The impact of aging on human brain network target controllability

Giulia Bassignana1,2, Giordano Lacidogna3, Paolo Bartolomeo1 Olivier
Colliot1,2, Fabrizio De Vico Fallani1,2,*

1 Sorbonne Universites, Paris Brain Institute (ICM), Inserm, CNRS, APHP,
Hopital Pitie Salpetriere, Paris, France

2 Inria Paris, Aramis Project Team, Paris, France

3 Hospital, Tor Vergata Unviersity, Rome, Italy

* corresponding author<EMAIL_ADDRESS>

### Abstract

Understanding how few distributed areas can steer large-scale brain activity
is a fundamental question that has practical implications, which range from
inducing specific patterns of behavior to counteracting disease.

Recent endeavors based on network controllability provided fresh insights into
the potential ability of single regions to influence whole brain dynamics
through the underlying structural connectome. However, controlling the entire
brain activity is often unfeasible and might not always be necessary. The
question whether single areas can control specific target subsystems remains
crucial, albeit still poorly explored. Furthermore, the structure of the brain
network exhibits progressive changes across the lifespan, but little is known
about the possible consequences in the controllability properties.

To address these questions, we adopted a novel target controllability approach
that quantifies the centrality of brain nodes in controlling specific target
anatomo-functional systems. We then studied such target control centrality in
human connectomes obtained from healthy individuals aged from $5$ to $85$.
Main results showed that the sensorimotor system has a high influencing
capacity, but it is difficult for other areas to influence it. Furthermore, we
reported that target control centrality varies with age and that temporal-
parietal regions, whose cortical thinning is crucial in dementia-related
diseases, exhibit lower values in older people. By simulating targeted
attacks, such as those occurring in focal stroke, we showed that the
ipsilesional hemisphere is the most affected one regardless of the damaged
area. Notably, such degradation in target control centrality was more evident
in younger people, thus supporting early-vulnerability hypotheses after
stroke.

##  1 Introduction

Brain controllability refers to the possibility to induce specific functional
states, or configurations, by means of internal or external control. In
neuroscience, this capability is associated with cognitive control [1], which
can be qualitatively assessed by measuring dynamic cooperation and competition
between different neural systems during goal-directed tasks [2].

Recently, the adoption of a control theoretic perspective has started to
provide quantitative insights on how functional brain states can be predicted
by the underlying brain network structure[3]. In particular, network
controllability - i.e., the theoretical ability to guide a system’s state by
operating on few driver nodes - has received a growing interest in several
biological applications [4, 5, 6, 7]. Controllability of brain networks has
been specifically explored to understand how the brain is able to endogenously
modify its dynamics and if it is possible to steer it from exogenus stimuli
[8, 9]. Hence, a crucial step in network controllability is to determine the
nodes that are the best-suited to drive the system’s state. To this end,
centrality measures based on controllability were developed and applied to
brain networks for identifying potential driver nodes and measuring the size
of the network that they can control [3, 10].  These studies can have
practical implications because they can inform possible intervention
strategies to favor specific patterns of behavior or treat brain diseases, by
means of brain stimulation technology [11, 12, 8].

While promising, controllability of the brain is still an underexplored
research field due to the different ways in which controllability can be
implemented [13]. In practice, controlling the whole system from a single node
could not be feasible due to algorithmic imprecision even in relatively small
networks of hundreds of nodes. Furthermore, the magnitude of the control
signal might be too high to be generated physically or destructive for the
network functioning itself [14, 15]. A possible solution would be to focus on
specific parts of the network so as to reduce the overall computational
complexity [4, 16, 17]. Determining how single nodes can affect parts of the
network still remains poorly understood in the human brain. Furthermore,
current results have neglected the fact that brain controllability could vary
with aging, which induces progressive changes in the underlying network
structure or with focal brain damages [18, 19].

To address these questions, we used cross-sectional anatomical $DTI$ and
functional MRI (fMRI) data from $171$ healthy individuals aged between $5$ and
$85$ [20]. Then, we evaluated the ability of single brain nodes to control
specific target subsystems, i.e., frontal, limbic, temporal, sensorimotor,
parietal, and occipital systems (Fig. S1). To do so, we adopted an exact
network controllability approach [21], which gives for each tested driver the
number of nodes in a predetermined target that it can control (Fig. 1).

