CN111612746A - A dynamic detection method of functional brain network hub nodes based on graph theory - Google Patents

Abstract

The invention discloses a dynamic detection method of a functional brain network center node based on graph theory. The invention improves on the multivariate central node detection method, so that a more reliable central node conforming to neuroscience cognitive activities can be detected. First, using the sliding window technique, the blood oxygen signal is evenly divided into several segments in the time dimension. In the sliding window of each period of time, the central node in the corresponding time window is detected, so as to obtain a change trajectory of the central node moving with the sliding window. Finally, this change trajectory is used as a constraint on the multivariate detection method, so that more reliable and accurate central nodes can be dynamically detected.

Description

A dynamic detection method of functional brain network hub nodes based on graph theory

technical field

The invention relates to the field of neuroscience brain network research, in particular to a dynamic detection method of a functional brain network center node based on graph theory.

Background technique

Resting-state magnetic resonance imaging (fMRI) provides a non-invasive method to measure changes in cerebral blood oxygen. In the resting state, the subjects did not perform any definite task, and the subjects spontaneously generated neural activity, and the fluctuation of neural activity was related to the change of the blood oxygen concentration signal. Therefore, the blood oxygen concentration signal is used to calculate the connectivity between brain regions, thereby constructing a functional brain network.

The brain network can be divided into many modules, some modules are responsible for vision, some are responsible for hearing, etc. The modular structure allows us to distinguish the different roles and positions of nodes in the brain region in more detail. For example, some nodes are very important in their module, but not necessarily important to the whole network, such nodes are called regional core nodes (provincialhub), while other nodes have a limited role in their own module , but they are connected to different modules and maintain the connectivity of the entire network. Such nodes are called hub nodes (connectorhub). The central node connects different functional modules in the brain network. Due to the high centrality of the central node, it plays a very important role in the connection within the brain, such as the integration of information and participation in a variety of cognitive activities. The properties suggest that when a central node is attacked by a disease, it is more likely to have an impact on our brains. Recently, there has been a consensus in the field of neuroimaging that the functional network of the brain changes throughout the scan time, even in a task-free environment. Many studies have shown that dynamic patterns are more relevant to certain neurological disorders. Therefore, if we can dynamically detect the central nodes more accurately, it can help us better analyze and understand the pathological mechanism of these diseases, and can also be used to assist the early diagnosis and treatment of diseases.

At present, the detection methods of central nodes are all designed for static functional brain networks, and it is difficult to capture the dynamic changes of central nodes over time, resulting in the inconsistency of the detection results and the inability to guarantee the changes of central nodes and functional brain networks. Cognitive changes presented in

SUMMARY OF THE INVENTION

The invention proposes a dynamic detection method for the central node of a functional brain network based on graph theory. Aiming at the shortcomings of the above static detection methods, this paper improves the multivariate central node detection method, so that it can detect more reliable central nodes in line with neuroscience cognitive activities.

First, using the sliding window technique, the blood oxygen signal is evenly divided into several segments in the time dimension. In the sliding window of each period of time, the central node in the corresponding time window is detected, so as to obtain a change trajectory of the central node moving with the sliding window (as shown in Figure 1). Finally, this change trajectory is used as a constraint on the multivariate detection method, so that more reliable and accurate central nodes can be dynamically detected.

The scheme adopted by the present invention to solve its technical problems is as follows:

Step 1: Construct a functional brain network in each sliding window using Pearson correlation;

是一个N×N的邻接矩阵。w_ij表示邻接矩阵W中第i行第j列的元素，具体表示第i个脑区节点和第j个脑区节点的关联度；接下来计算拉普拉斯矩阵L＝D-W，其中D是一个对角元素为

的度矩阵。Basic data format and preliminary knowledge, use G=(V, W) to represent a functional brain network, where V represents the set of N brain area nodes,

is an N×N adjacency matrix. w _ij represents the element of the i-th row and the j-th column in the adjacency matrix W, specifically representing the degree of correlation between the i-th brain area node and the j-th brain area node; then calculate the Laplace matrix L=DW, where D is A diagonal element is

degree matrix.

In graph theory, a graph G is formed by a set of orthonormal vectors Φ obtained by eigendecomposition of the Laplacian matrix L, that is, L=Φ ^T ΛΦ, the diagonal matrix Λ=diag[λ ₁ , λ ₂ , . . . , λ _N ], and λ ₁ , λ ₂ , . . . , λ _N represent the eigenvalues sorted in ascending order.

