Controllability in attention deficit hyperactivity disorder brains

Abstract

The role of network metrics in exploring brain networks of mental illness is crucial. This study focuses on quantifying a node controllability index (CA-scores) and developing a novel framework for studying the dysfunction of attention deficit hyperactivity disorder (ADHD) brains. By analyzing fMRI data from 143 healthy controls and 102 ADHD patients, the controllability metric reveals distinct differences in nodes (brain regions) and subsystems (functional modules). There are significantly atypical CA-scores in the Rolandic operculum, superior medial orbitofrontal cortex, insula, posterior cingulate gyrus, supramarginal gyrus, angular gyrus, precuneus, heschl gyrus, and superior temporal gyrus of ADHD patients. A comparison with measures of connection strength, eigenvector centrality, and topology entropy suggests that the controllability index may be more effective in identifying abnormal regions in ADHD brains. Furthermore, our controllability index could be extended to investigate functional networks associated with other psychiatric disorders.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-023-10063-z.

Introduction

As the abnormal neurodevelopment underlying genes, environment and social factors, attention deficit hyperactivity disorder (ADHD) is a mental disorder affecting school-age children, which may be associated with substantial lifelong impairment (Thapar et al. 2012; Friedman and Rapoport 2015; Sathyanesan et al. 2019). Monoamine neurotransmitter contents are the key explanations for biochemistry reasons, which are regarded norepinephrine, dopamine concentration, 5-hydroxytryptamine or serotonin hypotheses. From the molecular genetic viewpoint, these catecholamine metabolizing enzyme genes are tyrosine hydroxylase, dopamine $\beta$ hydroxylase, catechinol oxidative methyltransferase and monoamine oxidase genes. Neuroimaging analyses in the ADHD cases focus on abnormal dispersive brain structures and their dysfunctions (Hoogman et al. 2019). Intuitively, the ADHD patients are characterized by symptoms of inattention and hyperactivity impulsivity which are not appropriate for the person’s age. Presently, the therapeutic effect for the ADHD patients has been unsatisfactory because the causes of brain disease are not clear. In addition, the understanding of diversity, selection and mutation of brain cognitive functions is still limited.

Complex networks have been proved as a powerful tool to study brain systems, especially for mental illness (Bullmore and Sporns 2009; Zhou et al. 2010; Power et al. 2011; Ansari et al. 2022). For example, Bullmore and Sporns proposed network methods to characterize abnormal regions in the diseased brains (Sporns and Zwi 2009). For exploring the abnormal regions of Autism brain, our previous work applies the eigenvector centrality to investigate functional networks constructed by fMRI data (Chen 2022). Existing results show that combining edge, node, module and whole network indices would provide more comprehensive descriptions of altered brain connectivity (Bassett and Sporns 2017). To characterize the functional connections of the schizophrenia brains, Liu et al. proposed more network indices, including degree strength, clustering coefficient and small-worldness (Liu et al. 2008). It is noted that conventional methods can not reveal time-varying characteristics of brain networks. Sizemore and Bassett put forward the dynamic functional network methods to study the regions of time-varying connectivity (Sizemore and Bassett 2018). Based on a time-sliding windows method, we originally proposed a new index (topology entropy) to find the abnormal cortices of schizophrenia brains (Chen 2019).

Controllability of a dynamical system in control theory characterizes the cost of system’s initial state to final one in finite time (Xiang et al. 2019). As a control index, the controllability investigates the quantified cost of state transitions based on network configurations. In the linear time-invariant systems, the controllability has been established as many criteria, such as the Popov–Belevitch–Hautus (PBH) rank condition, Kalman rank conditions and graphic characteristics (Lin 1974; Kalman 1963). When the system is modeled as a network, the driver nodes by maximum matching are introduced to compute control cost based on the PBH rank condition (Liu et al. 2011; Ruths and Ruths 2014). Furthermore, the controllability concept has been applied in diverse networks in neuroscience, such as the mesoscopic neural circuits of elegans (Yan et al. 2017), macroscopical regional DTI network (Gu et al. 2015; Kim et al. 2018), gray structural network (Jamalabadi et al. 2021) and functional network (Yuan et al. 2022). However, reviewing the literature indicates that the average controllability has been focused and there is a lack of understanding the node controllability.

