BRAIN NETWORKS

JAVIER M. BULDÚ
UNIVERSIDAD REY JUAN CARLOS (MÓSTOLES)
CENTRO DE TECNOLOGÍA BIOMÉDICA (POZUELO)
COMPLEJIMAD (MADRID)
(… A MINEFIELD!)

OUTLINE
❑ Functional Brain Networks
❑ Measuring Brain Activity
❑Time Series & Network Construction
❑ Network Analysis
❑ Risks & Challenges
❑ Functional Networks are alive
❑ Living in Hillsville
❑ Conclusions
❑ Brain Networks
❑ Anatomical Networks
❑ Functional Networks

COMPLEX SYSTEMS
A complex system is composed of interrelated parts which, as a whole, exhibit
properties and behaviors that can not be explained analyzing each of the individual parts
separately:
A neuron A brain

What if we apply network science to the most challenging
system we are facing?
APPLYING NETWORK SCIENCE TO THE BRAIN

APPLYING NETWORK SCIENCE TO THE BRAIN
In brief, (main) types of brain networks
From Bullmore & Sporns, Nature Rev. 10, 186 (2009)
anatomical functional

M. Rubinov and O. Sporns,
NeuroImage 52, 1059–1069 (2010)
Appendix B. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.neuroimage.2009.10.003.
References
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Bassett, D.S., Meyer-Lindenberg, A., Achard, S., Duke, T., Bullmore, E., 2006. Adaptive
reconfiguration of fractal small-world human brain functional networks. Proc. Natl.
Acad. Sci. U. S. A. 103, 19518–19523.
Bassett, D.S., Bullmore, E., Verchinski, B.A.,Mattay,V.S.,Weinberger, D.R., Meyer-Lindenberg,
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M., Mut
Blondel, V.D
commun
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Brandes, U.,
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structur
Butts, C.T., 2
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cortical
1, 16.
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Table A1 (continued)
Measure Binary and undirected definitions W
Measures of resilience
Degree distribution Cumulative degree distribution of the network
(Barabasi and Albert, 1999),
P kð Þ =
X
kVzk
p kVð Þ;
where p(k′) is the probability of a node having degree k′.
C
C
C
Average neighbor
degree
Average degree of neighbors of node i (Pastor-Satorras et al., 2001),
knn;i =
P
jaN aijkj
ki
:
A
B
k
A
k
Assortativity coefficient Assortativity coefficient of the network (Newman, 2002),
r =
l
−1 P
i;jð ÞaL kikj − l
− 1 P
i;jð ÞaL
1
2 ki + kj
h i2
l−1
P
i;jð ÞaL
1
2 k2
i + k2
j

− l−1
P
i;jð ÞaL
1
2 ki + kj
h i2
:
W
L
r
D
r
Other concepts
Degree distribution
preserving network
randomization.
Degree-distribution preserving randomization is implemented by
iteratively choosing four distinct nodes i1, j1, i2, j2 ∈ N at random,
such that links (i1, j1), (i2, j2) ∈ L, while links (i1, j2), (i2, j1) ∉ L.
The links are then rewired such that (i1, j2), (i2, j1) ∈ L and (i1, j1),
(i2, j2) ∉ L, (Maslov and Sneppen, 2002).
“Latticization” (a lattice-like topology) results if an additional
constraint is imposed, |i1+j2| + |i2+j1| b |i1+j1| + |i2+j2|
(Sporns and Kotter, 2004).
T
In
li
p
to
sc
Measure òf network
small-worldness.
Network small-worldness (Humphries and Gurney, 2008),
S =
C = Crand
L = Lrand
;
where C and Crand are the clustering coefficients, and L and Lrand are
the characteristic path lengths of the respective tested network and
a random network. Small-world networks often have S ≫ 1.
W
D
In
All binary and undirected measures are accompanied by their weighted and directed generalizations. G
knowledge) are marked with an asterisk (⁎). The Brain Connectivity Toolbox contains Matlab functions
Table A1 (continued)
Measure Binary and undirected definitions Weighted and directed definitions
Modularity Modularity of the network (Newman, 2004b),
Q =
X
uaM
euu −
X
vaM
euv
!2
#
;
where the network is fully subdivided into a set of nonoverlapping
modules M, and euv is the proportion of all links that connect nodes
in module u with nodes in module v.
An equivalent alternative formulation of the modularity
(Newman, 2006) is given by Q = 1
l
P
i;jaN aij −
ki kj
l

δmi ;mj
,
where mi is the module containing node i, and δmi,mj = 1 if mi = mj,
and 0 otherwise.
Weighted modularity (Newman, 2004),
Qw
= 1
lw
P
i;jaN wij −
k
w
i k
w
j
l
w
!
δmi ;mj
:
Directed modularity (Leicht and Newman, 2008),
QY
= 1
l
P
i;jaN aij −
k
out
i k
in
i
l
!
δmi ;mj
:
Measures of centrality
Closeness centrality Closeness centrality of node i (e.g. Freeman, 1978),
L
−1
i =
n − 1
P
jaN;j≠i
dij
:
Weighted closeness centrality, Lw
i
À Á− 1
= n − 1P
jaN; j≠i
d
w
ij
.
Directed closeness centrality, LY
i
À Á−1
= n − 1P
jaN; j≠i
d
Y
ij
.
Betweenness centrality Betweenness centrality of node i (e.g., Freeman, 1978),
bi =
1
n − 1ð Þ n − 2ð Þ
P
h;jaN
h≠j;h≠i;j≠i;
ρ
hj ið Þ
ρ
hj
;
where ρhj is the number of shortest paths between h and j, and ρhj (i)
is the number of shortest paths between h and j that pass through i.
Betweenness centrality is computed equivalently on
weighted and directed networks, provided that path lengths
are computed on respective weighted or directed paths.
Within-module degree
z-score
Within-module degree z-score of node i
(Guimera and Amaral, 2005),
zi =
ki mið Þ − k mið Þ
σk mið Þ
;
where mi is the module containing node i, ki (mi) is the
within-module degree of i (the number of links between i and all
other nodes in mi), and k mið Þ and σk(mi)
are the respective mean
and standard deviation of the within-module mi degree distribution.
Weighted within-module degree z-score, zw
i =
k
w
i mið Þ − kw
mið Þ
σkw mið Þ
.
Within-module out-degree z-score, zout
i =
k
out
i mið Þ − kout
mið Þ
σ
kout mið Þ
.
Within-module in-degree z-score, zin
i =
k
in
i mið Þ − kin
mið Þ
σ
kin mið Þ
.
Participation coefficient Participation coefficient of node i (Guimera and Amaral, 2005),
yi = 1 −
X
maM
ki mð Þ
ki
2
;
where M is the set of modules (see modularity), and ki (m) is the
number of links between i and all nodes in module m.
Weighted participation coefficient, yw
i = 1 −
P
maM
kw
i
mð Þ
kw
i
2
.
Out-degree participation coefficient, yout
i = 1 −
P
maM
kout
i
mð Þ
kout
i
2
.
In-degree participation coefficient, yin
i = 1 −
P
maM
kin
i
ðmÞ
kin
i
2
.
Network motifs
Anatomical and
functional motifs
Jh is the number of occurrences of motif h in all subsets of the
network (subnetworks). h is an nh node, lh link, directed connected
pattern. h will occur as an anatomical motif in an nh node, lh link
subnetwork, if links in the subnetwork match links in h
(Milo et al., 2002). h will occur (possibly more than once) as a
functional motif in an nh node, lh′ ≥ lh link subnetwork, if at least one
combination of lh links in the subnetwork matches links in h
(Sporns and Kotter, 2004).
(Weighted) intensity of h (Onnela et al., 2005),
Ih =
P
u Π i;jð ÞaLhu
wij
1
lh ;
where the sum is over all occurrences of h in the network,
and L
hu
is the set of links in the uth occurrence of h.
Note that motifs are directed by definition.
Motif z-score z-Score of motif h (Milo et al., 2002),
zh =
Jh − h Jrand;hi
σ Jrand;h
;
where 〈Jrand,h〉 and σ Jrand,h
are the respective mean and standard
deviation for the number of occurrences of h in an ensemble of
random networks.
Intensity z-score of motif h (Onnela et al., 2005),
zI
h =
Ih − hIrand;h i
σ
Irand;h
;
where 〈Irand,h〉 and σ Irand,h
are the respective
mean and standard deviation for the intensity of h in an
ensemble of random networks.
Motif fingerprint nh node motif fingerprint of the network (Sporns and Kotter, 2004),
Fnh
hVð Þ =
X
iaN
Fnh;i hVð Þ =
X
iaN
JhV;i;
where h′ is any nh node motif, Fnh,i (h′) is the nh node motif
fingerprint for node i, and Jh′,i is the number of occurrences of
motif h′ around node i.
nh node motif intensity fingerprint of the network,
FI
nh
hVð Þ =
P
iaNFI
nh;i hV
À Á
=
P
iaN IhV; i,
where h′ is any nh node motif, FI
nh,i (h′) is the nh node motif
intensity fingerprint for node i, and Ih′,i is the intensity of
motif h′ around node i.
(continued on next page)
Mathematical definitions of complex network measures (see supplementary information for a self-contained version of this table).
Measure Binary and undirected definitions Weighted and directed definitions
Basic concepts and measures
Basic concepts and
notation
N is the set of all nodes in the network, and n is the number of nodes.
L is the set of all links in the network, and l is number of links.
(i, j) is a link between nodes i and j, (i, j ∈ N).
aij is the connection status between i and j: aij = 1 when link (i, j)
exists (when i and j are neighbors); aij = 0 otherwise (aii = 0 for all i).
We compute the number of links as l = ∑i,j∈N aij (to avoid
ambiguity with directed links we count each undirected link twice,
as aij and as aji).
Links (i, j) are associated with connection weights wij.
Henceforth, we assume that weights are normalized,
such that 0 ≤ wij ≤ 1 for all i and j.
lw
is the sum of all weights in the network, computed
as lw
= ∑i,j∈N wij.
Directed links (i, j) are ordered from i to j. Consequently,
in directed networks aij does not necessarily equal aji.
Degree: number of links
connected to a node
Degree of a node i,
ki =
X
jaN
aij:
Weighted degree of i, ki
w
= ∑j∈Nwij.
(Directed) out-degree of i, ki
out
= ∑j∈Naij.
(Directed) in-degree of i, ki
in
= ∑j∈Naji.
Shortest path length:
a basis for measuring
integration
Shortest path length (distance), between nodes i and j,
dij =
X
auv agi X j
auv;
where gi↔j is the shortest path (geodesic) between i and j. Note
that dij = ∞ for all disconnected pairs i, j.
Shortest weighted path length between i and j,
dij
w
= ∑auv∈gi↔j
w
f(wuv), where f is a map (e.g., an inverse)
from weight to length and gi↔j
w
is the shortest weighted
path between i and j.
Shortest directed path length from i to j, dij
→
= ∑aij∈gi→j
aij,
where gi→j is the directed shortest path from i to j.
Number of triangles: a
basis for measuring
segregation
Number of triangles around a node i,
ti =
1
2
X
j;haN
aijaihajh:
(Weighted) geometric mean of triangles around i,
tw
i = 1
2
P
j;haN wijwihwjh
À Á1=3
:
Number of directed triangles around i,
tY
i = 1
2
P
j;haN aij + aji
À Á
aih + ahið Þ ajh + ahj
À Á
.
Measures of integration
Characteristic path
length
Characteristic path length of the network
(e.g., Watts and Strogatz, 1998),
L =
1
n
X
iaN
Li =
1
n
X
iaN
P
jaN;j≠i dij
n − 1
;
where Li is the average distance between node i and all other nodes.
Weighted characteristic path length, Lw
= 1
n
P
iaN
P
jaN; j≠i
d
w
ij
n − 1 .
Directed characteristic path length, LY
= 1
n
P
iaN
P
jaN; j≠i
d
Y
ij
n − 1 .
Global efficiency Global efficiency of the network (Latora and Marchiori, 2001),
E =
1
n
X
iaN
Ei =
1
n
X
iaN
P
jaN;j≠i d
−1
ij
n − 1
;
where Ei is the efficiency of node i.
Weighted global efficiency, Ew
= 1
n
P
iaN
P
jaN; j≠i
dw
ij
− 1
n − 1 .
Directed global efficiency, EY
= 1
n
P
iaN
P
jaN; j≠i
dY
ij
− 1
n − 1 .
Measures of segregation
Clustering coefficient Clustering coefficient of the network (Watts and Strogatz, 1998),
C =
1
n
X
iaN
Ci =
1
n
X
iaN
2ti
ki ki − 1ð Þ
;
where Ci is the clustering coefficient of node i (Ci = 0 for ki b 2).
Weighted clustering coefficient (Onnela et al., 2005),
Cw
= 1
n
P
iaN
2tw
i
ki ki − 1ð Þ
. See Saramaki et al. (2007) for
other variants.
Directed clustering coefficient (Fagiolo, 2007),
CY
= 1
n
P
iaN
tY
i
kout
i
+ kin
ið Þ kout
i
+ kin
i
− 1ð Þ − 2
P
jaN
aijaji
:
Transitivity Transitivity of the network (e.g., Newman, 2003),
T =
P
iaN 2ti
P
iaN ki ki − 1ð Þ
:
Note that transitivity is not defined for individual nodes.
Weighted transitivity⁎, Tw
=
P
iaN
2tw
iP
iaN
ki ki − 1ð Þ
.
Directed transitivity⁎,
TY
=
P
iaN
tY
i
P
iaN
k
out
i + k
in
i
À Á
k
out
i + k
in
i − 1
À Á
− v2
P
jaN
aijaji
h i :
Local efficiency Local efficiency of the network (Latora and Marchiori, 2001),
Eloc =
1
n
X
iaN
Eloc;i =
1
n
X
iaN
P
j;haN;j≠i aijaih djh Nið Þ
h i−1
ki ki − 1ð Þ
;
where Eloc,i is the local efficiency of node i, and djh (Ni) is the
length of the shortest path between j and h, that contains only
neighbors of i.
Weighted local efficiency⁎,
Ew
loc = 1
2
P
iaN
P
j;haN; j≠i
wijwih d
w
jh Nið Þ½ Š
− 1
1 = 3
ki ki − 1ð Þ
:
Directed local efficiency⁎,
EY
loc = 1
2n
P
iaN
P
j;haN;j≠i
aij + ajið Þ aih + ahið Þ d
Y
jh Nið Þ
Â Ã− 1
+ d
Y
hj Nið Þ
Â Ã− 1