The eigenvalues are in one-to-one correspondence with the orthogonal vectors, and are the eigenvalues of the Laplace matrix L;

If all brain area nodes are connected in _a functional brain network with no discrete parts, then the smallest eigenvalue λ1 is zero. The number of zero eigenvalues is equal to the number of subgraphs. That is, when the values of λ ₁ and λ ₂ are both zero, the number of subgraphs is 2.

Step 2: Improved Multivariate Hub Node Detection

2-1. Each brain area node v _i in the functional brain network is associated with a binary label s _i in a selection vector, where s _i =0 indicates that the brain area node v _i is a central node, and s _i =1 Indicates that it is not a hub node. Whether it is the desired central node is determined by the degree of damage to the functional brain network after removing the selected brain area node, where the degree of damage is determined by the number of zero eigenvalues of the remaining functional brain network after removing the node in the brain area. judge.

The judgment criterion is as follows: if the number of zero eigenvalues of the remaining functional brain network increases more than other nodes after deleting the selected node, the node is regarded as a candidate hub node.

2-2. Since multiple hub nodes need to be selected at one time, the estimation of the selection vector s=[s ₁ , s ₂ , . . . , s _n ] is an NP-hard problem. According to KyFan's theorem, transform the problem into minimizing the sum of the first K smallest eigenvalues,

Assuming that there are a total of K central nodes in the N brain area nodes of the functional brain network, the calculation is as follows:

Then further derivation to get the final objective function:

表示一个矩阵；Among them, λ _i represents the ith eigenvalue;

represents a matrix;

表示剩余功能脑网络中每个脑区节点的K维矩阵，并受F^TF＝I正交约束，因此需要优化求解F和S：2-3. Place each element of the selection vector s on the diagonal of the diagonal matrix S. where L _s =DS ^T WS represents the Laplacian matrix of the residual functional brain network, and the subscript s is the mark used to distinguish;

represents the K-dimensional matrix of each brain area node in the residual functional brain network, and is constrained by the orthogonality of F ^T F=I, so it needs to be optimized to solve F and S:

Optimize F: fix the diagonal matrix S, and obtain the closed-form solution of F. The closed-form solution is the K orthogonal vectors corresponding to the minimum eigenvalues of the first K items of L _s .

Optimize S: fix F, and the objective function of optimizing the diagonal matrix S becomes:

是一个元素a_ij＝w_ij||f_i-f_j||²的N×N的矩阵，且f_i和f_j分别表示F矩阵的第i行向量和第j行向量。in

is an N×N matrix with elements a _ij =w _ij ||f _i -f _j || ² , and f _i and f _j represent the i-th row vector and the j-th row vector of the F matrix, respectively.

并将上述优化后的目标函数化简为:2-4. Since the optimized objective function is not strictly convex, s is a binary selection vector, which is not conducive to optimization, so an auxiliary vector is introduced

And simplify the above optimized objective function to:

where P is another diagonal matrix derived from the auxiliary vector p. Intuitively, the selection vector s is the binarization result of the auxiliary vector p.

Also solve for P and S alternately in equation (4):

Optimize F: fix the diagonal matrix S, and obtain the closed-form solution of F. The closed-form solution is the K orthogonal vectors corresponding to the minimum eigenvalues of the first K items of L _s .

Optimize S and P: fix F, use the augmented Lagrange multiplier method to optimize the diagonal matrix S and the auxiliary matrix P.

Step 3: Dynamic detection of functional brain network hub nodes

Fig. 2 shows the algorithm frame of the dynamic function network of the central node detection of the present invention. Specifically as follows:

3-1. Segment the entire blood oxygen signal into T groups of overlapping sliding windows. At each time t, estimate pi corresponding to each brain region node v _i , where _pi is the _ith element in the auxiliary vector p.

其中

是连续函数p_i(τ)在时间点τ_t的离散采样。因此连续函数p_i(τ)在任意其它时间点τ的值都可以使用径向基函数(RBFs)计算：3-2. Connect _pi to form a trajectory

in

is the discrete sampling of the continuous function p _i (τ) at time τ _t . Therefore the value of the continuous function p _i (τ) at any other time point τ can be calculated using radial basis functions (RBFs):

表示第i个脑区节点在时刻t的辅助向量的权值，即

表示p_i在时刻t的权值；τ-τ_t表示时间差。where the parameter σ is used to control the strength of the trajectory smoothing,

Represents the weight of the auxiliary vector of the i-th brain area node at time t, namely

Represents the weight of _pi at time t; τ-τ _t represents the time difference.

径向基函数集合

通过以下方法计算：3-3. Given track

Radial Basis Function Collection

Calculated by:

where the parameter λ is used to control the strength of the time dependence of the continuous function p _i (τ).