In this paper, a new node controllability (named by CA-score) is proposed to study the edges, regions and subsystems of functional networks in the ADHD brains. In particular, we find atypical CA-score happening in the regions of the ADHD disease (e.g., ROL, ORBsupmed) and the DMN subsystem. Based on previous studies, these cortex play a key role in the ADHD disease. Compared to existing network indices, the CA-score is more effective to find abnormal cortices of the ADHD brains and would be extended to investigate other psychiatric disorders.

The following sections of the present article are organized as follows: In “Materials and methods” section, the functional network and controllability index are defined and existing indices are given. Then, the results are presented in “Results” section, and the discussion of controllability index is listed in “Discussion” section. Finally, the main achievements and conclusions are summarized in “Conclusion” section.

Materials and methods

Demographic characteristics, imaging acquisition and data preprocessing

The dataset was obtained from the Institute of Mental Health and National Key Laboratory of Cognitive Neuroscience and Learning at Peking University. Further details about the dataset acquisition can be found in Table 1. The data are presented as the range of minimum-maximum (mean SD), and the P-value is calculated using a two-sample two-tailed T-test. All research was approved by the Research Ethics Review Board at the Institute of Mental Health, Peking University, and informed consent was obtained from all participants. Supplementary A contains materials detailing demographic characteristics, clinical symptoms, imaging acquisition, and data preprocessing.

Table 1.

The acquisition of dataset

Variable	ADHD (Mean ± SD (range))	HCs (Mean ± SD (range))	P value
Sample size	102	143
Gender(male/female)	90/12	84/59
Current age	12.09±2.03(8.33–17.33)	11.44± 1.86(8.08–15.17)
Inattentive	27.91±4.15(10–36)	15.80±3.94(9–26)	8.6404e−57
Hyper/Impulsive	22.43±6.47(9–36)	13.56±3.65(9–23)	2.3188e−28
ADHD Index	50.23±8.33(35–68)	29.36±6.41(18–46)	9.3088e−54

Definition of functional connections

R-fMRI signal $X_s$ denotes the s-th subject and $X_{i,t,s}$ is the time series $X_{i,s}$ at time point t of i-th region in s-th subject. For every pair of time series $X_{i,s}$ and $X_{j,s}$ from the i and j regions, the functional connection coefficient matrix $F_s$ of s-th subject is obtained by the Pearson correlation (Chen 2022). Each edge $F_{i,j,s}$ is calculated in the following form:

$$\begin{array}{c}\hfill F_{i,j,s}={\displaystyle \frac{\sum (X_{i,s}-\overline{X_{i,s}})(X_{j,s}-\overline{X_{j,s}})}{\sqrt{\sum _{t=1}^n{(X_{i,t,s}-\overline{X_{i,s}})}^2}\sqrt{\sum _{i=1}^n{(X_{j,t,s}-\overline{X_{j,s}})}^2}}},\end{array}$$ 1

where $i,j=1,\dots ,N$, $\overline{X_{i,s}}$ is mean value of time series $X_{i,s}$, and N = 116 is the ALL-116 based region number. It is obvious that the edge connections strength is $F_{i,j,s}$. The abbreviation of AAL-116 based regions and their systems are listed in the Supplementary B.

Controllability of the regions and subsystems

According to the control theory, a dynamical system is controllable if, with a suitable choice of inputs, it can be driven from any initial state to any desired final state within finite time (Lin 1974; Kalman 1963). Control theory probes a system’s ability to drive its output towards a desired outcome through the application of suitable input signals to select driver nodes (Liu et al. 2011; Ruths and Ruths 2014). The dynamical system is defined as

$$\begin{array}{c}\hfill Y(t+1)=AY(t)+BU(t),\end{array}$$ 2

where the vector $Y={(y_1,\dots ,y_N)}^T$ stands for the states of nodes, $A=(a_{\mathit{ij}})\in R^{N\times N}$ is a symmetric matrix and denotes the coupling matrix of the system, in which $a_{\mathit{ij}}$ depicts the link from node j to node i. The vector $U={(u_1,\dots ,u_m)}^T$ contains m controllers and B is the $N\times m$ controller configuration matrix.