We then studied how control centrality is distributed in the brain and which
are the most controllable target systems. We assessed how control centrality
is altered by age and whether there are specific regions that better explain
the aging process. Finally, we tested how control centrality changes in
presence of simulated targeted attacks and how the overall damage depends on
the participant’s age.

##  2 Results

###  2.1 Sensorimotor areas have high control centrality

We first studied how target control centrality is distributed across different
functional systems of the human brain (Fig. 2, Methods). Namely, we analyzed
the ability of a single node in controlling different cortical systems by
computing the target control centrality $\tau$, the number of target nodes
that this driver can control (Methods). The group-averaged results showed that
$\tau$ values significantly vary across regions (one-way ANOVA,
$df=5,\,F>969,\,p<10^{-6}$) as well as depending on the targeted system. In
particular, the sensorimotor regions had a high tendency to control the nodes
in other systems, but they were also difficult to control from other areas
(Fig S2). By pooling the results obtained with all different target systems,
it became clearer that the sensorimotor nodes have on average the largest
centrality values, while the limbic nodes have the lowest ones (Fig. 2a).

Next, we evaluated the tendency of single brain areas to control the system
they belong to as compared to other systems. To this end, we calculated the
self-regulation score of a node as the ratio of $\tau$ when targeting its own
system and the average obtained when it targets all the other systems
(Methods). Results showed that on average brain regions tend to have a high
self-regulation so that they control more nodes in their own system than those
in other systems (one-way ANOVA, $df=4,\,F>1405,\,p<10^{-6}$). That was
particularly true for the sensorimotor system which consistently presented the
highest values of the self-regulation score. On the contrary, the nodes in the
limbic system exhibited a very low self-regulation (Fig. 2b).

###  2.2 Control centrality decreases with age

By exploiting the high variability in the age of the participants (mean $35.8$
years, std $20.0$ years), we asked whether and how target control centrality
varied with age. To this end, we computed the Spearman partial correlation
between the $\tau$ values and age of the participants, correcting for the
outdegree of the nodes. This correction was needed to exclude the existence of
significant associations merely due to the presence of nodes with a high
number of outgoing links. Results show a general tendency of the brain areas
to be negatively correlated ($|R|>0.15$,$p<0.05$, FDR-corrected for multiple
comparisons) with age regardless of the targeted system (Fig. S3). The
presence of positive correlations is instead less consistent and weakly
concentrated in the frontal and central areas of the brain.

By focusing on the ROIs that presented a significant association with the same
sign across all targets, four regions emerged with a consistent negative
correlation with age (Fig. 3). Among those, the right lateral occipital
inferior area (RLOCid3) in the occipital system is known to be associated with
attentional processes and related to the dorsal attention network (Fig.
3b)[22]. The other regions were in the temporal system and three of them (left
middle temporal anterior gyrus (LMTGad), left middle temporal temporo-
occipital gyrus (LMTGtp), right middle temporal posterior gyrus (RMTGpd)
belonged to the default mode network (DMN) [23, 24]. The left middle temporal
temporo-occipital gyrus (LMTGtp) had the strongest negative correlation,
possibly related to hippocampal degeneration with age [25](Fig. 3a). Taken
together, these results suggested that most brain regions, and in particular
those located in the temporal system, tend to decline with age.

###  2.3 Targeted attacks lead to greater control centrality loss in younger
brains

Finally, we asked to what extent the target control centrality was impacted by
attacks to specific brain systems, like those occuring after stroke, traumatic
brain injury or tumor resection [26, 27]. To answer this question, we
simulated lesions to different target systems by removing the nodes and the
links from only one hemisphere. Then we evaluated the ability of all the other
nodes to control the contralesional part of the target in the intact
hemisphere, and we computed the difference with the original values
($\Delta_{\\_}{attack}=\tau_{\\_}{lesion}-\tau$) to quantify the impact of the
damage.

As expected, network attacks led to decreases of control centrality, with
greater losses in the ipsilesional hemisphere as compared to the unaffected
one (two-ways ANOVA, $df=4,\,F>15,\,p<10^{-6}$, Fig. 4 , Fig. S4). In
particular, results showed that control centrality losses
$\Delta_{\\_}{attack}$ were globally small when the sensorimotor system was
lesioned. Instead, when other systems were damaged, the sensorimotor system
exhibited larger $\Delta_{\\_}{attack}$ decrements. Conversely, the limbic
system was relatively mildly impacted by any attack. In terms of difference
between hemispheres we did not report a clear pattern across damaged systems.
However, we consistently observed a greater ipsilesional centrality loss when
the parietal system was attacked on the right hemisphere (two-ways ANOVA,
$df=4,\,F>204,\,p<10^{-6}$, Fig. 5), which would reflect the the ability of
the right parietal lobe to be involved in several high order cognitive
processes [28, 29, 30] and interacting more with the other hemisphere as
compared to the left one [31, 32].

