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INVESTIGACIÓN

The missing role of gray matter in studying
brain controllability

Hamidreza Jamalabadi1∗

, Agnieszka Zuberer1,2,3,7∗
Meng Li6,7, Sarah Alizadeh1, Ali Moradi Amani8, Christian Gaser7,9,
Michael Esterman2,3,10, and Martin Walter1,4,5,6,7

, Vinod Jangir Kumar6,

1Department of Psychiatry and Psychotherapy, University of Tübingen, Tübingen, Alemania
2Boston University School of Medicine, Department of Psychiatry, Bostón, MAMÁ, EE.UU
3Boston Attention and Learning Laboratory, VA Boston Healthcare System, Bostón, MAMÁ, EE.UU
4Clinical Affective Neuroimaging Laboratory, Magdeburg, Alemania
5Leibniz Institute for Neurobiology, Magdeburg, Alemania
6Max Planck Institute for Biological Cybernetics, Tübingen, Alemania
7Department of Psychiatry and Psychotherapy, Jena University Hospital, Jena, Alemania
8School of Engineering, RMIT University, Melbourne, Victoria, Australia
9Department of Neurology, Jena University Hospital, Jena, Alemania
10National Center for PTSD, VA Boston Healthcare System, Bostón, MAMÁ, EE.UU
*These authors contributed equally.

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Palabras clave: Network control theory, Gray matter, Brain controllability

ABSTRACTO

Brain controllability properties are normally derived from the white matter ﬁber tracts in
which the neural substrate of the actual energy consumption, namely the gray matter, tiene
been widely ignored. Aquí, we study the relationship between gray matter volume of regions
across the whole cortex and their respective control properties derived from the structural
architecture of the white matter ﬁber tracts. The data suggests that the ability of white ﬁber
tracts to exhibit control at speciﬁc nodes not only depends on the connection strength of the
structural connectome but additionally depends on gray matter volume at the host nodes.
Our data indicate that connectivity strength and gray matter volume interact with respect to
the brain’s control properties. Disentangling effects of the regional gray matter volume and
connectivity strength, we found that frontal and sensory areas play crucial roles in
controllability. Together these results suggest that structural and regional properties of the
white matter and gray matter provide complementary information in studying the control
properties of the intrinsic structural and functional architecture of the brain.

RESUMEN DEL AUTOR

Network control theory suggests that the functions of large-scale brain circuits can be
partially described with respect to the ability of brain regions to steer brain activity to
different states. This ability, often quantiﬁed in terms of controllability metrics, has normally
been derived from the structural architecture of the white matter ﬁber tracts. Sin embargo, gray
matter as the substrate that engenders much of the neural processes is widely ignored in this
contexto. In the present work, we study the relationship between regional gray matter volume
and control properties across the whole cortex and provide evidence that control properties
not only depend on the connection strength of the structural connectome but also depend on
sufﬁcient gray matter volume at the host nodes.

un acceso abierto

diario

Citación: Jamalabadi, h., Zuberer, A.,
Kumar, V. J., li, METRO., Alizadeh, S., Amani,
A. METRO., … Walter, METRO. (2021). The missing
role of gray matter in studying brain
controllability. Neurociencia en red,
5(1), 198–210. https://doi.org/10.1162
/netn_a_00174

DOI:
https://doi.org/10.1162/netn_a_00174

Supporting Information:
https://doi.org/10.1162/netn_a_00174

Recibió: 7 Abril 2020
Aceptado: 23 Octubre 2020

Conflicto de intereses: Los autores tienen
declaró que no hay intereses en competencia
existir.

Autores correspondientes:
Hamidreza Jamalabadi
hamidreza.jamalabadi@uni-tuebingen.de
Agnieszka Zuberer
azuberer@bu.edu

Editor de manejo:
Olaf Sporns

Derechos de autor: © 2020
Instituto de Tecnología de Massachusetts
Publicado bajo Creative Commons
Atribución 4.0 Internacional
(CC POR 4.0) licencia

La prensa del MIT

The missing role of gray matter in studying brain controllability

Network control theory:
A mathematical approach to
combine control and network theory.
Network control theory can be used
to describe dynamics of any system
that involves interconnected units,
and it allows making predictions
about the behavior of the system after
perturbation.

Controllability:
The property of a dynamical system
that quantiﬁes the ability of external
inputs to steer the system dynamics
toward any desired state.

