Random matrix theory for analyzing the brain functional network in attention deficit hyperactivity disorder

Attention deficit hyperactivity disorder (ADHD) is the most common childhood neuropsychiatric disorder and affects approximately $6–7\%$ of children worldwide. Here, we investigate the statistical properties of undirected and directed brain functional networks in ADHD patients based on random matrix theory (RMT), in which the undirected functional connectivity is constructed based on correlation coefficient and the directed functional connectivity is measured based on cross-correlation coefficient and mutual information. We first analyze the functional connectivity and the eigenvalues of the brain functional network. We find that ADHD patients have increased undirected functional connectivity, reflecting a higher degree of linear dependence between regions, and increased directed functional connectivity, indicating stronger causality and more transmission of information among brain regions. More importantly, we explore the randomness of the undirected and directed functional networks using RMT. We find that for ADHD patients, the undirected functional network is more orderly than that for normal subjects, which indicates an abnormal increase in undirected functional connectivity. In addition, we find that the directed functional networks are more random, which reveals greater disorder in causality and more chaotic information flow among brain regions in ADHD patients. Our results not only further confirm the efficacy of RMT in characterizing the intrinsic properties of brain functional networks but also provide insights into the possibilities RMT offers for improving clinical diagnoses and treatment evaluations for ADHD patients.

Figure 1

Flowchart of the methods pipeline: overview of the data processing and analysis pipeline. Resting-state fMRI data were acquired from 24 individual subjects in an ADHD group and a group of 24 healthy controls. Time courses were extracted from the 444 cerebral regions comprising the Multiscale Functional Brain Parcellations template, and the $444\times 444$ connectivity matrices were built using CF, CC, and MI. The resulting connectivity matrices were then investigated to identify between-group differences by analyzing the functional connectivity distribution as well as the global and local eigenvalue properties.

Figure 2

An analysis of functional connectivity, in which the undirected functional connectivity is measured by CF and the directed functional connectivity is measured by CC and MI. The pdfs of the (a) CF, (b) CC, and (c) MI were calculated individually for the control group and the ADHD group.

Figure 3

Global eigenvalue property analysis: the eigenvalue distribution. The eigenvalues were first calculated from the (a) CF, (b) CC, and (c) MI functional networks for the control group and the ADHD group, and the pdfs of the eigenvalues were then calculated.

Figure 4

Global eigenvalue property analysis: the largest eigenvalue. The largest eigenvalues of the brain functional network were first calculated for each subject, and the box plots in each group were then plotted for the (a) CF, (b) CC, and (c) MI functional networks, individually. In the box plot, the bottom and top of the box are the first and third quartiles, the band inside the box is the median, the ends of the whiskers are the minimum and maximum values, and the $\square$ is the mean value.

Figure 5

Local eigenvalue property analysis: the short-range correlations among eigenvalues measured by the NNSD. The NNSD were first calculated for each subject, and the probability distributions were then obtained for each group. The probability distributions of the NNSD for the two groups are plotted separately for the (a) CF, (b) CC, and (c) MI functional networks, where the blue lines represent the empirical values predicted by GOE and Poisson statistics.

Figure 6

Local eigenvalue property analysis: long-range correlations among eigenvalues measured by the spectral rigidity and number variance. The spectral rigidity (upper panel) and number variance (lower panel) were first calculated for each subject and then averaged over each group. The mean spectral rigidity and number variance for the two groups are plotted individually for the (a) and (d) CF, (b) and (e) CC, and (c) and (f) MI functional networks. The blue lines represent the empirical values predicted by GOE and Poisson statistics.

Figure 7

The pdfs of the (a) CF, (b) CC, and (c) MI for the control group and the ADHD group.

Figure 8

The pdfs of the eigenvalues for the (a) CF, (b) CC, and (c) MI functional networks in the control group and the ADHD group.

Figure 9

The box plots of largest eigenvalues for the (a) CF, (b) CC, and (c) MI functional networks in the control group and the ADHD group.

Figure 10

The probability distributions of the NNSD for the (a) CF, (b) CC, and (c) MI functional networks in the control group and the ADHD group, where the blue lines represent the empirical values predicted by GOE and Poisson statistics.

Figure 11

The spectral rigidity (upper panel) and number variance (lower panel) for the (a) and (d) CF, (b) and (e) CC, and (c) and (f) MI functional networks in the control group and the ADHD group. The blue lines represent the empirical values predicted by GOE and Poisson statistics.