TYPE Methods
PUBLISHED 09 February 2023
DOI 10.3389/fnins.2023.1025428
OPEN ACCESS
EDITED BY
Xi-Nian Zuo,
Beijing Normal University, China
REVIEWED BY
Zhiqiang Sha,
Max Planck Institute for Psycholinguistics,
Netherlands
Valeria Sacca,
Massachusetts General Hospital and Harvard
Medical School, United States
Tingting Wu,
Capital Normal University, China
*CORRESPONDENCE
Narges Chinichian
†These authors share senior authorship
SPECIALTY SECTION
This article was submitted to
Brain Imaging Methods,
a section of the journal
Frontiers in Neuroscience
RECEIVED 22 August 2022
ACCEPTED 04 January 2023
PUBLISHED 09 February 2023
CITATION
Chinichian N, Kruschwitz JD, Reinhardt P,
Palm M, Wellan SA, Erk S, Heinz A, Walter H and
Veer IM (2023) A fast and intuitive method for
calculating dynamic network reconfiguration
and node flexibility.
Front. Neurosci. 17:1025428.
doi: 10.3389/fnins.2023.1025428
COPYRIGHT
©2023 Chinichian, Kruschwitz, Reinhardt,
Palm, Wellan, Erk, Heinz, Walter and Veer. This
is an open-access article distributed under the
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License (CC BY). The use, distribution or
reproduction in other forums is permitted,
provided the original author(s) and the
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original publication in this journal is cited, in
accordance with accepted academic practice.
No use, distribution or reproduction is
permitted which does not comply with these
terms.
A fast and intuitive method for
calculating dynamic network
reconfiguration and node flexibility
Narges Chinichian1,2,3*, Johann D. Kruschwitz2,4, Pablo Reinhardt2,
Maximilian Palm5,6, Sarah A. Wellan2,7, Susanne Erk2, Andreas Heinz2,
Henrik Walter2† and Ilya M. Veer2,8†
1Institute for Theoretical Physics, Technical University of Berlin, Berlin, Germany, 2Department of Psychiatry
and Psychotherapy, Charité Campus Mitte (CCM), Charité-Universitätsmedizin Berlin, Freie Universität Berlin
and Humboldt-Universität zu Berlin, Berlin, Germany, 3Bernstein Center for Computational Neuroscience,
Berlin, Germany, 4Research Centre (SFB 940) “Volition and Cognitive Control”, Technische Universität
Dresden, Dresden, Germany, 5Department of Philosophy and Humanities, Freie Universität Berlin, Berlin,
Germany, 6Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany,
7Faculty of Philosophy, Berlin School of Mind and Brain, Humboldt-Universität zu Berlin, Berlin, Germany,
8Department of Developmental Psychology, University of Amsterdam, Amsterdam, Netherlands
Dynamic interactions between brain regions, either during rest or performance of
cognitive tasks, have been studied extensively using a wide variance of methods.
Although some of these methods allow elegant mathematical interpretations of
the data, they can easily become computationally expensive or difficult to interpret
and compare between subjects or groups. Here, we propose an intuitive and
computationally efficient method to measure dynamic reconfiguration of brain
regions, also termed flexibility. Our flexibility measure is defined in relation to an
a-priori set of biologically plausible brain modules (or networks) and does not rely on
a stochastic data-driven module estimation, which, in turn, minimizes computational
burden. The change of affiliation of brain regions over time with respect to these
a-priori template modules is used as an indicator of brain network flexibility. We
demonstrate that our proposed method yields highly similar patterns of whole-brain
network reconfiguration (i.e., flexibility) during a working memory task as compared to
a previous study that uses a data-driven, but computationally more expensive method.
This result illustrates that the use of a fixed modular framework allows for valid, yet
more efficient estimation of whole-brain flexibility, while the method additionally
supports more fine-grained (e.g. node and group of nodes scale) flexibility analyses
restricted to biologically plausible brain networks.
KEYWORDS
task-based fMRI, dynamic functional connectivity, network neuroscience, template-based
flexibility, community detection, dynamical network analysis, modular structure
1. Introduction
Over the past decades, a paradigm shift has taken place in studying the human brain,
moving from a local to a more network-based perspective, giving rise to the field of network
neuroscience. This evolution has, in part, been driven by the concept of graphs in math. A
graph (network) consists of a set of vertices (nodes), which are connected by edges (links).
In neuroimaging-based network neuroscience, brain regions identified by any given method
of parcellation are considered the nodes of the network, while links can either be defined
as white matter connections between brain regions (structural networks) or as statistical
interdependencies between the time series of brain regions (functional networks) (Bondy and
Murty, 2008;Fair et al., 2009;Power et al., 2010;Rubinov and Sporns, 2010;Sporns, 2010,2012;
Fornito et al., 2016).