Based on the Popov–Belevitch–Hautus (PBH) rank condition, the system is fully controllable if and only if $rank(c{I}_{N\times N}-A,B)=N$ is satisfied for any complex number c, where $I_{N\times N}$ is an identity matrix of dimension N. The Gramian matrix of the system is mathematically expressed as follows:

$$\begin{array}{c}\hfill W=\sum _{\tau =0}^{\infty }{A}^{\tau }B{B}^T{A}^{\tau }.\end{array}$$ 3

Let A be a correlation matrix, each element $a_{\mathit{ij}}$ of s-subject is $F_{i,j,s}$, and $A^{\tau }$ refers to the high order adjacent matrix, and B is the input configuration. Correlation matrix means information transmission configuration in the networks of brain regions.

The average controllability of a network equals the average input energy from a set of control nodes and over all possible target states. It is known that the average input energy is proportional to the trace of the controllability Gramian matrix (Gu et al. 2015), which is given by

$$\begin{array}{c}\hfill tr(W)=\sum _{j=1}^N{\lambda }_j.\end{array}$$ 4

A region with a highest average controllability is the most influential over all different target states. The existing controllability provides a system-level approach to analyze the perturbation of ADHD brains. In the sequel, we propose the node controllability.

For testing the control cost of each region contributing to whole controllability of the correlation matrix A, we choose every region as the only drive input site in the controller configuration matrix B by 116 computing enumeration experiments, that is,

$$\begin{array}{c}\hfill B_i={[\underset{i-1}{\underbrace{0,\dots ,0}},1,0,\dots ,0]}^T.\end{array}$$ 5

In particular, the order of configuration matrix B of region i is $N\times 1$, where the i-th element is 1 and other elements are 0. Then, the control cost of a region indicates its information reachability ability to all regions in order to drive the whole network. The “region CA-score” can be mathematically expressed in the following form:

$$\begin{array}{c}\hfill C{A}_{i,s}=tr(\sum _{\tau =0}^{\infty }{A}^{\tau }{B}_i{B}_i^T{A}^{\tau }).\end{array}$$ 6

In theoretical calculations, $\tau$ represents the sum that starts from zero and goes up to infinity. When programming a calculator, it is common to choose t as 1 because if $\tau$ is a large number, the sum of all results will tend to infinity through accumulation. Then, we set the $\tau =1.$ It is from Table 3 that more abnormal regions are identified by the controllability, which shows that the CA-score is more effective than traditional indexes.

Table 3.

Comparison of the regions by different network indexes are shown

	Regions	CA	TE	EC	FCS		Regions	CA	TE	EC	FCS
2	PreCG.R				↓	47	LING.L				↓
9	ORBmid.L			↑		48	LING.R				↓
17	ROL.L	↓**	↓**	↓**		49	SOG.L				↓
18	ROL.R	↓**	↓**	↓**		57	PoCG.L				↓**
24	SFGmed.R		↓			63	SMG.L	↓
25	ORBsupmed.L	↓	↓			64	SMG.R	↓	↓
26	ORBsupmed.R	↓**	↓**			65	ANG.L	↓	↓
29	INS.L	↓	↓**			66	ANG.R	↓**	↓**
30	INS.R		↓			67	PCUN.L		↓
35	PCG.L	↓**	↓**			68	PCUN.R	↓**	↓**	↓
36	PCG.R	↓**	↓**			74	PUT.R		↓
43	CAL.L				↓	79	HES.L	↓
44	CAL.R				↓	81	STG.L	↓	↓
45	CUN.L				↓	105	CERC9.L				↑

Furthermore, the modular CA-score is given by

$$\begin{array}{c}\hfill C{A}_{M,s}=\sum _{i\in M}C{A}_{i,s},\end{array}$$ 7

which is the sum of controllability values of regions included in the specific subsystems. The M stands for three types of subsystems, such as the top or bottom hemisphere (Katsuki and Constantinidis 2014; Long and Kuhl 2018), the left or right hemisphere (Wu et al. 2022) and the anatomical parcellation subsystems.