INTRODUCCIÓN

Network control theory, as recently applied to white matter (W.M.) ﬁber tracts in the human
cerebro, provides a novel mechanistic framework to describe the ease of switching between
different dynamical functional brain states, and the regions that best drive these dynamics
(bassett & despreciar, 2017; Medalla, 2019; Medalla, Pasqualetti, hamilton, Thompson-Schill,
& bassett, 2017). This approach has the potential to inform theories of dynamic cognitive
procesos, clinical neuroscience, neurodegeneration, and brain reserve. Específicamente, hay
evidence that these global brain state transitions are impaired in clinical populations (Braun
et al., 2019; Jeganathan et al., 2018; Kenett, Beaty, & Medalla, 2018) and that such impair-
ments can be traced back to speciﬁc driver nodes (Jeganathan et al., 2018; Kenett,
Beaty, et al., 2018; Muldoon et al., 2016; Zoeller et al., 2019). Sin embargo, thus far, estos
control properties have been exclusively derived from WM ﬁber tracts without the consider-
ation of gray matter (GM) propiedades. Given the importance of GM properties for cognitive
functioning and brain health, and the established interrelationships between white and gray
asunto, it has been suggested that regional gray matter integrity may be a critical contributor
and proxy for network and node controllability (Medalla, Pasqualetti, et al., 2017; Medalla,
Zurn, Sinnott-Armstrong, & bassett, 2017).

Several lines of research suggest that GM may be essential to understanding brain control-
lability. Primero, GM is a proxy for the quantity of neurons and synaptic densities in a particular
región (Lüders, Steinmetz, & Jäncke, 2002), and metabolic energy expenditure is primarily re-
alized through the gray matter cell bodies that scaffold white matter tracts (Zhu et al., 2012). En
some neurodegenerative disorders, region-speciﬁc lesions of GM only partially agree with cor-
responding lesions in WM (Agosta et al., 2011; Bodini et al., 2009; Douaud et al., 2007; Raine,
Lencz, Bihrle, LaCasse, & Colletti, 2000; Villain et al., 2008), suggesting that GM reserve and
WM may provide independent additional information with respect to controllability properties
of the structural connectome. Tomados juntos, these studies motivate the hypothesis that the
controllability properties suggested by the WM should be partially related to or even predicted
by GM integrity. críticamente, it has been argued that including GM metrics in control theory will
extend traditional volumetrics into network neuroscience (Medalla, Pasqualetti, et al., 2017).
Sin embargo, to our knowledge the nature of the interdependence of controllability properties
and GM properties has not been addressed empirically yet.

To tackle this issue, we used two independent datasets to investigate whether—and if so,
how—control properties extracted from the structural connectome relate to properties of the
gray matter, eso es, GM volume that engenders other GM metrics such as surface and thickness
(Kong et al., 2015; Winkler et al., 2010). Since previous studies have shown that brain control-
lability can be largely explained by the connectivity strength of the structural connectome, nosotros
also considered whether GM volume could explain additional variance in controllability not
accounted for by white matter connectivity. Initially, we investigated how WM and GM fac-
tors affect brain controllability on a whole-brain level. In a further step, we identiﬁed the brain
regions for which controllability was most sensitive to GM and/or WM properties. We discuss
our ﬁndings with respect to their potential relevance to cognitive and clinical neuroscience.

METHODS AND MATERIALS

Data Acquisition

The structural and diffusion datasets are from 65 healthy subjects with the age range of 22 a
36 (28 METRO, mean age 29.2), which were taken from the Human Connectome Project (HCP,

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The missing role of gray matter in studying brain controllability

principal investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657; VanEssen
et al., 2012). While HCP offers more than 1,100 subjects, the data in the present study are
limited by the resources necessary for preprocessing. We have tried to lift the potential bias by
including an independent dataset (see the Replication Study section).

Structural images were acquired with the following speciﬁcation: T1w
MRI data speciﬁcation.
MPRAGE, TR 2,400 EM, EL 2.14 EM, TI 1,000 EM, ﬂip angle 8 degrees, ﬁeld of view (FOV)
224 × 224, 256 slices, voxel size 0.7 mm isotropic, bandwidth 210 Hz/Px, IPAT 2, adquisición
tiempo 7:40 mín..

