On the Influence of Amplitude on the Connectivity between Phases.

Abstract

In recent studies, functional connectivities have been reported to display characteristics of complex networks that have been suggested to concur with those of the underlying structural, i.e., anatomical, networks. Do functional networks always agree with structural ones? In all generality, this question can be answered with "no": for instance, a fully synchronized state would imply isotropic homogeneous functional connections irrespective of the "real" underlying structure. A proper inference of structure from function and vice versa requires more than a sole focus on phase synchronization. We show that functional connectivity critically depends on amplitude variations, which implies that, in general, phase patterns should be analyzed in conjunction with the corresponding amplitude. We discuss this issue by comparing the phase synchronization patterns of interconnected Wilson-Cowan models vis-à-vis Kuramoto networks of phase oscillators. For the interconnected Wilson-Cowan models we derive analytically how connectivity between phases explicitly depends on the generating oscillators' amplitudes. In consequence, the link between neurophysiological studies and computational models always requires the incorporation of the amplitude dynamics. Supplementing synchronization characteristics by amplitude patterns, as captured by, e.g., spectral power in M/EEG recordings, will certainly aid our understanding of the relation between structural and functional organizations in neural networks at large.

Full text

Page 1

NEUROINFORMATICS
discriminating between frequency and phase-locked activity (Baker
et al., 1999; Mima and Hallett, 1999; Salinas and Sejnowski, 2001;
Fries, 2005; Womelsdorf et al., 2007). It is usually assumed that
amplitude or power variations take place on long time scales when
compared to the phase dynamics and are therefore considered neg-
ligible. The coupling that does, or does not, yield synchrony between
oscillators hence exclusively depends on the phase. Here we ask
whether this assumption is valid, and by this, tackle if a sole focus
on phase really covers all functional characteristics of networks. In
the present study we describe the dynamics of neural populations
at every node as a neural mass model (Wilson and Cowan, 1972;
Lopes Da Silva et al., 1974, 1976; Freeman, 1975; Lopes Da Silva,
1991; Jansen and Rit, 1995; Deco et al., 2008) that can behave like
weakly coupled self-sustained non-linear oscillators. This descrip-
tion generally allows for deducing the corresponding phase dynam-
ics (Schuster and Wagner, 1990a,b; Aoyagi, 1995; Tass, 1999) and,
by this, to investigate how amplitude affects the phase dynamics
in neural networks. The phase dynamics is indeed influenced by
the amplitudes of the individual oscillators as we show analytically.
In a nutshell, we start off with a network of N Wilson–Cowan
neural mass models (Wilson and Cowan, 1972) that are each located at
network nodes k = 1, 2, …, N and linked solely through excitatory con-
nections. Every model displays self-sustained oscillations with slightly
different natural frequencies. Given a certain structural connectivity
between the oscillators denoted by Ckl, we discuss how the connec-
tivity Dkl between phases explicitly depends on the oscillators ampli-
tudes Rk. The expression Dkl∝(Rl/Rk)Ckl can be derived analytically
IntroductIon
The interplay between structural and functional brain networks has
become a popular topic of research in recent years. It is currently
believed that the topologies of structural and functional networks in
various empirical systems may disagree (Sporns and Kötter, 2004)
but systematic analyses tackling this issue are few and far between.
In a combined neural mass and graph theoretical model of electro-
encephalographic signals, it was found that patterns of functional
connectivity are influenced by – but not identical to – those of
the corresponding structural level (Ponten et al., 2010). In this
and many other studies, functional connectivity has been defined
through the synchronization between activities at different nodes.
Neurons synchronize their firing pattern in accordance with
different behavioral states. On a larger scale, synchronous activi-
ties are considered to stem from meso-scale neural populations
that oscillate at certain frequencies with certain amplitudes. That
is, oscillatory activity may yield synchronization characteristics
within a neural population or between populations (Salenius and
Hari, 2003). The amplitude of a single oscillatory neural popula-
tion reflects the degree of synchronization of its neurons, that is,
it measures local synchrony. By contrast, synchronization between
two or more oscillatory neural populations is typically defined by
their (relative) phase variance. Changes in instantaneous phase
locking or coherence reflect changes in more global, distributed
synchronization, i.e., between ensembles or between areas. In fact,
synchronized activity across neural networks is believed to offer
an effective mechanism for information transfer, especially when
On the influence of amplitude on the connectivity between
phases
Andreas Daffertshofer* and Bernadette C. M. van Wijk
Research Institute MOVE, VU University Amsterdam, Amsterdam, Netherlands
In recent studies, functional connectivities have been reported to display characteristics of
complex networks that have been suggested to concur with those of the underlying structural,
i.e., anatomical, networks. Do functional networks always agree with structural ones? In all
generality, this question can be answered with “no”: for instance, a fully synchronized state
would imply isotropic homogeneous functional connections irrespective of the “real” underlying
structure. A proper inference of structure from function and vice versa requires more than a
sole focus on phase synchronization. We show that functional connectivity critically depends
on amplitude variations, which implies that, in general, phase patterns should be analyzed in
conjunction with the corresponding amplitude. We discuss this issue by comparing the phase
synchronization patterns of interconnected Wilson–Cowan models vis-à-vis Kuramoto networks
of phase oscillators. For the interconnected Wilson–Cowan models we derive analytically how
connectivity between phases explicitly depends on the generating oscillators’ amplitudes.
In consequence, the link between neurophysiological studies and computational models
always requires the incorporation of the amplitude dynamics. Supplementing synchronization
characteristics by amplitude patterns, as captured by, e.g., spectral power in M/EEG recordings,
will certainly aid our understanding of the relation between structural and functional organizations
in neural networks at large.
Keywords: connectivity, phase synchronization, Kuramoto network, Wilson–Cowan model, amplitude dependency
Edited by:
Olaf Sporns, Indiana University, USA
Reviewed by:
Joana R. B. Cabral, Universitat Pompeu
Fabra, Spain
Juergen Kurths, Humboldt Universität,
Germany
*Correspondence:
Andreas Daffertshofer, Research
Institute MOVE, VU University
Amsterdam, Van der Boechorststraat 9,
1081 BT Amsterdam, Netherlands.
e-mail: a.daffertshofer@vu.nl
Frontiers in Neuroinformatics www.frontiersin.org July 2011 | Volume 5 | Article 6 | 1
Original research article
published: 15 July 2011
doi: 10.3389/fninf.2011.00006