k
out
i + k
in
i
À Á
k
out
i + k
in
i − 1
À Á
− 2
P
jaN
aijaji
:
… and many more!!

ANATOMICAL BRAIN NETWORKS
The connectome is a comprehensive map of neural connections in the brain.The production
and study of connectomes, known as connectomics, may range in scale from a detailed map
of the full set of neurons and synapses of an organism to a macro scale description of
the structural connectivity between all cortical areas and subcortical structures.

Do we have a complete connectome?Yes, we do!
C. Elegans: connectivity matrix.
(O. Sporns,“The Networks of the Brain”)
• C. Elegans, a nematode.
• 302 neurons (hermaphrodite),
383 neurones (male).
• We now all neurons and
connections between them.
THE CONNECTOME: A NECESSARY SUBSTRATE

The wiring diagram is an starting point for making hypothesis but…
A) … it cannot reveal how neurons behave in real time, nor does it account
for the many “mysterious” ways that neurons regulate one another's behavior.
B) … the dynamics is quite unpredictable when studying complex movements.
C) … we don't have a comprehensive model of how the worm's nervous system
actually produces the behaviors.
D) … the strength of the synaptic connections changes due to dynamics, also
the amount of neurotransmitters…

Overall structure of PVX, a male-speciﬁc interneuron, showing distribution of synapses. (G) Detail
of individual synapses. Width of lines indicates synapse size. Edge weights are determined by
counting the number of 70- to 90-nm serial sections crossed by individual synapses and summing
over all the synapses.
onJuly26,2012www.sciencemag.orgDownloadedfrom
tween each pair of cells (16). (The resulting struc-
tural weight adjacency matrices for the chemical
S8 and S9). Individual presynaptic densities
varied in size over a 40-fold range, whereas
30-fold range (fig. S2). As a result of the vari-
ation in both number of synapses between pairs
Fig. 1. Specializations of the C. elegans adult male tail for mating. (A) The
substeps of mating. (B) Ventral view of the adult male tail showing mating
structures with five types of sensilla. (C) Overall structure of the male ner-
vous system. (D) Ganglia in the tail containing the neuron cell bodies, con-
nected through commissures. Most synaptic connectivity occurs in the
preanal ganglion (PAG). DNC, dorsal nerve cord; VNC, ventral nerve cord;
DRG, dorsorectal ganglion; LG, lumbar ganglion (left and right); CG, cloacal
ganglion (left and right). (E) An example of a male-specific interneuron,
PVX, which has a cell body and extensive sensory input in the PAG, and a
pr
an
(d
ju
ch
si
po
(d
m
27 JULY 2012 VOL 337 SCIEN438
each module correlate well with experimental
evidence (21–26).
Sensory neurons are recurrently connected.
Whereas much of the information flow through the
network from sensory neurons to end organs—
either in monosynaptic pathways or through
type Ia interneurons in disynaptic pathways—
is feedforward, the 52 sensory neurons are ex-
tensively reciprocally and recurrently connected
by both chemical and gap junction synapses.
Forty-nine percent of the chemical synaptic out-
put of sensory neurons is onto other sensory
neurons, and this constitutes input to th
neurons that is seven times the feedb
type Ib and type Ic interneurons. Ninet
the 36 ray sensory neurons make auta
stituting 6.9% of their input from sen
rons. Fifty-eight percent of the gap
connectivity of the sensory neurons is
sensory neurons.
Only on the basis of the recurren
tivity of the sensory neurons and the
tions to the type Ib interneurons, the n
sensory neurons could be partitioned
DOI: 10.1126/science.1221762
, 437 (2012);337Science
et al.Travis A. Jarrell
The Connectome of a Decision-Making Neural Network
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(II-III)
Functional Brain Networks

BEFORE BEGINNING….
Functional networks are not functional networks!
Just a secret* only shared between scientists working on
functional networks…
* keep the secret, please.

IT’S A LONG ROAD… FULL OF TROUBLE!
Obtaining a functional brain network in three steps:
Measuring Brain Activity
STEP 1
Time Series Analysis
Network Construction
Network Analysis
STEP 2 STEP 3

OBTAINING FUNCTIONAL BRAIN NETWORKS
STEP 1: Measuring Brain Activity
Functional MRI (fMRI). The detection of changes in regional brain activity through their effects on
blood ﬂow and blood oxygenation (which, in turn, affect magnetic susceptibility and tissue contrast in
magnetic resonance images). High spatial resolution (~mm3) but low temporal resolution
(~seconds).
Electroencephalography (EEG). A technique used to measure neural activity by monitoring
electrical signals from the brain, usually through scalp electrodes. EEG has good temporal resolution
but relatively poor spatial resolution.
Magnetoencephalography (MEG). A method of measuring brain activity by detecting
perturbations in the extracranial magnetic ﬁeld that are generated by the electrical activity of
neuronal populations. Like EEG, it has good temporal resolution but relatively poor spatial resolution.
It has better resolution than EEG.
Others…

Functional MRI (fMRI). The detection of changes in regional brain activity through their effects on blood ﬂow and blood
oxygenation (which, in turn, affect magnetic susceptibility and tissue contrast in magnetic resonance images). High spatial
resolution (~mm3) but low temporal resolution (~seconds).
Typical activation maps.Magnetic Resonance

