INTRODUCTION
The spontaneous activity of the brain at rest is spatially and
temporally organized in large-scale networks of cortical and
subcortical regions, denoted Resting State Networks (RSNs). The
topography of RSNs is similar to that of brain networks recruited
by different cognitive tasks (Biswal et al., 1995; Attwell and
Laughlin, 2001; Fox et al., 2005), and for this reason RSNs have
been named according to their putative function and pattern
of task activation: dorsal and ventral attention, visual, somatomotor, auditory, language, executive control, and default systems

(Doucet et al., 2011; Hacker et al., 2013; Glasser et al., 2016).
As behavior unfolds, these functionally specific systems
must integrate to ensure efficient processing and transfer of
information in the brain. This seems to be achieved through
a specialized architecture of brain regions, as shown by fMRI,
DTI, EEG and MEG studies (van den Heuvel and Sporns, 2013;
de Pasquale et al., 2018). A possible mechanism allowing for
the dynamic integration of activity in different brain regions
is the existence of structural and functional ‘hub’ regions
that, through the architecture of their interactions, structural,
as shown by DTI studies (van den Heuvel et al., 2012),
and functional, as shown by fMRI studies (Zuo et al., 2012;
Power et al., 2013), act as waystations of integration thus
facilitating the communication within/across RSNs. Studies at
higher temporal resolution, using for instance MEG, revealed
a more complex scenario where nodes of the Default Mode,
Dorsal Attention and Somato-Motor Networks act as dynamic
cortical cores of integration in the beta and alpha bands

(de Pasquale et al., 2010, 2012, 2018). Furthermore, these
areas are not independent but their centrality tend to cofluctuate, hence forming a so called “dynamic core network of
interaction” (de Pasquale et al., 2016). These findings nicely
link with other MEG studies, showing a temporally varying
organization of brain subnetworks – MEG states- (Baker et al.,
2014), and with the DTI-supported notion of a structural
Rich Club organization within the brain connectome. Many
of these functional and structural models strongly depend on
which measures of connectivity are adopted, e.g., correlationbased measures of interaction, and on which measures are
used to analyze graph properties. In the case of MEG,
even though neuromagnetic signals have broader frequency
content and higher temporal resolution than fMRI, and are
not influenced by neurovascular coupling, a serious drawback
is the inherent “spatial leakage” which generates a spurious
codependence among the reconstructed activity of distinct
sources. In fact, in order to solve the ill-posed inverse problem,
linear source projection schemes such as MNE (Hamalainen
and Ilmoniemi, 1994) or Beamformers (Van Veen et al., 1997;
Hillebrand et al., 2005) will inevitably yield a spatially blurred
representation of the underlying source distribution, with signals
reconstructed at different locations affected by activity from
neighboring brain areas. These effects will largely affect the
connectivity estimation. To overcome this problem, several
measures insensitive to spatial leakage have been introduced:
the imaginary coherence (Nolte et al., 2004), the multivariate
interaction measure (Marzetti et al., 2013) or the phase lag
index (Stam and Reijneveld, 2007; Hillebrand et al., 2012) for
phase coupling on the fast signal (activity), the orthogonalized
correlation (Brookes et al., 2012; Hipp et al., 2012; O’Neill
et al., 2015) and the symmetrical multivariate leakage corrections
(Colclough et al., 2015) for amplitude coupling on the slow
signal (activity envelope). The common idea behind all these
correction schemes is that spatial leakage can only induce zerolag linear spurious coupling, which can be in turn eliminated
by an appropriate regression model (Wens, 2015), and that it
does not affect non-zero-lag connectivity (see Palva et al., 2018
for a critical overview on this assumption). However, a potential
issue with these approaches is that physiological interactions
involving zero-lag linear coupling may be suppressed as well.
This is particularly important since, with synaptic delays in the
range of 5–25 ms from neighboring to remote regions, zerolag interactions are expected to be physiologically dominant.
This has been widely documented in empirical data as well as
modeling studies (Gollo et al., 2014). In fact, these mechanisms
have been ascribed to a range of crucial neuronal functions, from
perceptual integration to the execution of coordinated motor
behaviors (Roelfsema et al., 1997; Singer, 1999; Varela et al., 2001;
Uhlhaas et al., 2009). A recent work also suggests the existence of
zero-lag correlations at rest, specifically within the Default Mode
Network (Sjogard et al., 2019).
A possible alternative method for preserving zero-phase lag
correlation is the Geometric Correction Scheme (GCS) proposed
by Wens et al. (2015), which models spatial leakage from
a seed location based on the forward and inverse models.
The fundamental theoretical aspects of the GCS as well as
simulation- and data-based proof-of-concept were developed in
Wens (2015) and Wens et al. (2015), but they were limited
to the study of RSN topographies. Here, we investigate the
effect of this leakage correction on a dense connectome (155
nodes, involving 9 separate RSNs) as a function of the oscillatory
band and the eventual impact on the estimated topological
features. Specifically, we estimated the dense connectome based
on band-limited power (BLP) computed in the theta, alpha,
beta and gamma bands without leakage correction and with
the GCS. We first compared the overall topology through