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Mesoscopic structures or groups formed by interactions between
nodes of a network, called modules, clusters or communities, can
be quantified by a variety of detection methods (Fortunato, 2010).
Nodal interactions are typically represented by an adjacency matrix
(A) of the network, where each element i,jof A(called aij) is
the weight of the connection or strength of interaction between
nodes i and j. Modules are usually determined based on the
general idea of maximizing the number/weight of within-group and
minimizing the number/weight of between-group links. Modules can
then be considered as entities in the network that can be modified
individually without affecting the rest of the network. Modularity
measures have been shown to be useful as a biomarker of disease,
including epilepsy (Chavez et al., 2010), Alzheimer’s disease (Brier
et al., 2014), schizophrenia, bipolar, and major depressive disorder
(Ma et al., 2020). However, brain modularity has also been associated
with normal variation in cognition: Individuals with lower whole-
brain modularity performed better in complex tasks, while those with
higher modularity showed an advantage in simple tasks (Yue et al.,
2017). Whereas the ‘static’ community detection methods employed
in the above-mentioned studies consider the brain’s connectivity
averaged over time (based on only one adjacency matrix per subject
as a single-layer network), other methods have assessed changes
in community structure over time (Meunier et al., 2009;Newell
et al., 2009;Bassett et al., 2011;Calhoun et al., 2014;Alavash et al.,
2015;Sporns and Betzel, 2016). These dynamic approaches take into
account that a node can frequently change its connections depending
on which state the brain is in, both during resting-state (RS) and
during the performance of tasks. Here, changes in modular structure
are captured by a sequence of adjacency matrices (At), thus creating
multi-layer networks. The adjacency matrices are typically calculated
using a sliding-window approach on nodal time series, in which
the window length reflects the time scale of interest (Fornito et al.,
2016). Subsequently, dynamic module detection methods can be
applied to these time-dependent multi-layer networks to not only
characterize changes of modules over time, but also to determine
how nodes change their affiliation [the module/group they belong to]
as a function of time. The latter can be thought of as the flexibility
of a node (Bassett et al., 2011;Betzel and Bassett, 2017) and is
defined based on the consecutive presence of nodes in different
modules over time (Meunier et al., 2010;Calhoun et al., 2014). These
measures of flexibility enable us to track time-dependent changes
and thereby track phenomena of both integration and segregation
in the brain (Bassett et al., 2011;Braun et al., 2015). It offers the
opportunity to study which brain nodes are more likely to change
their affiliation over time and thereby which brain regions are
rather consistently associated with a certain brain module, forming a
backbone for the constantly changing network. For example, a recent
study by Harlalka et al. (2019) suggested higher symptom severity in
autism spectrum disorder to be associated with more connectivity
flexibility in visual and sensorimotor areas during rest. Braun
et al. (2015) demonstrated that individuals with more connectivity
flexibility in frontal cortices have enhanced memory performance
and score better on neuropsychological tests measuring cognitive
flexibility, suggesting that dynamic network reconfiguration may
form a fundamental mechanism underlying executive function. For
a broader discussion on modularity and flexibility findings, see
Karwowski et al. (2019).
A data driven widely used method to calculate brain network
flexibility is based on the Louvain community detection algorithm
by Blondel et al. (2008). This algorithm aims to optimize the variable
Q, initially introduced for a single layer network by Newman (2006),
and later modified for multi-layer networks by others (Mucha et al.,
2010;Bazzi et al., 2016;Vaiana and Muldoon, 2018).
Q=1
µX
ijsr Aijs −γs
kiskjs
2msδsr +δijCjsrδ(cis,cjr)(1)
More specifically: Where Ais the Adjacency matrix of the
network, Aijs is the weight of connection between nodes iand jin
layer s.γsis the resolution parameter for layer s,iand jare indices of
nodes, and sand rindices of layers. kis is the degree of node iin layer
s.msis proportional to the sum of weights in layer s.Cjsr refers to
the connection of node jto itself in different layers. cis is the defined
module/cluster of node iin layer s. Finally, Qcaptures how good the
grouping is compared to a null-model (here random).
Although, this and similar methods have undoubtedly
contributed to our understanding of brain dynamics, these come
with a cost: Given the random nature of algorithms like Louvain,
the resulting clusters may differ each time the algorithm is run on
the same adjacency matrix. As such, brain modules show variation
within and across participants, which is overcome by running the
algorithm multiple times to reach a consensus on the modular
structure (Lancichinetti and Fortunato, 2012). However, this can be
a computationally expensive process, while the identified modules
may in the end have low biological plausibility or at least cannot be
interpreted straightforwardly.
Here, we introduce a new method to capture nodal flexibility
and brain network reconfiguration using a fast and intuitive method
based on a set of template modules. This offers three main advantages
over the existing methods:
1. It is computationally more efficient and deterministic compared
to the Louvain (and similar) algorithm.
