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\Title{Supplementary Material for "EEG-Driven Arm Movement Decoding: Combining Connectivity and Amplitude Features for Enhanced Brain-Computer Interface Performance"}

\TitleCitation{Supplementary Material: EEG-Driven Arm Movement Decoding}

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% Authors, for the paper (add full first names)
\Author{Hamidreza Darvishi $^{1}$\orcidA{}, Ahmadreza Mohammadi $^{1}$, Mohammad Hossein Maghami $^{2,}$*, Meysam Sadeghi $^{1}$ and Mohamad Sawan $^{3,}$*}

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\AuthorNames{Hamidreza Darvishi, Ahmadreza Mohammadi, Mohammad Hossein Maghami, Meysam Sadeghi, and Mohamad Sawan}

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\AuthorCitation{Darvishi, H.; Mohammadi, A.; Maghami, M.H.; Sadeghi, M.; Sawan, M.}


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\address{%
$^{1}$ \quad Department of Cognitive Psychology, Institute for Cognitive Science Studies (ICSS), Tehran, Iran; darvishi\_hr@icss.ac.ir (H.D.); mohammadi\_a@icss.ac.ir (A.M.); m.sadeghi@icss.ac.ir (M.S.)\\
$^{2}$ \quad Research Laboratory for Integrated Circuits, Faculty of Electrical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran; mhmaghami@sru.ac.ir (M.H.M.)\\
$^{3}$ \quad Center of Excellence in Biomedical Research on Advanced Integrated-on-chips Neurotechnologies (CenBRAIN Neurotech), School of Engineering, Westlake University, Hangzhou, China; sawan@westlake.edu.cn (M.S.)\\
$^{*}$ \quad Correspondence: mhmaghami@sru.ac.ir (M.H.M.); sawan@westlake.edu.cn (M.S.)}


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\begin{document}

\section*{S0. Preamble}
\label{sec:sup_preamble}

This document provides supplementary information to the main manuscript titled "EEG-Driven Arm Movement Decoding: Combining Connectivity and Amplitude Features for Enhanced Brain-Computer Interface Performance." It details the architectures, hyperparameters, training procedures, overfitting avoidance strategies, and performance results for the Long Short-Term Memory (LSTM) network and the ICNN-inspired Convolutional Neural Network (CNN) used for comparative analysis, as referenced in Sections 2.8, 3.5, and 4.6 of the main manuscript. A comparative summary of performance including the proposed feature-based method from the main manuscript is also provided.

\section*{S1. End-to-End Deep Learning Models: Architectures and Hyperparameters}
\label{sec:sup_e2e_models_params}

\subsection*{S1.1. Data Preparation for End-to-End Models}
\label{sec:sup_data_prep_e2e}

As briefly described in Section 2.8 of the main manuscript, the input data for the end-to-end deep learning models (LSTM and ICNN-inspired CNN) consisted of EEG data from 32 channels. The raw EEG data, originally sampled at the subject-specific rate (e.g., 512 Hz as mentioned in main paper, though participant files show varying rates), was resampled to 200 Hz using a 10th order anti-aliasing filter via the MATLAB \texttt{resample} function. Preprocessing then involved bandpass filtering between 0.18 Hz (1st order Butterworth high-pass) and 40 Hz (4th order Butterworth low-pass), followed by Common Average Referencing (CAR). Input segments for the models were 1.0-second chunks (200 time points $\times$ 32 channels), extracted with a stride of 0.1 seconds (20 samples). These chunks were derived from 4-second epochs aligned to movement events, with each epoch starting 0.5 seconds before the recorded event onset. Prior to input into the neural networks, each EEG chunk was z-score normalized channel-wise, based on the mean and standard deviation of that channel across the entire recording for that participant. The target variable for regression was the EMG signal from channel 33. EMG data was also resampled to 200 Hz. It was then bandpass filtered between 40 Hz and 95 Hz (4th order Butterworth), full-wave rectified, and subsequently smoothed with a 2 Hz low-pass filter (4th order Butterworth). Finally, the smoothed EMG envelope was z-score normalized using its mean and standard deviation across the entire recording for that participant. The EMG target corresponding to each 1-second EEG chunk was the normalized EMG value at the last sample of that chunk.