Finally, we evaluated whether the age of the participants had an impact on the
observed node control centrality loss after the simulated network attacks.
Results showed a global positive correlation between $\Delta_{\\_}{attack}$
values and age (Spearman correlation, $|R|>0.15$, $p<0.05$, FDR-corrected for
multiple comparisons), Fig. 6, Fig. S5), suggesting a more important effect in
younger participants compared to adults. Notably, these associations were
consistently reported in the same temporal DMN areas, for which we also
observed a significant positive correlation between control centrality values
and age (Fig. 3).

##  3 Discussion

###  3.1 Driver nodes in the human brain

Thanks to its ability to establish theoretical relationship between structure
and dynamics, control network theory has been increasingly adopted to study
human brain functioning. Control centrality, measuring how important a network
node is in influencing other nodes’ states, has been shown to provide useful
markers of brain dynamics during resting states [3] and executive functioning
[33], as well as to predict responses to brain stimulation in practice [34]
and in theory [9]. These findings establish an intuitive link between the
brain regions identified as potential control drivers and the current
knowledge in cognitive neuroscience. Notably, large-scale brain activity has
been found deeply dependent on nodal controllability within specific systems,
such as default mode, fronto-parietal, cingulo-opercular, and attention [35].

While these results provide mechanistic insights on how single areas can steer
whole-brain activity, they do not inform on how different specialized brain
systems can influence each other. Is there a dominant subnetwork prone to
control all the other ones? Is there a subsystem that is instead more
controllable than others? These fundamental questions become particularly
pressing considering that one-to-whole control might be difficult to achieve
both from a theoretical and practical angle [14, 15]. To answer those
questions, we used the step-wise target control centrality, i.e. an optimized
Kalman-based heuristics that measures the ability of nodes to control
predetermined target parts of the network [21].

Our results showed that in general node control centrality varies depending on
the targeted system. Despite such variability, the sensorimotor areas always
exhibited the highest influencing capacity while being very hard to control.
Conversely, the limbic regions always had a scarce driving power and they were
on average easily controllable by all other nodes (Fig. 2). This tendency was
also confirmed when looking at the ability of those systems to control, or
self-regulate, themselves.

The sensorimotor system is known to be more densely connected, with many
anatomical long-distance interhemispheric connections, as compared to other
secondary motor-related systems [36]. Such hyperconnectivity, would constitute
therefore a structural prerequisite to reach and orchestrate different areas
during cognitive and motor functions [37, 38, 39, 40, 41]. From a functional
perspective the high control centrality of the sensorimotor system has been
previously associated with its ability to process information not only for
motor control but also for a broad range of recognition processes [42, 43,
44]. This is in line with the existence of gradients of cortical organization
through which the sensorimotor system could boost neural activity of other
association areas required for higher order functions such as cognitive
control, guided attention and motivation [45]. In line with such a cortical
organization gradient, we could have expected that the visual system -another
highly connected primary system- would have exhibited high control centrality,
too. However the control centrality of the nodes in the occipital lobe, where
main hubs of the visual system are localized, were rather low. Future works
will be crucial to assess whether the high control centrality of the
sensorimotor system is mainly due to its high connection density and/or
specific local synaptic properties and gradients of gene expression in the
excitation–inhibition balance of related interneurons [45].

While limbic areas constitute an important structural bridge for the
information transfer between cortical and subcortical regions [46, 47], their
control centrality was remarkably low regardless of the targeted cortical
system. This could be in part explained by the heterogenous nature the lymbic
system, which includes different anatomical and cytoarchitectonic components
[47, 48, 49]. However, it is also true that many important subcortical areas,
such as amygdala, caudate nucleus, and hypothalamus [46], were not available
in the dataset we used (see Methodological considerations for more details).
Future research will be crucial to elucidate the role of lymbic areas in terms
of target control centrality using more comprehensive and accurate
subcortical-cortical systems.