2. It offers high replicability, as it uses the same set of module
templates for all subjects and time scales. This ensures
comparability between subjects and studies, which is one of the
current concerns in the field (Hallquist and Hillary, 2018).
3. It gives researchers the opportunity to choose the best-fitting, or
biologically most relevant module templates for each study.
Although the exact computational complexity of the Louvain
algorithm is not mentioned in the literature, it is suggested to be
essentially linear in the number of links in the graph (Lancichinetti
and Fortunato, 2009).1But the complexity mentioned is regarding
the one time run of the greedy algorithm. The Louvain algorithm
starts with assigning a distinct community to each network node.
In the initial phase then, there are as many communities as nodes.
It then evaluates the gain in modularity [difference between Q
values for different cases] that would result from removing each
node i from its community and placing it in the community of
j for each of its neighbors j. The i-th node is then placed in the
community with the greatest positive gain. If there is no positive
gain, node i remains in its original community. This process is
repeated until no further improvement is possible, at which point
1 Other implementations of the Newman-Girvan Q optimization are
suggested to be of nlogn complexity (Lancichinetti and Fortunato, 2009;
Blondel, 2022)in networks with a clear modular structure.
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the first phase is finished. This first phase concludes when a local
modularity maximum is reached and no individual move can
improve modularity. The output of the algorithm is dependent on
the order in which the nodes are considered. The second phase of the
algorithm involves the construction of a new network whose nodes
are the communities discovered in the first phase. To accomplish
this, the weights of the links between the new nodes are determined
by adding the weights of the links between nodes in the respective
two communities. In the new network, links between nodes of the
same community result in self-loops for this community. Once
this second phase is complete, the algorithm’s initial phase can be
reapplied to the resulting weighted network again. This 2-phase
process is repeated until there are no more modifications and
maximum modularity is achieved. A partitioning of the network
is achieved through this process of repeating the 2-phase until the
Q cannot be improved, but to find a reliable final representative
partition that doesn’t depend on the order in which the algorithm
chooses the nodes, this whole process is repeated several times until
a consensus is reached (Blondel et al., 2008). On the other hand,
our template-based method, gives the same deterministic value each
time and does not need repetition or a consensus-finding step.
We believe that the deterministic nature of the template-method
can be interpreted as the intrinsic “efficiency factor”. The sum
is always linear to the number of links and we need one time
of adding the weights to find the total weight of connections to
each module.
In this work we describe our proposed method in detail and apply
it to a real-life dataset that was previously assessed using a Louvain-
like locally greedy heuristic algorithm (Blondel et al., 2008;Braun
et al., 2015). Compared to the previous work, we demonstrate that
our method is equally successful in capturing a brain reconfiguration
pattern that mimics the stimulation periods of an externally-cued
working memory task, yet in our case can be directly related to
well-known functional brain networks as well.
2. Methods
2.1. Concept and steps
Before going into mathematical detail, let us first explain the
concept behind the method. Consider the brain as a network, in
which each region of the brain (defined by any arbitrary parcellation)
is a node, each co-activation between any two nodes is an edge,
and each node belongs to an a-priori defined set of nodes, termed a
module. As a first step, we consider that each node has an a-priori
affiliation to one of the predefined template modules or in other
words, belongs to an a-prioiri template module. The affiliation is
determined as the template module with which each node has the
largest spatial overlap. Next, the strengths of all edges between each
node and all members of every module are summed. When a node
is more strongly connected to nodes affiliated with another module
FIGURE 1
Schematic overview of the template-based flexibility method. (A) Each node has an a-priori affiliation to a template module, not allowing overlap. In this
paper, we use the Brainnetome atlas for node definition (Fan et al., 2016) and the FIND Lab network templates as predefined modules [http://findlab.
stanford.edu/;Shirer et al., 2012]. Importantly, matrix M, describing the a-priori module affiliation for each node, is predetermined and serves as a
reference. (B) Using a sliding-window approach, an adjacency matrix is constructed for each time window by calculating Pearson correlation coefficients
between the time series of all possible pairs of nodes. Then, for each node and time window the reference module receiving the highest normalized
connection weight will serve as the new modular affiliation for that node in that time window. (C) Last, the number of affiliation changes between
affiliation vector in tand its successive vector in t+1 is defined as the flexibility Ftof the network between two time points. The average of Ftacross
participants (called Ft) can be plotted for all consecutive time points (an example presented later in Figure 3).
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than to nodes of its own predefined module, then this node will
receive another affiliation than its a-priori one. This can now be
extended to a dynamic scenario, in which node affiliations can be
determined for a range of consecutive time points. Some nodes might
change their affiliation over time, while others do not. The ratio of
nodes changing affiliation with respect to all nodes is what we are
interested in. We understand this ratio as a measure of flexibility of
the brain. In other words, the more nodes switch affiliation between
consecutive time points, the more flexibility in network dynamics we
assume. See Figure 1 for a summary of these steps.