The schematic diagram of our method is depicted in Fig. 1. In our quest to uncover the pathophysiological origins of ADHD, a pivotal question involves identifying the essential target nucleus using control theory methods applied to fMRI data. Through 116 computational enumeration experiments where a region is chosen as the sole driving input site, the controllability index reveals the node characteristics. This index reflects regional centrality and the underlying information reachability, shedding light on how the activities of other regions are influenced in the process.

Fig. 1.

A schematic diagram of the controllability index for the regions and subsystems for a given ADHD network

Existing measures of brain functional networks

In this subsection, three types of indices of brain functional networks are shown. The topology entropy (TE) index is

$$\begin{array}{c}\hfill T{E}_{i,s}={log}_{10}{\lambda }_{i,s},\end{array}$$ 8

where ${\lambda }_{i,s}$ is the largest eigenvalue of a state-transition matrix based on a dynamic functional network (Chen 2019). For reducing random factors of a binarization process, the original connectivity matrix is used for some primary network indices, such as FCs strength ($FC{S}_{i,s}$) and eigenvector centrality ($E{C}_{i,s}$).

The FCs strength is given by

$$\begin{array}{c}\hfill FC{S}_{i,s}=\sum _{j=1}^{116}{F}_{i,j,s}.\end{array}$$ 9

The eigenvector centrality(Bullmore and Sporns 2009) $E{C}_{i,s}$ of node i is defined as the i-th entry in the normalized eigenvector belonging to the largest eigenvalue of $F_s$. It is defined as follows:

$$\begin{array}{c}\hfill E{C}_{i,s}={\displaystyle \frac{1}{\lambda }}\sum _{t\in n(i)}{e}_t,\end{array}$$ 10

where n(i) is the neighbors of region i. With a small rearrangement, $E{C}_{i,s}$ can be rewritten in a vector notation as the eigenvector equation $F_{s}e=\lambda e$ with proportionality factor $\lambda$.

Results

Regions with different CA-score between ADHD patients and HCs

Nonlocal brain injury during critical periods is the principal cause of developmental disorders. This subsection uses the CA-score in Eq. (6) to investigate the abnormalities of overall region attributes. The obtained results show there are significantly atypical CA-score in the ROL, ORBsupmed, INS.L, PCG, SMG, ANG, PCUN, HES.L, and STG.L of ADHD patients. These regions may play a key role in the pathophysiology of ADHD disease because previous studies have reported similar results by other network indices (Friedman and Rapoport 2015; Zepf et al. 2019). To show the effectiveness of our controllability index, comparisons of existing results from the structure, activity and connectivity indexes at AAL-based and standard voxel levels are summarized in Table 2. The abnormal regions demonstrated by reviewing existing research findings are provided in the Supplementary C.

Table 2.