\subsection*{S1.2. Long Short-Term Memory (LSTM) Network}
\label{sec:sup_lstm}

Long Short-Term Memory (LSTM) networks, a type of recurrent neural network (RNN), were originally proposed by Hochreiter and Schmidhuber (1997) \citep{Hochreiter1997lstm} and are particularly well-suited for learning from sequential data due to their ability to capture long-range temporal dependencies. They have been widely applied in EEG signal processing for tasks such as motor imagery classification and continuous decoding of movement parameters \citep{Pancholi2022arxiv, Roy2019deepreview}. Our LSTM network was designed to leverage these capabilities for predicting EMG amplitudes from EEG time series.

\subsubsection*{S1.2.1. Architecture}
The LSTM model implemented in MATLAB using the Deep Learning Toolbox is detailed in Table~\ref{tab:lstm_architecture}.

\begin{table}[h!]
\centering
\caption{Architecture of the LSTM Network.}
\label{tab:lstm_architecture}
\renewcommand{\arraystretch}{1.2}
\footnotesize
\begin{tabular}{@{}lll@{}}
\toprule
\textbf{Layer Type} & \textbf{Key Parameters/Configuration} & \textbf{MATLAB Name} \\
\midrule
Sequence Input Layer & 32 features (EEG channels) per time step & \texttt{input} \\
LSTM Layer & 128 hidden units (\texttt{num\_lstm\_units}); \texttt{OutputMode = 'last'} & \texttt{lstm} \\
Dropout Layer & Dropout rate = 0.5 (\texttt{dropout\_rate\_lstm}) & \texttt{dropout\_lstm} \\
Fully Connected Layer & 1 output unit (for EMG prediction) & \texttt{fc\_out} \\
Regression Layer & Standard regression output & \texttt{output} \\
\bottomrule
\end{tabular}
\end{table}
Input data for the \texttt{sequenceInputLayer} was formatted as a cell array, where each cell contained a [32 features $\times$ 200 time steps] matrix representing one EEG chunk.

\subsubsection*{S1.2.2. Hyperparameters and Training}
The training hyperparameters for the LSTM model are listed in Table~\ref{tab:lstm_training_params}.

\begin{table}[h!]
\centering
\caption{Training Hyperparameters for the LSTM Network.}
\label{tab:lstm_training_params}
\renewcommand{\arraystretch}{1.2}
\footnotesize
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{Parameter} & \textbf{Value} \\
\midrule
Optimizer & Adam \\
Initial Learning Rate & $1 \times 10^{-4}$ (\texttt{learn\_rate}) \\
Loss Function & Mean Squared Error (MSE) \\
Maximum Epochs & 250 (\texttt{max\_epochs}) \\Mini-Batch Size & 64 (\texttt{batch\_size}) \\
Validation Data Split & 15\% of training data in each fold (\texttt{validation\_fraction}) \\
Validation Patience & 50 epochs (\texttt{patience}) \\
Output Network Selection & Best validation loss \\Data Shuffling & Every epoch \\
Cross-Validation & 6-fold (\texttt{num\_folds}) per participant \\
\bottomrule
\end{tabular}
\end{table}

\subsection*{S1.3. ICNN-inspired Convolutional Neural Network (CNN)}
\label{sec:sup_icnn}

Convolutional Neural Networks (CNNs) \citep{LeCun1998gradient} have become a cornerstone in deep learning for image and signal processing. For EEG analysis, various CNN architectures have been proposed to automatically learn relevant spatial, spectral, and temporal features, such as DeepConvNet, ShallowConvNet \citep{Schirrmeister2017deepconv}, and EEGNet \citep{Lawhern2018eegnet}. More recently, interpretable CNNs (ICNNs) for EEG, like Sinc-ShallowNet and MS-Sinc-ShallowNet by Borra et al. \citep{Borra2020sincshallow, Borra2023compbiomed}, aim to enhance model transparency by incorporating layers whose learned parameters (e.g., filter cut-off frequencies from SincNet layers \citep{Ravanelli2018sincnet}) are directly interpretable. Our CNN architecture was inspired by the multi-scale temporal feature extraction and the conceptual interpretable spectral-spatial feature extraction stages of MS-Sinc-ShallowNet \citep{Borra2023compbiomed}. However, due to MATLAB Deep Learning Toolbox constraints regarding custom SincNet layers, our implementation uses standard convolutional layers while retaining the block-wise and multi-scale processing philosophy. The network aims to learn relevant spatio-temporal features from the EEG segments for EMG regression.