Diffusion-weighted imaging (DWI) data were acquired by using a spin-echo EPI sequence
with TR 5,520 EM, EL 89.5 EM, ﬂip angle 78 degrees, voxel size, 1.25 mm isotropic, 111
slices, multiband factor, 3, echo spacing, 0.78 EM, b-values 1,000, 2,000, y 3,000 s/mm2.
Para más detalles, see Glasser et al. (2013) and Van Essen et al. (2012).

The 3D
Automated anatomical labeling (AAL) mask deﬁnitions and native space transformation.
anatomy atlas of the AAL2 was acquired from the Neurofunctional Imaging Group (http://www
.gin.cnrs.fr/en/tools/aal-aal2/; Tzourio-Mazoyer et al., 2002). It contains 120 regiones, cual
include subcortical structures such as thalamus, caudate, putamen, and pallidum. Sin embargo, él
misses the brain stem. The 12-parameter afﬁne transformation (Jenkinson, Bannister, Brady, &
Herrero, 2002; Jenkinson & Herrero, 2001) was computed for each volunteer’s T1 and nondiffusion
image and the MNI space standard brain. The resulting transformation matrix was applied to
the left and right AAL brain regions to transform them into the native structural and diffusion
espacio.

The tissue-type segmentation employed SPM12 uniﬁed segmenta-
Structural volume analysis.
tion approach. The process resulted in segmented gray, white, and cerebrospinal ﬂuid (CSF)
volumes.
In the next step, we determined the volume of the brain, gray matter, and under
each AAL atlas region for all subjects. The skull-extracted AC-PC aligned native space NIFTI
structural scans were obtained from the Human Connectome database. In the next step, el
tissue-type segmentation was applied to delineate the gray matter within the brain using the
SPM12 uniﬁed segmentation approach (Ashburner & Friston, 2005). This segmentation ap-
proach employs a generative model that combines nonlinear registration, tissue classiﬁcation,
and bias correction.

The obtained HCP diffusion data were reconstructed using a
Preprocessing and diffusion ﬁt.
SENSE1 algorithm (Sotiropoulos et al., 2013). The DWI data were corrected for motion and
distorsión (andersson, Skare, & Ashburner, 2003; andersson & Sotiropoulos, 2015, 2016).
Además, preprocessing included unringing, denoising, and tensor analysis implemented
in MRtrix (Tournier, Calamante, & connelly, 2012).

The data were reconstructed using the multi-shell multi-tissue constrained spherical decon-
volution (Jeurissen, Tournier, Dhollander, connelly, & Sijbers, 2014). The resulting orientation
distribution function (ODF) was registered to the structural space. The initial tractogram was
generated using mrtrix-tckgen, Resultando en 100 million streamlines within each subject. En el
next step, we applied spherical deconvolution informed ﬁltering of tractograms (SIFT) to reduce
the streamline count to 10 millón. In the ﬁnal step, the number of streamlines was determined
between AAL brain regions to produce a connectome. The analysis steps are documented in

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The missing role of gray matter in studying brain controllability

more detail at the MRtrix docs (https://mrtrix.readthedocs.io/en/latest/quantitative_structural
_connectivity/structural_connectome.html).

Network Control Framework

Controllability is one of the fundamental concepts in the control theory. The notion of control-
lability of a dynamical system was ﬁrst introduced in Kalman (1963). Estado (producción) controlla-
bility of a dynamical system is deﬁned as the possibility of driving states (outputs) del sistema
from an arbitrary initial condition to any desired values in ﬁnite time by applying appropriate
control signals (Kailath, 1980). The most famous classic method to ensure state controllability
of a dynamical system deﬁned by the noise-free linear discrete-time and time-invariant network
model says that the system

X(k + 1) = Ax(k) + Bu(k)

y(k) = Cx(k) + Du(k)

(1)

