Magnetoencephalogram Network Analysis of Depression Based on Polynomial Kernel Granger Causality

1. Introduction

Depression is a kind of emotional disorder, which is mainly characterized by persistent depression and loss of interest, accompanied by changes in cognitive function and biology, failure to regulate negative emotion is one of the main factors of depression[1,2,3].Nowadays, depression has become one of the most common neuropsychiatric disorders characterized by anxiety, pessimism and suicidal thoughts, seriously affecting people’s quality of life and social development[4,5].

The human brain is an extremely complex system, and its precise structure and efficient functional connection mode provide the basis for brain information differentiation and integration[6]. Complex network based on graph theory can effectively describe and evaluate the function and structure of the brain, and it is widely used in the study of the pathogenesis of mental illness. The researchers found that the construction of brain functional network and the analysis of topological properties by using the data obtained by EEG, MEG and fMRI can provide brain network imaging markers for the diagnosis and treatment of mental illness[7,8,9].Depression is a common mental disease, which may be closely related to the abnormality of brain function and structure[10]. Previous studies have shown that there are abnormalities in the global topological properties and regional connections of the brain complex network in patients with depression. the brain complex network may provide us with quantitative indicators for the diagnosis and treatment of depression and provide an objective basis for the physiological and pathological mechanism of depression.

Granger causality is a common tool for analyzing linear time series. After Granger causality was proposed in 1969, it has been widely used in economics, physiology, neuroscience and other fields[11,12,13,14]. In the field of neuroscience, the actual physiological signals are often nonlinear[15]. The traditional Granger causality can not accurately analyze the internal relationship between nonlinear signals. Mapping low-dimensional nonlinear data to high-dimensional space to make it linearly separable can effectively solve this problem, and it also causes "dimension disaster" in high-dimensional space computing. Kernel function is a method to calculate the similarity measure in feature space by using the dot product of vectors, which can solve complex calculation problems in high-dimensional space[16]. Therefore, the combination of kernel function and linear Granger causality can effectively analyze the internal relationship of nonlinear time series.

MEG is a kind of physiological signal with nonlinear characteristics, so this paper takes the nonlinear kernel Granger causality value as a measure of the connection edge between different brain nodes, constructs brain networks for patients with depression and healthy people, and analyzes their topological properties. the aim is to help clinical diagnosis and treatment of depression.

2. Methods

2.1. Granger causality basic theory

In order to complete the transformation from linear Granger causality to non-linearity, it is necessary to embed the data into Hilbert space for operation[17]. It is specifically expressed as follows:

Suppose there are two random variables X,Y,${\mathrm{X}}_i={\left({\xi }_i,\dots ,{\xi }_{i+m-1}\right)}^T$,$Y_i={\left({\eta }_i,\dots ,{\eta }_{i+m-1}\right)}^T$, $x_i={\xi }_{i+m}$,${\mathrm{X}}_i$,${\mathrm{Y}}_i$,${\mathrm{x}}_i$ is the realization of X,Y and x respectively.${\mathrm{X}}_i$ is the column vector of X,X is a $m*N$ matrix.$Z_i=\left(X_i^T,Y_i^T\right)$,and ${\mathrm{Z}}_i$ is a $2m*N$ matrix.Suppose $x^{T}x=1$,and the mean of each component of random variables X and Y is 0.For $i=1,2,...,N$,we have:

$${\tilde{x}}_i=\sum _{j=1}^m{A}_j{\xi }_{i+m-j}$$

(1)

$${\mathbf{x}}_i=\sum _{j=1}^m{A}_j{\xi }_{i+m-j}+\sum _{j=1}^m{B}_j{\eta }_{i+m-j}$$

(2)

The values of vectors $\tilde{x}={\left({\tilde{x}}_1,\dots ,\widetilde{x_N}\right)}^T$ and can be estimated by linear regression method.$\mathrm{K}=X^{T}X$ is an $N*N$ matrix,let $\mathrm{H}\subseteq {\Re }^N$ be the range of K.Let the eigenvector of K be $v_1,...,v_n$ and satisfy the orthogonal relation,For this eigenvector, there are:

$$\mathrm{P}=\sum _{i=1}^m{v}_i{v}_i^T$$

(3)

$x^{\prime }$ is the projection of x on H,we have $\tilde{x}=Px$,$y=x-Px$.Similarly,$H^{\prime }$ is the range of $K^{\prime }=Z^{T}Z$ and $x^{\prime }=P^{\prime }x$. Here,$H\subseteq H^{\prime }$,decompose $H^{\prime }$ into $H\oplus H^{\perp }$,$H^{\perp }$is all spaces in space $H^{\prime }$ that are orthogonal vectors in space H. The projection of P on space $H^{\perp }$ is expressed as $P^{\perp }$, and the Granger causality intensity is obtained:

$$\delta =\frac{{\parallel P-y\parallel }^2}{1-{\tilde{x}}^T\tilde{x}}$$

(4)