###  3.2 Aging and control centrality

Brain aging is a highly heterogeneous and dynamic process that involves
structural and functional changes both at individual and group level. Among
structural changes, cortical thinning and regional atrophy [50], white matter
loss of integrity [51], neuronal loss and degeneration [52] and
neurotransmitters depletion have been detected at varying degrees among older
subjects [52]. Functional alterations refers to maladaptive, age-related brain
activity, detected in neuroimaging studies, including decreased specificity of
ventral-visual and motor areas [53, 54], decreased memory-related recruitment
of medial temporal lobe regions [55] and dysregulation of the default mode
network [56].

Network approaches investigating brain reorganization across the lifespan have
constantly reported whole-brain intrinsic connectivity changes by using
standard centrality metrics such as node degree [57], strength [58] and
betwenness [59]. Compared to young adults, older ones exhibit reduced local-
efficiency and modularity as well as connectivity changes within and between
specific subnetworks [60, 61, 62, 63]. Only few studies have attempted to
study how network controllability evolves with age. In a recent study, authors
showed that regional controllability increases with development [64], but no
information is available on how it evolves across the lifespan. To address
this question, we combined information from both structural (DTI) and
functional (fMRI) neuroimaging data, and we analyzed the effect of aging on a
new measure of centrality based on network controllability.

Main results showed that control centrality was negatively correlated with age
indicating a global trend of node controllability reduction in older brains.
The presence of sporadic positive correlations in the frontal and central
areas should be further investigated for possible compensatory mechanisms
occuring in later age as well as in mild cognitive impairment and Alzheimer’s
disease [65, 66, 67]. Instead, brain regions in the middle temporal gyrus
(MTG) were significantly impacted by aging, in terms of relative loss of
control centrality (Fig. 3). Those areas were previously found to be less
activated in elderly people, reflecting a lower semantic retrieval control
process [24, 68]. Such a decrement could be associated with the the more
general age-dependent DMN functional rearrangement [69, 70].

According to the “network dedifferentiation” hypothesis [71], DMN regions in
older adults progressively present a reduction in the communication with other
systems, such as the dorsal attention network (DAN) or the frontoparietal
network[63, 72]. Thus, age-related reduced control centrality could be a
possible reorganizational mechanism subserving the failure to deactivate
neural systems that are unrelated to the task [73] and leading to abnormal
increased brain activation [74, 75] and negative correlation with task
performance [76, 77].

Our results showed that while there is a common distribution between the age-
centrality correlations in the left and right hemisphere, there are also few
notable differences. The specific asymmetric involvement of the right lateral
occipital inferior cortex (RLOCid3) could be intriguingly related to a
progressive inability to mediate the interaction between the right ventral
attention system, involved in saliency analysis, and the bilateral dorsal
attention system, mostly involved in attention shifting [78, 79, 30]. The
interaction between the two attentional systems finally influences the
processing of attended visual stimuli in the primary visual cortex [80, 72].
While asymmetry patterns could be interpreted in the light of age dependent
loss of lateralization in specific cognitive processes, such as semantic
control [73, 81], more investigation is needed to better disentangle this
aspect.

Finally, recent studies suggest that long-range connections may be more
vulnerable to aging effects than short-range connections in both DMN and DAN
[82]. A future perspective would be to check to what extent the observed
controllability changes depend on the relative spatial distance between the
driver and target nodes.

Overall, our results indicated that age related reduction in controllability
in specific nodes at the crossroad of different brain systems relevant for
cognitive control efficiency, could be viewed as a yet another evidence of a
functional “reshaping” of brain networks along the lifespan [83, 73].

###  3.3 Control centrality markers of brain lesions

Network control theory not only informs on the basic brain functioning but it
can also offer new analytical ways for quantifying dysfunctions in
neurological diseases [84]. After a focal damage, such as in stroke, traumatic
injuries and tumors, the brain typically loses the functions associated with
the lesioned area and with those connected to it. Quantifying the effects of
such local destruction on the rest of the network is therefore crucial to
predict the extent to which the brain will recover its functions through a
reorganizational process, i.e. plasticity [85, 86].

Here, we studied how control centrality changed when simulating unilateral
attacks to the target systems. Results showed that the nodes in the
ipsilesional hemisphere globally underwent larger losses in control
centrality. That was particularly true for the sensorimotor regions regardless
of the system attacked. From a mechanistic perspective, the ability of the
nodes in the damaged hemisphere to control the unaffected side of the target,
would be compromised by the removal of the interhemispheric homotopic
connections, i.e. the links bridging the homologous regions in the two
hemispheres [87, 88]. The interruption of such homotopic bridge, which is
typically stronger compared to heterotopic connectivity, eventually made the
nodes in the intact target less reachable from the nodes in the lesioned
hemisphere. The extent to which this also affects possible compensatory
actions from the contralesional “intact” hemisphere, remains a question to be
elucidated with more clinical and longitudinal data [89, 90, 91, 92, 93].