Page 2

by characterizing every oscillator via its amplitude and phase and
formulating for the latter the dynamics in terms of a Kuramoto net-
work (Kuramoto, 1984; Strogatz, 2000; Acebron et al., 2005).
The discussed structural connectivities differ qualitatively in
their topology. In detail, we consider the fully connected isotropic
network, a network with small-world topology generated by the
Watts–Strogatz model (Watts and Strogatz, 1998), and an anatomi-
cal network reported by Hagmann et al. (2008). Capitalizing on
the derived analytical expression for Dkl, we show how the ampli-
tude dependency can alter the topology of connectivity in the
network of Wilson–Cowan oscillators when reducing them to the
Kuramoto-like network of mere phase oscillators. The connectivity
at the level of phase dynamics, Dkl, largely prescribes the func-
tional connectivity as quantified by the resulting synchronization
patters. We illustrate this numerically using the aforementioned
network topologies that are known to influence synchronizability
(Watts and Strogatz, 1998; Barahona and Pecora, 2002; Achard and
Bullmore, 2007; Brede, 2008).
MaterIals and Methods
network Models
To understand the qualitative relationship between macroscopically
defined functional networks and the (underlying) structural con-
nectivity, modeling local populations of neurons in terms of aver-
aged properties like their mean voltage and/or firing rates appears
very efficient. This mean-field-like approach has a long tradition
and is typically referred to as neural mass modeling (Wilson and
Cowan, 1972; Lopes Da Silva et al., 1974, 1976; Freeman, 1975;
Lopes Da Silva, 1991; Jansen and Rit, 1995; Deco et al., 2008). Neural
mass models have been used to study the origin of alpha rhythm,
evoked potentials, pathological brain rhythms, and the transition
between normal and epileptic activity (Lopes Da Silva et al., 1974;
Jansen and Rit, 1995; Stam et al., 1999a,b; Valdes et al., 1999; David
et al., 2005). Several studies considered small networks of two or
three interconnected neural mass models (Van Rotterdam et al.,
1982; Schuster and Wagner, 1990a,b; Wendling et al., 2001; David
and Friston, 2003; Ursino et al., 2007) as well as larger networks
of interconnected models (Sotero et al., 2007; Ponten et al., 2010).
Here we chose for Wilson–Cowan as seminal neural mass model
because it can readily be derived from microscopic descriptions
like integrate-and-fire neurons, but also from more general models
like Haken (2002) pulse-coupled neurons. By the same token, the
Wilson–Cowan model provides a comprehensive link toward an
even more macroscopic description as its continuum limit resembles
by now well-established neural field equations (Jirsa and Haken,
1996). That is, Wilson–Cowan units may be viewed as an interme-
diate but in some sense generic description of densely connected
neural populations.
Network of Wilson–Cowan models
As said, we are going to put individual Wilson–Cowan models at
every node k of the network under study. Every model contains
distinct populations of excitatory and inhibitory neurons that are
described by their firing rates. If en denotes the firing rate of an
excitatory neuron and in the firing rate of an inhibitory neuron,
then a neural mass description can be obtained by averaging over
the neural population in terms of Ee
N
n
N
n
e
e
=∑=
1
1
and, Ii
N
n
N
n
i
i
=∑=
1
1
where Ne and Ni are the numbers of excitatory and inhibitory neu-
rons. By this averaging, E and I represent the mean firing rates of
all excitatory and inhibitory neurons, respectively, of the neural
population in question, i.e., that at node k.
Within that population, every neuron receives input from all
other neurons of the population. Furthermore, the excitatory units
individually receive constant external inputs pn, whose average is
given by Pp
N
nn
e
1
. The sum of all inputs is (instantaneously)
integrated in time when it exceeds some threshold u. This thresh-
olding is realized by means of a sigmoid function S. Without loss of
generality we here chose S[x] = (1 + e−x)−1; we note that, in general,
the thresholds may differ between excitatory and inhibitory units1.
In consequence, the mean firing rates of the neural populations can
be cast in the following dynamical system
N
e
=∑=
1
d
dtE
d
dtI
ES a c E
E

c I
IE
P
IS a c E
I

c I
II
EEE
EII
= − +−−+
(
)

= − +−−
(
)
u
u  
The characteristics of this dynamical system range from a mere
fixed-point relaxation to limit cycle oscillations (self-sustained
oscillations) depending on parameter settings (Wilson and Cowan,
1972), in particular on the choice of the external input P. That input
is usually chosen at random. In the current study, we restrict all
parameter values to the regime within which the dynamics displays
self-sustained oscillations; see Appendix.
To combine Wilson–Cowan models in a network, different pop-
ulations are now connected via their excitatory units by virtue of
the sum of all El in the dynamics of Ek (see Figure 1). The dynamics
at node k then becomes
d
dtE
ES a

c E
EE
c I
IE
P
N
C E
kl
d
dt
kkEkkEkl
l
N
=+−++









=∑
−−
u
h
1
I IIS a c E
I

c I
II kkkEIkI
= − +−−
()

u
(1)
In words, all Wilson–Cowan oscillators, located at nodes l in the
network drive the change of the firing rate of the excitatory units
Ek. The connectivity is given by the real-valued matrix Ckl that has
vanishing diagonal elements, i.e., Ckk = 0. That connectivity matrix
is scaled via the overall coupling strength h. It is important to note
that the Ckl connectivity matrix is here always identified as the
structural connectivity.
As the different Wilson–Cowan models display self-sustained
oscillations, it seems obvious to describe them using their ampli-
tude and phase dynamics. The required transforms and approxima-
tions are summarized in the Appendix and the outcomes reveal a
phase dynamics similar to the seminal Kuramoto network of phase
1At the individual neuron level, the dynamics reads:
(

∑
(

∑
d
dte
d
d dti
eS au e
mn m
v i
mn m
p
nne
N
m
N
N
n
e
m
N
n
e
e
=
i
i
=
= −+−−+
)



∑
∑
1
1
1
1
u
iS aw e
mn m
z i
mn mnni
N
m
N
N
n
i
m
N
e
e
=
i
i
=
= − +−−
)



1
1
1
1
u
where u, v, w, and z are positive constants representing coupling matrices within
the local neural population – see, e.g., Schuster and Wagner (1990a,b) for details.
Daffertshofer and van Wijk Amplitude influences phase connectivity
Frontiers in Neuroinformatics www.frontiersin.org July 2011 | Volume 5 | Article 6 | 2