All the non-zero eigenvectors of the matrix $\tilde{K}$ are extended, and the eigenvectors are expressed as $t_1,...,t_m$, and $H^{\perp }$ is the set of the eigenvectors. The range of values for $\tilde{K}=K^{\prime }-P{K}^{\prime }-K^{\prime }P-P{K}^{\prime }P$,$H^{\perp }$is the range of $\tilde{K}$.R is The Pearson correlation coefficient of y and $t_i$ is $r_i$, then we have:

$$\delta =\sum _{i=1}^m{r}_i^2$$

(5)

Finally, the Granger causality index with filtering property is obtained by FDR correction:

$${\delta }_{\mathrm{F}}=\sum _i{r}_i^2$$

(6)

2.2. Granger causality based on polynomial Kernel

The kernel function of a nonhomogeneous polynomial of order p (p is an integer) is defined as:

$$K_p\left(X,X^{\prime }\right)={\left(1+X^T{X}^{\prime }\right)}^p$$

(7)

In this case, the characteristic function is a unary function, and the input variable is up to order p. The dimension of H space is $m_1=1/B(p+1,m+1)-1$, when p=1, it is a linear regression. The dimension of $H^{\prime }$space is $m_2=1/B(p+1,2m+1)-1$.Here,$H\subseteq H^{\prime }$,$H^{\prime }$ into $H\oplus H^{\perp }$.The calculation formula is $\tilde{K}=K^{\prime }-P{K}^{\prime }-K^{\prime }P-P{K}^{\prime }P$. Construct the kernel Granger causality corrected by FDR to get:

$${\delta }_F^K=\sum _{i^{\prime }}{r}_{i^{\prime }}^2$$

(8)

2.3. Introduction to topology Properties of brain network

• Degree and Degree distribution:
  The degree $D_i$of a node is defined as the number of edges connected to node i in the network, and the degree of the I node can be expressed as:

  $$D_i=\sum _i{C}_{ij}$$

  (9)

  where $C_{ij}$ is the connection state between nodes i and j, when there is a connection between i and j, $C_{ij}$=1, when there is no connection between i and j, $C_{ij}$=0. When the value of i increased, the more connected edges of the node, representing the node is more important. The average degree of all nodes in the network is the average degree of the network, which can directly reflect the connection level between the nodes in the network and measure the complexity of the network.
• Clustering coefficient:
  The clustering coefficient C represents the clustering situation of nodes in the network. Generally speaking, C represents the probability that the neighbors of nodes are also neighbors to each other. Cluster coefficient $C_i$ of node i can be defined as:

  $$C_i=\frac{2e^i}{k_i\left(k_i-1\right)}$$

  (10)

  where $e_i$ represents the number of connections between $k_i$ nodes connected to node i, $k_i(k_i-1)$ represents the maximum number of connected edges between $k_i$ nodes connected to node i. The average clustering coefficient of a network is the average clustering coefficient of all nodes in the network, which can be expressed as:

  $$C=\frac{{\sum }_i^N{C}_i}{N}$$

  (11)
  N is the number of nodes in the network, and the value of $C_i$ is between 0 and 1. When there is a connection between two nodes in the network, $C=1$, and when all nodes in the network do not have a connection, $C=0$. C can reflect the degree of collectivization of the network, which is the tightness of connections between nodes.
• Average path length The calculation formula for the average path length L of the network is:

  $$L=\frac{1}{N(N-1)}\sum _{i\ne j\in N}{d}_{ij}$$

  (12)

  where N is the number of nodes in the network, $d_{ij}$ is the shortest distance between node i and node j, representing the minimum number of connection edges required for connectivity between node i and node j. In small-scale networks, we usually use the Floyd algorithm to calculate $d_{ij}$. The average path length is an important indicator to measure the transmission efficiency of a network. Random networks typically have shorter average lengths, while regular networks typically have longer average path lengths.