Example:
The correlation matrix is calculated and then used to define the network among the highest
correlated nodes. Top four images represent snapshots of activity and the three traces
correspond to selected voxels from visual (V1), motor (M1) and posterio-parietal (PP)
cortices. 147456 voxels.
Scale-Free Brain Functional Networks
Victor M. Eguı´luz,1
Dante R. Chialvo,2
Guillermo A. Cecchi,3
Marwan Baliki,2
and A. Vania Apkarian2
1
Instituto Mediterrańeo de Estudios Avanzados, IMEDEA (CSIC-UIB), E07122 Palma de Mallorca, Spain
2
Department of Physiology, Northwestern University, Chicago, Illinois, 60611, USA
3
IBM T.J. Watson Research Center, 1101 Kitchawan Rd., Yorktown Heights, New York 10598, USA
(Received 13 January 2004; published 6 January 2005)
Functional magnetic resonance imaging is used to extract functional networks connecting correlated
human brain sites. Analysis of the resulting networks in different tasks shows that (a) the distribution of
functional connections, and the probability of finding a link versus distance are both scale-free, (b) the
characteristic path length is small and comparable with those of equivalent random networks, and (c) the
clustering coefficient is orders of magnitude larger than those of equivalent random networks. All these
properties, typical of scale-free small-world networks, reflect important functional information about
brain states.
DOI: 10.1103/PhysRevLett.94.018102 PACS numbers: 87.18.Sn, 87.19.La, 89.75.Da, 89.75.Hc
Recent work has shown that disparate systems can be
described as complex networks, that is, assemblies of
nodes and links with nontrivial topological properties,
examples of which include technological, biological and
social systems [1]. The brain is inherently a dynamic sys-
tem, in which the traffic between regions, during behavior
or even at rest, creates and reshapes continuously com-
plex functional networks of correlated dynamics. An im-
portant goal in neuroscience is to understand these spatio-
temporal patterns of brain activity. This Letter proposes a
method to extract functional networks, as revealed by func-
tional magnetic resonance imaging (FMRI) in humans, and
analyze them in the context of the current understanding of
complex networks (for reviews see [1–3]).
Figure 1 shows how underlying functional networks are
exposed during any given task. In these experiments, at
each time step (typically 400 spaced 2.5 sec.), magnetic
resonance brain activity is measured in 36 64 64 brain
sites (so-called ‘‘voxels’’ of dimension 3 3:475
3:475 mm3
). The activity of voxel x at time t is denoted
as V x; t . We define that two voxels are functionally con-
nected if their temporal correlation exceeds a positive
predetermined value rc, regardless of their anatomical
connectivity [4,5]. Specifically, we calculate the linear
correlation coefficient between any pair of voxels, x1 and
x2, as
r x1; x2
hV x1; t V x2; t i hV x1; t ihV x2; t i
V x1 V x2
; (1)
where 2 V x hV x; t 2i hV x; t i2, and h i repre-
sents temporal averages.
Figure 2 shows the degree distributions of networks
extracted using this method. The data were collected while
the subject was opposing fingers one and two during
10 sec, and then resting during 10 sec. We find a skewed
distribution of links with a tail approaching a distribution
p k k , with around 2. This power law is more
evident for networks constructed with higher thresholds
rc (more correlated conditions). For decreasing rc, a maxi-
mum appears which shifts to the right. Despite changes in
parameters, networks remain clearly defined indicating
that the main conclusions are robust with respect to the
selection of parameters. The small inset in Fig. 2 shows the
distribution of links of a network constructed from the
randomly shuffled (in time) voxels’ signal. This network
displays a Gaussian degree distribution in which the mean
and width depend on rc. The largest values of the correla-
tion thresholds used to construct the random networks are
usually extremely low (rc 0:1) compared to that used to
FIG. 1 (color). Methodology used to extract functional net-
works from the signals. The correlation matrix is calculated and
then used to define the network among the highest correlated
nodes. Top four images represent snapshots of activity and the
three traces correspond to selected voxels from visual (V1),
motor (M1) and posterio-parietal (PP) cortices.
PRL 94, 018102 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending
14 JANUARY 2005
0031-9007=05=94(1)=018102(4)$23.00 018102-1 © 2005 The American Physical Society

Electroencephalography (EEG). A technique used to measure neural activity by monitoring
electrical signals from the brain, usually through scalp electrodes. EEG has good temporal
resolution (up to kHz) but relatively poor spatial resolution (typically between 19-256
electrodes).
There are approximately 10 times more glial cells than neurons.
The general shape of neurons is longer and thinner than other cells of the body.
They transmit information and communicate with each other using a combination of
chemical and electrical signals. Figure 2.1 shows a simplified picture of communica-
tion between three neurons. Neuron 2 receives a chemical input from neuron 1, which
causes an electrical signal along its length and generates a chemical signal, which is
transmitted to neuron 3. The chemical signals are in the form of neurotransmitters,
which are released by one neuron and detected by another. The electrical signals are in
the form of action potentials, which travel along neuronal axons. While neurons have
a variety of shapes and sizes, the most general features are dendrites, which receive
input from other neurons, a soma or cell body, and the axon, which conveys electrical
information.
In the brain, each neuron influences and is influenced by many other neurons. An
action potential occurs when inputs at a neuron are summed and the threshold is ex-
ceeded. When the action potential reaches the axon terminal of the neuron, it results
in the release of neurotransmitter from the presynaptic membrane, which causes post-
synaptic potentials (PSP).
Two types of PSP’s can be classified. The Excitatory postsynaptic potential (E)PSP,
causes a depolarization of the cell due to the arrival of action potentials at a synapse.
Neuron 1
(Action potential)
Output
chemical
transmission
(Synapse)(Synapse)
Dendrites
Axon
Soma
Neuron 2 Neuron 3
Electrical
transmission
Input
chemical
transmission
Figure 2.1: A simple picture of signal transmission in neurons.
Oh my god! A syringe!

The sample of human EEG with prominent resting state activity - alpha-rhythm. Left - EEG traces (horizontal
- time in seconds; vertical - amplitudes, scale 100uV). Right - power spectra of shown signals (vertical lines -
10 and 20 Hz, scale is linear).Alpha-rhythm consists of sinusoidal-like waves with frequencies in 8–12 Hz
range (11 Hz in this case) more prominent in posterior sites.Alpha range is red at power spectrum graph.
We are basically recording time series:

The samples of main types of artifacts in human EEG. 1 - Electrooculographic artifact caused by the
excitation of eyeball's muscles (related to blinking, for example). Big amplitude, slow, positive wave
prominent in frontal electrodes. 2 - Electrode's artifact caused by bad contact (and thus bigger
impedance) between P3 electrode and skin. 3 - Swallowing artifact. 4 - Common reference electrode's
artifact caused by bad contact between reference electrode and skin. Huge wave similar in all channels.
Unavoidable artifacts:

Magnetoencephalography (MEG). A method of measuring brain activity by detecting
perturbations in the extracranial magnetic ﬁeld that are generated by the electrical activity of
neuronal populations. Like EEG, it has good temporal resolution but relatively poor spatial
resolution. It has better resolution than EEG.
EEG and MEG in brain research
+++
++++++
+ +
++++++
...
...
.
...
.....
..
..
.....
......
.
...
.....
...
Extracellular current
Volume
currents
Magnetic
field
lines
Electric
isopotential
lines
+
++++
+ + +
--
-
-
-
- --
c)
Propagation
+
a+
Dendrite
Na+
channels in the neuronal membrane open in response to a
of the membrane potential. The leading edge of the depolarization
y Na+
channels and a wave of depolarization spreads from the
ction potentials move in one direction. This is achieved because
iod of the Na+
channels. After activation Na+
channels do not
ures that the action potential is propagated in only one direction
intracellular current and oppositely directed extracellular current
rcuit. Within the relatively long time during, which the current
ically tens of milliseconds or more), it is reasonable to consider
femtoTeslas… yes: 10^(-15)…
… Earth doesn’t help (microTeslas)

Low spatial resolution (we have ~10
11
neurons)
Measurements are overlapped
In EEG and MEG, we only measure cortical activity
Deﬁning the nodes is a complex task
Brain is not an isolated system
High variability in the results

Brain parcellation: what is a node?

Source reconstruction: inverse problem
Source reconstruction tries to identify the sources of the magnetic
ﬁeld, but ….
… every inverse methods makes speciﬁc assumptions…
… (ideally) performs well if assumptions are met. … 
… but there are no method that performs well in general.
Inverse methods
MNE
MCE
WMNE
Loreta
sLORETA
eLORETA
Laura
Electra
WROP
DICS
LCMV-Beamformer
Nulling Beamformer
FOCUSS
Champagne
Minimum Entropy
Dipole Modeling
Multipole Modeling
MUSICRAP-MUSIC
S-FLEX
DCM
Different methodologies,
the majority are black
boxes for the user.

IT’S A LONG ROAD… FULL OF TROUBLE!
Obtaining a functional brain network in three steps:
Time Series Analysis
Network Construction
STEP 2
Measuring Brain Activity
STEP 1
Network Analysis
STEP 3

STEP 1I: Time Series Analysis Network Construction
How to measure coordination
between brain regions?
Cross-correlation
Wavelet coherence
Synchronization Likelihood
Generalized Synchronization
Phase Synchronization
Mutual Information
Granger Causality
Once coordination is evaluated, we
construct the functional network.
* For a review: Pereda et al, Prog. Neurobiol, 77 (2005)

Linear: Evaluate correlation between time series.They are the simplest and,
sometimes, good enough.
Two groups: MCI (21) and Control (21).
MEG. Memory task. Classiﬁcation
algorithms: Multi-Layer Perceptron
(MLP), Probabilistic Neural Networks
(PNN), Decision Tree (DT), y K Nearest
Neighbours (KNN).M. Zanin, PhDThesis

Non-Linear: Based on a nonlinear function between x(t) and y(t).They
also include phase synchronization indexes.
0 50 100 150
time [sec]
−5
0
5
10
15
ϕ1,1/2π
0 50 100 150
time [sec]
0 50 100 150
time [sec]
yx
(a) (b) (c)
0 50 100 150
time [sec]
−5
0
5
10
15
ϕ1,1/2π
0 50 100 150
time [sec]
0 50 100 150
time [sec]
yx
(a) (b) (c)
Fig. 7. Stabilograms of a neurological patient for EO (a), EC (b), and A
x(t)=f(y(t))
f(y(t))?
−5
0
5
10
15
ϕ1,1/2πy
(a) (b)
0 50 100 150
−5
0
5
10ϕ1,1/2π
0 50 100 150
yx

Spectral: Based on the analysis of the spectrum of the time series. Also
include different linear/nonlinear ways of comparing spectra.
Spectra of EEG electrodes. M.G. Knyazeva et al., Journal of Neurophysiology

And now, what matrix do I analyze?
“coordination” matrix adjacency matrix normalized matrix

Percentage ThresholdWeighted
Basically, you can choose between three options:

Binarize the matrix by selecting an adequate threshold:
Figure3. Effectsoftaskdifficultyonworkspaceconfigurationofbrainfunctionalnetworksatdifferentfrequencyintervalsoverarangeofnetworkconnectiondensities.A–E,Overallfrequencies
(A)andineachfrequencyinterval(B–E),meanbrainnetworkmetricswith95%confidenceinterval(dottedlines)forzero-back(redline),one-back(greenline),andtwo-back(blueline)taskstend
toconvergeontheirvaluesinsurrogatenetworks(grayline)asconnectiondensityisincreasedfrom2%to20%ofpossibleedges.Theverticaldottedlinesindicatestheconnectiondensityof10%
chosen for ANOVA modeling. Asterisks denote significant difference at p Ͻ 0.05 between two-back and surrogate networks.
Kitzbichler et al. • Workspace Configuration of MEG Networks J. Neurosci., June 1, 2011 • 31(22):8259–8270 • 8265
Kitzblicher et al., J. Neurosci. 2011

STEP 1I: Time Series Network Construction
It is difﬁcult to evaluate weights and causality in the interactions
There is not a unique way of interacting
No clear way of deﬁning the network (threshold problem spurious links)
Functional networks are not static

−5
0
5
10
15
ϕ1,1/2πy
(a)(b)(c)
Example 1: Functional networks are virtual
REAL FUNCTIONAL
0 50 100 150
time [sec]
−5
0
5
10
15
ϕ1,1/2π
0 50 100 150
time [sec]
0 50 100 150
time [sec]
yx
(a) (b) (c)
0 50 100 150
time [sec]
0 50 100 150
time [sec]
(b) (c)
is it correct?

