2.1.1

Application in the experiment

This study proposes a method based on Bayesian Network (BN) to construct brain effective connectivity networks, represented by a graph structure matrix A ∈ R^n×n, with the aim of exploring causal relationships between brain regions. We employ a heuristic search algorithm, the HC algorithm, to search for optimal network structures within the possible network structure space. The Bayesian Information Criterion (BIC) is adopted as our scoring function to evaluate the goodness of fit of different network structures. Usually the structure with the lowest BIC value is selected as the current optimal solution. Then we continue to search for a better solution of BN (Bayesian network) through iterative local optimization strategy.

2.1.2

Bayesian networkV = {X₁, X₂,…,X_n}E ∈ V × V

BN (Bayesian Network) is a probabilistic graph model used to describe the joint probability distribution of a set of random variables [3]. BN is a directed acyclic graph (DAG) of structure G=(V, E), consisting of a set of nodes and a set of edges.The set of nodes V ={X₁, X₂, …, X_n} corresponds to the random variables in the system, while the directed edges (or arcs) in E ∈ V × V represent causal or conditional dependencies between these variables.

For each node X_i ∈ V in the network, a conditional probability distribution P(X_i|pa(X_i)) is defined, where pa(X_i) denotes any combination of values of the parent nodes pa(X_i). These conditional probability distributions are central to the network, as they quantify the strength of dependency between nodes and their parents. In addition, the joint probability distribution of the whole network can be reconstructed by Markov conditions.[4]. We use BIC (Bayesian Information Criterion) as an effective model selection criterion for Bayesian networks. The BIC, also known as the Schwartz Information Criterion, aims to evaluate the balance between goodness of fit and complexity of a model through comprehensive measures. Lower BIC values mean that the model fits the data better, while avoiding overfitting.The calculation of BIC involves the model’s log-likelihood function (LL), a dataset (D) in the BN, the pair of nodes and edges in the BN (T), and the observed data (N)[5].

2.1.3

Hill climbing algorithm

HC algorithm is an optimization algorithm based on heuristic search. Its core mechanism involves iteratively evaluating and refining the current network structure to find a statistically optimal model within the possible network structure space. The HC algorithm starts with an initial network structure and then applies a predefined scoring function to evaluate the fit of that structure. During each iteration, the algorithm searches the neighborhood of the current structure, incrementally building the network structure. If a structure with a higher score is found in the neighborhood, it is adopted as the new current structure, and the search continues. If no higher-scoring structure is found, the algorithm terminates, and the current structure is considered a local optimum [6]. In constructing brain effective connectivity networks, the HC algorithm is employed to identify causal interactions between brain regions. These interactions are crucial for understanding the pathophysiological processes of neuropsychiatric disorders.

2.3.1

Application of LSTM in this experiment

In the experiment, LSTM networks are employed to process and analyze time series data. The specific application process is as follows:

First, 120 continuous time series data points are sequentially input into the encoder. The LSTM network is used to predict the characteristics of the next time point iteratively. This step-by-step prediction method enables the model to continuously update its internal hidden states and memory cells, effectively capturing long-term dependency information and enhancing the model’s ability to handle complex time series data.

The decoder then receives the memory cell states and hidden layer representations output by the encoder to iteratively predict the subsequent 120 time series features. During the prediction process, the decoder utilizes a threshold mechanism to select inputs [8]. The application process of LSTM in the experiment is shown in the Fig.3.

Figure 3.

Architecture of the Proposed LSTM

Finally, the predicted data is compared with the actual observed data, the loss function is calculated, and the model parameters are optimized using the backpropagation algorithm.This process effectively reduces prediction error and improves model accuracy. The loss function used here is the Mean Squared Error (MSE):

2.3.2

Long short-term memory

Recurrent Neural Networks (RNNs) [9] are a class of deep neural networks that can model the dynamic temporal characteristics of sequences by maintaining internal hidden states. However, basic RNNs encounter the problems of vanishing and exploding gradients when dealing with long sequences. LSTM networks address these issues by adding three gating mechanisms: forget gate, input gate, and output gate [10]. These mechanisms extend the basic RNN, enabling LSTMs to capture long-term dependencies in sequences and make the optimization process easier.