3. Experimental results and analysis

3.1. Experimental data

The experimental data used in this experiment comes from the magnetoencephalography center of the Brain Hospital of Nanjing Medical University. The equipment for collecting MEG is the Canadian CTF275 full-head magnetoencephalogram acquisition system, and the sampling frequency is 1200 hz. This data includes a total of 16 participants, including 11 healthy individuals and 5 depression patients. All participants had no bad habits, all physiological indicators were normal, the age was between 20 and 30, and the average age was 25 ± 2. There were no other mental illnesses except for depression. Before the experiment, the experimental subjects were introduced to the equipment to ensure that they had no fear of the device. 240 images were obtained from the International Emotional Image Library (IAPS), including 80 positive, neutral, and negative emotional images each. The selected image parameters remained consistent. In the experiment, three types of images were played consecutively in order, and healthy subjects and depression patients were asked to observe these three types of emotional stimuli. The CTF 275 full-head magnetoencephalography system was used to record the brain magnetic signals of the subjects under different stimuli.

3.2. Experimental process and steps

The data processing software used in this experiment is MATLAB, and the sampling frequency of the data is 1200$hz$. The MEG data are pre-processed by artifact removal, baseline correction, and spm8 filtering. The specific processing was as follows: for each subject’s acquired data, a band-pass filter was used to obtain a signal from 0.1$hz$-200$hz$, remove 50$hz$ of industrial frequency noise, and finally obtain a three-dimensional matrix of 275*161*80. 275 represents the 275 channels of magnetoencephalography acquisition, 161 represents the data sampling points, and 80 represents the number of each picture type. The duration of the stimulus was 600ms, with intervals of 650ms to 800ms, and the 3 types of pictures were randomly displayed on the screen in front of the subjects. In this experiment, we first analyzed the subjects’ $\delta$-wave (0.5-4$hz$), $\theta$-wave (4-8$hz$), $\alpha$-wave (8-13$hz$),$\beta$-wave (13-30$hz$) and full frequency band using nuclear Granger causality, and finally found significant differences in the $\beta$-band brain magnetic signals (13-30$hz$) of the two groups. All the MEG signals in this paper were collected under positive, neutral, and negative emotional picture stimuli given to the subjects. In the study by Bahar Güntekin [18,19] et al, they found that on the response amplitude of different bands in human electroencephalogram a significant increase in the response amplitude in the $\beta$ band when processing facial emotional picture stimuli, which was the most active compared to other bands. In our study, we found that the $\beta$-band response amplitude was significantly higher when subjects processed facial emotional picture stimuli and triggered higher beta oscillations in response to negative emotional stimuli. Based on this, our studies in the following are based on $\beta$-band MEG signals.

The brain can be divided into five brain regions, namely: frontal region (F), parietal region (P), occipital region (O), temporal region (T), and central region (C). As shown in Figure 1, the CTF275 magnetoencephalography system further divides these 5 brain regions into left and right, or left, middle, and right.

When selecting the embedding dimension m of polynomial kernel function, since Glenmorangie distillery causality is based on the theory of regression model, we use the AIC(Akaike information criteria) information criterion when selecting the embedding dimension m. AIC is a method proposed by Akaike[20] et al. to obtain the order of a time series analysis model. While encouraging good data fitting, it is necessary to avoid overfitting as much as possible, so that the model with the minimum AIC function is considered the optimal model. The expression of the AIC criterion used in this experiment is[21]:

$$AIC\left(m\right)=ln\left(\sum \right)+\frac{2p^{2}m}{n}$$

(13)

where, p is the dimension of the variable, ∑ is the Covariance matrix of the prediction error obtained by m-order regression, and n is the length of the time series. In this experiment, we determined the order m of the model using the AIC information criterion, and calculated the model order from 1 to 20. We found that when m=4, both groups of people had smaller AIC values. Therefore, we chose to embed the dimension $m=4$. When determining the value of p, two groups of channels were randomly selected in the occipital and frontal regions of depressed patients and healthy people, and p=1-6 was taken for calculation. It was found that when p=2, the polynomial Glenmorangie distillery causality strength between the two groups of channels reached the peak. In order to avoid over fitting problems due to excessive index selection, we selected p=2 in the subsequent analysis.

After averaging the channels in 14 brain regions, the polynomial kernel Granger causality parameters m=4 and p=2 were fixed, and calculate the Granger causality values of polynomial kernels between two of 14 brain regions in 11 depressed patients and 5 healthy individuals under 3 mood stimuli:positive, neutral, and negative, respectively, to obtain a 14*14 polynomial kernel Granger causality intensity matrix. The polynomial kernel Granger causal index matrix was averaged separately for the 2 populations,then we obtained the average Granger causal weight matrix for the 2-class population under the three emotional stimuli. The independent sample t-test was then conducted, and the results are shown in Table 1. From the table, it can be seen that under negative and neutral stimuli, the p-value is less than 0.05, and there is a significant difference between depressed patients and healthy people, specifically, the mean Granger causality values between 14 brain regions of depressed patients are greater than those of healthy people, and under positive stimuli, there is no significant difference between the two groups of people.