3-4. Use the improved diagonal matrix P composed of auxiliary vectors to initialize the diagonal matrix S composed of the selection vector s of each sliding window, as shown by the arrows in Figure 2.

3-5. Bring the improved diagonal matrix P composed of auxiliary vectors into step 2-3 and repeat the optimization solution until the target converges.

The intentional effect of the present invention is as follows:

The output of the method of the present invention is the central nodes that change with the changes of the brain network, and these central nodes are not only more accurate than the traditional method at each time point, but also retain the correlation information between the time points.

Description of drawings

Fig. 1 is the overall framework of the dynamic detection method of the central node;

Fig. 2 is the algorithm framework of the dynamic detection method of the central node;

Fig. 3 is the analysis diagram of traditional method and the stability of this method;

Fig. 4 is a visualization result diagram of the detection of a normal person's central node by the traditional method and the present method;

Fig. 5 is a visualization result diagram of the detection of the central node of a patient with obsessive-compulsive disorder by the traditional method and the present method;

Detailed ways

The present invention will be further described in detail below in conjunction with the accompanying drawings and experiments.

The present invention uses the sliding window technology to divide the blood oxygen signal into several segments evenly in the dimension of time. In the sliding window of each period of time, the central node in the corresponding time window is detected, so as to obtain a change trajectory of the central node moving with the sliding window (as shown in Figure 1). Finally, this change trajectory is used as a constraint on the multivariate detection method, so that more reliable and accurate central nodes can be dynamically detected.

The main steps realized by the embodiment of the present invention:

Step (1): Select the data of 63 normal people and 62 obsessive-compulsive disorder patients for the experiment. Each subject had T1-weighted magnetic resonance images (specific parameters were TR = 8 ms, TE = 1.7 ms ms, flip angle = 20°, resolution = 1.0 x 1.0 x 1.0 mm ² ) and resting-state fMRI Resonance data (specific parameters are TR=2s, TE=60ms milliseconds, flip angle=90°, resolution=3.0×3.0×4.0mm ² ), each subject contains 230 detection time points.

Step (2): Register all these experimental data into the AAL template, which is divided into 116 brain regions. And the correlation between these brain regions is calculated to obtain the corresponding functional brain network connection matrix W, which is used as the input of the experiment.

Step (3): set the experimental parameters.

The size of the sliding window is set to 10% of the entire time point; the detection number of the central node is set to 12; the parameter σ of the radial basis RBF is set to 0.7; the optimal λ of the objective function (2) is found by the grid search method The parameter is 0.6.

依次优化F和S，直到收敛，得到最终的结果。Step (4): Finally, on the objective function

Optimize F and S in turn until convergence and get the final result.

Analysis of results:

For each individual, the number of detected pivot nodes in each sliding window is counted, the corresponding histogram is constructed, and the corresponding entropy value is finally calculated.

The lower the entropy, the less the change of the central node in the whole detection process. As shown in Figure 3, we can see that the entropy value is above the diagonal line, indicating that the entropy value detected by our method is lower and more Stablize.

We further visualized the change results of the hub nodes for a period of time. As shown in Figure 4 and Figure 5, the hub nodes detected by our method did not change within 3 TRs (about 6 seconds), while the traditional methods detected no changes. The central node jumped at the second TR moment, and returned to the previous state at the next moment. This situation is unusual and does not conform to the law of cognitive change. Therefore, it can be seen that our method is more stable and reliable than traditional methods.

Claims (5)

1. a kind of dynamic detection method for the functional brain network center node based on graph theory, it is characterized in that utilizing the technique of sliding window, the blood oxygen signal is divided into several sections on average with time as dimension; In the sliding window of each period of time, The pivot node in the corresponding time window is detected, and a change trajectory of the pivot node moving with the sliding window is obtained; finally, the change trajectory is used as a constraint on the multivariate detection method, so that the dynamic detection can be performed. More reliable and accurate hub nodes. 2. a kind of dynamic detection method to the functional brain network center node based on graph theory according to claim 1, is characterized in that comprising the steps: Step 1: Construct a functional brain network in each sliding window using Pearson correlation; A functional brain network is represented by G=(V, W), where V represents the set of N brain area nodes,

is an N×N adjacency matrix; w _ij represents the element of the i-th row and j-th column in the adjacency matrix W, specifically representing the correlation between the i-th brain area node and the j-th brain area node; then calculate the Lapla The Si matrix L=DW, where D is a diagonal element of