\subsubsection*{S1.3.1. Architecture}The network graph, detailed in Table~\ref{tab:icnn_architecture}, was constructed with an initial feature extraction block (ISS-like) followed by two parallel multi-scale temporal convolution branches (DST-like), a concatenation layer, and a final regression output.

\begin{table}[h!]
\begin{adjustwidth}{-\extralength}{0cm}
\centering
\caption{Architecture of the ICNN-inspired Network.}
\label{tab:icnn_architecture}
\renewcommand{\arraystretch}{1.2}
\footnotesize
\begin{tabular}{@{}llll@{}}
\toprule
\textbf{Block} & \textbf{Layer Type} & \textbf{Key Parameters/Configuration} & \textbf{MATLAB Name} \\
\midrule
Input & Image Input & Size: [32$\times$200$\times$1]; Norm: \texttt{'none'} & \texttt{input} \\
\midrule
ISS-like & Temporal Conv2D & Kernel: [1$\times$51] (\texttt{F0\_ISS\_temporal}); Filters: 16 (\texttt{K0\_ISS}); Pad: \texttt{'same'} & \texttt{temporal\_conv} \\
& Batch Norm & - & \texttt{bn\_sinc} \\
& Spatial Conv2D & Kernel: [32$\times$1]; Filters: 32 ($16 \times 2$, \texttt{D1\_ISS}=2); Pad: \texttt{'same'} & \texttt{spatial\_conv} \\
& Batch Norm & - & \texttt{bn\_spatial} \\
& ELU Activation & - & \texttt{elu\_iss} \\
& Dropout & Rate: 0.5 (\texttt{dropout\_rate\_iss}) & \texttt{dropout\_iss} \\
\midrule
DST-like & \multicolumn{3}{l}{\textit{Output of ISS-like block ($L_{\text{ISS\_out}}$) feeds into two parallel branches:}} \\
\textit{(Large Scale)} & Temporal Conv2D (1) & Kernel: [1$\times$51] (\texttt{F0\_DST\_large}); Filters: 32; Pad: \texttt{'same'} & \texttt{dw\_conv\_large} \\
& Batch Norm & - & \texttt{bn\_dw\_large} \\
& Pointwise Conv2D (2) & Kernel: [1$\times$1]; Filters: 32 (\texttt{K1\_DST}) & \texttt{pw\_conv\_large} \\
& Batch Norm & - & \texttt{bn\_pw\_large} \\& ELU Activation & - & \texttt{elu\_large} \\
& Average Pooling & Pool: [1$\times$10]; Stride: [1$\times$10] & \texttt{pool\_large} \\
& Dropout & Rate: 0.5 (\texttt{dropout\_rate\_dst}) & \texttt{dropout\_large} \\
\midrule
\textit{(Small Scale)} & Temporal Conv2D (1) & Kernel: [1$\times$25] (\texttt{F0\_DST\_small}); Filters: 32; Pad: \texttt{'same'} & \texttt{dw\_conv\_small} \\& Batch Norm & - & \texttt{bn\_dw\_small} \\
& Pointwise Conv2D (2) & Kernel: [1$\times$1]; Filters: 32 (\texttt{K1\_DST}) & \texttt{pw\_conv\_small} \\
& Batch Norm & - & \texttt{bn\_pw\_small} \\
& ELU Activation & - & \texttt{elu\_small} \\
& Average Pooling & Pool: [1$\times$10]; Stride: [1$\times$10] & \texttt{pool\_small} \\
& Dropout & Rate: 0.5 (\texttt{dropout\_rate\_dst}) & \texttt{dropout\_small} \\
\midrule
Output & Depth Concatenation & Inputs: 2 (from Large and Small Scale branches) & \texttt{concat\_dst} \\
& Fully Connected & Output Units: 1 & \texttt{fc\_out} \\
& Regression Layer & Standard regression output & \texttt{output} \\
\bottomrule
\end{tabular}
\end{adjustwidth}
\end{table}

\subsubsection*{S1.3.2. Hyperparameters and Training}
The training hyperparameters for the ICNN-inspired model are listed in Table~\ref{tab:icnn_training_params}.