Table 1. Independent sample t-test of Glenmorangie distillery causality mean of polynomial kernel between 14 brain regions of two groups.

	Depression	Normal	t	p
]1*Positive stimulus	0.1155±0.0041	0.1158±0.0041	0.040	0.968
]1*Neutral stimulus	0.1198±0.0041	0.1154±0.0041	20.158	0.000
]1*Negative stimulus	0.1211±0.0041	0.1157±0.0041	21.573	0.000

Table 1. Independent sample t-test of Glenmorangie distillery causality mean of polynomial kernel between 14 brain regions of two groups.

	Depression	Normal	t	p
]1*Positive stimulus	0.1155±0.0041	0.1158±0.0041	0.040	0.968
]1*Neutral stimulus	0.1198±0.0041	0.1154±0.0041	20.158	0.000
]1*Negative stimulus	0.1211±0.0041	0.1157±0.0041	21.573	0.000

Before constructing a brain network, we need to select a suitable threshold value. The selection of threshold T has been a difficult point in brain network research, which can directly affect the statistics and analysis of topological properties in brain networks. There is no universally applicable threshold selection method, and the following three principles will be referred to for threshold selection in this paper [22,23]: (1)The threshold value shall be chosen to ensure as much as possible the complete connectivity of the network, meaning that there are no or very few isolated nodes in the network; (2)The threshold needs to be chosen to ensure as much as possible the small-world nature of the network, so that it has a high global efficiency and a low local efficiency; (3)Thresholds are selected to reflect differences between groups whenever possible. The threshold range was set between 0.11 and 0.13, and a threshold was taken every 0.001. According to the above principles of threshold selection, the two groups of people were analyzed, and the final selected threshold was T=0.115 to generate the adjacency matrix. To further observe the specific differential brain areas of the two populations under negative and neutral stimuli, we drew the adjacency matrix of the difference between healthy and depressed individuals. As shown in Figure 3, where blue squares represent the existence of causal effects between the two brain regions corresponding to the coordinates at that location in depressed individuals, while no connections exist in healthy individuals; yellow squares represent the existence of causal effects between the two brain regions corresponding to the coordinates at that location in healthy individuals, while no connections exist at that location in depressed individuals, and green squares represent the consistency between the two populations in the two brain regions corresponding to the coordinates at that location. i.e., both or neither has a causal effect.

As can be seen from the figure, under negative stimulation, depressed patients had more causal interactions with other brain regions in RF (right frontal), LO (left occipital), and RO (right occipital) brain regions and higher degrees of connectivity than healthy individuals and healthy individuals had higher degrees of connectivity in LP (left parietal) and LC (left central) regions than depressed patients; under neutral stimulation, depressed patients had higher degrees of connectivity in LO (left occipital), RF (right frontal) and RO (right occipital ) and other brain regions in depressed patients, and in LP (left parietal) and ZC (central region) in healthy subjects. This result suggests that under non-positive emotional stimuli, depressed individuals show more causal connections and closer information exchange with other brain regions in the frontal and occipital regions compared to healthy individuals, who show more causal effects in the parietal and central regions. The frontal lobe is closely related to the regulation of human emotions, and depressed patients have significant abnormalities in the frontal lobe compared to healthy individuals when processing negative stimuli[24]. The occipital lobe, as the primary processing area of the brain for visual stimuli, is mainly involved in the processing of facial emotions, and it has been shown that depressed patients have abnormal activation of the occipital lobe in response to negative facial emotional stimuli[25] and that depressed patients have hyperfocused occipital areas on negative stimuli. The parietal region is mainly responsible for the integration of visual and spatial information, and some studies have shown that healthy people have higher neurotransmitter activation in the parietal region than depressed patients, and neurotransmitters are the main substances that transmit information between neurons, and high neurotransmitter activation in the parietal region of healthy people indicates that healthy people have more information exchange and more intense activity in the parietal region with other brain regions[26], which is consistent with the conclusion obtained in this paper that healthy people have more causal role consistent with the conclusion obtained in this paper that healthy individuals have a more causal role in parietal regions. Based on the above basis, we constructed brain networks at a threshold value of T=0.115 for the magnetic brain signals of depressed patients and healthy individuals, respectively, as shown in Figure 4, Figure 5, and Figure 6. From the figure, we can see that the brain network connections are tighter in depressed patients compared with healthy individuals under negative and neutral stimuli. Due to a large number of nodes, we cannot visually observe the topological properties of the network from the network diagram, so we will find the changes in topological properties of the two groups under different emotional stimuli by analyzing the average degree of the network, the average clustering coefficient, and the characteristic path length index in the small-world property.