1 4 7 10 13 16 19
40
50
60
70
80
Classificationscore(%)
Deleted node
0.775
0.7875
19 16 13 10 7 4
40
50
60
70
80
Classificationscore(%)
Number of nodes
F3
P8
T7
T8
P7
O2
FZ
FP1
F8
T7
FZ
O2
P8
T8
F3
F8
P7
FP1
Figure 14: Discarding nodes from the networks. (Top Left) Classification score as a function of the node discarded. The dashed
horizontal line represents the best classification score with the complete network (77.5%). (Top Right) Classification score as
a function of the number of surviving nodes. (Bottom) The networks of Fig. 5 when only the 9 most important nodes are
included.
these topological features yielded a rather low score (⇡ 60%). This suggests that these differences were not
as important as initially thought, possibly because the analysis (and specifically, its parameters) was not
properly tuned. The same data mining techniques were the instrument to increase the significance of the
networks, and to point us towards the synchronisation metrics and brain regions most relevant. The upshot
is that we end up with higher prognostic capabilities and better understanding of the pathology at hand,
EEG (19 electrodes), image recognition task. 40 controls alcoholic individuals. Discarding nodes from the networks. (Top
Left) Classification score as a function of the node discarded.The dashed horizontal line represents the best classification
score with the complete network (77.5%). (Top Right) Classification score as a function of the number of surviving nodes.
(Bottom)The networks with only the 9 most important nodes. (Zanin et al., Phys. Rep. 2016)
Example 1I: where to put the threshold?

Suppose we have the network…. let’s analyze it!
a Healthy volunteers b People with schizophrenia
44'
46
37
45
22
45'
39'
39'
22 44
44
44'
8
34'
19 36
36
22
37
9
47
43'
43
39
42'47'
43
21'
20'
21'
19'
10
10'
8'
45'
10
8
9'
39
40
20 19'
19
40'
40
40'
8'
9'
10'
37'
20
12
21
20
36'
37'
43'
46'
47'
47
46
45
Frontal
Temporal
Parietal
Occipital
High clustering
Low clustering
Degree
ANALYZING FUNCTIONAL BRAIN NETWORKS

A. Characterize the topology of brain functional networks and its
inﬂuence in the processes occurring in them.
B. Identify differences between healthy brains and those with a
certain pathology.
C. Develop models in order to explain the changes found in impaired
functional networks.
Network Analysis: Why?

A. Characterize the topology of brain functional networks*
and its inﬂuence in the processes occurring in them:
• Heterogeneous - Crucial nodes.
• High clustering - Good local resilience?
• Small-world topology - High efﬁciency in information transmission?
• Modularity - Segregation integration of information?
• Others: Assortative, degree-degree correlations, rich-clubs, hierarchical
structure,…
* “All generalizations are false, including this one”, MarkTwain (probably…)

Hubs unavoidably appear in functional networks:
❑ Two activities: music and finger tapping
❑ fMRI
❑ 36 x 64 x 64 regione (147456 voxels)
❑ Linear correlation between voxels:
❑ Matrix is thresholded
Music Finger tapping
FUNCTIONAL NETWORKS ARE HETEROGENEOUS

Highly (functionally) connected nodes:
Two different tasks (Eguíluz et al., PRL 2005)
define the functional networks (rc 0:7). Our data were
also compared with values from a randomly rewired net-
work, where nodes keep their degree by permuting links
(i.e., the link connecting nodes i, jis permuted with that
connecting nodes k, l) [6] (see below). In this control the
degree of each node is maintained but all other correlations
(including clustering) are destroyed.
To test the generality of these findings the same analysis
10
0
10
1
10
2
10
3
Degree K
10
0
10
1
10
2
10
3
10
4
10
5
)k(stnuoC
rc
= 0.6
rc
= 0.7
rc
= 0.8
10
0
10
1
10
2
(mm)
10
-4
10
-3
10
-2
10
-1
10
0
k)(.borP
∆
∆
10
0
10
1
10
2
10
3
Degree K
10
0
10
1
10
2
10
3
10
4
Counts(k)
rc
= 0.5
rc
= 0.6
rc
= 0.7
700 800
Degree K
0
500
Counts(k)
FIG. 2 (color online). Degree distribution for three values of
the correlation threshold. The inset depicts the degree distribu-
tion for an equivalent randomly connected network.
PRL 94, 018102 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending
14 JANUARY 2005
fMRI, finger tapping, different thresholds
“…scale-free complex networks are known to
show resistance to failure, facility of
synchronization, and fast signal processing… ”

It is common to observe an exponential cut-off:
The degree distribution of all networks at all frequency bands
and both behavioral states was best described by a truncated
power law, given in the form P(k)∼Ak^(λ−1) e^(k/kc), where A is
the coefficient, λ describes the power law, and kc is the
exponential parameter.
Table 2. Parameters of exponentially
truncated power law degree distribution
A kc
Resting
1 0.8 ± 0.2 1.5 ± 0.4 8 ± 4
2 0.9 ± 0.3 1.6 ± 0.5 5 ± 3
3 0.9 ± 0.3 1.5 ± 0.5 8 ±15
4 0.9 ± 0.3 1.6 ± 0.6 6 ± 4
5 0.8 ± 0.2 1.6 ± 0.4 5 ± 2
6 1.0 ± 0.2 1.4 ± 0.3 8 ± 6
Tapping
1 0.9 ± 0.1 1.4 ± 0.2 9 ± 4
2 0.8 ± 0.2 1.7 ± 0.4 5 ± 1
3 0.8 ± 0.3 1.7 ± 0.5 5 ± 2
4 0.8 ± 0.3 1.7 ± 0.5 6 ± 4
5 0.9 ± 0.2 1.5 ± 0.5 10 ±14
6 1.0 ± 0.1 1.2 ± 0.3 14 ±20
The degree distribution of all networks at all fre-
quency bands and both behavioral states was best
described by a truncated power law, given in the
form P(k) ⇠ Ak 1
ek/kc
, where A is the coe cient,
describes the power law, and kc is the exponential
parameter. These three parameters are given here
along with their standard deviation and show a large
which reflects the distinctive topological properties (greater
density and clustering) of the ␥ network.
Spatial Configuration of Scale-Specific Networks. The spatial distri-
bution of network hubs was also broadly similar across scales and
states (see Fig. 2 and SI Fig. 5). See SI Fig. 6 for average hub
distributions across all scales in both states. However, there were
striking differences between scales and states in the physical
distance between functionally connected network nodes (see
Fig. 3).
In the resting state, long-range functional connectivity be-
tween brain regions was stronger at low frequencies. At higher
frequencies (␤, ␥), long-range connectivity was weaker, and most
of the edges in the graph represented high-density local con-
nections (see Figs. 1E and 4), shown by the increase in charac-
teristic length scale of network edges ␨, going from high to low
frequency scales; and by the increasing number of connector
compared with provincial nodes at low frequencies (see SI Fig.
7 for a schematic and SI Fig. 8 for distributions of provincial and
connector hubs in both states and all frequency bands).
In the finger-tapping state, long-range functional connections
emerged more strongly at high frequencies (␤, ␥), shown by the
significant motor task-related increases in characteristic length
scale of edges in high-frequency motor networks. It is also
represented by the shift from resting-state ␥ networks dominated
by provincial hubs (predominantly connected to locally neigh-
boring regions of bilateral occipital, parietal, and central cortex)
to motor ␥ networks with a larger number of connector hubs in
medial premotor and bilateral prefrontal cortex. Some of the
new long-range connections engendered by task performance at
high frequencies link to topologically pivotal nodes in right
medial premotor and prefrontal cortex with high betweenness
scores (see Fig. 3; and see SI Fig. 9 for betweenness distributions
at all frequencies). This indicates that task performance is
associated with reconfiguration of high-frequency networks to
favor long-distance connections between prefrontal and premo-
Fig. 2. Self-similarity of spatial distribution of highly connected network nodes or ‘‘hubs’’ in the frequency range 2–38 Hz (64). Each column shows the surface
distribution of the degree of network nodes in frequency bands ␤ to ␦: red represents nodes with high degree. The last column shows the spatial distribution
of degree averaged over these four frequency bands, which emphasizes the similarity of spatial conﬁgurations across scales. See SI Fig. 5 for the hub distributions
in both states at all frequency bands.
Fig. 3. State-related differences in spatial conﬁguration of the highest frequency ␥ network. The top row shows the degree distribution and betweenness scores
for the resting state ␥ network; the middle row shows the same maps for the motor ␥ network; the bottom row shows the between-state differences in degree
and betweenness. It is clear that motor task performance is associated with emergence of greater connectivity in bilateral prefrontal and premotor nodes, and
appearance of topologically pivotal nodes (with high betweenness scores) in medial premotor, right prefrontal, and parietal areas. See SI Fig. 7 for the
betweenness distributions in both states at all frequency bands.
19520 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0606005103 Bassett et al.
MEG (275 channels), 11 resting and 11 finger tapping. Node
degree and betweenness. (Basset et al., PNAS 2006)