INTRODUCTION
The spontaneous activity of the brain at rest is spatially and
temporally organized in large-scale networks of cortical and
subcortical regions, denoted Resting State Networks (RSNs). The
topography of RSNs is similar to that of brain networks recruited
by different cognitive tasks (Biswal et al., 1995; Attwell and
Laughlin, 2001; Fox et al., 2005), and for this reason RSNs have
been named according to their putative function and pattern
of task activation: dorsal and ventral attention, visual, somatomotor, auditory, language, executive control, and default systems
(Doucet et al., 2011; Hacker et al., 2013; Glasser et al., 2016).
As behavior unfolds, these functionally specific systems
must integrate to ensure efficient processing and transfer of
information in the brain. This seems to be achieved through
a specialized architecture of brain regions, as shown by fMRI,
DTI, EEG and MEG studies (van den Heuvel and Sporns, 2013;
de Pasquale et al., 2018). A possible mechanism allowing for
the dynamic integration of activity in different brain regions
is the existence of structural and functional ‘hub’ regions
that, through the architecture of their interactions, structural,
as shown by DTI studies (van den Heuvel et al., 2012),
and functional, as shown by fMRI studies (Zuo et al., 2012;
Power et al., 2013), act as waystations of integration thus
facilitating the communication within/across RSNs. Studies at
higher temporal resolution, using for instance MEG, revealed
a more complex scenario where nodes of the Default Mode,
Dorsal Attention and Somato-Motor Networks act as dynamic
cortical cores of integration in the beta and alpha bands
(de Pasquale et al., 2010, 2012, 2018). Furthermore, these
areas are not independent but their centrality tend to cofluctuate, hence forming a so called “dynamic core network of
interaction” (de Pasquale et al., 2016). These findings nicely
link with other MEG studies, showing a temporally varying
organization of brain subnetworks – MEG states- (Baker et al.,
2014), and with the DTI-supported notion of a structural
Rich Club organization within the brain connectome. Many
of these functional and structural models strongly depend on
which measures of connectivity are adopted, e.g., correlationbased measures of interaction, and on which measures are
used to analyze graph properties. In the case of MEG,
even though neuromagnetic signals have broader frequency
content and higher temporal resolution than fMRI, and are
not influenced by neurovascular coupling, a serious drawback
is the inherent “spatial leakage” which generates a spurious
codependence among the reconstructed activity of distinct
sources. In fact, in order to solve the ill-posed inverse problem,
linear source projection schemes such as MNE (Hamalainen
and Ilmoniemi, 1994) or Beamformers (Van Veen et al., 1997;
Hillebrand et al., 2005) will inevitably yield a spatially blurred
representation of the underlying source distribution, with signals
reconstructed at different locations affected by activity from
neighboring brain areas. These effects will largely affect the
connectivity estimation. To overcome this problem, several
measures insensitive to spatial leakage have been introduced:
the imaginary coherence (Nolte et al., 2004), the multivariate
interaction measure (Marzetti et al., 2013) or the phase lag
index (Stam and Reijneveld, 2007; Hillebrand et al., 2012) for
phase coupling on the fast signal (activity), the orthogonalized
correlation (Brookes et al., 2012; Hipp et al., 2012; O’Neill
et al., 2015) and the symmetrical multivariate leakage corrections
(Colclough et al., 2015) for amplitude coupling on the slow
signal (activity envelope). The common idea behind all these
correction schemes is that spatial leakage can only induce zerolag linear spurious coupling, which can be in turn eliminated
by an appropriate regression model (Wens, 2015), and that it
does not affect non-zero-lag connectivity (see Palva et al., 2018
for a critical overview on this assumption). However, a potential
issue with these approaches is that physiological interactions
involving zero-lag linear coupling may be suppressed as well.
This is particularly important since, with synaptic delays in the
range of 5–25 ms from neighboring to remote regions, zerolag interactions are expected to be physiologically dominant.
This has been widely documented in empirical data as well as
modeling studies (Gollo et al., 2014). In fact, these mechanisms
have been ascribed to a range of crucial neuronal functions, from
perceptual integration to the execution of coordinated motor
behaviors (Roelfsema et al., 1997; Singer, 1999; Varela et al., 2001;
Uhlhaas et al., 2009). A recent work also suggests the existence of
zero-lag correlations at rest, specifically within the Default Mode
Network (Sjogard et al., 2019).
A possible alternative method for preserving zero-phase lag
correlation is the Geometric Correction Scheme (GCS) proposed
by Wens et al. (2015), which models spatial leakage from
a seed location based on the forward and inverse models.
The fundamental theoretical aspects of the GCS as well as
simulation- and data-based proof-of-concept were developed in
Wens (2015) and Wens et al. (2015), but they were limited
to the study of RSN topographies. Here, we investigate the
effect of this leakage correction on a dense connectome (155
nodes, involving 9 separate RSNs) as a function of the oscillatory
band and the eventual impact on the estimated topological
features. Specifically, we estimated the dense connectome based
on band-limited power (BLP) computed in the theta, alpha,
beta and gamma bands without leakage correction and with
the GCS. We first compared the overall topology through