Page 3

d
dtN
kk
l
N
lk
wv
h
ww
=+−
()
=∑sin
1
For the sake of legibility, however, we here refer to (2) also as
the Kuramoto model.
As mentioned above, the effect of increasing h in the isotropic
case is to increase the phase synchrony amongst the oscillators.
Suppose the coupling is weak (i.e., smaller than the critical value,
or h  hc, then the oscillators’ phases disperse, whereas for strong
coupling h ? hc the oscillators become synchronous, i.e., the phases
are locked at fixed differences. In the intermediate case h ≈ hc , clus-
ters of synchronous oscillators may emerge. However, many other
oscillators, whose natural frequencies are at the tails of the distribu-
tion, are not locked into a cluster. In other words, as h increases, the
interaction functions overcome the dispersion of natural frequencies
vn resulting in a transition from incoherence, to partial and then
full synchronization (Acebron et al., 2005; Breakspear et al., 2010).
Linking neural mass models to phase oscillators
When deriving the Kuramoto network from the Wilson–Cowan oscil-
lator network, the major ingredient is to average every oscillator over
one cycle when assuming that its amplitude and phase change slowly
as compared to the oscillator’s frequency. That is, time-dependent
amplitude and phase are fixed, the system is integrated over one period
to remove all harmonic oscillations, and, subsequently, amplitude and
phase are again considered to be time-dependent (Guckenheimer and
Holmes, 1990) – this procedure is also referred to as a combination of
rotating wave approximation and slowly varying amplitude approxi-
mation (Haken, 1974). As shown in more detail in the Appendix, the
phase dynamics of the system (1) can in this way be approximated as
d
dtN
∑
C a S
kl
R
R
N
C a
kl
kkE
l
N
E k
,
l
k
lk
wv
h
xww
h
=+

−
()
+
=
( )
0
∑
2
16
1
′
sin
E E
l
N
E k
,
kl
lk
))
SR Rcc
c c2
3
1
0
2
EE
2
IE
3
=
( )

+
()
(
−
()
+
″′ xww
sin
c
EE IEo os ww
lk
−
(
with S′ and S′′′ referring to the first and third derivative of the
sigmoid function S. The parameter xE k ,
( ) 0 is given by
xu
h
N
E k,EkkEkl
l
N
ac E
EE
c I
IE
P C E
kl
00
00
1
( )( )( )( )
=
=−−++




∑
with (
the Wilson–Cowan model (1) and at network node k. For more
details including the definition of the natural frequency we refer
to the Appendix. Considering the case that the amplitudes Rk are
reasonably small, this phase dynamics can be further simplified to
,)
( )
k
( )
k
EI
00 defining the unstable node within the limit cycle of
d
dtN
a S
E
R
R
C
kkE k,
l
N
l
k
kllk
wv
h
xww
≈+
 
−
()
( )
0
=∑
2
1
′
sin
which does resemble a Kuramoto network. In fact, by comparing
this form with the dynamics (2) we find
Da S
E
R
R
C
klE k
l
k
kl
=

( )
0
1
2
′ x,

(3)
oscillators. The Kuramoto model and its link to the here-discussed
network of Wilson–Cowan models will be briefly sketched in the
following two sub-sections.
Kuramoto network of phase oscillators
The collective behavior of a network of oscillators, whose states are
captured by a single scalar phase wk each, can, in first approximation,
be represented by the set of N coupled differential Eq.
d
dtN
D
kk
i
N
lk
wv
h
ww
=+−
()
=∑
klsin
1

(2)
That is, the k-th oscillator, with natural frequency vk, adjusts its
phase according to input from other oscillators through a pair-wise
phase interaction function sin(wl – wk). The connectivity matrix
Dkl is again scaled by an overall coupling strength, h. As will be
sketched below, h serves as a bifurcation parameter in that small
values of h yield a network behavior that essentially agrees with
the entirely uncoupled case (i.e., the phases are not synchronized),
whereas h larger than a certain critical value hc causes the phases
to synchronize. The frequencies vk are distributed according to
a specified probability density usually taken to be a symmetric,
unimodal distribution (e.g., Lorentzian or Gaussian distributions)
with mean v0. Although the sinusoidal interaction function is an
approximation, it still permits a variety of highly non-trivial solu-
tions. As such the model (2) can be viewed as the canonical form
for synchronization in extended, oscillatory media. We note that
the connectivity matrix Dkl represents also a structural connectiv-
ity that does not necessarily agree with that of the Wilson–Cowan
model – see below.
Strictly speaking the system (2) does not represent the Kuramoto
model in its original form as there the coupling between nodes k
and l was considered isotropic and homogeneous, i.e., Dkl = 1 for
all connections, by which the model reduces to
cEI
cEE
CkI EI
cIE
cII
Ik
Pk
Ek
Figure 1 | Network of Wilson–Cowan models. At each node k a neural
population containing excitatory and inhibitory units (Ek and Ik, respectively)
yields self-sustained oscillations. Other nodes are connected to the excitatory
unit by means of SCklEl. Note that this (mean-field) coupling is scaled by a
scalar h – see Eq. 1 for details.
Daffertshofer and van Wijk Amplitude influences phase connectivity
Frontiers in Neuroinformatics www.frontiersin.org July 2011 | Volume 5 | Article 6 | 3

Page 4

strength h we induced qualitative differences in synchronization as
the order parameter was expected to undergo well-defined bifurca-
tions from an unlocked state to in-phase locking. We simulated both
the network of Wilson–Cowan oscillators as well as the Kuramoto
network. For the Wilson–Cowan model, we defined the phase as the
quadrant-corrected inverse tangents of the ratio of excitatory and
inhibitory units at node k, i.e., wk = arctan(Ek/Ik) – this phase largely
agreed with the Hilbert-phase of Ek because of the smoothness of
the Wilson–Cowan limit cycle. For the Kuramoto network, the phase
was, of course, the state variable under study, which did not require
any further definition. In all simulations the primary outcome vari-
able in all simulations was, hence, r(h) for different network types
and, in the case of the Wilson–Cowan network, distinct ranges of
input values Pk as will be explained below in all detail. In addition,
we computed the phase locking index of the pair-wise relative phases
between nodes which served as definition of the functional networks.
The precise transform of the Kuramoto network dynamics to the
dynamics of relative phases is beyond the scope of the current paper.
To study potentially “erroneous” simulations of the phase
dynamics – and thus possible “misinterpretations” of structural
connectivity when solely looking at functional networks defined
via phase synchrony – we ignored for the Kuramoto network the
amplitude dependency (3) of the connectivity matrix and simply
identified Dkl by Ckl. We further accelerated numerical simulations
by adding some small dynamic noise (Stratonovich, 1963; Risken,
1989; Daffertshofer, 1998), so-called Langevin forces Gk(t), in
the form of mean-centered Gaussian white noise. The simulated
dynamics hence looked like
d
dtE
E S a