\begin{table}[h!]
\centering
\caption{Training Hyperparameters for the ICNN-inspired Network.}
\label{tab:icnn_training_params}
\renewcommand{\arraystretch}{1.2}
\footnotesize
\begin{tabular}{@{}ll@{}}
\toprule\textbf{Parameter} & \textbf{Value} \\
\midrule
Optimizer & Adam \\
Initial Learning Rate & $1 \times 10^{-4}$ (\texttt{learn\_rate}) \\
Loss Function & Mean Squared Error (MSE) \\Maximum Epochs & 250 (\texttt{max\_epochs}) \\
Mini-Batch Size & 64 (\texttt{batch\_size}) \\
Validation Data Split & 15\% of training data in each fold (\texttt{validation\_fraction}) \\
Validation Patience & 50 epochs (\texttt{patience}) \\
Output Network Selection & Best validation loss \\
Data Shuffling & Every epoch \\
Cross-Validation & 6-fold (\texttt{num\_folds}) per participant \\
\bottomrule
\end{tabular}
\end{table}

\subsection*{S1.4. Overfitting Avoidance Strategies}
\label{sec:sup_overfitting_avoidance}

To ensure the generalization capabilities of the LSTM and ICNN-inspired models and to mitigate the risk of overfitting to the training data, several strategies were employed:

\begin{itemize}
    \item \textbf{Cross-Validation:} A 6-fold cross-validation scheme was implemented for each participant, as detailed in Sections S1.2.2 and S1.3.2. This ensures that the model's performance is evaluated on unseen data portions within each subject's dataset, providing a more robust estimate of generalization ability than a single train-test split. The final reported metrics are averaged across these folds.

    \item \textbf{Validation Set and Early Stopping:} Within each training fold of the cross-validation, 15\% of the training data was held out as a distinct validation set. The model's performance (validation loss) was monitored on this set after each epoch. Training was halted if the validation loss did not improve for a specified number of consecutive epochs (patience = 50). The model parameters that yielded the best performance on the validation set were then saved and used for testing. This prevents the model from continuing to train and potentially memorize the training data once its performance on unseen validation data starts to degrade.

    \item \textbf{Dropout:} Dropout layers were incorporated into both the LSTM and ICNN-inspired architectures (see Tables~\ref{tab:lstm_architecture} and \ref{tab:icnn_architecture}). During training, dropout randomly sets a fraction (here, 0.5) of neuron activations to zero at each update. This prevents neurons from co-adapting too much and encourages the network to learn more robust features that are not overly reliant on any single neuron \citep{Srivastava2014dropout}.

    \item \textbf{Data Shuffling:} The training data was shuffled before each training epoch. This helps to break any spurious correlations that might arise from the order of data presentation and ensures that the model sees varied mini-batches, contributing to better generalization.

    \item \textbf{Model Complexity and Regularization (Implicit):}
        \begin{itemize}
            \item \textbf{LSTM:} While the LSTM model uses 128 hidden units, this number was chosen to balance representational capacity with the amount of training data available per subject, aiming to avoid excessive complexity that could lead to overfitting.
            \item \textbf{ICNN-inspired:} The design of the ICNN-inspired model, while incorporating multiple layers, also employs techniques like pooling which help manage the number of parameters. The architectural choices, inspired by Borra et al. (2023) \citep{Borra2023compbiomed} who used kernel max-norm constraints for regularization, were aimed at balancing performance with model parsimony, although direct kernel max-norm was not explicitly implemented in our MATLAB version.
        \end{itemize}
    \item \textbf{Sufficient and Diverse Training Data (within folds):} By using a substantial portion of the available data for training within each cross-validation fold (after reserving for testing and validation), the models are exposed to a reasonable amount of variability, which aids in learning generalizable patterns.
\end{itemize}
These combined strategies aim to produce models that not only perform well on the training data but also generalize effectively to new, unseen EEG segments, which is critical for practical BCI applications.

\section*{S2. Comparative Performance of Decoding Models}
\label{sec:sup_comparative_performance}