While the controllability power of ipsilesional sensorimotor regions was
particularly impacted when damaging the targets, the other regions appeared to
be less affected when the sensorimotor target was attacked. This weak effect
was in part due to the relatively low values of control centrality of those
regions, which limited the possible range of change after the damage. More
generally, this might be related to a putative high reachability of the
sensorimotor system given its high connection density [94, 36, 95] and the
presence of alternative pathways between frontal, parietal and limbic areas
[96, 97].

We observed a progressive decrease of control centrality with age, mostly
localized in the temporal lobes (Fig. 6). We then asked whether the control
centrality losses induced by the lesions, regardless of the targeted system,
could also depend on the age of individuals. Results confirmed that the
damage-related control centrality reductions in the temporal lobes were larger
in younger brains, which have in general higher baseline values of
controllability.

From a biological perspective, it’s well established that changes in
neuroplasticity occur lifelong through many age-specific processes [98],
ultimately affecting cerebral network maturation [99, 100] and controllability
[64]. Likewise, brain lesions could impact brain organizational properties in
an age-dependent manner. In this direction, two seemingly contradictory
explanations have emerged: first, “early plasticity”, arguing for the greater
flexibility of the immature brain, and associated good recovery and outcome
[101, 102]; and secondly, “early vulnerability” referring to the young brain’s
unique susceptibility and subsequent poor outcome [103, 104, 105].

Our results indicated that target control centrality follows an age-dependent
trend that mirrors an early vulnerability condition, so that networks of
younger participants are more impacted by focal lesions, possibly reflecting a
reduced resiliency to damages in the early stages. Shedding a light on
functional reorganization and recovery after brain injuries or stroke might
enable better prediction and prevention of clinical outcome. However, the lack
of clinical data for our population allows us neither to validate this result
from a clinical perspective nor to assess the early-plasticity hypothesis
which would require longitudinal clinical data associated with the recovery of
patients.

Future research is needed to better identify possibly longitudinal changes in
network controllability related to specific clinical outcome in both earlier
and later stages of brain reorganizaiton after brain damages [103].

###  3.4 Methodological considerations

In this work, we exploited the step-wise target controllability framework
[21], which relies on a linear time invariant (LTI)dynamics and is based on
the Kalman rank condition. While it is known that the brain presents a
nonlinear dynamics, the study of linear models has proved to be beneficial in
improving our understanding [3, 106, 8]. Specifically, the controllability of
a linearized model can inform on the controllability of the nonlinear model
[107].

In order to study the controllability of brain networks efficiently, we gave a
directionality to the connectomes. Despite the fact that it is not currently
possible for neuroimaging techniques to discern the directionality of bundles
of axons, it is known that each neuron propagates signals through a well-
defined direction from the soma to the axon terminal. Previous efforts to
direct a connectome relied on the hypothesis that, given a set of brain
networks, edges present for all participants are the oldest, and any new edge
would be directed from the new node to the existing cluster [108]. However, a
limitation of this method is that it is strictly dependent on the initial set
of networks, and the procedure would not be easily scalable. Other methods
based on local navigation and communicability were devised to infer the
directionality of neural signaling, but since they operate best for nodes
connected by longer paths, they are not well-suited to perform inference for
structurally connected nodes [109]. Our approach was instead inspired by the
diffusion processes taking place in the connectome [110, 111, 112, 113, 114]
and preserves the hierarchical and modular properties of the anatomical
pathways.

Finally, it is important to mention that the obtained results refers to the
specific way we have selected the target systems and and ranked the nodes
therein contained, and should not be generalized to other possible choices. By
adopting an “anatomical proximity” criterion, some regions that could be
functionally associated to the frontal lobe, e.g. the anterior cingulate
cortex ACC, have been instead assigned to the lymbic system. The original
dataset did not contain subcortical areas, such such as amygdala, caudate
nucleus, and hypothalamus [47], which are known to be tightly related to the
lymbic system. Future research will therefore be important to validate the
obtained results on different brain atlases and target system selection.
Furthermore, more rigorous anatomical and circuit-level information from
histological and causal experiments can better elucidate the robustness of our
results obtained from noninvasive neuroimaging [115, 116].