Let’s see what happens at the local level: clustering coefficient
Scales 1– 6 denote progressively lower-frequency intervals (Hz). r is the mean inter-regional correlation, and R is the
correlation threshold. (N-1 nodes) Lnet
and Cnet
are the mean path length and clustering coefficient, respectively, of the
thresholded network.The lambda︎ and gamma︎ are ratios of brain network path length and clustering coefficient, respectively,
to comparable random network metrics. Sigma︎ is a scalar measure of “small-worldness.” (Achard et al., J. Neurosci. 2006)
FUNCTIONAL NETWORKS HAVE HIGH CLUSTERING
tions between fMRI time series measured in multiple cortical and
subcortical human brain regions in normal human volunteers
scanned during “rest” or a no-task condition. Analysis of resting-
state data has the advantage that it may focus attention on endog-
stituted the set of regional mean time series used for wavelet correlation
analysis.
Wavelet correlation analysis. Wavelet transforms effect a time-scale
decomposition that partitions the total energy of a signal over a set of
compactly supported basis functions, or little waves, each of which is
Table 1. Wavelet scale dependency of functional connectivity and small-world parameters for an entire human brain network
Scale Hz r R Lnet Cnet ␭ ␥ ␴
1 0.23–0.45 0.12 0.13 2.9 0.534 1.28 1.81 1.42
2 0.11–0.23 0.21 0.2 2.6 0.566 1.12 2.14 1.92
3 0.06–0.11 0.39 0.39 2.69 0.555 1.16 2.22 1.91
4 0.03–0.06 0.45 0.44 2.49 0.525 1.08 2.38 2.19
5 0.01–0.03 0.44 0.35 2.4 0.554 1.04 2.39 2.30
6 0.007–0.01 0.41 0.17 2.65 0.515 1.15 2.15 1.88
Scales1–6oftheMODWTdenoteprogressivelylower-frequencyintervals(Hz).risthemeaninter-regionalcorrelation,andRisthecorrelationthreshold.Lnet andCnet arethemeanpathlengthandclusteringcoefficient,respectively,ofthe
thresholded network. The ␭ and ␥ are ratios of brain network path length and clustering coefficient, respectively, to comparable random network metrics. The equation ␴ ϭ ␥/␭ is a scalar measure of “small-worldness.”
64 • J. Neurosci., January 4, 2006 • 26(1):63–72 Achard et al. • A Small-World Human Brain Functional Network
Behavioral/Systems/Cognitive
A Resilient, Low-Frequency, Small-World Human Brain
Functional Network with Highly Connected Association
Cortical Hubs
Sophie Achard,1 Raymond Salvador,1,2 Brandon Whitcher,3 John Suckling,1 and Ed Bullmore1,3
1Brain Mapping Unit and Wolfson Brain Imaging Centre, University of Cambridge, Departments of Psychiatry and Clinical Neurosciences, Addenbrooke’s
Hospital, Cambridge CB2 2QQ, United Kingdom, 2Sant Joan de Deu–Serveis de Salut Mental, Sant Boi de Llobregat, Spain, and 3Translational Medicine and
Genetics, GlaxoSmithKline, Cambridge CB2 2QQ, United Kingdom
Small-world properties have been demonstrated for many complex networks. Here, we applied the discrete wavelet transform to func-
tional magnetic resonance imaging (fMRI) time series, acquired from healthy volunteers in the resting state, to estimate frequency-
dependent correlation matrices characterizing functional connectivity between 90 cortical and subcortical regions. After thresholding
the wavelet correlation matrices to create undirected graphs of brain functional networks, we found a small-world topology of sparse
connections most salient in the low-frequency interval 0.03–0.06 Hz. Global mean path length (2.49) was approximately equivalent to a
comparable random network, whereas clustering (0.53) was two times greater; similar parameters have been reported for the network of
anatomical connections in the macaque cortex. The human functional network was dominated by a neocortical core of highly connected
hubs and had an exponentially truncated power law degree distribution. Hubs included recently evolved regions of the heteromodal
association cortex, with long-distance connections to other regions, and more cliquishly connected regions of the unimodal association
and primary cortices; paralimbic and limbic regions were topologically more peripheral. The network was more resilient to targeted
attack on its hubs than a comparable scale-free network, but about equally resilient to random error. We conclude that correlated,
low-frequency oscillations in human fMRI data have a small-world architecture that probably reflects underlying anatomical connectiv-
ity of the cortex. Because the major hubs of this network are critical for cognition, its slow dynamics could provide a physiological
substrate for segregated and distributed information processing.
Key words: small world; fMRI; wavelet; functional connectivity; association cortex; neuroimaging
Introduction
A remarkable variety of social, economic, and biological net-
works demonstrate “small-world” properties (Strogatz, 2001),
meaning the nodes of the network have greater local interconnec-
tivity or cliquishness than a random network, but the minimum
path length between any pair of nodes is smaller than would be
expected in a regular network or lattice (Watts and Strogatz,
1998). Some small-world networks, such as the worldwide web or
the Hollywood network of movie actors, include a small number
of “hubs,” nodes with an unusually large number of connections
(or large degree) (Baraba´si and Albert, 1999; Baraba´si, 2003). The
degree distribution of the worldwide web obeys a power law,
implying no meaningful “average” number of links to each site,
and for this reason, it has been described as a scale-free network
(Baraba´si and Albert, 1999). Many other small-world networks,
including Hollywood, have exponential or exponentially trun-
cated power law distributions, implying relatively reduced prob-
abilities of huge hubs (Amaral et al., 2000; Albert and Baraba´si,
2002).
Small-worlds are attractive models for connectivity of nervous
systems because the combination of high clustering and short
path length confers a capability for both specialized or modular
processing in local neighborhoods and distributed or integrated
processing over the entire network (Sporns et al., 2004). It has
been demonstrated that the neuronal network of Caenorhabditis
elegans has small-world topology at a microscopic anatomical
scale (Watts and Strogatz, 1998). Anatomical connectivity matri-
ces from tract-tracing studies of cat and monkey cortices show
small-world properties at a macroscopic scale (Hilgetag et al.,
2000). There is evidence also for small-world properties of graphs
inferred from functional connectivity matrices measured at a
macroscopic (regional) scale in monkey and human neurophys-
iological data (Stephan et al., 2000; Stam, 2004; Salvador et al.,
2005b) and at a mesoscopic (voxel) scale in human functional
magnetic resonance imaging (fMRI) data (Eguı´luz et al., 2005).
Here, we used wavelets to decompose the pairwise correla-
Received May 17, 2005; revised Oct. 26, 2005; accepted Oct. 27, 2005.
This neuroinformatics research was supported by a Human Brain Project grant from the National Institute of
Bioengineering and Biomedical Imaging and the National Institute of Mental Health and was conducted in the
Medical Research Council (MRC)/Wellcome Trust Behavioral and Clinical Neurosciences Institute (Cambridge, UK).
The Wolfson Brain Imaging Centre was supported by an MRC Cooperative Group grant. This work was presented in
part at the Brain Connectivity 2005 Workshop at Florida Atlantic University (Boca Raton, FL), April 15–16, 2005.
CorrespondenceshouldbeaddressedtoProf.EdBullmore,BrainMappingUnit,UniversityofCambridge,Depart-
ment of Psychiatry, Addenbrooke’s Hospital, Cambridge CB2 2QQ, UK. E-mail: etb23@cam.ac.uk.
DOI:10.1523/JNEUROSCI.3874-05.2006
Copyright © 2006 Society for Neuroscience 0270-6474/06/260063-10$15.00/0
The Journal of Neuroscience, January 4, 2006 • 26(1):63–72 • 63
rldpropertiesofbrainnetworksasafunctionofcorrelationthreshold.a,
shold R is increased, mean degree k monotonically decreases (the net-
arselyconnected)atallscalesofthewavelettransform:blacklines,scale
en lines, scale 3; dark blue lines, scale 4; light blue lines, scale 5; purple
World Human Brain Functional Network J. Neurosci., January 4, 2006 • 26(1):63–72 • 67
fMRI, 5 individuals,
resting state, eyes closed.
Brain image: Scale 4
Figure2. Small-worldpropertiesofbrainnetworksasafunctionofcorrelationthreshold.a,
As the correlation threshold R is increased, mean degree k monotonically decreases (the net-

Small-world everywhere! (I know you already realized)
FUNCTIONAL NETWORKS ARE SMALL-WORLD