c E
EE
c I
IE k
P
N
C E
klkkEkEkl
l
N
k
= −+−−+++








=∑
u
h
e
2
1
G
 
= −+−−
()

d
dtI
I S a c E
I
c I
II kkkEIkIu


and
(4)
d
dtN
C
kkkllkk
l
N
wv
h
wwe
=+−
()+
=∑
sin2
1
G

(5)
Recall that the connectivity in (5) differs from (2) by means
of Dkl → Ckl.
Throughout simulations we fixed parameter settings as: aE = 1.2,
aI = 2, cEE = 5, cII = 1, cIE = 6, cEI = 10, uE = 2, uI = 3.5. The strength
of the dynamical noise was always considered very small (it only
served to accelerate numerics and not to discuss impact of stochas-
tic forces). It was set to e = 10−4 for all simulations of the Wilson–
Cowan network (4) and to e = 10−2 for the Kuramoto network (5).
Simulations were realized using a simple Euler-forward scheme
with step-size 10−2. Per run a total number of 105 samples were
simulated. For each network, simulations were repeated with 10
different realizations of constant but random inputs Pk (Wilson–
Cowan oscillators) or constant but random natural frequencies vn
(Kuramoto oscillators). In addition, for the small-world network,
new Ckl matrices were generated with different rewiring pattern for
each realization. Each of these 10 realizations was again repeated
five times with different initial values of Ek and Ik, or wk. The resulting
r(h) values were computed over the final 100 samples of every run
In sum, the phase dynamics can, in good approximation, be cast
into the form of a Kuramoto network provided the connectivity matrix
is corrected by means of (3). This correction yields a non-trivial ampli-
tude dependence of the connectivity at the level of the phase dynam-
ics. Since S is a sigmoid function, S′ becomes bell-shaped implying
a change in connectivity Dkl whenever the parameter xE k ,
e.g., by shifting the center of the Wilson–Cowan limit cycle at node
k and/or l. This probably more global dependence is supplemented
by the here more important node-by-node dependence. When the
amplitudes Rk differ per node, the ratio Rl/Rk in (3) directly affects the
value of Dkl, which can, strictly speaking, be entirely independent on
the choice of the connectivity matrix Ckl. Put differently, the structural
connectivity at the neural mass level does not necessarily agree with
the structural connectivity at the phase dynamics level.
Given our interest in amplitude dependency, we finally add a
note about “large” amplitudes. In line with the Appendix Eq. A.7
including larger amplitudes yields a slight modification of the phase
dynamics that we here abbreviate as
( ) 0 is altered,
d
dtN
D
kkkl
l
N
lkkl
wv
h
wwa
=+−−
()
=∑
1
sin
Interestingly, the presence of large amplitudes yields, apart from
slightly different coupling coefficients Dkl, phase shifts akl that
translate to finite transmission delays. Prior studies that incorporate
transmission delays into phase oscillators have revealed elaborate
synchronization behaviors (Zanette, 2000; Jeong et al., 2002). The
more complex dynamics due to a suggests the notion of frustra-
tion, whereby the interaction functions require some finite phase
offset in order to vanish (Acebron et al., 2005). For a more detailed
discussion we refer to a recent review by Breakspear et al. (2010).
Note that for our analytical estimates we always consider the case
in which Eq. (2) and (3) apply to good approximation.
sIMulatIons
More recently, several research groups started investigating the
relationship between structural and functional connectivity, sug-
gesting that functional connectivity may indeed resemble aspects
of structural connectivity, at least to some extent (Lebeau and
Whittington, 2005; Ingram et al., 2006; Honey et al., 2007, 2009,
2010; Voss and Schiff, 2009; D’angelo et al., 2010). In most stud-
ies, a fixed structural architecture was implemented based on, for
instance, the cortical structure of the cat (Zhou et al., 2007), or
the macaque neo-cortex (Honey et al., 2007). Yet it is unclear how
variations in the network properties at the structural level or fixed
network properties with variations by means of (node-dependent)
amplitudes may affect the synchronization strength and more
global network characteristics at the functional level.
Synchronization was quantified via the phase locking index or
the phase uniformity r, defined as (Mardia and Jupp, 2000)
r
w
=
=∑
k
1
N
1
ei
N
k
This index agrees with the so-called Kuramoto order parameter
and reflects the degree of divergence of the different phases in the
network (not the relative phases). By varying the overall coupling
Daffertshofer and van Wijk Amplitude influences phase connectivity
Frontiers in Neuroinformatics www.frontiersin.org July 2011 | Volume 5 | Article 6 | 4