##  4 Conclusions

In this study, we presented a method to quantify the ability of candidate
driver nodes to drive the state of a target set in directed brain networks.
The obtained results revealed that sensorimotor areas are theoretically
inclined to control different target systems, while regions in the temporal
lobe were negatively impacted by age and by simulated damages to the network.
These results are in line with the general claim of a dominant gradient of
cortical organization with sensory-motor and association regions at opposing
ends [117, 118]. We hope that further developments of network controllability
measures will contribute to the identification of the key nodes in biological
networks to better identify targets of brain stimulation to counteract human
diseases.

##  5 Methods

#### Structural and functional brain networks

We used already processed brain network data from the NKI-Rockland database
[119], and selected $171$ participants, aged from $5$ to $85$ [120]. For each
participant we had access to both structural (DTI) and functional (fMRI) data.
Both structural and functional networks consisted of $188$ nodes corresponding
to functional regions of interest (ROIs) established with a spatially
constrained spectral clustering method [121]. All related information
including node labels and spatial position can be accessed via the USC
Multimodal Connectivity Database http://umcd.humanconnectomeproject.org/umcd/.
We also provided a compacted version of this information in a dedicated
supplementary file (File S1). Note that some ROIs had been split in different
parts. In this case, we added a suffix number at the end of the label to
specify its relative position along the longitudinal axis of the brain (i.e.
higher numbers, more caudal positions).

#### Directing structural brain networks

We directed the structural connectomes in order to study their controllability
properties in a more efficient way, and we used the information from fMRI
brain networks to establish a hierarchy among the nodes in the target systems.
Starting from the DTI network, first we applied a logarithmic transformation
($\log{(w+1)}$) to make the weights more homogeneous. Then, we performed a
biased random walk [122] to direct each edge from the node with lower strength
to the node with higher strength. The probability $P_{\\_}{ij}$ to go from
node $i$ to node $j$, can be computed as

$P_{\\_}{ij}=\frac{w_{\\_}{ij}}{\sum_{\\_}h{w_{\\_}{ih}}}$ (1)

where $[w]_{\\_}{ij}$ is the symmetric, weighted adjacency matrix of the
structural network. For each pair of nodes $i$ and $j$, we had two directed
edges with weights $P_{\\_}{ij}$ and $P_{\\_}{ji}$. We chose one direction by
keeping only the highest probability (we kept both if they were equal), thus
drastically reducing the presence of loops in the network.

The choice of removing some links was inspired by the works on the minimum
spanning tree (MST) procedure applied to brain networks [123]. Similarly to
our strategy, the MST filtering procedure eliminates any loop in the network,
yet important information can be obtained on the brain network structure and
function with implication in aging [124]. Despite such background, our choice
still remains a modeling strategy that favors information propagation towards
higher degree nodes. Note that the resulting effect is in line with the
evidence that hubs in real networks tend to have less control power
(mechanistically exerted by outgoing links), as they instead need to be
efficiently accessed to control the rest of the network [125]. As for the
reduction of loops, it is important to stress that this might lead to
underestimate the number of controllable nodes, but not to introduce false
positive results. This is because of the underlying k-walk theory by which a
driver-target configuration is controllable if the length of the path from the
driver to each target is unique [16].

#### Step-wise target controllability

We implemented the step-wise target controllability framework [21], that
analyses the single-input target controllability problem, in which the
interest is to study the role of a single driver node in controlling a target
set of the system. We assumed the linear time invariant (LTI)dynamics

$\dot{\textbf{x}}(n)=A\textbf{x}(n)+B\textbf{u}(n),\qquad\textbf{y}(n)=C\textbf{x}(n)$
(2)

where $\textbf{x}(n)\in\mathbb{R}^{N}$ describes the state of each node at
time $n$, $A$ is the adjacency matrix of the network, $B\in\mathbb{R}^{N}$
specifies the driver node that will receive an external input,
$\textbf{u}(n)\in\mathbb{R}^{N}$ is its external input (or control) signal,
$\textbf{y}(n)\in\mathbb{R}^{S}$ is the output vector, and
$C\in\mathbb{R}^{S\times N}$ is the output matrix identifying the target
nodes.

Such a system is controllable if it can be guided from any initial state to
any desired final state in finite time, with a suitable choice of input.