The steps to calculate this flexibility measure are listed below
in detail:
1. An a-priori affiliation is assigned to each node to form the
following matrix M:
M=
00010... 0
01000... 0
.
.
.
10000... 0
Nreg ×Nmod
(2)
Where Nreg is the number of regions (nodes) and Nmod number of
a-priori modules. Each row of this matrix belongs to a node and, in
the first-approximation case in this paper, has only one non-zero
element that indicates the a-priori modular affiliation of the node.
For example, in row 1 the fourth column is 1, which means that
the first node has an a-priori affiliation to template module 4.
Note that we assign all nodes that do not show any overlap
with the template modules to a last, artificial module to not exclude
these nodes in calculating the flexibility metric.
2. Next, for each node we extract the mean time series across
all volumes of the fMRI scan. We then divide our time-series
into smaller windows using a sliding-window approach. For
each time window, an adjacency matrix is constructed using
Pearson correlation coefficients between all possible node pairs.
The adjacency matrix at time-window tis defined as Atof shape
Nreg ×Nreg:
At=Weighted Adjacency Matrix
of Time Window t(3)
3. Now, we want to calculate how each node is connected to the
nodes that are the predefined members of each of the template
modules, as defined in M. To this end, we sum the absolute values
of all the weights from one node to all the nodes affiliated to each
of the modules, so that each node has Nmod [in our subsection 2.2
analysis: 15] different values (one weighted sum for links to each
module), indicating the strength of its links with the predefined
members of each of the template modules. In mathematical terms,
we calculate the matrix S′as follows:
S′Nreg×Nmod = |A|t×M(4)
where |A|tmatrix elements are the absolute values of Atelements
and the matrix has the dimension Nreg ×Nreg. Row i of S′belongs
to the region iand each column jshows the sum of absolute
connection weights of ito the members of j-th module. As the
predefined modules differ in size, the S′matrix elements are
FIGURE 2
Task and signals. (A) Example of the N-back working memory task with a 0-back and 2-back condition, during which participants were asked to choose
the value that was either shown at the current step or 2 steps ago, respectively. (B) Four blocks of each condition were presented in alternated fashion for
30 s. (C) After preprocessing, mean time courses were extracted from 246 Brainnetome atlas regions (Fan et al., 2016). (D) Windowed time series were
extracted using a sliding-window approach, moving a window of 15 time points over the time series one volume at a time.
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then normalized to the number of regions affiliated by template
definition to the modules, creating a new matrix called S[dividing
each matrix element S′
ij by the number of regions affiliated to
the jth template module]. Importantly, to be able to compare the
elements of S, we normalize it in a way that the sum of each
row is one. This normalization step has no effect on the output
of the next steps but is rather to increase the interpretability at
this stage. The normalized numbers thus represent which portion
of each node’s connections is to which module. We call this new
matrix, S.
SNreg ×Nmod =Normal(SNreg ×Nmod ) (5)
4. With S, we have the ratio of affiliations to each module calculated
for all nodes. From these, the strongest module affiliation per node
is chosen as the winner which together form an affiliation vector
for time window t; we call this vector t:
t=
ArgMax(S1∗)
ArgMax(S2∗)
ArgMax(S3∗)
ArgMax(S4∗)
..
ArgMax(Si∗)
...
ArgMax(SNreg ∗)
Nreg ×1
(6)
where ArgMax(Si∗) points to the name/number (argument) of
the winner module in row i of matrix S.
5. Following steps 2-4 for consecutive time windows, we calculate
one tfor each window t. The flexibility of the network denoted by
Fis then defined as the ratio of regions that change their affiliation
from one window to the next to the total number of network
regions, or:
Ft=1−1
Nreg
N
X
i=1
δωt
i,ωt−1
i, (7)
TABLE 1 Region a-priori Affiliation, columns marked “R” are region numbers and “M” columns are a-priori modular affiliations.