The abnormal regions are identified by different indices

Regions	Structure	FCs	activity in rest-stating	network indexes
ROL		FCs ↑ (Sörös et al. 2019)	activity ↑ (Sörös et al. 2019)	DC ↓ (Griffiths et al. 2016; Saad et al. 2017) BC ↓ (Griffiths et al. 2016)
ORBsupmed	CT ↓ (Fernández-Jén et al. 2014) GMV ↓ (Klein et al. 2021)	FCs ↑ (Sörös et al. 2019)	activity ↑ (Sörös et al. 2019)	DC ↓ (Griffiths et al. 2016; Saad et al. 2017 BC↓ (Griffiths et al. 2016)
SFG	CT ↓ (Makris et al. 2007) GMV ↓ (Friedman and Rapoport 2015)	abnormal FCs (Cao et al. 2009)	ALFF ↑ (Yang et al. 2011) activity ↓ (McCarthy et al. 2014)
INS	CT ↓ (Baribeau et al. 2019) GMV ↓ (Baribeau et al. 2019)	abnormal FCs (Tian et al. 2006; Zhao et al. 2017) FCs ↑ (Sripada et al. 2014) FCs ↓ (Zepf et al. 2019)	activity ↑ (Sörös et al. 2019)
PCG	GMV ↓ (Nakao et al. 2011)	FCs ↓ (Castellanos et al. 2008; Sun et al. 2012) FCs ↑ (Sripada et al. 2014)	activity ↑ (Sörös et al. 2019)	DC ↓ (Griffiths et al. 2016) BC↓ (Griffiths et al. 2016)
SMG	CT ↓ (Mclaughlin et al. 2014)	FCs ↑ (Sripada et al. 2014)	ALFF ↑ (Yang et al. 2011)	DC ↓ (Xia et al. 2014; Park and Park 2016)
ANG	GMV ↓ (Chaim et al. 2014)	FCs ↑ (Barber et al. 2015)	ReHo ↓ (Yu et al. 2016)
PCUN	CT ↓ (Mclaughlin et al. 2014) GMV ↓ (Carmona et al. 2005; Yang et al. 2008)	abnormal FCs (Cao et al. 2009; Zhao et al. 2017) FCs ↑ (Barber et al. 2015) FCs ↓ (Castellanos et al. 2008)	ReHo ↓ (Uddin et al. 2008; Yu et al. 2016)
STG		abnormal FCs (Cao et al. 2009) FCs ↑ (Barber et al. 2015; Gao et al. 2019)	fALFF ↑ (Shang et al. 2016)	DC ↑ (Xia et al. 2014) BC ↑ (Xia et al. 2014)

The ↑ denotes the higher mean values of the corresponding indices in the ADHD brains, and the ↓ is the opposite. The GM volume (GMV) and cortical thickness (CT) belong to the structural indices. The activity, ALFF and ReHo char time/frequency-domain features of fMRI time series. The degree centrality (DC) and betweenness centrality (BC) are network indices

Here the T-test of controllability experiments are executed to seek out the significant regions in the dataset. The pairwise T-test of node CA-score between ADHD patients and matched HCs for all 116 brain regions are performed. The significant regions (passing the FDR correction ($q<$0.05)) are: ROL.L** (p = 3.6705e−05), ROL.R** (p = 0.00010), ORBsupmed.L (p = 0.00071), ORBsupmed.R** (p = 3.5357e−05), INS.L (p = 0.00066), PCG.L** (p = 9.0164e−05), PCG.R** (p = 0.00019), SMG.L (p = 0.00183), SMG.R (p = 0.00560), ANG.L (p = 0.00141), ANG.R** (p = 0.00014), PCUN.R** (p = 0.00033), HES.L (p = 0.00188) and STG.L (p = 0.00485). The double asterisk ** means that the regions pass the Bonferroni correction for a probability p = 0.05. Compared to the heathy controls, the mean CA-score of all above regions in the ADHD patients are lower. The whole brain significant CA-score topography is shown in Fig. 2.

Fig. 2.

Abnormal regions are displayed by different colors in the brain topology using the significant p-values of the CA-score between HCs and ADHD patients. Comparisons of the CA-score between HCs and ADHD patients are shown in the bottom part

It is noted that the regions ROL and ORBsupmed have the most significant p-values in this paper. Previous studies have reported that the ROL and ORB are abnormal in the ADHD brains (Griffiths et al. 2016; Saad et al. 2017; Sörös et al. 2019; Itami and Uno 2002; Fernández-Jén et al. 2014; Klein et al. 2021). The ROL is related with language and emotion functions, which belongs to the frontal lobe and locates in the intersection of frontal lobe, parietal lobe, temporal lobe and subcortical cortex. The ORBsupmed is an emotional region which interfaces between the emotional response and control complex motivational behaviors (e.g., regret, happy, angry, embarrassment and sadness).