A necessary and sufficient condition to assess the controllability of Eq. 2,
is that the controllability matrix $Q$

$Q=\begin{bmatrix}B&AB&A^{2}B&\cdots&A^{N-1}B\end{bmatrix}$ (3)

has full row rank, i.e. rank$(Q)=N$. That is the Kalman rank condition, which
basically verifies the existence of linearly independent rows in $Q$ [126,
127]. If so, the driver node can reach and control the dynamics of all the
other nodes through independent walks of length $N-1$ at maximum.

If it is of interest to control only a target set $\mathcal{T}$ of the
network, specified in $C$ and consisting of $S\leq N$ nodes, then Eq. 2 can be
reduced into a target controllability matrix $Q_{\\_}\mathcal{T}=CQ$, where
$C$ filters the rows of interest corresponding to the targets. Now, the rank
of $Q_{\\_}\mathcal{T}$ gives the number $\tau\leq S$ of nodes in the target
set that can be controlled by the driver.

To identify a driver-target configuration, we further introduce a hierarchy
among the target nodes, so that we can order and relabel them from the most
important one to the least, i.e. $t_{\\_}1\succ t_{\\_}2\succ...\succ
t_{\\_}S$. Then we perform the following step-wise procedure for each
candidate driver node.

  * •

Step 1. Initialization

    * –

Create a temporary empty target set $\mathcal{T^{\prime}}\leftarrow\\{\\}$

    * –

Set the number of controllable targets $\tau\leftarrow 0$

  * •

Step 2. Repeat until termination criteria are met. For $j\leftarrow 1,...,S$
do

    * –

Add the $j$-th target node to the target set
$\mathcal{T^{\prime}}\leftarrow\mathcal{T^{\prime}}\cup\\{t_{\\_}j\\}$

    * –

Build the subgraph containing the nodes on walks from the driver to the
targets in $\mathcal{T^{\prime}}$

    * –

Compute the rank of the target controllability matrix
$Q_{\\_}\mathcal{T^{\prime}}$

    * –

If rank($Q_{\\_}\mathcal{T^{\prime}}$) is full then $\tau\leftarrow\tau+1$
else
$\mathcal{T^{\prime}}\leftarrow\mathcal{T^{\prime}}\setminus\\{t_{\\_}j\\}$

    * –

$j\leftarrow j+1$

  * •

Step 3. Output $\tau$ and $\mathcal{T^{\prime}}$

Eventually, the target control centrality $\tau$ is the number of controllable
targets in $\mathcal{T}$, and the set $\mathcal{T}^{\prime}$ contains the
$\tau$ controllable targets with highest ranking.

#### Application to brain networks

In this specific application, we assumed that the states of the nodes/ROIs
were influenced by the adjacency matrix corresponding to the directed
structural connectome. The target sets were the structural systems (frontal,
limbic, temporal, sensorimotor, parietal, occipital, Fig. S1). ROIs in the
target system were ranked according to the group-averaged node strength
obtained from the fMRI functional brain network (Fig. 1, File S1).

More precisely, we ranked the nodes in the target set $\mathcal{T}$ in a
descending order from the quantity $\frac{1}{M}\sum_{\\_}ms_{\\_}{i}^{(m)}$,
where $s_{\\_}{i}^{(m)}$ is the node strength (i.e. the weighted sum of all
the connections) of the node $i$ obtained from the fMRI functional network of
the participant $m$, and $M$ is the number of participants in the study. Node
strengths quantify the tendency of brain areas to act as hubs which are
crucial constituents of the overall information integration [128].

The target control centrality $\tau$ of each node is obtained via the stepwise
procedure described above and gives the corresponding highest-ranked
controllable configuration of target nodes. To obtain a more robust
estimation, we derived an intergated measure of target control centrality by
averaging the results from the connectomes thresholded with different
connection densities [129]. Specifically, for each participant, we retained
the strongest links so to have networks with mean node degree $k$ ranging from
$1$ to $14$. This upper limit corresponded to the lowest number of links found
after directing the connectomes.

####  System regulation score

To quantify how the target control centrality of a system is globally
distributed across the other target systems we introduced the so-called system
regulation score $\mathcal{R}$. Given two brain systems $i$ and $j$ the
regulation score reads as

$\mathcal{R}_{\\_}{ij}=\frac{\frac{1}{|\mathcal{S}^{(i)}|}\sum_{\\_}{k\in\mathcal{S}^{(i)}}\tau^{(j)}_{\\_}k}{\frac{1}{N}\sum_{\\_}{k\in\mathcal{V}}\tau^{(j)}_{\\_}k}$
(4)

where $\mathcal{S}^{(i)}$ is the reference system $i$, $\mathcal{V}$ is the
set of all $N$ nodes in the brain network, and $\tau_{\\_}k^{(j)}$ is the
target control centrality of each node in the system $i$ over the target
system $j$. When $i=j$, we specifically talk about self-regulation.