R M R M R M R M R M R M R M R M R M R M
1 1 26 11 51 6 76 2 101 15 126 13 151 9 176 4 201 5 226 3
2 1 27 7 52 11 77 1 102 15 127 13 152 9 177 9 202 5 227 3
3 7 28 11 53 12 78 8 103 13 128 14 153 4 178 4 203 5 228 3
4 11 29 14 54 12 79 2 104 13 129 14 154 4 179 4 204 5 229 3
5 4 30 14 55 12 80 6 105 5 130 14 155 12 180 1 205 5 230 3
6 4 31 14 56 12 81 7 106 5 131 8 156 12 181 13 206 5 231 3
7 1 32 11 57 12 82 15 107 14 132 8 157 2 182 13 207 9 232 3
8 1 33 6 58 12 83 6 108 14 133 14 158 2 183 1 208 9 233 3
9 1 34 11 59 12 84 15 109 15 134 14 159 14 184 1 209 14 234 3
10 1 35 6 60 12 85 6 110 15 135 13 160 14 185 4 210 14 235 8
11 1 36 6 61 2 86 6 111 13 136 13 161 12 186 4 211 15 236 8
12 1 37 1 62 2 87 6 112 13 137 7 162 12 187 4 212 15 237 4
13 4 38 1 63 14 88 6 113 13 138 11 163 8 188 4 213 15 238 4
14 4 39 6 64 14 89 15 114 13 139 14 164 15 189 5 214 15 239 8
15 1 40 1 65 13 90 15 115 15 140 14 165 1 190 5 215 4 240 3
16 11 41 4 66 8 91 14 116 15 141 8 166 1 191 10 216 4 241 4
17 7 42 4 67 12 92 14 117 15 142 11 167 1 192 10 217 4 242 3
18 11 43 6 68 12 93 15 118 15 143 6 168 1 193 5 218 4 243 3
19 1 44 11 69 15 94 15 119 13 144 6 169 8 194 10 219 3 244 3
20 1 45 15 70 15 95 7 120 15 145 8 170 8 195 10 220 3 245 3
21 1 46 15 71 2 96 15 121 6 146 8 171 15 196 5 221 3 246 3
22 11 47 4 72 2 97 14 122 6 147 13 172 15 197 10 222 3
23 7 48 4 73 2 98 14 123 6 148 13 173 1 198 10 223 8
24 11 49 4 74 2 99 7 124 6 149 13 174 1 199 5 224 15
25 14 50 15 75 6 100 15 125 13 150 13 175 4 200 5 225 3
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where ωdenotes an element of vector . The Kronecker delta
δωs
i,ωt
jis 1 if ωs
i=ωt
jand 0 otherwise. The Pthen counts the
number of nodes that did not change their affiliation between
windows tand t+1. Note that as a side-product of calculating ,
we can output a vector describing the affiliations over time for each
node separately as well by making a vector of the same element
in t=1,..,Nt:
hωt=1
i,ωt=2
i,ωt=3
i...,ωt=Nt
ii1×Nt
(8)
Where Ntis the total number of time windows. This output can be
used for further region-specific analysis.
6. Where we apply the method to real-life data (see subsection 2.2)
we also calculate the average flexibility over time for a sample
(cohort of subjects), Ft, by simply summing the flexibility over all
participants and divide it by the sample size (Nsub).
2.2. Application on a previously studied
dataset
In our application study, we used 331 participants of the 344
participants included in Braun et al. (2015): Thirteen subjects
were excluded due to scanning artifacts, exceeding movement or
insufficient image quality. Functional MRI data were acquired at
three sites during performance of an N-back task: the Life and Brain
Center of the University of Bonn, the Central Institute of Mental
Health Mannheim, and Charité - Universitätsmedizin Berlin. The
study was approved by the Medical Ethics Committee of the three
study sites and all participants provided written informed consent.
At all sites, a Siemens Trio 3T MRI scanner (Siemens Healthcare,
Erlangen, Germany) was used with identical sequences: gradient-
echo EPI, 28 slices, slice thickness 4mm (1mm gap), field of view 192 x
192 x 140 mm, acquisition matrix 64 x 64, TR (repetition time) 2s, TE
(echo time) 30 ms, flip angle 80◦. The task was presented in a blocked
fashion. Four blocks of 0-back and 2-back each (30s duration)
were alternated, starting with the 0-back condition. Participants
were asked to either press the button corresponding to the number
shown on the screen (0-back) or the number that was shown 2
steps ago (2-back). See Figure 2 for more information on the task.
Python packages nilearn, Scikit-learn and matplotlib are used for
visualization purposes in this manuscript (Hunter, 2007;Pedregosa
et al., 2011). Standard preprocessing was conducted using SPM8
(Penny et al., 2011) and included motion correction (participants
with >3mm translation and >1.7◦rotation between volumes were
excluded), slice-time correction, spatial smoothing with a FWHM
of 9 mm, high-pass temporal filtering with a 128s cutoff, and
normalization to the Montreal Neurological Institute (MNI) template
space with 3 mm isotropic voxel size. A detailed description of
data acquisition and preprocessing is provided in Esslinger et al.
(2009). Mean time-courses of the 246 Brainnetome Atlas regions (Fan
et al., 2016) were extracted from the preprocessed data of the 331
subjects. In line with Braun et al. (2015), a 15-volume window length
with 14 volumes overlap was chosen for the sliding-window analysis
(Figures 2C,D), generating in total 114 windows for each subject.