Table~\ref{tab:sup_comparative_metrics_avg} provides a summary of the average decoding performance across all 10 participants for the proposed feature-based method (FBCSP+PLV+NN) from the main manuscript, alongside the LSTM and ICNN-inspired end-to-end models detailed in this supplementary material. This allows for a direct comparison of their overall efficacy on this dataset. \begin{table}[h!]
\centering
\caption{Average Performance Metrics Across All Participants for Different EMG Decoding Models. Values are Mean $\pm$ SD.}
\label{tab:sup_comparative_metrics_avg}
\renewcommand{\arraystretch}{1.3}
\footnotesize
\begin{tabular}{@{}lccc@{}}
\toprule
\textbf{Model} & \textbf{RMSE} & \textbf{R\textsuperscript{2}} & \textbf{Pearson R} \\
\midrule
Proposed (FBCSP+PLV+NN) & $0.579 \pm 0.098$ & $0.675 \pm 0.126$ & $0.829 \pm 0.077$ \\
LSTM (End-to-End) & $\mathbf{0.4231 \pm 0.0640}$ & $\mathbf{0.8744 \pm 0.0371}$ & $\mathbf{0.9345 \pm 0.0199}$ \\
ICNN-inspired (End-to-End) & $0.6013 \pm 0.1159$ & $0.7407 \pm 0.1031$ & $0.8603 \pm 0.0700$ \\\bottomrule
\end{tabular}
\flushleft
\scriptsize{\textit{Note:} Best performance for each metric is highlighted in bold. The "Proposed" method results are from Table 2 in the main manuscript.}
\end{table} As shown in Table~\ref{tab:sup_comparative_metrics_avg}, the LSTM model achieved the highest average decoding accuracy across all metrics. The ICNN-inspired model also demonstrated strong performance, outperforming the proposed feature-based method in terms of R\textsuperscript{2} and Pearson R, though with a slightly higher RMSE. These results are discussed in Section 4.6 of the main manuscript, considering trade-offs between accuracy, interpretability, and computational demands.

\subsection*{S2.1. Participant-Specific Decoding Performance (End-to-End Models)}
\label{sec:sup_participant_perf_e2e}

Table~\ref{tab:sup_participant_metrics_e2e} below presents the participant-specific decoding performance metrics for the LSTM and ICNN-inspired models, based on the aggregated results from the 6-fold cross-validation for each participant.

\begin{table}[h!]
\centering
\caption{Participant-Specific Performance Metrics for LSTM and ICNN-inspired End-to-End Models.}
\label{tab:sup_participant_metrics_e2e}
\renewcommand{\arraystretch}{1.2}
\footnotesize
\begin{tabular}{@{}lcccccc@{}}\toprule
\textbf{Participant} & \multicolumn{3}{c}{\textbf{LSTM}} & \multicolumn{3}{c}{\textbf{ICNN-inspired}} \\
\cmidrule(lr){2-4} \cmidrule(lr){5-7}
 & \textbf{RMSE} & \textbf{R\textsuperscript{2}} & \textbf{Pearson R} & \textbf{RMSE} & \textbf{R\textsuperscript{2}} & \textbf{Pearson R} \\
\midrule
1  & 0.4420 & 0.8631 & 0.9301 & 0.8607 & 0.4809 & 0.6945 \\
2  & 0.3951 & 0.8864 & 0.9416 & 0.5690 & 0.7644 & 0.8775 \\
3  & 0.5784 & 0.7799 & 0.8833 & 0.5004 & 0.8352 & 0.9147 \\
4  & 0.3974 & 0.8904 & 0.9436 & 0.7144 & 0.6458 & 0.8050 \\
5  & 0.3537 & 0.9130 & 0.9556 & 0.6309 & 0.7232 & 0.8529 \\
6  & 0.4368 & 0.8707 & 0.9333 & 0.5568 & 0.7900 & 0.8941 \\
7  & 0.4115 & 0.8820 & 0.9392 & 0.5203 & 0.8113 & 0.9011 \\
8  & 0.4172 & 0.8772 & 0.9366 & 0.5490 & 0.7874 & 0.8877 \\
9  & 0.3672 & 0.9054 & 0.9515 & 0.5215 & 0.8091 & 0.9057 \\
10 & 0.4320 & 0.8754 & 0.9300 & 0.5900 & 0.7600 & 0.8700 \\
\bottomrule
\end{tabular}
\end{table}


\section*{S3. References}
\label{sec:sup_references}

Key references for the deep learning methodologies discussed in this supplementary material include foundational works and specific applications in EEG analysis.


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\bibitem[S2]{Pancholi2022arxiv}
Pancholi, S.; Giri, A.; Jain, A.; Kumar, L.; Roy, S. Source Aware Deep Learning Framework for Hand Kinematic Reconstruction using EEG Signal. \textit{arXiv preprint arXiv:2103.13862}, 2022.

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\bibitem[S8]{Borra2020sincshallow}
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\bibitem[S9]{Ravanelli2018sincnet}
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\bibitem[S10]{Craik2019deepreview}
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\end{thebibliography}

\end{document}