Edges with different FCs between ADHD patients and HCs

Apart from above studies on the region CA-score, this subsection uses the edge connections strength $F_{(i,j,s)}$ in Eq. (1) to investigate abnormal edges between ADHD patients and HCs. Calculations show that there are 12 different edges out of 4005 ones passing the Bonferroni correction (p = 0.05), which are presented in the form: Region — Region (p-value). They are CUN.L—SFGmed.L (p = 2.9039e−06), CUN.L—SFGmed.R (p = 3.632e−06), CUN.L—PoCG.L (p = 2.8777e−06), PCUN.R—ROL.L (p = 3.1606e−06), PCUN.R—PoCG.L (p = 6.0196e−09), PCUN.R—PoCG.R (p = 1.7555e−06), PUT.R—PoCG.L (p = 1.0577e−06), HES.L—PCG.R (p = 1.0992e−06), CER6.L—PoCG.L (p = 1.261e−06), CER6.R—PoCG.L (p = 1.6079e−06), CER10.L—CUN.R (p = 3.5516e−06) and VER6—CERC2.L (p = 1.3582e−06). The abnormal edges are distributed in these regions-CUN, SFGmed, PoCG and PCG. The regional pairwise mean matrix ($116\times 116$) and significant edges between HCs and ADHD patients are shown in Fig. 3.

Fig. 3.

The abnormal edges calculated by the FCs are shown in the brain topology whose values passing the Bonferroni correction (p = 0.05) under T-test. The weight values of abnormal edges are set as $-lo{g}_{10}$ (p value)

Subsystems with different CA-score between ADHD patients and HCs

This subsection calculates the modular CA-score in Eq. (7) including the top or bottom hemisphere, the left or right hemisphere and the anatomical parcellation subsystems. The default mode network (DMN) and parietal lobe have distinct CA-score in the ADHD brains. It is found that the DMN dysfunction has been generally identified in the literature (Castellanos et al. 2008; Fair et al. 2010; Sutcubasi et al. 2020; Agoalikum et al. 2021). The independent pairwise T-test of modular CA-score of each subsystem are carried out between ADHD patients and HCs. If the p value $<0.05$, it is regarded as significant. The results are as follows: left hemisphere (p = 0.2125), right hemisphere (p = 0.2565), top hemisphere (p = 0.0256), bottom hemisphere (p = 0.1913), DMN (p = 0.0094), prefrontal lobe (p = 0.0582), other parts of frontal lobe (p = 0.0678), parietal lobe (p = 7.8348e−04), temporal lobe (p = 0.2136), occipital lobe (p = 0.9760), corpus striatum (p = 0.4896), cerebellum (p = 0.2793) and vermis (p = 0.1213).

In many neuropsychiatric and neurodegenerative diseases, the DMN dysfunction has been identified using the atypical activation of regions during the resting state and various tasks (Broyd et al. 2009; Hart et al. 2013; Salavert et al. 2018). The critical regions of the DMN (e.g., SFGdor, ACG, PCG and PCUN) display the abnormal FCs in the ADHD patients (Castellanos et al. 2008; Fair et al. 2010; Sripada et al. 2014; Barber et al. 2015). Based on the dynamic functional connections, the connectivity differences of higher-order or multi-layer networks were found in the DMN subsystem (Agoalikum et al. 2021; Yin et al. 2022).

Comparisons between CA-score and existing network indices

This subsection compares our proposed CA-score index with node centrality indices including eigenvector centrality (EC), topology entropy (TE) and FCs strength (FCS). The results show that our controllability index is more effective to identify more abnormal regions than the FCS and EC in the ADHD brains. Table 3 summarizes the corresponding results among CA, TE, EC and FCS. It is noted that the controllability index captures the eigenvalues of the Gramian matrix in Eq. (3), while the topology entropy index surveys the eigenvalues of the state transition matrix and the eigenvector centrality studies the eigenvalues of the adjacency matrix. Finally, the FCs strength accumulates the edges of the adjacency matrix.