For every window, we calculated an adjacency matrix using Pearson
correlation coefficients between all possible pairs of the 246 regions
mean time series [using scipy.stats.pearsonr Virtanen et al. (2020)].
Considering that the N-back working memory task consisted of 30 s
alternating blocks of 0-back and 2-back, the 15-volume window (30 s
TABLE 2 Findlab-based modules (Shirer et al., 2012)used in our application
section.
Number Name
Module 1 Anterior Salience
Module 2 Auditory
Module 3 Basal Ganglia
Module 4 Dorsal Default Mode Network (dDMN)
Module 5 High Visual
Module 6 Language
Module 7 Left Executive Control (LECN)
Module 8 Posterior Salience
Module 9 Precuneus
Module 10 Prim Visual
Module 11 Right Executive Control (RECN)
Module 12 Sensorimotor
Module 13 Ventral Default Mode Network (vDMN)
Module 14 Task Positive
Module 15 Undefined
length) allows for one window purely reflecting a single condition
block. For more information on selection of the window length see
Braun et al. (2015) and Leonardi and Ville (2015).
The a-priori modules (Matrix M) were selected based on 14 well-
described functional connectivity template networks (modules) in
Shirer et al. (2012) by the FIND lab (http://findlab.stanford.edu/). As
described before, a 15th (artificial) module was added comprising
all atlas regions that did not overlap with any of the 14 template
networks. The a-priori affiliations of all atlas regions can be found
in Table 1 and the labels of the FIND lab templates in Table 2.
To obtain a broader view of the meso-scale dynamics, the
modular allegiance matrix T and integration matrix R were calculated
using the methods from Braun et al. (2015). Each element ti,jof
modular allegiance matrix T shows the ratio of windows where node
i and j were present in the same module relative to all windows. To
calculate the T for each condition, we separated windows with 80% of
their time-points in one condition and ignored the others.
To calculate the integration matrix R with elements rk,l, which
show the strength of co-working between modules k and l, when we
have Nmod modules {M1,M2,...MNmod }, we first use all the T matrix
elements [link between two regions] with one end (region) in module
k and the other end (region) in module l to extract I matrix elements
(ik,l). It can be written as:
ik,l=Pi∈Mk,j∈MlTi,j
|Mk||Ml|, (9)
where k and l are two modules, |Mk|shows the size of module
Mk. Then we normalize the I elements with division by internal
connections of both modules and call the resulting elements elements
of matrix R:
rk,l=ik,l
pik,kil,l
, (10)
R is the integration matrix.
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FIGURE 3
Comparison of flexibility generated by the generalized Louvain-like locally greedy heuristic algorithm (Blondel et al., 2008;Jeub et al., 2022) and the
template-based method during an N-back working memory task. (A) Flexibility plot from Braun et al. (2015) illustrating the probability that a brain region
changes its modular allegiance between two consecutive windows in a sample of 344 healthy subjects. The original plot is used with permission of the
publisher. (B) Flexibility plot generated by the template-based method. Here, the flexibility number in each time-window is the fraction of regions that
change their affiliation from one time window to the next (i.e., the number of changed regions divided by the total number of nodes). The plots are
generated using a subset of 331 subjects from the same cohort as used in Braun et al. (2015). Note that in both plots a time window covers 15 EPI
volumes with a TR of 2 s, corresponding to a window length of 30 s. The window was shifted with one volume at a time, allowing for 14 EPI volumes
overlap between consecutive windows, which yielded 114 windows in total.
FIGURE 4
Brainnetome atlas brain regions switching. Number of affiliation switches between consecutive windows for regions of the Brainnetome Atlas, averaged
across all subjects and normalized to the most frequently switching node to yield values between 0 and 1. The visualized regions are those with values
higher than 0.7.
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FIGURE 5
Findlab brain areas switching. (A) Average number of affiliation switches between consecutive windows for each FIND lab template network, averaged
across all subjects. Abbreviations are listed in Table 2.(B) Illustration of the four template networks for which its constituent nodes demonstrated the
highest flexibility [http://findlab.stanford.edu/;Shirer et al. (2012)]. See Figure 6 for more statistics.
3. Results
Figure 3A shows the N-back flexibility pattern across all nodes
from Braun et al. (2015), while Figure 3B shows the pattern generated
by our method when applied to the same dataset (331/344 subjects
of the same sample). Similar to Figure 3A, the peaks illustrate
maximum flexibility of the brain during performance of both the 0-
and 2-back condition. In contrast, the transitions between the two
task conditions coincide with troughs when applying our method,
whereas Braun et al. (2015) described additional, yet smaller peaks
during these transition phases when using the generalized Louvain
algorithm. On average, higher flexibility is observed during the 2-
back than 0-back blocks, although the difference is relatively small
(t= −2.9, p=0.03).
