## 3 Materials and Methods
### DL Model for Schizophrenia Detection Using TE Matrices  
One key innovation in this study is the development of a DL-based model trained on TE matrices derived from EEG data in order to support the diagnosis or prediction of schizophrenia-associated abnormalities. For this purpose, an openly available EEG dataset [56] was used that is specifically focused on schizophrenia and contains preprocessed EEG data for analysis with various methods such as artifact removal, baseline correction, and epoching in the task time frame. The derived ERPs are obtained across multiple trials and conditions, resulting in a reliable dataset for DL applications. The proposed DL model focused on nine electrode data available in the dataset (Fz, FCz, Cz, FC3, FC4, C3, C4, CP3, CP4) in order to distinguish schizophrenia-related abnormalities from HCs. The collected data were converted into TE matrices containing information on the direction of flow between different electrode sites and were represented as inputs to the DL model. Throughout this study, we refer to TE matrices as representations of directional connectivity between electrode pairs. Each TE matrix reflects the directional influence of brain activity from one region to another, where values are not necessarily \begin{table}
\begin{tabular}{c c c} \hline \hline
**Band** & **Frequency** & **Brain State** \\ \hline Delta (\(\delta\)) & 1–4 Hz & Dreamless sleep (deep meditation) \\ Theta (\(\theta\)) & 4–8 Hz & Deeply relaxed, inward focused (drowsiness, vivid dreams) \\ Alpha (\(\alpha\)) & 7.5–13 Hz & Very relaxed, passive attention (thoughtful \\  & & times, reflective, restful) \\ Beta (\(\beta\)) & 13–30 Hz & Anxiety dominant, active, external attention, relaxed (busy mind) \\ Gamma (\(\gamma\)) & 30–40 Hz & Concentration (higher levels of \\  & & consciousness, problem solving) \\ Delta (\(\delta\)) & 1–4 Hz & Dreamless sleep (deep meditation) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Brain wave bands, frequency ranges, and associated cognitive states. symmetrical across pairs due to the directional nature of TE. These matrices are visualized as heatmaps in figures, capturing asymmetric connectivity patterns between brain regions, rather than mutual correlations. Due to the characteristics of EEG data and the difficulties of diagnosing neuropsychiatric disorders, such as schizophrenia, a solution that integrates CNNs with BiLSTM networks is proposed. The CNN component aims to obtain spatial features from the TE matrices, and the BiLSTM component considers the temporal relations of these features. Such an architecture allows identifying intricate details of neural connections that typically might escape normal diagnostic procedures. The CNN-BiLSTM model combines a convolutional neural network (CNN) and a Bidirectional Long Short-Term Memory (BiLSTM) network to extract spatial and temporal patterns from the TE matrices. The CNN component processes the spatial structure within the TE matrix. For a given input X\(\in\)\(\mathbb{R}^{\mathrm{nxn}}\), where \(n\) is the number of electrodes, the CNN layer applies a convolution operation as follows:  
\[X^{\prime}_{i,\ j}=f\ (\sum_{m=1}^{M}\sum_{n=1}^{N}X_{i}+m-1,j+n-1\ \cdot W_{m,\ n}+b)\]  
where \(W\) is the convolutional filter, \(b\) is the bias term, and \(f\) is an activation function (e.g., ReLU). The BiLSTM network then processes the sequential patterns extracted from the CNN output by maintaining both forward and backward temporal dependencies and is ideal for capturing temporal dynamics in brain connectivity. \[h_{t}=\sigma(\ W_{x}X_{t}+W_{h}h_{t-1}+b_{h})\]  
where \(h_{t}\) represents the hidden state at time \(t\), \(W_{x}\) and \(W_{h}\) are weights, and \(\sigma\) is the activation function. By combining these approaches, the model learns both spatial and temporal patterns, enabling robust classification between schizophrenia patients and HCs.