30 individuals (17 young/13 old). Functional network during resting state (fMRI). Modularity optimization through a greedy
algorithm. 5% of the total number of links is considered. 5 functional modules are detected. Meunier et al., Front.
Neuroinformatics 3:37 (2009).
Anatomical or functional, in any case they are modular:
FUNCTIONAL NETWORKS ARE MODULAR
719D. Meunier et al. / NeuroImage 44 (2009) 715–723
D. Meunier et al. / NeuroImage 4
Age-related changes in modular organization of human brain functional networks
David Meunier a,b
, Sophie Achard a,b,c
, Alexa Morcom d
, Ed Bullmore a,b,
⁎
a
Brain Mapping Unit, University of Cambridge, Cambridge, UK
b
Behavioural and Clinical Neurosciences Institute, University of Cambridge, Cambridge, UK
c
Grenoble Image Parole Signal Automatique, Centre National de la Recherche Scientifique, Grenoble, France
d
Centre for Cognitive and Neural Systems, University of Edinburgh, Edinburgh, UK
a b s t r a c ta r t i c l e i n f o
Article history:
Received 21 May 2008
Revised 4 September 2008
Accepted 30 September 2008
Available online 5 November 2008
Keywords:
Aging
Complex networks
Functional MRI
Modularity
Graph theory allows us to quantify any complex system, e.g., in social sciences, biology or technology, that
can be abstractly described as a set of nodes and links. Here we derived human brain functional networks
from fMRI measurements of endogenous, low frequency, correlated oscillations in 90 cortical and subcortical
regions for two groups of healthy (young and older) participants. We investigated the modular structure of
these networks and tested the hypothesis that normal brain aging might be associated with changes in
modularity of sparse networks. Newman's modularity metric was maximised and topological roles were
assigned to brain regions depending on their specific contributions to intra- and inter-modular connectivity.
Both young and older brain networks demonstrated significantly non-random modularity. The young brain
network was decomposed into 3 major modules: central and posterior modules, which comprised mainly
nodes with few inter-modular connections, and a dorsal fronto-cingulo-parietal module, which comprised
mainly nodes with extensive inter-modular connections. The mean network in the older group also included
posterior, superior central and dorsal fronto-striato-thalamic modules but the number of intermodular
connections to frontal modular regions was significantly reduced, whereas the number of connector nodes in
posterior and central modules was increased.
Crown Copyright © 2008 Published by Elsevier Inc. All rights reserved.
Introduction
Modularity is a word with many meanings in neuroscience (Fodor,
1983; Zeki and Bartels, 1998; Redies and Puelles, 2001; Callebaut and
Rasskin-Gutman, 2005). Here we are concerned with the topological
organization of whole human brain functional networks and the
partitioning of these networks into a set of modules, each module
being defined by dense internal or intra-modular connectivity and
relatively sparse external or inter-modular connectivity (Newman and
Girvan, 2004); see Fig. 1. This pattern of complex network organiza-
tion, also sometimes described as a community structure, is wide-
spread in biochemical, social and infrastructural networks (Guimerà
et al., 2005). A key advantage of modular organization, which may
explain its ubiquity in diverse systems, is that it favours evolutionary
and developmental optimization of multiple or changing selection
criteria (Redies and Puelles, 2001; Slotine and Lohmiller, 2001;
Kashtan and Alon, 2005; Pan and Sinha, 2007): a modular network
can evolve or grow one module at a time, without risking loss of
function in other modules.
Mathematical tools have recently been developed to quantify the
modularity of any network that can be abstractly described as a set of
nodes and links (Newman and Girvan, 2004; Newman, 2004a; Danon
et al., 2005; Newman, 2006). Once the modules have been identified,
this information can be further used to refine the definition of the
topological role of any particular node. For example, the global air
transportation network has a modular organization (Guimerà and
Amaral, 2005b), broadly conforming to geopolitical constraints, which
informed the assignment of distinct roles to the component nodes
(cities) based on the ratio of intra- and inter-modular links (flights)
connecting each node to the rest of the network. Thus a highly-
connected city, like London, with many long-haul flights to other
modules (different continents), was designated a connector hub;
whereas a regionally important city, like Barcelona, with relatively few
long-haul flights outside Europe and North Africa, was designated a
provincial hub.
Here we extend the analysis of modularity and topological roles in
functional brain networks, using tools drawn from the literature on
physics of complex networks (Newman and Girvan, 2004; Guimerà
and Amaral, 2005b) that have not been previously applied to analysis
of human functional neuroimaging data. However, we note that there
have been several prior studies using conceptually related multi-
variate or graph theoretical methods to explore the clustered or
modular organization of mammalian cortex. Young (1992) applied
non-metric multidimensional scaling (MDS) to anatomical connectiv-
ity matrices to demonstrate dorsal and ventral “streams” of inter-
regional connectivity, and a predominance of local neighbourhood
NeuroImage 44 (2009) 715–723
⁎ Corresponding author. Brain Mapping Unit, University of Cambridge, Department of
Psychiatry, Addenbrooke's Hospital, Cambridge CB2 2QQ, UK. Fax: +44 1223 336581.
E-mail address: etb23@cam.ac.uk (E. Bullmore).
1053-8119/$ – see front matter. Crown Copyright © 2008 Published by Elsevier Inc. All rights reserved.
doi:10.1016/j.neuroimage.2008.09.062
Contents lists available at ScienceDirect
NeuroImage
journal homepage: www.elsevier.com/locate/ynimg
5 modules were detected in
the young individuals.
5 modules were detected in the young individuals.

Representation in the topological space. Left:Young adults (18-33); Right: Older adults (62-73)
The modular organization changes with age:
FUNCTIONAL NETWORKS ARE MODULAR

Assortativity in functional brain networks:
More (functionally) connected regions are prone
to be connected between them. (ﬁnger tapping)
music ﬁnger tapping
FUNCTIONAL NETWORKS ARE ASSORTATIVE

B. Identify differences between healthy brains and those
with a certain pathology:
• Identify differences with respect to a control group.
• Evaluate the effects of a certain disease in the functional network.
• Quantify evolution towards “impaired” topologies.
• Evaluate the loss of segregation/integration in the functional networks.
• Quantify the increase of energy expenses.

Self-portrait of William Utermohlen (american painter (1993-2007)). In
1995 (62 years old) he began to suffer problems with memory and writing.
FROM A HEALTHY TO AN IMPAIRED FUNCTIONAL NETWORK

Fig. 5 Mean PLI averaged over all pairs of MEG sensors for
Alzheimer’s disease patients and controls in six frequency
bands. Error bars are SDs. The mean PLI was significantly lower
in Alzheimer’s disease patients compared to controls in the
lower alpha band (two-tailed t-test, P50.022) and the beta
band (two-tailed t-test, P = 0.036).
Fig. 4 Average weighted graphs of Alzheimer’s disease patients and controls in six frequency bands. The value of the PLI for all
individual pairs of MEG sensors is indicated in colour (blue: low PLI; red: high PLI).
Fig. 3 Damage modelling procedure. The mean PLI of a
control subject network is lowered by randomly weakening
edges in the network, until it reaches the same value as in a
Alzheimer’s disease patient network. The effect of this damage
is then examined by comparing the network characteristics of
the damaged network to the Alzheimer’s disease patient net-
work characteristics. RF = Random Failure, TA = Targeted
Attack, Cw = mean weighted clustering coefficient, Lw = mean
weighted path length.
byguestonApril7,2011brain.oxfordjournals.orgnloadedfrom
The non-parametric Mann–Whitney U-test for independent
samples revealed that Cw was lower in Alzheimer’s disease
subjects in the 8–10 Hz band (U = 89.5; P = 0.022), but not in
network in the Alzheimer’s disease group. Please note that, by
definition, the average PLI of both models is the same as the
average PLI of the Alzheimer’s disease data.
Further analysis of the model data compared with the real data
is shown in Fig. 8. For the Random Failure model the ^Cw was not
different from the control data, and significantly higher than ^Cw of
the Alzheimer’s disease group (Mann–Whitney U-test, U = 76.5;
P = 0.007). In contrast, ^Cw of the Targeted Attack model was
not significantly different from the Alzheimer’s disease group,
but significantly lower than ^Cw of the control group (U = 87.0;
P = 0.018). The weighted path length ^Lw showed a decreasing
trend going from controls to Random Failure, Targeted Failure
and controls (Fig. 8, right panel). ^Lw of both models did not
differ significantly from control data.
Correlation with MMSE
No significant correlations between MMSE and PLI or network
measures were found in the Alzheimer’s disease patient group
(Fig. 9). When correlation with MMSE was analysed for all subjects
(Alzheimer’s disease and control) put together in one group,
we found significant effects between MMSE and mean PLI in the
beta band (Spearman’s r = 0.570, P = 0.001) and between MMSE
and ^Cw in the lower alpha band (Spearman’s r = 0.475, P = 0.008).
Table 1 Results of weighted graph analysis for Alzheimer’s disease patients and controls in six frequency bands
Cw Lw
^Cw
^Lw
Alzheimer’s
disease
Control Alzheimer’s
disease
disease
disease
Control
0.5–4 Hz 0.12
(0.10–0.32)
0.12
(0.10–0.17)
4.05
(1.69–4.40)
3.92
(2.89–4.59)
1.04
(1.03–1.12)
1.04
(1.02–1.11)
1.09
(1.06–1.33)
1.08
(1.05–1.34)
4–8 Hz 0.11
(0.09–0.20)
0.10
(0.09–0.15)
4.23
(2.48–4.99)
4.44
(3.22–5.01)
1.05
(1.03–1.17)
1.04
(1.03–1.13)
1.14
(1.04–1.41)
1.15
(1.05–1.43)
8–10 Hz 0.15
(0.12–0.21)
0.17
(0.13–0.29)
3.27
(2.25–3.76)
2.69
(1.80–3.73)
1.04
(1.02–1.12)
1.07
(1.04–1.13)
1.08
(1.05–1.32)
1.19
(1.07–1.30)
10–13 Hz 0.12
(0.11–0.14)
0.13
(0.11–0.22)
3.83
(3.28–4.36)
3.72
(2.36–4.30)
1.04
(1.03–1.10)
1.04
(1.03–1.21)
1.10
(1.05–1.35)
1.12
(1.04–1.45)
13–30 Hz 0.06
(0.05–0.06)
0.06
(0.05–0.08)
7.97
(6.44–9.24)
7.61
(5.18–9.35)
1.04
(1.02–1.07)
1.04
(1.03–1.16)
1.11
(1.05–1.50)
1.12
(1.04–1.50)
30–45 Hz 0.05
(0.05–0.09)
0.05
(0.05–0.08)
8.70
(5.17–9.07)
8.54
(6.06–9.14)
1.02
(1.02–1.07)
1.02
(1.02–1.07)
1.09
(1.06–1.33)
1.04
(1.02–1.30)
Values are medians, with range printed between parentheses. Cw = mean weighted clustering coefficient; Lw = mean weighted path length; ^Cw = mean normalized
average weighted clustering coefficient (see Materials and Methods section), ^Lw = mean normalized average weighted path length. Significant differences between
Alzheimer’s disease and controls with non-parametric testing (Mann–Whitney U-test, P50.05) are given in bold.
Fig. 6 Schematic illustration of significant differences in long
distance (indicated by arrows) and short distance (indicated by
filled squares) PLI in the 8–10 Hz and 13–30 Hz band.
Alzheimer’s disease patients had lower left sided fronto-
temporal, fronto-parietal, temporo-occipital and parieto-
occipital PLI in the 8–10 Hz band. Local left frontal and tem-
poral, and right parietal PLI were also decreased in Alzheimer’s
disease patients (A). For the 13–30 Hz band, Alzheimer’s dis-
ease patients had lower inter hemispheric frontal, right fronto-
parietal and bilateral frontal PLI (B).
byguestonApril7,2011brain.oxfordjournals.orgDownloadedfrom
BRAINA JOURNAL OF NEUROLOGY
Graph theoretical analysis of
magnetoencephalographic functional
connectivity in Alzheimer’s disease
C. J. Stam,1
W. de Haan,2
A. Daffertshofer,3
B. F. Jones,4
I. Manshanden,1
A. M. van Cappellen van Walsum,5,6
T. Montez,7
J. P. A. Verbunt,1,8
J. C. de Munck,8
B. W. van Dijk,1,8
H. W. Berendse2
and P. Scheltens2
1 Department of Clinical Neurophysiology and MEG, Amsterdam, The Netherlands
2 Department of Neurology, Alzheimer Center, VU University Medical Center, Amsterdam, The Netherlands
3 Research Institute MOVE, VU University, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands
4 Dementia Research Centre, Institute of Neurology, UCL, London, UK
5 Department of Anatomy, Radboud University Nijmegen Medical Centre, Nijmegen, The Netherlands
6 Institute of Technical Medicine, University of Twente, Enschede, The Netherlands
7 Institute of Biophysics and Biomedical Engineering, Faculty of Sciences, University of Lisbon, Portugal
8 Department of Physics and Medical Technology, VU University Medical Center, Amsterdam, The Netherlands
Correspondence to: Willem de Haan,
Department of Neurology, Alzheimer Center,
VU University Medical Center, PO Box 7057,
1007 MB Amsterdam, the Netherlands
E-mail: w.dehaan@vumc.nl
In this study we examined changes in the large-scale structure of resting-state brain networks in patients with Alzheimer’s
disease compared with non-demented controls, using concepts from graph theory. Magneto-encephalograms (MEG) were
recorded in 18 Alzheimer’s disease patients and 18 non-demented control subjects in a no-task, eyes-closed condition. For
the main frequency bands, synchronization between all pairs of MEG channels was assessed using a phase lag index (PLI,
a synchronization measure insensitive to volume conduction). PLI-weighted connectivity networks were calculated, and char-
acterized by a mean clustering coefficient and path length. Alzheimer’s disease patients showed a decrease of mean PLI in the
lower alpha and beta band. In the lower alpha band, the clustering coefficient and path length were both decreased in
Alzheimer’s disease patients. Network changes in the lower alpha band were better explained by a ‘Targeted Attack’ model
than by a ‘Random Failure’ model. Thus, Alzheimer’s disease patients display a loss of resting-state functional connectivity
in lower alpha and beta bands even when a measure insensitive to volume conduction effects is used. Moreover, the large-scale
structure of lower alpha band functional networks in Alzheimer’s disease is more random. The modelling results suggest
that highly connected neural network ‘hubs’ may be especially at risk in Alzheimer’s disease.
Keywords: Alzheimer’s disease; functional connectivity; MEG; synchronization; small-world networks
Abbreviations: EEG = electro-encephalography; MEG = Magneto-encephalography; MMSE = mini mental state examination;
PLI = phase lag index; SL = synchronization likelihood
doi:10.1093/brain/awn262 Brain 2009: 132; 213–224 | 213
Received May 5, 2008. Revised September 12, 2008. Accepted September 18, 2008. Advance Access publication October 24, 2008
ß The Author (2008). Published by Oxford University Press on behalf of the Guarantors of Brain. All rights reserved.
For Permissions, please email: journals.permissions@oxfordjournals.org
byguestonApril7,2011brain.oxfordjournals.orgDownloadedfrom
MEG, resting state. 18 AD patients and 18 controls. (Phase Lag Index).Weighted Clustering coefficient (Cw) and
shortest path (Lw). Only one frequency band showed statistically significant differences (pval0.05)