It is from Table 3 that more abnormal regions are identified by the controllability and topology entropy, which shows that the CA-score is more effective. Using the EC index, there are only four prominent regions—ORBmid.L(p = 0.00154), ROL.L**(p = 2.6151e−04), ROL.R**(p = 1.0982e−04) and PCUN.R (p = 7.3393e−04). The ** means the regions pass the Bonferroni correction (p = 0.05). While there are more abnormal regions identified by the CE-score and TE indices, which are ORBsupmed.L, ORBsupmed.R, INS.L, PCG, SMG.R and ANG.L.

Compare to the CA, TE and EC indices, there are different abnormal regions using the FCS index. These different regions are PreCG.R (p = 9.6515e−04), CAL.L (p = 4.9730e−04), CAL.R (p = 0.0015), CUN.L (p = 0.0023), LING.L (p = 9.6545e−04), LING.R (p = 0.0029), SOG.L (p = 0.0023), PoCG.L**(p = 1.8428e−04) and CERC9.L(p = 0.0027). In this manner, combining the eigenvalue indices (CA, TE and EC) and the functional connections (FCs) would identify more abnormal regions in the ADHD case.

Discussion

Controlling a system refers to drive the evolution behaviors from the original state to the goal (Liu et al. 2011). The CA-score index investigates the region importance of information processing (including integration and specialization) in the sense of network topology. For example, the small-world properties assure the brain generates and integrates information with a high efficiency. A possible explanation of this controllability index is shown in Fig. 4.

Fig. 4.

A possible explanation of the node controllability. When a region is selected as a drive node, it causes the transition of the functional network and further changes the network topology

From the existing fMRI data of ADHD patients, there lacks the classified multimodal data and the developmental longitudinal data. In this way, the functional networks are only founded at a macroscopical region level. Furture works regarding the controllability index of brain networks at microcosmic and mesoscopic levels are underway. It is noted that this manuscript only counts the eigenvalues of the Gramian matrix and other indices in the control theory should be introduced to explore physiological markers.

Network indices include node centrality indices (e.g., degree centrality, betweenness centrality, closeness centrality and eigenvector centrality), modularity (e.g., motif and community) and structural properties (e.g., small-world, scale-free and heterogeneity) (Bassett and Sporns 2017; Chen 2019). The results obtained in this paper show that combining eigenvalue indices and functional connections is a better way to find more abnormal regions for the ADHD case. On the other hand, other eigenvalue indices should be introduced to explore functional networks of the ADHD brains, such as network coherence and mean first-passage time defined by the sum of reciprocals of nonzero Laplacian eigenvalues (Sun et al. 2020; Chen et al. 2023), the number of spanning trees by the product of all nonzero Laplacian eigenvalues and synchronizability by the proportion of the maximum Laplacian eigenvalue to the smallest nonzero one (Zhang et al. 2022).

Conclusions

To explore the pathophysiological origins of ADHD brains, a crucial question is to propose network indices to identify abnormal regions from a network perspective. In this paper, the concept of controllability has been applied to functional networks constructed using fMRI data from ADHD patients. Compared to average controllability, a new node controllability index is introduced to examine the unique regions and subsystems of ADHD brains. The obtained results demonstrate that this index is an effective biomarker for excavating ADHD disease. It should be noted that while the ADHD case is studied, whether our results from this paper hold for other psychiatric disorders such as schizophrenia, Autism, Alzheimer’s disease, requires further investigation.

Supplementary Information

Below is the link to the electronic supplementary material.

Author Contributions

B.C., W.S. and C.Y. contributed to the conception and design of the study. B.C. and W.S. performed the numerical results. B.C., W.S. and C.Y. wrote the manuscript.

Funding

This work was supported by Guangxi Key Laboratory of Trusted Software (No. KX202309).

Availability of data and materials

The datasets of the current study are available at http://fcon_1000.projects.nitrc.org/indi/adhd200/index.html.

Code availability

The MATLAB code here is available from the corresponding author, Dr. Weigang Sun, upon reasonable request.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

All research was approved by the Research Ethics Review Board, the Institute of Mental Health, Peking University.