Dear Reviewer Dcww,

We are grateful for your insightful and valuable comments. We have carefully addressed every concern you raised. Please see our itemized responses below.

1. W1: The integration of scalp EEG and intracranial EEG (iEEG) represents an innovative contribution, supported by a well-developed theoretical framework. However, the use of state-space models and Expectation-Maximization (EM) algorithms is less novel, as similar approaches have been explored previously. For instance, a 2016 NeurIPS paper, "A State-Space Model of Cross-Region Dynamic Connectivity in MEG/EEG," proposed a one-step state-space model with EM algorithms for single-modality data, indicating that the primary innovation of the current approach is somewhat limited.

We appreciate the reviewers' thoughtful feedback and the opportunity to clarify our work's contributions. Our method introduces a novel framework for estimating whole-brain electrophysiological networks by integrating EEG and iEEG through EM. We highlight the key advantages and distinctions of our approach from the Yang et al. 2016 (NeurIPS 2016 paper) below.

The distinguishing features of our method are:

1. Multimodal data integration: Unlike the Yang et al. 2016 paper that relies only on MEG/EEG data, our approach harnesses the complementary advantages of EEG and iEEG. This fusion significantly enhances connectivity estimation, particularly for deep brain structures, making it highly applicable to clinical scenarios.

2. Comprehensive whole-brain modeling including cortical and subcortical spaces: Our method directly models the entire source space based on the Harvard-Oxford atlas (N = 69), capturing global brain network dynamics, an extension from the important work of Yang et al. 2016 with predefined regions of interest with the experiments on two brain regional activation. The inclusion of iEEG electrodes made the whole brain network partially observable, thus its estimation at whole brain scale is achievable. Our whole-brain perspective is essential for studying intricate cognitive processes and neurological disorders requiring comprehensive network analysis at the whole brain scale.

3. Principled algorithm under the new framework of noninvasive and invasive source integration: We combine EM with FIS, incorporating forward filtering and backward smoothing to refine source signal estimates. Additionally, our implementation of sparsity priors effectively mitigates overfitting while reducing computational complexity, achieving efficient performance even for larger source spaces. Agreeably, we leveraged the EM optimization framework (most of the non-convex optimization problems falls into this umbrella) for the estimation of the state-space model, however, the EM algorithm is derived uniquely from our problem setup. We would like to highlight the innovations in modeling and practical applications of this framework, as the EM framework is a practical approach to solving this problem. We also identify similar EM frameworks for state and parameter estimation of state-space dynamic systems (Yang et al. 2016 NeurIPS, and Lamus et al. 2018 NeuroImage, and Sanchez Bornot et al. 2024 NeuroImage). It is the problem settings that make different variants of EM algorithms.

We believe these contributions represent a meaningful step forward in the field. We will revise our discussion to better highlight the novelty and contrast with prior work. We will acknowledge early important work, including Yang et al 2016 NeurIPS, and Lamus et al. 2018 NeuroImage, and Sanchez Bornot et al. 2024 NeuroImage, all used EM algorithm based on state-space models. We hope this addresses the reviewer’s concern and underscores the innovative aspects of our approach.

4. Enhanced Robustness and Clinical Applicability: Through multimodal data integration and sparsity constraints, our method reduces dependence on restrictive assumptions that may limit the previous approaches. This enhanced robustness ensures adaptability across diverse scenarios, particularly in clinical environments where precise connectivity assessments are critical.

2. W2 & Q1: The baseline algorithms used for comparison, such as MNE (1994), sLORETA (2002), dSPM (2000), and eLORETA (1997), are somewhat dated. Including comparisons with more recent methods could strengthen the experimental results and provide a more comprehensive evaluation of the proposed framework's performance relative to current state-of-the-art approaches. Could the study include comparisons with more recent methods in its experimental evaluation? Adding such comparisons might offer a more comprehensive assessment of the proposed framework’s performance against current state-of-the-art approaches, further strengthening the findings.

We thank the reviewer for this valuable feedback regarding the inclusion of more recent baseline methods in our experimental evaluation. We completely agree that incorporating more recent state-of-the-art methods would significantly strengthen our experimental evaluation and provide a more comprehensive assessment of our framework's performance. In the revised manuscript, we will add a comprehensive comparison with recent state-of-the-art (SOTA) methods. We expanded the baselines to include recent SOTA methods, which are illustrated in the Table below. The result shows that our methods consistently outperform both SOTA and classical baselines.

Table: Evaluation of performance with different levels of scalp EEG SNR.

References:

[1] Yektaeian & Makkiabadi, Accelerated algorithms for source orientation detection and spatiotemporal LCMV beamforming in EEG source localization. Frontiers in Neuroscience, 18, 1505017. 2025.

[2] Wan et al. Electrophysiological brain source imaging via combinatorial search with provable optimality. AAAI 2023.

[3] Liu et al. Imaging brain extended sources from EEG/MEG based on variation sparsity using automatic relevance determination. Neurocomputing, 389, 132-145, 2020.