(2)

is full state controllable if and only if the Kalman’s controllability matrix [B, AB, . . . , An−1B]
In the system represented in Equations 1 y 2, x ∈ Rn and
has full rank (Kailath, 1980).
u ∈ R p are state and input signals, respectivamente. A, B, C, and D are matrices with appropriate
dimensions, where A and B are called state and input matrices, respectivamente. When applied in
the context of brain controllability, x describes the activity of brain regions. A is an adjacency
matrix that represents the interactions between brain regions, and its elements are often the
strength of the white tracts connecting two areas (see the Statistical Analysis section below for
details). The input matrix B identiﬁes the control nodes in the brain that may be conﬁned to one
or more brain areas, whose activities are denoted by the corresponding elements of x. Mientras
the controllability matrix is a valuable metric to study the overall character of a system, it does
not directly quantify the potential ability of different nodes of the system to act as driver nodes.
To achieve this, a common practice is to use Tr(Wk), which is the trace of the controllability
Gramian Wk = ∑∞
i=0 AiBBT(AT)i when the system is controlled from node k. Referred to as
average controllability (AC), this metric is the most commonly used controllability measure in
the neuroimaging literature (Gu et al., 2015; Medalla, 2019); it is a measure of the average
energy required for node k to steer the brain into all possible output states (see Tang & bassett,
2018, for a formal deﬁnition). In addition to AC that quantiﬁes the ability of the nodes to drive
the system into all potential target states, modal controllability (MC) is another commonly used
métrico; it is a measure of the ability of the nodes to push the system toward difﬁcult-to-reach
estados. Formally deﬁned as φk = ∑N
j (A)]v2
kj, MC is a scaled measure of difﬁculty of
driving the system toward all N modes of A from node k (Pasqualetti, Zampieri, & Bullo, 2014).

j [1 − ξ2

Statistical Analysis

Linear mixed-effects (LME) regression (Baayen, Davidson, & Bates, 2008) allows us to model
the interrelationship among multiple variables and has the ability to accommodate various ex-
perimental designs, including repeated measurements, subject variability, and grouping struc-
turas, in one uniﬁed implementation (Boisgontier & Cheval, 2016). en este documento, we model
the interrelationship between brain controllability (AC and MC), GM volume, and connectiv-
ity strength, for which we built multiple LMEs. En particular, we include regional gray matter
(rGM) and total intracranial volume (TIV) in the LMEs.

To predict brain controllability metrics based on structural measures of the brain, we built
a linear mixed-effects regression (Baayen et al., 2008) using a stepwise approach, retaining
an effect only if there was a signiﬁcant difference between the log-likelihood ratio of the two

Average controllability:
A controllability metric that
quantiﬁes the ability of nodes to steer
the network dynamics toward
easy-to-reach brain states.

Modal controllability:
A controllability metric that
quantiﬁes the ability of nodes to steer
the network dynamics toward
difﬁcult-to-reach states.

Linear mixed-effects model:
A form of linear data modeling that
allows a hierarchical data structure
and dependency between
observations by concurrently
mapping random and ﬁxed effects.

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The missing role of gray matter in studying brain controllability

