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. 2020 Feb 9;10(2):92.
doi: 10.3390/brainsci10020092.

The Maximum Eigenvalue of the Brain Functional Network Adjacency Matrix: Meaning and Application in Mental Fatigue Evaluation

Affiliations

The Maximum Eigenvalue of the Brain Functional Network Adjacency Matrix: Meaning and Application in Mental Fatigue Evaluation

Gang Li et al. Brain Sci. .

Abstract

The maximum eigenvalue of the adjacency matrix (AM) has been supposed to contain rich information about the corresponding network. An experimental study focused on revealing the meaning and application of the maximum eigenvalue is missing. To this end, AM was constructed using mutual information (MI) to determine the functional connectivity with electroencephalogram (EEG) data recorded with a mental fatigue model, and then was converted into both binary and weighted brain functional network (BFN) and corresponding random networks (RNs). Both maximum eigenvalue and corresponding network characters in BFNs and RNs were considered to explore the changes during the formation of mental fatigue. The results indicated that large maximum eigenvalue means more edges in the corresponding network, along with a high degree and a short characteristic path length both in weighted and binary BFNs. Interestingly, the maximum eigenvalue of AM was always a little larger than that of the corresponding random matrix (RM), and had an obvious linearity with the sum of the AM elements, indicating that the maximum eigenvalue can be able to distinguish the network structures which have the same mean degree. What is more, the maximum eigenvalue, which increased with the deepening of mental fatigue, can become a good indicator for mental fatigue estimation.

Keywords: adjacency matrix; brain functional network; electroencephalogram (EEG); maximum eigenvalue; mental fatigue; network characters.

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Conflict of interest statement

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Figures

Figure 1
Figure 1
Electroencephalogram data acquisition (EEG DAQ) procedures.
Figure 2
Figure 2
Mean maximum eigenvalue (Max-Eigen) of the adjacency matrix (AMs) for all EEG rhythms among T0, T1, T2, T3, and T4 during the formation of mental fatigue, with no threshold (T = 0). The bar in the figures means the standard deviation across the subjects. The p-values of the one-way ANOVA are also given in the figures. (A) Resting state. (B) Task state.
Figure 3
Figure 3
Relationship between the maximum eigenvalue and the sum of the AMs elements for alpha1 rhythm at task state. (A) Results of the AMs with no threshold (T = 0). (B) Results of the weighted AMs with the thresholds of T = 0.15, 0.20, 0.25, 0.30, 0.35. (C) Results of the binary AMs with the threshold of T = 0.15, 0.20, 0.25, 0.30, 0.35.
Figure 4
Figure 4
Differences of the maximum eigenvalue (Max-Eigen) between AM and random matrix (RM). (A) Results of the maximum eigenvalue of AM and RM with no threshold (T = 0) for alpha1 rhythm at task state. The results of the RM were acquired from averaging two-hundred RMs. The bars indicate the standard error of mean. (B) Networks converted by the AMs using a threshold of T = 0.35 for T0, T1, T2, T3, and T4. (C) Networks converted by the RMs using a threshold of T = 0.35, corresponding to the networks of the AMs. In (B) and (C), if the mutual information (MI) value between 2 electrodes is above the threshold, an edge is drawn between the 2 vertices, otherwise not. The boldness of the line indicated the size of the MI value. Networks were drawn by Pajek.
Figure 5
Figure 5
Results of the maximum eigenvalue (Max-Eigen) obtained from AM and RM for alpha1 rhythm at task state. The results of the RM were acquired from two-hundred RMs. Error bars correspond to standard error of the mean. (A) Results of the weighted matrices. (B) Results of the binary matrices.
Figure 6
Figure 6
Results of the degree obtained from AM and RM for alpha1 rhythm at task state. The results of the RM were acquired from two-hundred RMs. Error bars correspond to standard error of the mean. (A) Results of the weighted networks. (B) Results of the binary networks.
Figure 7
Figure 7
Results of the characteristic path length obtained from AM and RM for alpha1 rhythm at task state. The results of the RM were acquired from two-hundred RMs. Error bars correspond to standard error of the mean. (A) Results of the weighted networks. (B) Results of the binary networks.

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