degree matrix; In graph theory, a graph G is formed by a set of orthonormal vectors Φ obtained by eigendecomposition of the Laplacian matrix L, that is, L=Φ ^T ΛΦ, the diagonal matrix Λ=diag[λ ₁ , λ ₂ , ..., λ _N ], and λ ₁ , λ ₂ , ..., λ _N represent the eigenvalues sorted in ascending order; Step 2: Improved Multivariate Hub Node Detection 2-1. Each brain region node v _i in the functional brain network is associated with a binary label s _i in a selection vector, where s _i = instrument indicates that the brain region node v _i is a central node, and s _i =1 Indicates that it is not a central node; whether it is the desired central node is determined by the degree of damage to the functional brain network after removing the selected brain area node, and the degree of damage is determined by the zero feature of the remaining functional brain network after removing the node in this brain area. The number of values to judge; 2-2. Since multiple hub nodes need to be selected at one time, the estimation of the selection vector s=[s ₁ , s ₂ , ..., s _n ] is an NP-hard problem; according to Ky Fan's theorem, the problem is converted into Minimize the sum of the first K smallest eigenvalues; Assuming that there are a total of K central nodes in the N brain area nodes of the functional brain network, the calculation is as follows:

Then further derivation to get the final objective function:

Among them, λ _i represents the ith eigenvalue;

represents a matrix; 2-3. Put each element of the selection vector s on the diagonal of the diagonal matrix S; where L _s =DS ^T WS represents the Laplace matrix of the remaining functional brain network, and the subscript s is used to distinguish mark;

represents the K-dimensional matrix of each brain area node in the residual functional brain network, and is constrained by the orthogonality of F ^T F=I, so it needs to be optimized to solve F and S: Optimize F: fix the diagonal matrix S, and obtain the closed-form solution of F. The closed-form solution is the K orthogonal vectors corresponding to the minimum eigenvalues of the first K items of L _s ; Optimize S: fix F, and the objective function of optimizing the diagonal matrix S becomes:

in

is an N×N matrix with elements a _ij =w _ij ||f _i -f _j || ² , and f _i and f _j represent the i-th row vector and the j-th row vector of the F matrix, respectively; 2-4. Since the optimized objective function is not strictly convex, s is a binary selection vector, which is not conducive to optimization, so an auxiliary vector is introduced

And simplify the above optimized objective function to:

where P is another diagonal matrix derived from the auxiliary vector p; intuitively, the selection vector s is the binarization result of the auxiliary vector p; Also solve for P and S alternately in equation (4): Optimize F: fix the diagonal matrix S, and obtain the closed-form solution of F. The closed-form solution is the K orthogonal vectors corresponding to the minimum eigenvalues of the first K items of L _s ; Optimize S and P: fix F, and use the augmented Lagrange multiplier method to optimize the diagonal matrix S and the auxiliary matrix P; Step 3: Dynamic detection of functional brain network hub nodes 3-1. Segment the entire blood oxygen signal into T groups of overlapping sliding windows; at each time t, estimate the pi corresponding to each brain region node v _i , where _pi is the _ith element in the auxiliary vector p ; 3-2. Connect _pi to form a track

in

is a discrete sampling of the continuous function p _i (τ) at time points {τ _t }; thus the value of the continuous function p _i (τ) at any other time point τ can be computed using radial basis functions (RBFs):

where the parameter σ is used to control the strength of the trajectory smoothing,

Represents the weight of the auxiliary vector of the i-th brain area node at time t, namely

represents the weight of _pi at time t; τ-τ _t represents the time difference; 3-3. Given track

Radial Basis Function Collection

Calculated by:

Among them, the parameter λ is used to control the strength of the time correlation of the continuous function p _i (τ); 3-4. Use the improved diagonal matrix P composed of auxiliary vectors to initialize the diagonal matrix S composed of the selection vector s of each sliding window, as shown by the arrow in Figure 2; 3-5. Bring the improved diagonal matrix P composed of auxiliary vectors into step 2-3 and repeat the optimization solution until the target converges. 3. a kind of dynamic detection method to the functional brain network center node based on graph theory according to claim 2, it is characterized in that described eigenvalue and orthogonal vector one-to-one correspondence, is the Laplace matrix L Eigenvalues. 4. a kind of dynamic detection method for the functional brain network center node based on graph theory according to claim 2 or 3, it is characterized in that if all brain region nodes are all connected in a functional brain network, there is no separate part, Then the smallest eigenvalue λ1 is zero; the number _of zero eigenvalues is equal to the number of subgraphs; that is, when the values of both λ1 and _λ2 are zero, the number of subgraphs is ₂ . 5. a kind of dynamic detection method for the functional brain network center node based on graph theory according to claim 4, it is characterized in that described judgment criterion is as follows: if the quantity of the zero eigenvalues of the remaining functional brain network is deleted and selected After adding more nodes than deleting other nodes, this node is regarded as a candidate hub node.

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