modelos, based on an analysis of variance (ANOVA, pag < 0.05). Statistical analysis was per- formed using the lme4 package in R (Bates, Maechler, Bolker, & Walker, 2014). Speciﬁcally, ∼ two different models were constructed. The ﬁrst model is deﬁned by controllability (AC/MC) TIV + Regions + Nodal degree × rGM + (1|participants), where × denotes the interaction and (1|participants) assigns participants as a random intercept with a ﬁxed slope. The aim was to quantify the contribution of regional gray matter and nodal degree in explaining AC after con- trolling for the regional differences of AC. In a second model, which is deﬁned as AC/MC ∼ TIV + Regions × Nodal degree + Regions × rGM + (1|participants), we investigate the contri- bution of regional differences of regional gray matter and regional differences of nodal degree onto AC. In both models, the volume of the entire brain (TIV) was added as a covariate given its inter-relationship with GM volume (Lüders et al., 2002). In these models all the variables are centered around zero within each subject and normalized using z-transformation. Models were built with the lme4 package in R (Bates et al., 2014). In these models, the elements Aij of the structural connectivity matrix (i.e., A in Equation 1) represent the number of streamlines between regions i and j. To ensure robustness, we keep only 10% of the strongest connections using the Brain Connectivity Toolbox (Rubinov & Sporns, 2010). Within this scheme, ith node degree is estimated by the sum of all elements of A in the ith row. GM volume is estimated from the uniﬁed segmentation approach within SPM12 (see the section on structural volume analysis, above, for details). To further test our hypothesis, similar to Lee, Rodrigue, Glahn, Bassett, and Null models. Frangou (2020), we built random null models by randomizing the structural connectivity matrix (i.e., A in Equation 1) and estimated the interrelationship between controllability, gray matter, and degree distribution as explained in the section on LME formulation and statistical model comparison, above. Speciﬁcally, preserving its degree distribution, we randomized matrix A 1,000 times using the Brain Connectivity Toolbox (Rubinov & Sporns, 2010) and compared the beta values of rGM in the randomized networks with those obtained in the original network. RESULTS Effects of Gray Matter on Brain Controllability In the ﬁrst step, we investigated whether we could replicate previously reported ﬁndings that higher nodal degree relates to higher AC (see Figure 1A). We built a linear mixed-effects model to predict AC based on nodal degree while controlling for TIV and including sub- jects as a random intercept (for details, see the Supporting Information: Model comparisons, Table C1). Our results, summarized in Figure 1A, replicate previous ﬁndings (Gu et al., 2015), suggesting that structural connectivity strength quantiﬁed in terms of nodal degree across the whole brain is positively associated with nodal AC. In the second step we investigated whether, beyond this positive association between degree and AC, rGM explains additional variance of AC. To this aim, we extended our model by including GM volume as an ad- ditional predictor to nodal degree. Our results show that rGM and nodal degree are both critical to explain AC, and their respective sizes of effects are comparable (βdegree = 0.36, p value < 0.001; βrGM = 0.44, p value < 0.001). Next, we included regions as addi- tional predictors to further explain AC and to improve the ﬁtness of the model. Our results show that rGM and AC are signiﬁcantly positively associated (see Figure 1B) and interact with nodal degree (β = 0.04, 95% CI [0.01, 0.07], pbonf = 0.01), suggesting that the highest lev- els of average controllability were best explained with concurrent high rGM and high node Network Neuroscience 202 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / / t e d u n e n a r t i c e - p d l f / / / / / 5 1 1 9 8 1 8 8 9 6 8 9 n e n _ a _ 0 0 1 7 4 p d . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 The missing role of gray matter in studying brain controllability l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / / 5 1 1 9 8 1 8 8 9 6 8 9 n e n _ a _ 0 0 1 7 4 p d t . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 1. Visualization of interaction effect of nodal degree and rGM in the mixed-effects model predicting average controllability (AC). This effect was controlled for by the TIV and regional differ- ences of average controllability. The ﬁgure shows that AC is best explained by WM structure and rGM together. Each dot represents one region from one subject. The density bar shows where the (A) Association between nodal degree and AC. (B) Association majority of the data are located. between rGM and AC. (C) Interaction of rGM and degree on AC, suggesting that the highest levels of AC are reached when both degree and rGM are high together. For visualization, median split was used to classify rGM and degree into high and low. In the original model, both GM and nodal degree were preserved as continuous variables. degree (see Figure 1C). To verify that the AC cannot not be explained with simpler mod- els, we compared competing models (see the Supporting Information: Model comparisons, Table C1), suggesting that the ﬁnal model outperformed the competing models. Finally, we used randomized