Some general features of different brain diseases:
❑ Alzheimer’s Disease:
❑ The overall synchronization of the network is decreased.
❑The average path length increases (probably as a consequence of the reduction of
the synchronization).
❑The clustering coefﬁcient is signiﬁcantly reduced (the network evolves to random
topologies).
❑ Mild Cognitive Impairment:
❑ The average synchronization increases.
❑The network becomes more random.
❑ Network outreach increases as a consequence of an unbalanced increase of the
synchronization in the long-range connections.

❑ Schizophrenia:
❑ The small-world properties of the network are impaired (specially at low-frequency
bands).
❑ Clustering and average path length are shifted to random configurations.
❑The hierarchical configuration of the network is also affected.
❑ Epilepsia:
❑ Synchronization increases during the epileptic episodes.
❑ As a consequence, clustering coefficient increases and average path length
decreases.
❑ Changes are more significant at delta, theta and alpha bands.

A brain disorder in which thinking abilities are mildly impaired.
Individuals with MCI are able to function in everyday activities but
have difﬁculty with memory, trouble remembering the names of
people they met recently, the ﬂow of a conversation, and a tendency
to misplace things. Every year, around 10% of MCI patients develop
Alzheimer’s disease.
We perform magnetoencephalograms (MEG) to a group of 19 MCI's patients and 19 control
subjects during a memory task. By means of the synchronization likelihood (SL) we quantify the
interaction between the 148 channels of the MEG system and we obtained a weighted connectivity
matrix between cortical areas.
❑ What is Mild Cognitive Impairment (MCI)?
❑ The experiment
EXAMPLE 1: MILD COGNITIVE IMPAIRMENT
J.M. Buldú, R. Bajo, F. Maestú et al., Reorganization of Functional Networks in Mild Cognitive Impairment, PLoS ONE 6(5): e19584 (2011)

Topological analysis of the functional networks of both groups (Control and MCI):

Differences between the MCI and Control groups:
❑ Global Parameters:
❑The network strength K increases (+15.9%)
❑ Network outreach increases (+23.4%)
(and more than the increase in K)
❑The network modularity decreases (-13.5%)
❑ Normalized Parameters:
❑ Normalized clustering decreases (-13.6%):
CCONTROL =1.76 CMCI =1.52
❑ Normalized outreach increases (+6.7%):
OCONTROL =0.63 OMCI =0.67
CAUTION! The functional network is
becoming random
^ ^
^ ^

❑ Intra-lobe synchronization:
❑The intra-lobe synchronization increases
❑The inter-lobe synchronization increases
(more than the intra-lobe sync.)
❑ Modularity decreases
CAUTION! The segregated operation of
the brain is decreasing
In-strengthOut-strengthModularity
Differences between the MCI and Control groups, Inter-lobe connections:

Within module degree Participation coefﬁcient
From macroscopic (network) to microscopic (node) analysis:

Δ
Δ
Nodes increase their participation
From macroscopic (network) to microscopic (node) analysis:

MCI diagnostic must be done by analysing longitudinal recordings
Caution, GIGO is around...

Caution, GIGO is around...
“Lies, damned lies and statistics”
From :The Evolution of Adult Height in Europe: A Brief Note*
Jaume Garcia and Climent Quintana-Domeque
I’m Swedish!

?
Randomness
Networkstrength
Control
Alzheimer
M.C.I.
We n e e d l o n g i t u d i n a l
experiments in order to
understand the emergence of
MCI
The evolution of MCI to
Alzheimer is still unknown …
despite there are some clues
❑ High Synchronization
❑ Low clustering
❑ Higher outreach
❑ Low modularity
❑ Higher Randomness
❑ Low Synchronization
❑ Low clustering
❑ Higher Randomness
Some conclusions:

Another good candidate:Trauma recovering therapy
Accident Head Trauma Cognitive Therapy
MEG recording
(after the accident)
MEG recording
(9-14 months of therapy)
Comparison
between both
networks
N.P. Castellanos, I. Leyva, J.M. Buldú, et al., “Principles of recovery from traumatic brain injury: reorganization of functional
networks, Neuroimage, 55, 1189-1199 (2011).
EXAMPLE 1I: TRAUMATIC BRAIN INJURY

Band δ [1-4 Hz]
Band α [8-13 Hz]
Network changes:
❑ The delta band is overconnected
❑ The alfa band is underconnected
❑ The cognitive therapy shifts network
parameters towards control values
Black bars:After the TBI
Grey bars:After the therapy
EXAMPLE 1I: TRAUMATIC BRAIN INJURY
Another good candidate:Trauma recovering therapy

C. Develop models in order to explain the changes found in
impaired functional networks:
• Identify what are the rules that determine the network distortion.

Two speciﬁc applications of network modeling:
❑ Mild Cognitive Impairment
❑Traumatic Brain Injury
EVOLUTIONARY NETWORK MODELS

1) We select a link randomly.
2) We change the weight of the link according to a certain function:
w'ij=wij [1+λ+η] ξ(dij)
3) We normalize and recalculate the network parameters.
4) We go back to step 1.
w'ij= modiﬁed link weight
wij = previous link weight
λ=degradation rate (λ 0)
η= noise term
ξ(dij)= length dependence function
dij= link length
EVOLUTIONARY NETWORK MODELS: MCI
Develop models in order to explain the changes found in impaired
functional networks:

Mild Cognitive Impairment: Real data versus evolutionary models
Real data
Models

Healthy brain
Impaired brain
Length dependent
Length independent
Develop models in order to explain the changes found in impaired
functional networks:

The goal of this model is enhancing those links with higher initial weights.
This leads to an increase of the relative difference between higher and
lower weights along the evolution.
Modeling network recovery inTraumatic Brain Injury:
Contrasting model (T+):
Unifying model (T-):
The global average strength of the matrix decreases and, in addition, the
relative differences between link weights are reduced at each time step.
EVOLUTIONARY NETWORK MODELS: TBI

Post (after therapy)
*Pre (before therapy) Control (healthy subject)
Alpha band
Contrasting model Unifying model
EVOLUTIONARY NETWORK MODELS: TBI

We are accumulating errors from the previous two steps
Functional networks are not static
(Functional networks do not evaluate function)
But… above all…
STEP III: Network Analysis

… NETWORK MEASURES ARE COMMONLY
MISINTERPRETED….
… SINCE WE NORMALLY FORGET THAT WE ARE
ANALYZINGTHE BRAIN!

THE WHOLE PROCESS IS A MINELAND
2.4 The Brain as a Complex Network 39
0MROW
*MPXIVMRK1IXVMGW
7XEXMWXMGW
(ITIRHIRGMIW2SHIW
Brain
activity
Recorded
signals
Connectivity
Matrix
Graphs
Topological
properties
Neuromarkers
Healthy vs. Diseased
Rest vs. Task
Figure 2.5: The general framework of brain networks. Clockwise guideline. Nodes can be
regarded as sensor or electrodes recording the electromagnetic signals of the brain, which may
contain dependencies based on correlation or causality. These interdependencies, or link weights,
lead to a weighted connectivity matrix, which is the mathematical representation of a network. This

HOW CAN I COMPARE NETWORKS BETWEEN THEM?
Anatomical network (Hagmann et aI., 2008) and functional network (Honey et aI., 2009) of the
same group of individuals. 998 regions of interest (ROIs).The anatomical matrix is positive while
the functional one has both positive/negative values. RH: right hemisphere, LH: left hemisphere.
Anatomical Network (DTI) Functional Network (fMRI)

A SOCIAL ANALOGY:
Facebook: Four views of the
same Facebook network.
Respectively: friendship network,
proﬁles visited, unidirectional
c o m m u n i c a t i o n a n d
bidirectional communication.
Same network, different levels of information.
D. Easley J. Kleinberg, Networks, crowds and markets.
HOW CAN I COMPARE NETWORKS BETWEEN THEM?

fMRI in (A) resting state and (B) during a memory task. Functional relations between
the most active nodes. Node description: rMTL, right medial temporal lobe; IMTL, left medial
temporal lobe; dmPFC, dorsomedial prefrontal cortex; vmPFC, ventro medial prefrontal cortex; rTC,
right temporal cortex; lTC, left temporal cortex; rIPL, right inferior parietal lobe; lIPL, left inferior
parietal lobe. Fransson et al., Neuroimage (2008).
Functional networks adopt different conﬁgurations
depending on the task you are carrying out:
PROBLEM: FUNCTIONAL NETWORKS CHANGE

Functional network (fMRI) for groups of different ages.. In the picture, nodes are grouped following
a spring algorithm.The frontal region is depicted in blue. We can observe how it segregates with
the maturity of the functional network. Fair et al. PLoS Comp. Bio.(2009).
They also change with age:
PROBLEM: FUNCTIONAL NETWORKS CHANGE

The underlying anatomical network inﬂuences the dynamics but, in turn, the dynamics
inﬂuences the anatomical network. For example, hebbian learning reinforces
connexions between regions that are usually coordinated. Sporns, The networks of the Brain.
Functional networks do not evolve…. they co-evolve!
determina
afecta
evolución topológica
afecta
dinámica neuronal
topología
estado
determina
PROBLEM: TOPOLOGY AND DYNAMICS ARE INTERRELATED

FUNCTIONAL BRAIN NETWORKS: RISKS CHALLENGES
When projecting the brain activity into a network, we are
loosing a lot of information…
… and we may forget what is behind…

How to interpret the results of the network analysis?