In addition to calculating flexibility across all nodes, we can use
the information captured in the fifth step to describe the affiliation
changes of each individual node. This allows us to have a closer look
at which nodes switch their affiliation over time most frequently, or at
how often the a-priori constituents of each of the template networks
switch their affiliation. Figure 4 illustrates how many times each node
(Brainnetome regions in our analysis) switches its affiliation between
two consecutive windows. Note that the number of switches was
normalized to the number of switches performed by the node that
switched most frequently, forcing the latter node to have a value of 1
and the other nodes to have a value between 0 and 1. Nodes within the
prefrontal cortex predominantly show affiliation changes over time
during execution of the N-back task. This is in agreement with the
previous findings (Owen et al., 2005;Cao et al., 2014;Braunlich et al.,
2015;Minamoto et al., 2015).
One level coarser at the module level, we can look at the
average switching ratio of template modules. The boxplots in Figure 5
demonstrate for each of the FIND lab template modules how
often their a-priori defined constituent nodes on average switch
their modular affiliation over time across participants. Additional
statistical analysis for modules in Figure 5 is provided in Figure 6.
Constituent nodes of the default mode network (DMN), salience
network (SN), left and right executive control network (L/RECN),
and language network seemingly switch their affiliation most often
during execution of the N-back task.
Figure 7 shows the result of modular allegiance and integration
analysis. We observe a general increase in integration values
in 2-back compared to 0-back except for three modules. This
overall increase in integration is in agreement with previous
findings (Finc et al., 2020).
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Chinichian et al. 10.3389/fnins.2023.1025428
FIGURE 6
Additional statistics for Figure 5. Independent t and p values between boxplot modules in Figure 5, shown as [t-value, p-value]. The last column called
“Comp. w Rest” calculates the t-test between the specific module and the whole brain. For visualization purpose the table is cut to two parts.
4. Discussion
In this work we introduce a new method to assess flexibility
in analyses of dynamic functional connectivity. In the application
section we set out to compare our method against the currently
most used data-driven method described in Braun et al. (2015),
in which the computationally more expensive generalized Louvain
algorithm was applied to derive the modular structure of the data
(Blondel et al., 2008;Mucha et al., 2010;Bassett et al., 2011;Jeub
et al., 2022). See Figure 8 for a schematic comparsion of steps in
standard vs. template flexibility calculations. We demonstrate that
our method is able to reveal a flexibility pattern during the N-back
working memory task that is highly similar to the pattern found in
Braun et al. (2015). The most notable difference between the results
obtained with our method and the Louvain algorithm was the absence
of the small increase in flexibility during the transition of the 0-
and 2-back blocks. Braun et al. (2015) interpret this to reflect “dual-
task” performance. We suggest an alternative explanation based on
the current results: increased flexibility may be needed for switching
tasks at the start of each new condition block (shown as a delayed
peak in the middle of the marked blocks), while less flexibility
may be needed during prolonged execution of the task in each
block (shown as a delayed trough exactly in between blocks). As
such, the periods of lower flexibility may show the preferred brain
configuration for the execution of the task blocks. A further more
theoretical analysis of a simulated BOLD signal with block induced
inputs might be helpful in interpreting the dual-task vs. no-dual-task
hypothesis.
As has been shown abundantly in the literature, the prefrontal
cortex plays an important role in the performance of working-
memory tasks (Owen et al., 2005;Cao et al., 2014;Braunlich et al.,
2015;Minamoto et al., 2015). Therefore, it is not surprising that
we found nodes in the prefrontal cortex to show the most flexible
behavior during execution of the N-back task. Moreover, at the
modular level we see the highest flexibility in nodes that have an a-
prori affiliation to the DMN, SN, L/RECN and language modules. The
DMN is known to have an antagonistic relation with fronto-parietal
networks, such as the L/RECN: when the latter is more active during
cognitively demanding tasks (such as the N-back) the DMN is less
active (Fox et al., 2005). Interestingly, a key role has been assigned
to the SN in allocating neural resources between more internally
(DMN) or externally (ECN) oriented processes (Uddin et al.,
2011). Taken together, we see these results as further proof of our
method’s validity.
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Chinichian et al. 10.3389/fnins.2023.1025428
FIGURE 7
Modular allegiance and integration. Diagonal elements of the matrices are set to be zero. (A) Modular allegiance of the two conditions 2-back and
0-back; to calculate a T matrix for one condition, we used only the windows with 80% of their time-points in that condition. (B) Integration matrix for
0-back and 2-back. (C) Change in the integration values R2−back −R0−back (left plot) and sum of rows (from the left plot matrix) as each modules
integration value (right plot).