null networks (for details, see the section on null models, above) to inves- tigate whether rGM would remain a signiﬁcant factor. Our results show that the contribution of rGM in the randomized networks is signiﬁcantly lower than those in the original networks (p value < 0.001). Taken together, our results stress the interdependence of nodal connectivity strength and GM volume for brain controllability. Finally, in a further step, we used the same model to assess the relationship between MC, rGM, and nodal degree (see the Supporting Information: Modal controllability, Figure B1). Replicating previously reported ﬁndings that MC and nodal degree relationships are negatively correlated (see Supporting Information: Modal controllability, Figure B1a), we ﬁnd that rGM explains a large part of MC variance and that the combination of nodal degree, rGM, and their interaction is necessary. Regional Distribution of Average Controllability Based on Gray Matter Volume Further, we investigated whether this global interdependence of WM and rGM (see the previous section) differs on a regional level. Given our previous results that MC and AC are strongly negatively correlated we focus on AC. Network Neuroscience 203 The missing role of gray matter in studying brain controllability Figure 2. Visualization of interaction effects of mixed-effects model predicting average control- lability (AC) based on regional GM (A) and regional nodal degree (B). For visualization, colors represent standardized beta coefﬁcients for effects of rGM and nodal degree, respectively, for each brain region. Higher values indicate a beneﬁcial effect and lower values indicate an impeding effect of rGM/nodal degree on AC. Our results (see the competing models Supporting Information: Model comparisons, Table C2, and the full outcomes of the winning model in Table S2) show that higher rGM and nodal degree concomitantly are associated with higher AC (see Figure 2; Table S2). Notably, the highest AC levels with higher nodal degree were exhibited in the left frontal middle gyrus (β = 15.11, 95% CI [4.09, 26.13], pbonf = 0.007) and the left superior frontal gyrus (β = 3.01, 95% CI [0.24, 5.78], pbonf = 0.033), which agrees with previous research that also locates driver nodes for AC in the frontal lobes. Further, higher levels of AC were linked to higher lev- els of nodal degree in the left calcarine (β = 1.78, 95% CI [0.69, 2.86], pbonf = 0.001). There were also regions where higher levels of nodal degree exacerbated AC, with the strongest effects located in the right and left cuneus (right cuneus: β = −1.34, 95% CI [−2.11, −0.57], < 0.001). When turning pbonf = 0.001; left cuneus: β = −2.70, 95% CI [−3.33, −2.08], pbonf to the relation of rGM and AC, higher rGM is associated with higher AC levels in the right < 0.001), right lingual area (β = 2.98, 95% calcarine (β = 5.61, 95% CI [4.50, 6.73], pbonf < 0.001), and the left and right anterior cingulate (left anterior cingu- CI [2.63, 3.33], pbonf < 0.001; right anterior cingulate: β = 2.88, 95% late: β = 3.76, 95% CI [2.61, 4.91], pbonf CI [2.28, 3.48], pbonf < 0.001). There were several regions exhibiting lower AC levels with higher rGM. The strongest effects < 0.001) were found in the right cuneus (β = −16.17, 95% CI [−18.46, −13.88], pbonf and the left frontal middle gyrus (β = −3.34, 95% CI [−6.62, −0.07], pbonf = 0.045). The ﬁnding suggests that although on a whole-brain level nodal degree and rGM are concomitantly associated with increased AC, some regions, most notably the left frontal middle gyrus, exhibit higher AC with higher nodal degree, and lower rGM, together (see the Supporting Information, Table S2). Replication Study To investigate whether the results in the effects of Gray Matter on Brain Controllability sec- tion (complementary effects of rGM and nodal degree and AC and MC) are replicable, we used data from a cohort of 48 subjects from another publicly available dataset, where we used a slightly different preprocessing pipeline (for details, see the Supporting Information: Network Neuroscience 204 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / / 5 1 1 9 8 1 8 8 9 6 8 9 n e n _ a _ 0 0 1 7 4 p d t . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 The missing role of gray matter in studying brain controllability l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / / 5 1 1 9 8 1 8 8 9 6 8 9 n e n _ a _ 0 0 1 7 4 p d . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 3. Replication sample. Average controllability (AC) is estimated based on WM structure but also relates to rGM. Each dot represents data from one region of one subject and density bar shows where the majority of data are located. (A) Effect of nodal degree on AC. (B) Effect of rGM on AC. (C) Interaction effect of rGM and nodal degree suggests that the highest levels of AC are reached when both degree and rGM are high together. For visualization, median split was used to classify rGM and degree into high and low. In the original model, both GM and nodal degree were preserved as continuous variables. Replication study methods). Also in this dataset, nodal degree and rGM increased AC (see Figure 3; for details see Table S3), while the highest AC levels were achieved when both nodal degree and higher rGM were high together (β = 0.08, 95% CI [0.04, 0.12], pbonf = 0.01). Furthermore, rGM and nodal degree both decrease MC, and