“…the analysis reported here looks at the
synchronizability from different perspective
and considers the synchronization properties
of the brain networks rather than looking for a
synchronous pattern in the original EEG signal…”
EXAMPLE 1: Synchronizability
M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011)1184
compared to other bands), where the synchroniz
Fig. 7. Measure of synchronizability of brain functional netw
index, i.e., the eigenratio (the largest eigenvalue of the Laplac
patients and normal controls. Other designations are as Fig.
M. Jalili, M.G. Kny1184
compared to other bands), where the synchronization properties
of the SZ networks were worse than those of controls.
Ref. [43]), decreased cortic
increased cell packing dens
neurons [40], suggest the d
dysconnection hypothesis
Fig. 7. Measure of synchronizability of brain functional networks in SZ patients compared to normal controls. The graphs
index, i.e., the eigenratio (the largest eigenvalue of the Laplacian matrix of the connection graph divided by its second sma
patients and normal controls. Other designations are as Fig. 3.
M. Jalili, M.G. Knyazeva / Computers in Biology and Medicine 41 (2011) 1178–11184
Synchronizability parameter for the
control and patient (schizophrenia)
group in the alpha band.
Fig. 2. Whole-head difference maps of nod
in Eq. (4)) for delta, theta, alpha, beta, and
with strength values significantly higher in
gray regions.
Fig. 3. Functional segregation and integra
worldness index as a function of the thres
brain functional networks were based on
EEG-based functional networks in schizophrenia
Mahdi Jalili a,n
, Maria G. Knyazeva b,c
a
Department of Computer Engineering, Sharif University of Technology, Tehran, Iran
b
Department of Clinical Neuroscience, Centre Hospitalier Universitaire Vaudois (CHUV), and University of Lausanne, Lausanne, Switzerland
c
Department of Radiology, Centre Hospitalier Universitaire Vaudois and University of Lausanne, Switzerland
a r t i c l e i n f o
Keywords:
EEG
Schizophrenia
Functional connectivity
Graph theory
Unpartial cross-correlation
Partial cross-correlation
a b s t r a c t
Schizophrenia is often considered as a dysconnection syndrome in which, abnormal interactions between
large-scale functional brain networks result in cognitive and perceptual deficits. In this article we apply
the graph theoretic measures to brain functional networks based on the resting EEGs of fourteen
schizophrenic patients in comparison with those of fourteen matched control subjects. The networks were
extracted from common-average-referenced EEG time-series through partial and unpartial cross-correla-
tion methods. Unpartial correlation detects functional connectivity based on direct and/or indirect links,
while partial correlation allows one to ignore indirect links. We quantified the network properties with the
graph metrics, including mall-worldness, vulnerability, modularity, assortativity, and synchronizability.
The schizophrenic patients showed method-specific and frequency-specific changes especially pro-
nounced for modularity, assortativity, and synchronizability measures. However, the differences between
schizophrenia patients and normal controls in terms of graph theory metrics were stronger for the
unpartial correlation method.
2011 Elsevier Ltd. All rights reserved.
1. Introduction
Techniques from graph theory are increasingly being applied to
model the functional and/or structural networks of the brain [1,2].
The brain networks can be studied at different levels ranging from
micro-scale containing a number of interconnected neurons to
macro-scale containing distributed brain regions. To construct the
large-scale networks, signals recorded from the brain via methods
such as electroencephalography (EEG), magnetocephalography
(MEG), functional magnetic resonance imaging (fMRI), or diffusion
tensor imaging (DTI), are used [3–7]. Often, binary (directed or
undirected) adjacency matrices are analyzed [1,2], where binary
links represent the presence or absence of a connection. The first
step in analyzing brain networks is to extract its structure from
the time-series. Possible methods are cross-correlation, coherence,
and synchronization likelihood [3–6]. The next step is to represent
it in a number of biologically meaningful measures. To this end,
measures such as characteristic path length, efficiency of connec-
tions, clustering coefficient, modularity, node degree and central-
free properties [9,10]. Graph theoretical analysis on anatomical
and functional networks of the brain have revealed its economical
small-world structure characterized by high clustering (transitiv-
ity) and a short characteristic path length [11]. The brain func-
tional networks are cost-efficient in the sense that they provide
efficient parallel processing for low connection cost [12]. Brain
disorders influence the anatomical and functional brain networks.
Brain wirings may show abnormal patterns in schizophrenia
(SZ). SZ symptoms affect the patients by manifesting as auditory
hallucinations, paranoid or bizarre delusions and/or disorganized
speech and thinking in the context of significant social and/or
occupational dysfunction. About 1% of the population worldwide
suffers from different forms of SZ [13]. Additionally, another 3% of
the population has SZ-type personality disorders. SZ is the fourth
leading cause of disability in the developed counties for people at
the age of 15–44.
Schizophrenic patients show the abnormal patterns of brain
connectivity. MRI-based studies on a large group of SZ patients
revealed the reduced hierarchy of multimodal networks and
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/cbm
Computers in Biology and Medicine
Computers in Biology and Medicine 41 (2011) 1178–1186

EXAMPLE 1: Synchronizability
60 120 180
time [s]
es during a seizure. Vertical broken lines
d Tend of the seizure. ͑a͒ Exemplary time
e cluster coefficient C͑w͒ of the epileptic
arger than the corresponding value of a
ure onset and attains a maximum devia-
the seizure. Already prior to seizure end,
15
25
λ
max
(w)
0.2
0.6
1
b
λ
min
(w)
20
60
100
c
S(w)
0 60
0.2
0.4
0.6
0.8
dε(w)
Evolving functional network properties and synchronizability
during human epileptic seizures
Kaspar A. Schindler,1,2,a͒
Stephan Bialonski,1,3
Marie-Therese Horstmann,1,3,4
Christian E. Elger,1
and Klaus Lehnertz1,3,4,b͒
1
Department of Epileptology, University of Bonn, Sigmund-Freud-Strasse 25, 53105 Bonn, Germany
2
Department of Neurology, Inselspital, Bern University Hospital and University of Bern, Switzerland
3
Helmholtz-Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14-16, 53115 Bonn,
Germany
4
Interdisciplinary Center for Complex Systems, University of Bonn, Römerstrasse 164, 53117 Bonn,
Germany
͑Received 21 May 2008; accepted 10 July 2008; published online 15 August 2008͒
We assess electrical brain dynamics before, during, and after 100 human epileptic seizures with
different anatomical onset locations by statistical and spectral properties of functionally defined
networks. We observe a concave-like temporal evolution of characteristic path length and cluster
coefficient indicative of a movement from a more random toward a more regular and then back
toward a more random functional topology. Surprisingly, synchronizability was significantly de-
creased during the seizure state but increased already prior to seizure end. Our findings underline
the high relevance of studying complex systems from the viewpoint of complex networks, which
may help to gain deeper insights into the complicated dynamics underlying epileptic seizures.
© 2008 American Institute of Physics. ͓DOI: 10.1063/1.2966112͔
Epilepsy represents one of the most common neurological
disorders, second only to stroke. Patients live with a con-
siderable risk to sustain serious or even fatal injury dur-
ing seizures. In order to develop more efficient therapies,
the pathophysiology underlying epileptic seizures should
be better understood. In human epilepsy, however, the
exact mechanisms underlying seizure termination are still
as uncertain as are mechanisms underlying seizure initia-
tion and spreading. There is now growing evidence that
an improved understanding of seizure dynamics can be
achieved when considering epileptic seizures as network
phenomena. By applying graph-theoretical concepts, we
analyzed seizures on the EEG from a large patient group
and observed that a global increase of neuronal synchro-
nization prior to seizure end may be promoted by the
underlying functional topology of epileptic brain dynam-
ics. This may be considered as an emergent self-
regulatory mechanism for seizure termination, providing
clues as to how to efficiently control seizure networks.
techniques.11–13
Then two nodes are connected by an edge, or
direct path, if the strength of their interaction increases above
some threshold. Among other structural ͑or statistical͒ pa-
rameters, the average shortest path length L and the cluster
coefficient C are important characteristics of a graph.1,14
L is
the average fewest number of steps it takes to get from each
node to every other, and is thus an emergent property of a
graph indicating how compactly its nodes are interconnected.
C is the average probability that any pair of nodes is linked
to a third common node by a single edge, and thus describes
the tendency of its nodes to form local clusters. High values
of both L and C are found in regular graphs, in which neigh-
boring nodes are always interconnected yet it takes many
steps to get from one node to the majority of other nodes,
which are not close neighbors. At the other extreme, if the
nodes are instead interconnected completely at random, both
L and C will be low.
Recently, the emergence of collective dynamics in com-
CHAOS 18, 033119 ͑2008͒
Evolving synchronizability during an epileptic
seizure. The synchronizability parameter increases,
thus being the network LESS synchronizable.
“…we observed a concave-like temporal
evolution, with highest values of S ︎i.e.,
lowest synchronizability︎ in the middle of
the seizure, followed by a decline ︎i.e., an
increasing synchronizability︎…”
“…while the aforementioned interpretation
WOULD indicate that the transient evolution
in graph properties is an active process of
the brain to abort a seizure, our findings could
also be understood as a passive
consequence of the seizure itself.”

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