# Network Entropy for the Sequence Analysis of Functional Connectivity Graphs of the Brain

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sequence Analysis of FCGBs

**F**, whose element

**F**(i, j), shows the measured edge between electrodes (nodes) i and j. There are large numbers of edge measurement methods divided into time domain methods such as cross-correlation function [37] and synchronization likelihood [38] and frequency or phase domain methods such as coherence, phase synchronization [39] and phase lag index [40]. Even though previous studies have shown that the correlation-based and coherence-based methods suffer from the primary and secondary leakage which may lead to false positives due to “ghost interactions” [26], the solution has already been proposed in the case of amplitude correlations [27]. Sliding-window correlation estimates have also been used to assess the uncertainty underlying dynamic connectivity by providing confidence bands [41]. Therefore, a sliding-window cross-correlation function was adopted in this study to reflect the dynamic adjacency relations among different node pairs.

_{i}and s

_{j}can be calculated by the following formulas:

_{ij}corresponds to the element of the functional connectivity matrix

**F**, which presents in i-th row and j-th column. To exclude self-connections of nodes, the elements on the main diagonal of

**F**were set to zero. The off-diagonal value of

**F**varies between 0, meaning no correlation and 1 meaning complete correlation and interdependency between pairs of EEG electrodes. Here, we employed the sliding window with the translation parameter to be sampling length (SL) to calculate the time-resolved cross-correlation matrices and form FCGB sequences at the second time scale. The window moved along the data points step by step and the functional connectivity matrices constituted a tensor

**F**, whose element

**F**(

**F**, k) shows the constructed FCGB in epoch k. The following step is in connection with the assessment and quantification of the

**F**properties.

_{l}, p

_{2}, ..., p

_{n}. Thus, we have

_{l}, x

_{2}, ..., x

_{n}denote the possible interactive activities. Their corresponding probabilities also vary between 0 (no occurrence) and 1 (inevitable occurrence). These probabilities are known for the brain signal recordings, when we combine the adjacency relations in FCGB and the probability set in SE through Equations (1) and (2). It is reasonable to assume that the stronger the correlation between nodes, the higher its potential for functional integration and, consequently, its probability. According to SE, NE can be calculated as follows

_{l}, p

_{2}, ..., p

_{n}} is replaced by the connection sequence distribution in

**F**. First, we create a correlation sequence {γ

_{l}, γ

_{2}, ..., γ

_{n}} from the upper triangular elements of

**F**, as the functional connectivity matrix is symmetric and the portion of

**F**convey all the interactive information. To fulfill the requirement of $\sum _{i=1}^{n}{p}_{i}=1$, the correlation sequence is normalized. Then we have

_{i}is an element in the correlation sequence $\sum _{i=1}^{n}\frac{{\gamma}_{i}}{{\displaystyle \sum _{j=1}^{n}{\gamma}_{i}}}=1$.

_{k}} from two dimensions to carry out the event-related analysis of the changes in oscillatory activity in the final step of the NE algorithm. The wavelet-based transform is an effective tool for localizing the time-frequency components of a non-stationary signal because of its attractive properties, such as sharper time resolution in high-frequency components and multi-rate filtering (differentiating the signals that have various frequencies) [44]. In this paper, the “dmey” mother wavelet was selected for its efficiency in local representation. The sequence features {NE

_{k}} were decomposed between 0 Hz and 0.5 Hz with the scales of 1024. The frequency band of interest is 0–0.1 Hz.

#### 2.2. Verification Experiment

**F**exceeded a threshold value T, they were set to 1 (otherwise 0). The threshold T was determined adaptively by choosing its maximal value such that the resulting network is connected (without an isolated node). Starting from T = 1, we gradually decreased the threshold (decrease step size = binarization precision 0.01) and we calculated, at each step, the second smallest eigenvalue λ

_{min}of the corresponding Laplacian matrix

**L**. In a connected graph, the decreased step size had a relationship with the variances of the measured edges in

**F**. If the decreased step size was larger than the variances’ order of magnitude (i.e., insufficient binarization precision), the graph probably would not be a connected graph, no matter how threshold value T changed. In this paper, the decreased step size was selected by trial and error. We began with the order of magnitude of the edges’ standard deviation (0.01). If the binarization precision was insufficient, it would be decreased by an order of magnitude. The elements of the Laplacian matrix were obtained by the following equation

_{i}represents the degree of node i. F

_{ij}denotes elements of the functional connectivity matrix

**F**. δ

_{ij}is the function of Kronecker delta.

_{min}became positive.

## 3. Results

## 4. Discussion

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Preprocessing of original electroencephalpgram (EEG) signal. (

**a**) Original EEG signal; (

**b**) Corrected EEG signal.

**Figure 3.**Event-related functional connectivity graphs of the brain (FCGB) sequences. (

**a**) FCGB sequence corresponding to the response of a short sigh; (

**b**) FCGB sequence corresponding to the response of a long sigh.

**Figure 4.**Comparison of event-related analysis results of a short sigh response. (

**a**) Network entropy (NE) event-related analysis in the single trial; (

**b**) Event-related analysis based on clustering coefficient in the single trial; (

**c**) Event-related analysis based on characteristic path length in the single trial; (

**d**) Event-related analysis based on global efficiency in the single trial; (

**e**) Event-related analysis based on vulnerability in the single trial.

**Figure 5.**Comparison of event-related analysis results of a long sigh response. (

**a**) NE event-related analysis in the single trial; (

**b**) Event-related analysis based on clustering coefficient in the single trial; (

**c**) Event-related analysis based on characteristic path length in the single trial; (

**d**) Event-related analysis based on global efficiency in the single trial; (

**e**) Event-related analysis based on vulnerability in the single trial.

**Figure 6.**Sequence analysis results of NE in the whole duration of driving task. (

**a**) NE versus time curve; (

**b**) NE time-frequency features.

**Figure 7.**Comparison between the extracted time based on NE and the recorded event time in the experiment. (

**a**) Correlation between the extracted event time and the recorded event time; (

**b**) Errors between the extracted event time and the recorded event time; (

**c**) Event time extraction based on the NE time-frequency features.

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**MDPI and ACS Style**

Zhang, C.; Cong, F.; Kujala, T.; Liu, W.; Liu, J.; Parviainen, T.; Ristaniemi, T. Network Entropy for the Sequence Analysis of Functional Connectivity Graphs of the Brain. *Entropy* **2018**, *20*, 311.
https://doi.org/10.3390/e20050311

**AMA Style**

Zhang C, Cong F, Kujala T, Liu W, Liu J, Parviainen T, Ristaniemi T. Network Entropy for the Sequence Analysis of Functional Connectivity Graphs of the Brain. *Entropy*. 2018; 20(5):311.
https://doi.org/10.3390/e20050311

**Chicago/Turabian Style**

Zhang, Chi, Fengyu Cong, Tuomo Kujala, Wenya Liu, Jia Liu, Tiina Parviainen, and Tapani Ristaniemi. 2018. "Network Entropy for the Sequence Analysis of Functional Connectivity Graphs of the Brain" *Entropy* 20, no. 5: 311.
https://doi.org/10.3390/e20050311