Figure 7 and Figure 8 show the error bar graphs of the mean degree and mean clustering coefficients of the two populations under the 3 emotional stimuli: negative, neutral, and positive, respectively. From Figure 7, it can be seen that the mean degree of depressed patients was higher than that of healthy individuals under negative and neutral stimuli, and both passed the independent samples t-test (p=0.008), while under positive stimuli, the two groups did not show significant differences. This result suggests that the connections between brain network nodes are stronger in depressed patients under non-positive emotional picture stimuli and that the causal effects between brain regions are significantly enhanced in depressed patients when processing negative stimuli. From Figure 8, we can find that the mean clustering coefficients were higher in healthy individuals under all three emotional stimuli, and all passed the independent samples t-test with p=0.034 under negative stimuli, p=0.028 under neutral stimuli, and p=0.011 under positive stimuli, indicating a higher degree of grouping in the brain network of healthy individuals.

From Figure 9, it can be seen that depressed patients had the greatest mean under negative stimuli and the least under positive stimuli and passed the independent sample t-test (p=0.003), while healthy individuals did not differ significantly under the 3 emotional stimuli. As can be seen in Figure 10, depressed patients had the largest mean clustering coefficient under negative stimuli and the smallest under positive stimuli, and passed the independent samples t-test (p=0.015), while the mean clustering coefficient did not change significantly in healthy individuals under different emotional stimuli. This result suggests that depressed patients differed more in different brain networks under different emotional stimuli, which may be due to the fact that depressed patients are more prone to emotions.

A small-world network is a network with a high clustering coefficient and short characteristic path length, and the small-world property can be characterized by calculating the small-world coefficient. The characteristic path length of a network is the average of the shortest path lengths that all nodes in the network need to travel to connect. In this paper, we observed the changes in the brain networks of depressed and healthy people by calculating the deviation of the network’s characteristic path length under positive, neutral, and negative stimuli compared to the characteristic path length of a random small-world network model. Fifty small-world networks were randomly constructed and their characteristic path lengths Ls were calculated. The mean path lengths L were calculated for depressed patients and healthy people under three emotional stimuli at different thresholds and the degree of deviation of the two populations under different emotional stimuli compared to the small-world networks was observed by calculating the value of L/Ls. If the value of L/Ls is closer to 1, it indicates that the characteristic path length of the network is close to that of the small-world network.

Figure 11 shows the comparison of the L/Ls values of the two groups under three different emotional stimuli, * represents p<0.05 after using the independent samples t-test. it can be seen from the figure that the L/Ls values of the brain network of depressed patients are greater than 1 under both negative and neutral stimuli at different thresholds and have a tendency to continue to increase, indicating that the characteristic path length of the brain network of depressed patients has a significant deviation from the small world property. In contrast, the L/Ls values of the brain networks of depressed patients and healthy individuals did not differ significantly under positive stimuli and were both close to 1, indicating that the brain networks of the two groups were close to the small-world network under positive stimuli. This result indicates that the brain network characteristic path lengths of depressed patients significantly deviated from the small-world property under negative and neutral stimuli, and the changes in the brain network of depressed patients were more pronounced, while the brain network of healthy individuals was close to the small-world network property under the three emotional stimuli.

4. Summary

In this paper, we used Granger causality based on polynomial kernel as the connecting edge between brain network nodes to construct brain networks for two groups of people, depressed patients and healthy people, and divided their topological properties, and obtained the following conclusions: (1) Under negative and neutral emotional stimuli, depressed patients exchanged information more closely between frontal and occipital regions than healthy people and other brain regions, and the causal role was weaker in parietal and central regions. (2) Depressed patients had higher mean degrees under negative and neutral emotional stimuli than healthy individuals, and healthy individuals had higher mean clustering coefficients under all three emotional stimuli than depressed patients. (3) Comparing the changes of mean degree and mean clustering coefficients in the same population under different stimuli, it was found that the mean degree and mean clustering coefficients were greater in depressed patients under negative stimuli than neutral and positive stimuli, indicating that depressed patients’ brain networks changed more when they were subjected to emotional stimuli. (4) The mean path length of depressed patients significantly deviated from the small-world property under negative and neutral stimuli. In conclusion, using Granger causal analysis of depression brain magnetograms based on polynomial kernels and constructing networks can effectively distinguish healthy individuals from depressed patients and facilitate the diagnosis of depression.