#### Statistical analysis

All statistical analysis were performed with a statistical threshold of
$p=0.05$, adjusted via a false-discovery rate (FDR) procedure in the case of
multiple tests [130].

## Funding and/or Conflicts of interests/Competing interests

Authors would like to acknowledge Thibault Rolland
(fr.linkedin.com/in/thibault-rolland-40b57419a) for the realization of Picture
1. The research leading to these results has received funding from the French
government under management of Agence Nationale de la Recherche as part of the
”Investissements d’avenir” program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA
Institute) and reference ANR-10-IAIHU-06 (Agence Nationale de la
Recherche-10-IA Institut Hospitalo-Universitaire-6), and from the Inria
Project Lab Program (project Neuromarkers), the European Research Council
(ERC) under the European Unions Horizon 2020 research and innovation programme
(Grant Agreement No. 864729). The content is solely the responsibility of the
authors and does not necessarily represent the official views of any of the
funding agencies. Authors declare no conflict of interest.

## Data availability statement

All the experimental data used in this work are fully accessible from the NKI-
Rockland database [119].

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Fig 1: Illustrating principle of the step-wise target control centrality. Left
panel: the lower triangle indicates the selected driver node, upper triangles
indicate the nodes in the chosen target system. The structural brain network
derived from DTI data is the connection matrix of the LTI control system.
Right panel: the step-wise algorithm identifies the controllable driver-target
configuration by visiting the target nodes according to their strength in the
fMRI functional network, i.e. from the highest to the lowest.

Fig 2: Group-averaged target control centrality and regulation score. Panel a)
cortical maps of $\tau$ values averaged across different target systems. Top
view on the left and lateral views on the right. L=left hemisphere, R=right
hemisphere. Panel b) Regulation scores between different systems (Methods).
Dark grey markers denote group-averaged self-regulation scores, i.e. the
ability of a system to control itself. Light grey markers denote group-
averaged regulation scores between different systems and are reported for the
sake of completeness. Error bars stand for standard deviation across
participants.

Fig 3: Statistical correlation between target control centrality and age.
Panels a) and b) show respectively the correlation maps for the left and right
hemisphere of the brain. The size and color of nodes code the number of times
that a node exhibit a significant ($p<0.05$) correlation (negative-blue or
positive-red) for different target systems. Statistical correlation is
computed by a partial Spearman correlation corrected for the out-degrees of
the nodes. The insets on the right show the population scatter plots for the
ROIs with highest correlation in each hemisphere.

Fig 4: Loss of target control centrality after simulated attacks. Panel a)
shows the cortical maps of the group-averaged $\Delta_{\\_}{attack}$ values
when attacking different target systems on the left hemisphere (see Fig. S4
for attacks to the right hemisphere). Size and color of the nodes are
proportional to the decrease of target control centrality with respect to the
healthy situation. Panel b) shows the values of the $\Delta_{\\_}{attack}$
averaged across the nodes of a same system. Both values for the systems in the
ipsilesional and contralesional hemisphere are illustrated. Error bars stand
for the standard error across participants.  Fig 5:  Comparison of target
control centrality loss between lesions to the right and left parietal system.
Left side panel illustrates the group-averaged values for the
$\Delta_{\\_}{attack}$ of the intact systems when attacking the ipsilateral
parietal system. Error bars stand for the standard error across participants.
Right panels show the cortical maps of the group-averaged
$\Delta_{\\_}{attack}$ values when attacking respectively the parietal system
in the left (L) and right (R) hemisphere.

Fig 6: Statistical correlation between target control centrality loss
($\Delta_{\\_}{attack}$) and age. Panels a) and b) show respectively the
correlation maps for the left and right hemisphere of the brain. The size and
color of nodes code the number of times that a node exhibit a significant
($p<0.05$) correlation (negative-blue or positive-red) for different target
systems. Statistical correlation is computed by a partial Spearman correlation
corrected for the out-degrees of the nodes. The insets on the right show the
population scatter plots for the ROIs with highest correlation in each
hemisphere.