Page 5

Small-world network
The model for generating small-world networks employed here
was introduced by Watts and Strogatz (1998) to generate graphs
with high clustering and low path length (high efficiency).
Starting from an ordered network on a ring lattice where nodes
are only connected to a small number of direct neighbors, con-
nections are subsequently rewired to a random (distant) node
with certain probability. The introduction of a few random con-
nections in an ordered network drastically increases the synchro-
nizability of the network (Watts and Strogatz, 1998; Barahona
and Pecora, 2002; Motter et al., 2005; Zhou and Kurths, 2006;
Stam and Reijneveld, 2007; Wu et al., 2008; Chen et al., 2009).
We used a network with an average degree of 10 and a rewir-
ing probability of 0.2. An example of a Ckl matrix is given in
Figure 5 below.
Hagmann network
Empirical networks are unlikely to have an organization that can
be exactly described by one of the theoretical network models.
To study a network that more realistically represents anatomical
connections in the human brain we repeated our simulations on
a network that was based on axonal pathways obtained by dif-
fusion spectrum imaging. This dataset has been used to identify
the so-called “structural core” of anatomical connections in the
human cerebral cortex as described by Hagmann et al. (2008),
which is accessible via http://www.connectomeviewer.org/viewer/
datasets. To reduce the size of the network and, by this, acceler-
ate simulation time, the original 998 regions were assigned to a
66-node parcellation scheme and averaged over all five subjects as
was also done in the original study (Hagmann et al., 2008). The
resulting weighted, undirected network was subsequently thresh-
olded to obtain a binary network with an average degree of 10.
This network served as our connectivity matrix Ckl; see Figure 2
and also Figure 6 below.
results
The changes in synchronization r as a function of overall coupling
strength h are summarized in Figure 3. First thing to notice is that,
for a critical h, the Wilson–Cowan model shows a brisk increase
in r after which maximal synchronization is reached. Increasing h
again after a critical value breaks down the synchronization as the
individual Wilson–Cowan oscillators leave the stable limit cycle
regime when their inputs exceed a certain value (Schuster and
Wagner, 1990a). That means, the neural masses at the different
nodes stop oscillating altogether if coupling is too strong. Of course,
this does not apply for the Kuramoto model since, by construction,
the phases keep oscillating. In consequence, r keeps increasing with
h and reaches asymptotically maximum synchronization (see bot-
tom row’s panels in Figure 3).
The different choices of Pk intervals result in altered synchro-
nization curves. This was most apparent for the [−0.8,…,−0.7]
and [0.7,…,0.8] intervals (blue solid lines in Figure 3, third row’s
panels) when amplitudes lie furthest apart. In general, different
Pk intervals caused a shift in critical h, with networks with larger
amplitudes reaching maximum synchronization for lower coupling
strength than oscillators with smaller amplitude. Interestingly, the
cases with bimodal amplitude distributions (dashed lines) were
and averaged over all simulations. Primary outcome variable was,
hence, r(h) for simulations of (4) and (5) using the three different
network types, and in the case of the Wilson–Cowan network (4),
using altered input-distributions to set Pk.
For the Kuramoto model the natural frequencies vn were ran-
domly drawn from a Cauchy–Lorentz distribution with width
g = 0.5 and initial wk values at time t = 0 were drawn from a uni-
form distribution over the interval [0, 2π). The initial Ek and Ik
values for the Wilson–Cowan oscillators were uniformly chosen
from the interval [0, 1].
By default, the constant input values Pk were drawn from a uni-
form distribution with −0.25 ≤ Pk ≤ 0.25 for every node k. In order
to tackle amplitude effects, however, we looked also at the case in
which (selected) nodes displayed oscillations with clearly different
amplitudes than others. For this we selected four different intervals
from which Pk was drawn: −0.25 ≤ Pk ≤ −0.20; 0.20 ≤ Pk ≤ 0.25;
−0.8 ≤ Pk ≤ −0.7; and 0.7 ≤ Pk ≤ 0.8. Simulations were performed
using either a single interval or a combination of two intervals
for which the first 50% of the nodes were assigned a Pk from the
first interval and the second 50% from the second interval. These
combinations of intervals were between similar ranges, hence:
−0.25 ≤ Pk ≤k −0.20 with 0.20 ≤ Pk ≤ 0.25 and −0.8 ≤ Pk ≤ −0.7
with 0.7 ≤ Pk ≤ 0.8. As shown in the final part of the Appendix
the stationary amplitude at network node k either vanishes, i.e.,
Rk,stationary = 0 or it obeys the form
R
a c S
E EE
) ′′′
a c S
I II
a c
E
k
E k
,
−
I k
,
,stationary=
−
′
x
+
′
2

) ′′′
( )
0
( )
0
8
2
3
xx
E EEEEIEE k,IIIIIEII k,
ccS a cccS
22
0
32
0
+
(
+
(
( )( )
x
  
that by virtue of xE k ,
dependency, varying the input Pk systematically could be used to
create different scenarios of amplitude effects, which – in particular
if selected nodes received significantly different input than others
– potentially caused pronounced, qualitative differences between
Ckl and Dkl. In these cases, the simulations of (4) and (5) were
expected to disagree.
The connectivity matrices Ckl were chosen as either a fully con-
nected isotropic network, as a network with small-world topology
generated by the Watts–Strogatz model (Watts and Strogatz, 1998),
or via an anatomical network reported by Hagmann et al. (2008).
For all the connectivities we estimated the functional networks via
phase locking between nodes.
( ) 0 explicitly depends on the input Pk. Given this
Fully connected homogeneous network
The original Kuramoto network comprises a fully connected homo-
geneous network – see above. Ckl in this case consists of an N × N
matrix containing ones everywhere except for the diagonal, where
all values were set to zero, i.e., we did not allow for self-connections.
We note that discarding diagonal elements is, strictly speaking, not
necessary for the phase dynamics (2) or (6) as the coupling via the
sine of relative phase vanishes, i.e., by construction (or symmetry)
there are no self-connections. This argument, however, does not
apply for the network (1) or (5), hence we always set Ckk = 0.
Although the Kuramoto network is usually studied for large
size networks, we chose a network of 66 nodes in order to make a
better comparison with the Hagmann dataset.
Daffertshofer and van Wijk Amplitude influences phase connectivity
Frontiers in Neuroinformatics www.frontiersin.org July 2011 | Volume 5 | Article 6 | 5