We discussed above how our method could be used to assess
flexibility. That is, both on the network (module) level and
at the regional (node) level, thereby extending the inferential
potential compared to the other widely-used algorithms. However,
our analytical procedure also offers possibilities for more fine-
grained investigations of modular affiliations. In the description of
our method and application analysis we determined the modular
affiliation for a particular node and window as the module with which
the node demonstrated the strongest connectivity in the affiliation
vector. Although this is arguably the easiest and most pragmatic
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Chinichian et al. 10.3389/fnins.2023.1025428
FIGURE 8
Schematic steps to calculate dynamical flexibility. The time series are extracted from brain scans. Selected sliding windows are used to generate
adjacency/connectivity matrices. The groups/clusters/modules are found in each matrix∗using a feasible clustering method. In this step, the method of
choice can be a well-known method like the optimization of Newman’s modularity Q using greedy Louvain algorithm or it can be our template-based
method that considers the a-priori information about brain as pre-assumption. Finally the assigned affiliations in windows are compared and the
differences are found. *In some methods, different sliding window matrices are put together to make a multi-layer network and then an adjusted version
of modularity optimization is employed to find module through all layers.
choice, it would also be possible to use the weighted affiliation
with each of the template modules in the affiliation vector [Method
section, step 3] to assess flexibility. Such a weighted approach may
ultimately prove to be even more informative in characterizing brain
flexibility. Another limitation of our method appears in the limits
of a-priori module sizes. If the template modules are significantly
different in size, a single division to the size of each module is not
enough to account for the difference in the size of modules. In
theory, one can define two a-priori modules of size 1 and Nreg −1,
but such a template definition would result in a flexibility which
is very sensitive to the connection weights from that one single
node. In spite of this, additional analysis revealed a comparable but
not easily interpretable flexibility result if the nodes are randomly
assigned to another similar-size a-priori module set and if, in a
simplified case, the Pearson distance between different windows is
used as a measure of flexibility [see chapter 6 of Chinichian (2022) for
details]. This demonstrates that, in a larger context, the connectivity
changes between successive windows can be tracked even before
using a template. The researcher’s choice of template provides an
additional degree of freedom to find a suitable match for the research
question. It allows researchers to focus on a subset of nodes that are
relevant relative to the rest of the network, but it also introduces
the limitation that results from different template selections may
not be easily comparable. We recommend that every report on the
template flexibility should include the exact template details [similar
to Table 1] to allow for a fair interpretation of the results.
The current manuscript, focuses on the block-designed task
fMRI which provides a fairly easy-to-interpret and comprehensible
application case. A further investigating of resting-state fMRI and the
changes in the flexibility during rest could provide more insight to
the different aspects of this method. A study of resting-state fMRI
from 95 subjects meeting criteria for Major Depressive Disorder
and/or common anxiety disorders from the Netherlands Study of
Depression and Anxiety (NESDA) is in preparation by a collaborator
team (Dickhoff, 2022).
In conclusion, the method proposed in the current study is able
to generate flexibility results that are highly comparable to the results
obtained with a more sophisticated data-driven method. Besides
having a much higher computational efficiency, our method also
promotes replicability across different samples and studies through
the use of biologically plausible template modules. We believe that
our approach can be a feasible choice for researchers aiming to study
dynamical reconfiguration at multiple scales of the brain, be it nodes,
modules, or the brain as a whole.
Data availability statement
The data analyzed in this study is subject to the following
licenses/restrictions: Special permission is required from the
project leaders. Participants’ fMRI data is to be treated as
confidential. Requests to access these datasets should be directed to
Ethics statement
The studies involving human participants were reviewed
and approved by Life and Brain Center of the University of
Bonn, the Central Institute of Mental Health Mannheim, and
Charité - Universitätsmedizin Berlin Medical Ethics Committee.
The patients/participants provided their written informed consent to
participate in this study.
Author contributions
NC designed the method, the computational framework,
analyzed the data, and wrote the manuscript. PR extracted the average
time-series from the pre-processed data. IV and HW supervised
the project. IV, HW, and JK contributed to research design and
discussions of the manuscript. All authors discussed the results and
commented on the manuscript.
Acknowledgments
This study was supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation)-
SPP2041, WA 1539/9-1/SPP2031, WA 1539/11-1, ERK 724/4-1,
and 337619223/RTG2386. We thank Prof. Eckehard Schöll and
Tilo Schwalger from TU Berlin together with their group members
for their constructive contributions to improve this method and
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Chinichian et al. 10.3389/fnins.2023.1025428
manuscript. We acknowledge support by the German Research
Foundation and the Open Access Publication Fund of TU Berlin.
Conflict of interest
The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could be
construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those
of the authors and do not necessarily represent those of
their affiliated organizations, or those of the publisher,
the editors and the reviewers. Any product that may be
evaluated in this article, or claim that may be made by
its manufacturer, is not guaranteed or endorsed by the
publisher.
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