Thank you for your constructive feedback. We sincerely hope our responses have addressed all concerns and demonstrate this work's important contributions to whole-brain electrophysiological analysis through innovative multimodal integration. We would very much appreciate it if you could reconsider your evaluation.

Dear Reviewer r5Ha,

We sincerely appreciate your time evaluating our paper, highlighting the importance of our 'integration of scalp EEG and iEEG' for 'whole-brain electrophysiological networks inference', and recognizing the 'reliability and practical applicability' of our validation approach.

1. Reply to W1:

Thank you for the thoughtful feedback on the manuscript’s structure and clarity. We agree that the current version leans too heavily on mathematical formulations. To improve readability, we’ll move the more technical material to the Appendix. Specifically, detailed derivations (Eqs. 5–12, 20–22) are moved to the Appendix.

2. Reply to W2:

We thank the reviewer for the insightful suggestion regarding enhancing the discussion of our findings. We will incorporate a discussion section that offers a detailed interpretation of the results and add the potential clinical impact of this research..

“Given that working memory (WM) is a resource-limited system, the brain likely engages mechanisms to optimize attention, temporary storage, and manipulation of task-relevant information [1, 2]. Our findings support prior research showing that the basal ganglia, particularly the caudate nucleus, play a key role in verbal WM. The sustained activation of the caudate during encoding suggests involvement in goal maintenance and distraction suppression, functions often impaired in conditions such as ADHD, schizophrenia, and other neuropsychiatric disorders [3, 4]."

References:

[1] Chang et al. Temporal dynamics of basal ganglia response and connectivity during verbal working memory. Neuroimage, 34(3), 1253-1269, 2007.

[2] Moore et al. Bilateral basal ganglia activity in verbal working memory. Brain and language, 125(3), 316-323, 2013.

[3] Castellanos, & Proal. Large-scale brain systems in ADHD. Trends in Cognitive Sciences, 16(1), 17–26, 2012.

[4] Minzenberg et al. Meta-analysis of executive function in schizophrenia. Archives of General Psychiatry, 66(8), 811–822, 2009.

3. Reply to W3:

We thank the reviewer for this insightful comment. In our original submission, we conducted a systematic analysis of our model's performance under varying conditions, including different numbers of sources, different levels of observable brain nodes, different noise levels, and network complexities (Appendix A4). As our framework is memoryless and fits the state-space model independently to each iEEG + EEG episode, increasing the sample size does not directly improve estimation accuracy for a single episode. Instead, a larger sample size may improve the reliability of group-level inferences or the robustness of summary statistics derived across episodes, but it has limited effect on the accuracy of individual model fits.

As shown in the real data experiments, aggregation of multiple sessions of EEG+iEEG data will enable group level statistics on the difference of cortical-subcortical information flows during the working memory tasks. For group level analysis (on behaviors, pathological patterns etc.), the sample size should be justified with sufficient statistical power.

To provide a better understanding of algorithm complexity, we will detail it in Appendix.

“The computational complexity of the proposed method is dominated by the FIS in the E-step. The forward Kalman filtering requires matrix operations at each time step, including the covariance update and matrix inversion, both involving O(N^3) operations, where N is the number of source regions.”

4. Reply to Q1:

We appreciate the reviewer's thorough assessment. In our experiments, we systematically evaluated different proportions of iEEG data, including 0% iEEG coverage, which means no iEEG signal is used for the brain network estimation. The results confirm that our approach benefits from both the information provided by iEEG and the effectiveness of integration strategy using our one-step method. To improve the clarity, we will add specific comments on 0% iEEG, meaning not using iEEG signal in the inference framework.

5. Reply to Q2:

We thank the reviewer for this thoughtful question. We conducted a comprehensive analysis of connectivity pattern consistency to validate the reliability of our derived brain connections across different subjects and experimental sessions using Pearson correlations.

Intra-subject consistency:

Encoding phase: 0.539 ± 0.062

Maintenance phase: 0.511 ± 0.043

Inter-subject consistency:

Encoding phase: 0.355 ± 0.037

Maintenance phase: 0.353 ± 0.059

These results reflect the expected balance between network stability and task-specific adaptability based on EEG data. The moderate intra-subject correlations are consistent with the functional flexibility of brain networks during Sternberg working memory tasks, where each trial involves a novel set of words followed by a recognition probe. Individual connectivity patterns are preserved across sessions while allowing for task-related adaptations to different verbal stimuli.

The inter-subject consistency reflects a biologically meaningful balance [1]. It demonstrates the presence of shared functional networks underlying verbal working memory while preserving the individual neural diversity that characterizes brain function [2]. This moderate correlation (compared to random correlation (~=0) between two vectors of 4761 dimensions (directed edges from a 69 by 69 matrix)) indicates our method successfully captures both the common neural mechanisms required for the task and the individual network signatures that contribute to cognitive variability across subjects.

References:

[1] Gratton et al. Functional brain networks are dominated by stable group and individual factors, not cognitive or daily variation. Neuron, 98(2), 439-452, 2018.