Page 6

synchronize for a combination of the two. A closer look at the func-
tional synchronization patterns between individual nodes of this
network revealed that two distinct clusters emerged corresponding
to the bimodal inputs and thus amplitude distribution (Figure 4).
less synchronizable than their unimodal counterparts. An exam-
ple of this phenomenon is the case of a fully connected network
that reaches global synchronization for each of the [−0.8,…,−0.7]
and [0.7,…,0.8] Pk intervals separately but appears unable to fully
lateralorbitofrontal
parsorbitalis
frontalpole
medialorbitofrontal
parstriangularis
parsopercularis
rostralmiddlefrontal
superiorfrontal
precentral
caudalmiddlefrontal
paracentral
rostralanteriorcingulate
posteriorcingulate
postcentral
caudalanteriorcingulate
isthmuscingulate
supramarginal
superiorparietal
precuneus
cuneus
pericalcarine
lingual
fusiform
parahippocampal
inferiortemporal
inferiorparietal
lateraloccipital
entorhinal
temporalpole
middletemporal
bankssts
superiortemporal
transversetemporal
lateralorbitofrontal
parsorbitalis
frontalpole
medialorbitofrontal
parstriangularis
parsopercularis
rostralmiddlefrontal
superiorfrontal
caudalmiddlefrontal
precentral
paracentral
rostralanteriorcingulate
caudalanteriorcingulate
posteriorcingulate
isthmuscingulate
postcentral
supramarginal
superiorparietal
inferiorparietal
precuneus
cuneus
pericalcarine
lateraloccipital
lingual
fusiform
parahippocampal
entorhinal
temporalpole
inferiortemporal
middletemporal
bankssts
superiortemporal
transversetemporal
RIGHT LEFT
Figure 2 | Plot of the Hagmann network (Hagmann et al., 2008). The original 998 regions were assigned to a 66-node parcellation scheme. For the sake of
visualization, all 66 nodes are located on the circle; see text for more details.
0246810
0
0.5
1
Fully Connected
ρ
02468 10
0
0.5
1
ρ
Wilson-
Cowan
0246810
0
0.5
1
ρ
024
η
68
0
0.5
1
ρ
Kuramoto
020 4060
0
0.5
1
Small-World
0 204060
0
0.5
1
0 2040 60
0
0.5
1
01020 30
0
0.5
1
η
0102030 4050
0
0.5
1
Hagmann
010 2030 40 50
0
0.5
1
-0.25 ≤ Pk ≤ -0.2
0.2 ≤ Pk ≤ 0.25
50/50
010 2030 4050
0
0.5
1
-0.8 ≤ Pk ≤ -0.7
0.7 ≤ Pk ≤ 0.8
50/50
020 4060
0
0.5
1
η
Figure 3 | The synchronization r as a function of overall coupling strength h for the network of Wilson–Cowan oscillators [eq. 4; three upper rows] and for
the Kuramoto network [eq. 5; bottom row]. For the upper row, Pk values were drawn from the interval [−0.25,…,0.25], for the second and third row from the
indicated intervals (see right-hand side).
Daffertshofer and van Wijk Amplitude influences phase connectivity
Frontiers in Neuroinformatics www.frontiersin.org July 2011 | Volume 5 | Article 6 | 6

Page 7

local clusters but the large difference between the input intervals pre-
vented them from synchronizing with one another. It is important
to note that, if amplitude effects were not taken into account, a full
synchronization of the network would have been found.
With the current parameter settings no full global synchroniza-
tion could be achieved in both the small-world and the Hagmann
network. However, partial synchronization patterns could be observed
that did not correspond with the structural connectivity but also not
with the distribution of amplitudes (Figures 5 and 6). These patterns
rapidly emerged and disappeared with varying h. Although the match
with the amplitude distribution was not as clear-cut as in the case of
the fully connected network (Figure 4), a similar clustering could be
observed, by which the functional connectivities turned out to dif-
fer not only quantitatively but also qualitatively from the underlying
structural connectivity – a fact that would be missed if relying on a
description of sole phase oscillators that show such partial synchroni-
zation patterns only in close vicinity of the critical coupling strength.
dIscussIon and conclusIon
The introduction of network analysis to neuroscience has paved new
ways for the study of neural network organizations. Particular focus
has been on the search for complex networks since many of these
networks – especially in the neuroinformatics context – are known


0
0.5
1
0.8
1
1.2
η = 5.5 ρ = 0.95
2040 60
20
40
60
20
40
60
2040 60
Figure 4 | Functional connectivity between the nodes of the fully
connected (isotropic and homogeneous) network. Ckl = 1 except for its
diagonal elements, Ckk = 0. Input values Pk for the first 33 nodes were drawn
from the interval [−0.8,…,−0.7] and the second 33 nodes from the interval
[0.7 ,…,0.8]. Due to this bimodal input distribution, the amplitudes at the nodes
and hence their ratios Rl/Rk differ between pairs of nodes (left panel), which
yields – by virtue of Eq. (3) – a change in the functional connectivity (right
panel). Here it can be seen that the phases of nodes within the same Pk range
are fully synchronized but fail to synchronize between clusters; coupling
strength was set to h = 5.5 resulting in a global phase synchrony of r = 0.95.
η = 21
ρ = 0.25
η = 25
ρ = 0.49
η = 29
ρ = 0.2

η = 33
ρ = 0.78
η = 37
ρ = 0.59
η = 41
ρ = 0.46
η = 45
ρ = 0.59
η = 49
ρ = 0.07


0
0.5
1
−1
0
1
204060
20
40
60
Ckl
Figure 5 | Amplitude ratio distribution [log(Rl/Rk); upper row] and
functional connectivity (lower row) of the small-world network that differs
from the underlying structural connectivity (Ckl, most left panel). As in
Figure 4 we here chose the inputs from a bimodal distribution, here the intervals
[−0.8,…,−0.7] and [0.7 ,…,0.8], which causes the amplitudes to differ between
the first and second half of the nodes and, consequently, the functional
connectivity matrix to disagree with Ckl. Similar to the homogeneous case in
Figure 4 the ratios Rl/Rk largely prescribe the functional connectivity, here,
however, mixed with the small-world structure Ckl – see, e.g., the pronounced
synchrony along the diagonal that is absent in the patterns of amplitude ratios. In
addition, dependent on the overall coupling strength h, pronounced clusters of
synchronized nodes appear – the corresponding coupling values, h = 25–45,
agree with the synchronization regime of this small-world network displayed in
Figure 3, middle column, third row, blue dashed line.
η = 22
ρ = 0.62
η = 24
ρ = 0.66
η = 26
ρ = 0.82

η = 28
ρ = 0.9
η = 30
ρ = 0.79
η = 32
ρ = 0.71
η = 34
ρ = 0.3
η = 36
ρ = 0.27


0
0.5
1
−1
0
1
204060
20
40
60
Ckl
Figure 6 | Amplitude ratios [upper row; log(Rl/Rk)] and functional
connectivity (lower row) of the Hagmann network using again a bimodal
distribution of Pk. The left most panel is again the structural network Ckl; we here
used Pk ∈ [−0.25,…,−0.20] and [0.20,…,0.25]. As in the small-world case, localized
clusters of synchronized nodes emerge dependent on the overall coupling
strength h. These clustered patterns apparently disagree with the underlying
anatomical network (most left panel); cf. Figure 3, right column, second row,
green dashed lines.
If the structural connectivity is isotropic, then amplitude
distribution largely (if not fully) prescribes the functional connectivity
pattern that thus clearly disagrees with the structural connectivity. In
consequence, the current example revealed two strongly synchronized
Daffertshofer and van Wijk Amplitude influences phase connectivity
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Page 8