the lowest values of MC were achieved only for the lowest levels of rGM and nodal degree (see the Supporting Information: Modal controllability, Figure B2). Taken together, these results suggest that this association between rGM and nodal degree is robust and not driven by individual differences in different datasets. DISCUSSION In this work, we investigated how brain volumetrics contribute to global network control prop- erties derived from the structural connectome composed of the white matter ﬁber tracts. In line with Medaglia, Pasqualetti, et al. (2017), we hypothesized that large-scale network dynamics derived from the structural connectome (here quantiﬁed by average and modal brain con- trollability) would be further explained by structural properties of GM. This work is, to our knowledge, the ﬁrst attempt to map the interdependence of both metrics, and we discuss our ﬁndings with respect to their clinical relevance. We show that, on average, the amount of rGM directly affects the brain’s availability to dy- namically transition between brain states and to adopt new modes of activity. However, levels of brain controllability were best explained when combining information from structural prop- erties of both WM and rGM, suggesting that volumetrics might provide additional information Network Neuroscience 205 The missing role of gray matter in studying brain controllability in relating brain controllability to understanding cognition, neurological and neuropsychiatric disorders, and the concept of brain reserve. Mediating Role of GM on the Relationship Between WM and Brain Controllability Our ﬁnding that nodal degree is highly predictive of brain controllability agrees with previous work (Gu et al., 2015; Medaglia, 2019), suggesting that the brain’s ability to traverse into easy and difﬁcult-to-reach brain states relies on the strength of structural connectivity, which might reﬂect the degrees of freedom to steer the transition of brain states. However, our ﬁndings sug- gest that this picture is incomplete. Structural connectivity relies on sufﬁcient support from GM reserves. The highest effects of AC were reached with enhanced nodal degree within frontal regions; this supports the rich literature showing that frontal brain networks play a central role in initiating dynamic reconﬁgurations during executive cognition. However, increased rGM within that very region was negatively related to brain controllability. While within clinical populations reduced rGM is generally related to neuropathology, there is research suggesting that within healthy subjects, rGM decreases with increases of WM density throughout devel- opment from adolescence to adulthood. This ﬁnding has been related to reduced quantity of synapses resulting from synaptic pruning (Giorgio et al., 2010), which has been predomi- nantly found in primary visual (calcarine sulcus) and prefrontal cortex (middle frontal gyrus) (Huttenlocher, 1979; Huttenlocher & Dabholkar, 1997). In our data, average brain control- lability was maximal when exactly these regions showed reduced rGM and increased con- nectivity of the white matter connectome. One could speculate that this ﬁnding reﬂects more efﬁcient and developmentally advanced brain functioning in a broad range of tasks potentially related to synchronizing the actions with intentions in a goal-directed way. Potential Contribution of Sensory Regions to Brain Controllability On a functional level, we ﬁnd several key visual areas that stand out with respect to both average as well as modal controllability. Enhanced rGM in the right cuneus has previously been reported to predict higher error rates in a response inhibition task in bipolar (Haldane, Cunningham, Androutsos, & Frangou, 2008) and has also been related to motor response in functional imaging studies (Booth et al., 2005; Matthews, Simmons, Arce, & Paulus, 2005). We believe that these ﬁndings suggest that the function of those primary visual areas goes far beyond unimodal information processing and that primary sensory cortices might occupy more “hub-like” positions in the brain through enhanced long-distance connectivity across brain-wide communities (Esfahlani, Bertolero, Bassett, & Betzel, 2020). Taken together, we speculate that sensory regions could be ideal hot spots for brain controllability nodes. Given their high global interconnectivity, these sensory nodes act potentially as the controllers with respect to the afferent inputs while the other regions act as controllers for efferent demands. Linking GM and WM in the Context of Controllability Cognitive functioning arises from complex reconﬁgurations across metabolically expensive large-scale networks, facing a trade-off between wiring cost (topological efﬁciency) and efﬁ- cient adaptation patterns between multiple neuronal populations (topological value; Bullmore & Sporns, 2012). Recent studies have suggested that the behavioral relevance of this trade-off between topological efﬁciency and topological value can be described by the brain’s energy expenditure to exhibit control along large-scale structural networks. The ratio of neuronal signaling- to nonsignaling-related metabolic energy expenditure has shown opposite direc- tionalities for white and gray matter (Yu, Herman, Rothman, Agarwal, & Hyder, 2018; Zhu Network Neuroscience 206 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / / t e d u n e n a r t i c e - p d l f / / / / / 5 1 1 9 8 1 8 8 9 6 8 9 n e n _ a _ 0 0 1 7 4 p d . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 The missing role of gray matter in studying brain controllability et al., 2012). Here, we speculate that energy expenditure could be one of the key factors linking GM and WM in the framework of controllability analysis. AC relates to the average en- ergy a brain region needs to exert to steer the brain dynamics into all possible brain states (Gu et al., 2015; Kenett, Medaglia, et al., 2018; Liu, Slotine, & Barabasi, 2011) and therefore, more regional gray matter volume is more likely to provide the sufﬁcient energy. In contrast, in ab- sence of sufﬁcient WM tracts, that is, lower nodal degree, rGM cannot fully force the transitions since the energy cannot be exerted. This conception must be expressed on a behavioral level, in that the brain system’s control capacity is especially sensitive to rGM. Indeed, a range of studies have suggested that rGM but not white matter changes relate to abnormal behavioral conditions, such as in antisocial personality disorder (Raine et al., 2000), medication-naive high-functioning children with autism spectrum disorder (Palmen et al., 2005), and alcohol- dependent individuals (Fein et al., 2002). Closely related, MC is strongest when nodal degree and rGM are simultaneously low. MC reﬂects the ability to drive the brain dynamics toward difﬁcult-to-reach states by changing the modes of activity on the whole-brain level. It is there- fore conceivable that similar to the relevance of nodal sparsity to enable optimal MC (Gu et al., 2015), the scarcity of rGM enhances the ability of the host node by targeting only a minimal set of other nodes. Limitations: Beyond Linear Full Controllability Our results in the current study warrant the conclusion that the interplay between gray matter and controllability is of complex nature. Although these results highlight the potential missing role of gray matter in studying brain controllability, there are important aspects that remain yet to be explored. The choice of nonlinear dynamics to deﬁne the range of controllability metrics could have considerable effects on our ﬁndings. For instance, it is suggested that importance of nodal geometry could actually follow opposite trends when nonlinear and control models are compared (Jiang & Lai, 2019). How the nonlinearity might (re)deﬁne the role of rGM for control is an interesting question. Brain controllability metrics considered in the current paper are trajectory unspeciﬁc. Studies of dynamical functional and structural connectivity and analysis of structural covariance have reliably shown that brain state trajectories are not random, but rather follow general rules (see Gu et al., 2017; Tang & Bassett, 2018, for recent attempts to accommodate trajectory dependence in the broader context of network control theory). Taken together, we suggest that future work should also include GM in studying brain controllability and further investigate non-linearity of brain controllability. ACKNOWLEDGMENTS HJ was supported by Fortüne grant of Medical Faculty of University of Tübingen (No. 2487-1- 0). AZ was supported by the Swiss National Science Foundation (P2ZHP1_181435). MW was supported by EU-ERA-Net: Neuromarket, EU-WIDESPREAD: Fat4BBrain, DFG Wa2673/10, and Neurobiologie motivierten Verhaltens (TPA06) from SFB 779. The authors declare no conﬂict of interest. DATA AVAILABILITY The data used in the current study are publicly available online. See the Methods section for detail. SUPPORTING INFORMATION Supporting information for this article is available at https://doi.org/10.1162/netn_a_00174. Network Neuroscience 207 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / / t e d u n e n a r t i c e - p d l f / / / / / 5 1 1 9 8 1 8 8 9 6 8 9 n e n _ a _ 0 0 1 7 4 p d . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 The missing role of gray matter in studying brain controllability AUTHOR CONTRIBUTIONS Hamidreza Jamalabadi: Conceptualization; Formal analysis; Investigation; Methodology; Proj- ect administration; Software; Validation; Writing – original draft; Writing – review & editing. Agnieszka Zuberer: Conceptualization; Formal analysis; Investigation; Methodology; Project administration; Software; Validation; Visualization; Writing – original draft; Writing – review & editing. Vinod Jangir Kumar: Data curation; Writing – review & editing. Meng Li: Data curation; Writing – review & editing. Sarah Alizadeh: Conceptualization; Validation; Writing – review & editing. Ali Moradi Amani: Methodology; Writing – review & editing. Christian Gaser: Investigation; Validation; Writing – review & editing. Michael Esterman: Investigation; Resources; Supervision; Writing – review & editing. Martin Walter: Investigation; Resources; Supervision; Validation; Writing – review & editing. FUNDING INFORMATION Hamidreza Jamalabadi, Fortüne Grant of the Medical Faculty of University of Tübingen, Award ID: 2487-1-0. Agnieszka Zuberer, Swiss National Science Foundation, Award ID: P2ZHP1_ 181435. Martin Walter, ERA-NET NEURON, Award ID: Neuromarket. Martin Walter, EU- WIDESPREAD, Award ID: Fat4BBrain. Martin Walter, DFG, Award ID: Wa2673/10. Martin Walter, Neurobiologie motivierten Verhaltens from SFB 779, Award ID: TPA06. 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