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acknowledgMents
We thank the Netherlands Organisation for Scientific Research for
financial support (NWO grant # 021-002-047).
for their efficiency when transferring and integrating information
from local, specialized brain areas, even when they are distant (Sporns
and Zwi, 2004). Over the years, small-world structural networks have
been found for C. Elegans (Watts and Strogatz, 1998), cat cortex,
and macaque (visual) cortex (Sporns and Zwi, 2004). In humans,
anatomical connectivity can be estimated in vivo indirectly via cross-
correlation analysis of cortical thickness in structural MRI (He et al.,
2007; Chen et al., 2008) and more directly using tractography based
on diffusion tensor imaging and diffusion spectrum imaging (Iturria-
Medina et al., 2007; Hagmann et al., 2008; Gong et al., 2009). Similar
to the structural, i.e. anatomical connections, functional connections
have also been found to display characteristics of complex networks,
especially by looking at human functional networks in resting state,
using either fMRI (Salvador et al., 2005; Achard et al., 2006; Van
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to neurological and psychiatric diseases like Alzheimer’s disease (Stam
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foot movements (De Vico Fallani et al., 2008) and finger tapping
(Bassett et al., 2006).
Do these functional networks precisely match their underlying
structural counterparts? In general, networks do not agree, espe-
cially when the functional networks are solely defined via (phase)
synchronization patterns, which is common practice when studying
Daffertshofer and van Wijk Amplitude influences phase connectivity
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Page 10

Conflict of Interest Statement: The
authors declare that the research was con-
ducted in the absence of any commercial
or financial relationships that could be
construed as a potential conflict of interest.
Received: 30 May 2011; accepted: 20 June
2011; published online: 15 July 2011.
Citation: Daffertshofer A and van Wijk
BCM (2011) On the influence of ampli-
tude on the connectivity between phases.
Front. Neuroinform. 5:6. doi: 10.3389/
fninf.2011.00006
Copyright © 2011 Daffertshofer and van
Wijk. This is an open-access article sub-
ject to a non-exclusive license between the
authors and Frontiers Media SA, which per-
mits use, distribution and reproduction in
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Page 11

d
dtE
ES a

c E
EE
c I
IE k
P
N
C E
klkkEEkl
l
N
( )0( )0 ( )
0
( )
0
1
= −+−−++
=∑
k
(0)
u
h
 

(







= −+−−
)
 
d
dtI

holds. In principle this can be any solution but here we identify
(,)EI
kk
cise) within the stable limit cycle (see the intersection point of
the nullclines in Figure A1). We investigate the deviation of this
solution by means of
IS a c E
I
c I
II kkkEIkI
( )0( )0( )0( )0
u

(A.1)
( )
0
( )
0 with the unstable fixed-point (unstable node to be pre-
d
dt
EEEE
SacEcI
N
C
kkkk
E k,EEEkIEk kl
( )
0
( )
0
( )0
+
()= −+
()
++−+
dd
xdd
h
d d
ddx
E
d
dt
IIIIS
l
l
N
kkkkI k
,
=∑










+
()= −+
()++
1
000( )( ) ( )
a a c
I
EcI
EIkIIk
dd
−
()
 
(A.2)
where we abbreviated
xu
h
N
x
E k,EEEkIE k
c I
Ekkl l
l
N
I
ac EPC E
( )0( )0( )
0
( )
0
,
k k
1
=−−++




=∑
IEIkII k
c I
I
a c E
(
( )0( )0( )0
=−−
)
u

(A.3)
As said, this “mean”-centering allows for expanding the sigmoid
function to the M-th order, S x[
0
+
here S(n) denotes the n-th derivative of S; see also Figure A2. Inserting
this expansion into (A.2) yields the following system differential Eq.
xS x[ ] Sxx
n
M
n
nn
][ ];
!
( )
01
1
0
≈+ ∑=
dd

d
dt
EE
n
SacEcI
N
CE
kk
n
E k
,
EEEk IE
k kll
l
N
dd
xdd
h
d
= −
+

−+
=
)
1
0
1
!
( )( )
∑ ∑∑
=
n








= −+

n
M
kk
n
I k
,
EEIk
d
dt
II
n
SacE
1
0
1
ddxd
!
( )( )
− −
()
(
=∑
n
cI
IIk
n
M
d
1


(A.4)
Here the zero-th order S(0) cancels because of (A.1). The sys-
tem (A.4) is weakly non-linear presuming M is small implying
the presence of only low-order polynomial terms. Put differ-
ently, the sigmoid function S is evaluated close to its threshold.
For the sake of simplicity we here use M = 3. Furthermore we
set overall coupling strength h to be small, i.e., we drop all
terms containing h2 or higher orders in h. By this (A.4) can
be reduced to
d
dt
EE
nS
a c
E
(
EcI
kk
n
E k
,
EEkIEk
n
n
M
=∑
dd
xdd
= −
+

−
()
)


1
0
1 !
( )( )
 

+




= −+

=∑
l
1
1
nS
!
1
0
,
h
N
d
ddx
CE
d
dt
IIa c
E
(
kll
N
kk
n
I k
( )( )
E EIkIIk
n
n
M
=∑
EcI
dd
−
()
)
1


(A.5)
appendIx
To show the link between the network of Wilson–Cowan models
(1) and the Kuramoto network of phase oscillators (2) we adopt
Schuster and Wagner’s derivation (Schuster and Wagner, 1990). In
contrast to their description of two coupled oscillators, however,
we explicitly account for a network structure containing N nodes.
When deriving the Kuramoto network, the strategy is to consider
the Wilson–Cowan model in the oscillatory regime, i.e., in the pres-
ence of a stable limit cycle (Figure A1), which is first “mean”-centered
simplifying the expansion of the sigmoid function S. Then, the oscil-
lator is averaged over one cycle when assuming that its amplitude
and phase change slowly as compared to the oscillator’s frequency.
That is, time-dependent amplitude and phase are fixed, the system
is integrated over one period to remove all harmonic oscillations,
and, subsequently, amplitude and phase are again considered to
be time-dependent (Guckenheimer and Holmes, 1990) – we note
that this procedure is also referred to as a combination of rotating
wave approximation and slowly varying amplitude approximation
(Haken, 1974). The averaging immediately results in the oscillator
network that, when assuming weak coupling and small amplitudes,
resembles the Kuramoto network.
More explicitly, let (,)EI
kk
( )
0
( )
0 be a known solution, for which
00.20.40.60.81
0.2
0.4
0
0.6
0.8
1
I
E
Figure A1 | Limit cycle oscillations of a single Wilson–Cowan oscillator,
d
dtE
d
dtI
ES a

c E
EE
(
c I
IE
P
IS a c E
I

c I
II
EE
EII
= − +−−+
)

= − +−−
(
)
u
u  
i.e., Eq. 1 with N = 1; the sigmoid function was set to S[x] = (1 + e−x)−1.
Dot-dashed lines represent the nullclines (temporal derivatives of E and I
vanish). At the intersection of the nullclines is an unstable node.
Parameter values:
aE = 1.2, aI = 2, cEE = cII = 10, cIE = 6, cEI = 1, uE = 2, uI = 3.5, P = 0.5.
Daffertshofer and van Wijk Amplitude influences phase connectivity
Frontiers in Neuroinformatics www.frontiersin.org July 2011 | Volume 5 | Article 6 | 11

Page 12

For the sake of completeness we also list the natural frequen-
cies Vk of the uncoupled and linearized Wilson–Cowan oscillators:
1
4
−
′
x
,,

a c S
x
with which V in (A.9) can be defined via averaging over nodes,
i.e., VV
=∑
=
N
kk1
.Furthermore the amplitude dynamics cor-
responding to the phase Eq. (A.7) reads
(
2
1
16
−+
(
x
VkE EE
a c S
E k,
(
I II
a c S
I k,
E IE
a c S
E k
200
2
=
′
+
′

)
()
xx
( )( )
( ( )
0
( )
0
E k
1

′
−
E EE

(A.11)
−
N1
d
dtR
a c S
E IE
(
a c S
I EI
R
a
kE kI k,k
E
c c
=
′
(
−
′
−
)
+
1
2
00
3
xx
,
( )( )
ccS
a c
I
ccS
EEEEIEE k
IIIIEII k
,
220
322
+
) ′′′
) ′′′

x
,
( )
(0 0
3
k
1
0
16
)
,
( )
E k
cos
)
−
′

−
()
+
=∑
l
h
R
N
C a S
kl
R
N
C a
kl
E
N
llk
h
xww
E E
l
N
E kkl EEIElk
EE I
c c2
SR Rcc
3
1
0222
3
=∑
′′′

+
()
−
()
(
+
xww
,
( )
cos
E Elk
sin ww
−
())(A.12)
When ignoring all coupling terms (i.e., setting h = 0), this ampli-
tude dynamics of such isolated Wilson–Cowan oscillators reduces to
(
2
1
16
d
dtR
a c S
E IE
a c S
I EI
R
a
k
(
E k,I k,k
E
c c
=
′
−
′
−
(
)
+
1
2
00
3
xx
( )( )
ccSa c
I
ccS
EEEEIEE k,IIIIEII k,
2203220
+
() ′′′
−
+
) ′′′
xx
( )( ) )

)Rk
3
(A.13)
Which has the stationary solutions Rk, stationary= 0 and
R
a c S
E EE
S
IE
) ′′′
a c S
I II
a c
III
a
k
E k
,
−
I k
,
c
EI
E
c c
,
( )
0
( )
0
stationary
= ±
−
′
x
+
′
2

) ′′′
8
2
+
3
xx
+
cccS
EEEEE k,III k,
220320
((
x
( )( ) )

(A.14)

provided the square-root exists; cf. Figure A2.
which represents a network of weakly non-linear, self-sustained
oscillators. Conventionally its characteristics are studied after trans-
forming the system into polar coordinates
(
=
cos
V
and
dwdw
ERtIRt
kkkkkk
=+
)
+
()
sin
V

(A.6)
where Rk and wk are the time-dependent amplitude and phase,
respectively, of the network node k, and V is a yet unknown (mean)
frequency. As said, we assume that amplitude and phase change
slowly with respect to V, and average the system (A.5) over a cycle
t = [0…2π/V). This averaging yields the phase dynamics as
d
dtN
C a S
kl
R
R
N
C a
kl
kkEE k
l
k
lk
l
N
wv
h
xww
h
=+
′

−
()
+
=∑
2
16
0
1
,
( )
sin
E E
l
N
E kklEEIElk
EE IE
c c2
+
SR Rcc
3
1
022
3
′′′

+
()
−
()
(
=∑
xww
,
( )
sin
c cos ww
lk
−
()) (A.7)
with S′ and S- referring to the first and third derivative of the sigmoid
function S, respectively (see Figure A.2), and the frequency being given by
(
V
2
1
16
As a last approximation, we consider the case in which all ampli-
tudes Rk are sufficiently small so that their quadratic and higher
orders can be ignored. We note that Rk are the amplitudes of the
limit cycles describing (dEk,dIk) which do not agree with the mean
“activities” of the Wilson–Cowan oscillators as they are shifted by
(,)EI
kk
(
V
2
and
vxx
kE IE
a c S
E k
,
I EI
a c S
I k
,
E IE
a cc
= − +
′
) ′′′
+
′

)
+
1
00
3
( )( )
E EEIEE k
,
IEIEIII
I k,
cSa cccS
2203220
+
(
+
+
() ′′′

xx
( )( ) 
()Rk
2(A.8)
( )
0
( )
0. Discarding these higher order terms finally leads to
vxx
kE IE
a c S
E k,I EI
a c S
I k,
≈ − +
′
+
′

)
1
00
( )( )

(A.9)
d
dtN
a S
E
R
R
C
kkE k
,
l
k
kllk
l
N
w
h
xww
≈+
′

−
()
=∑
ω
2
0
1
( )
sin

(A.10)
which is equivalent to (2).
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-6 -4-20
x
246-6-4-20
x
246
-6-4-2
-0.02
0
24
x
0.25
0.20
0.15
0.10
0.05
0.04
0.02
-0.04
-0.06
-0.08
-0.10
-0.12
S [x]
S' [x]
S''' [x]
6
Figure A2 | Shape of the sigmoid function S[x] = 1/(1 + e−x) and its first and third derivatives. In the vicinity of the threshold x = 0, the third derivative (right
panel) is negative allowing for the existence of the stationary amplitude Rk,stationary as given in (A.14) dependent on parameter settings.
Daffertshofer and van Wijk Amplitude influences phase connectivity
Frontiers in Neuroinformatics www.frontiersin.org July 2011 | Volume 5 | Article 6 | 12