[2] Finn et al. Functional connectome fingerprinting: identifying individuals using patterns of brain connectivity. Nature neuroscience, 18(11), 1664-1671, 2015.

6. Reply to Q3:

Thank you for this thoughtful question. As shown in Figure 4, two-step methods often produce more connections, but many of these are likely to be spurious. This is largely due to issues like spatial leakage and volume conduction. Because these methods estimate source activity and connectivity in separate steps, errors from the first stage can propagate into the second. This can lead to artificial connections, especially between nearby regions, which inflate the total count but don't correspond to true underlying interactions [1,2].

In addition, two-step methods often require more aggressive thresholding to limit false positives, which can unintentionally filter out real connections. As a result, despite detecting more links overall, fewer true connections are correctly identified (based on our sensitivity metric).

References:

[1] Schoffelen, J. M., & Gross, J. Source connectivity analysis with MEG and EEG. Human brain mapping, 30(6), 1857-1865, 2009.

[2] Colclough et al. A symmetric multivariate leakage correction for MEG connectomes. Neuroimage, 117, 439-448, 2015.

7. Reply to Q4:

We thank the reviewer for this important question regarding our independence assumption for EEG and iEEG noise sources. We would like to clarify our rationale with supporting evidence from the literature.

While EEG and iEEG recordings may share some common noise sources, their impact differs substantially between the two modalities. For myoelectric signals, McMenamin et al. demonstrated that cranial electromyographic signals are significantly larger than typical EEG signals, causing significant contamination in scalp recordings [1]. However, multiple studies have shown that iEEG recordings are "significantly less affected by artifacts" compared to MEG/EEG, especially in the deep brain [2]. For example, Ball et al. emphasized that a principal advantage of invasive EEG that has been repeatedly emphasized is that it is less susceptible to artifact contamination [3].

As for eye movements, they create large-amplitude artifacts, particularly in frontal electrodes on scalp EEG. However, for iEEG in deep brain, the influence of this noise is minimal, due to the long distance from the noise source and the fact that artifacts such as electrocardiogram, movement artifacts, and skin potentials are significantly attenuated or even absent in sEEG recordings [4]. When we process the scalp EEG, we conduct artifact removal, which will further reduce the noise correlations between scalp EEG and iEEG.

Therefore, from a signal processing perspective, our independence assumption can be a reasonable approximation. Even when common noise sources exist, the impact on iEEG is significantly lower than that of the EEG due to physical factors. In this case, the cross-correlation between noise components in the two modalities becomes negligible, and the independence assumption is practically justified.

References:

[1] McMenamin et al. Electromyogenic artifacts and electroencephalographic inferences revisited. Neuroimage, 54(1), 4-9, 2011.

[2] Muthukumaraswamy, S. D. High-frequency brain activity and muscle artifacts in MEG/EEG: a review and recommendations. Frontiers in human neuroscience, 7, 138, 2013.

[3] Ball et al. Signal quality of simultaneously recorded invasive and non-invasive EEG. Neuroimage, 46(3), 708-716, 2009.

[4] Herff et al. The potential of stereotactic-EEG for brain-computer interfaces: current progress and future directions. Frontiers in neuroscience, 14, 123, 2020.

8. Reply to Q5 & Q6:

We thank the reviewer for the careful review. Section 3.1, line 196 should be 0% to 60%, and iEEG coverage rate indicates what proportion of the activation source regions is covered by iEEG.

We are grateful for your thoughtful feedback. We hope that our responses have adequately addressed the concerns and helped to highlight the significance of our work.

Dear Reviewer BLF3,

We sincerely appreciate your recognition of our "first principled framework for integrating invasive and non-invasive EEG" addressing "a critical methodological gap" with "promise for clinical applications".

1. Reply to W1 & Q1 about computational complexity

We appreciate the reviewer's valuable feedback. The revised manuscript will incorporate a section analyzing the computational complexity of our proposed approach. Specifically, we will include:

“The computational complexity of the proposed method is dominated by the FIS in the E-step. The forward Kalman filtering requires matrix operations at each time step, including the covariance update and matrix inversion, both involving O(N^3) operations, where N is the number of source regions.

To align with established work on brain network analysis using fMRI [1], we also adopted an atlas-based approach where we estimate region-to-region connectivity. In this work, we used Harvard-Oxford atlas regions (N = 69). The computational cost for the atlas-based analysis will remain stable regardless of different dimensions of source space. Practically, the brain network estimation is computationally feasible (about 3 minutes wall clock for 1000 time points).

[1] Ibrahim et al. Diagnostic power of resting‐state fMRI for detection of network connectivity in Alzheimer's disease and mild cognitive impairment: A systematic review. Human brain mapping, 42(9), pp.2941-2968, 2021.

2. Reply to W2 about lack of rigorous statistical treatment for real data analysis

We thank the reviewer for the valuable feedback. We believe our method can address these issues through well-established statistical techniques, as detailed below.

Firstly, to control the False Discovery Rate (FDR), we will apply the Benjamini-Hochberg procedure, maintaining an FDR of $q = 0.05$. For each Granger causality test, we will compute p-values using a de-biased test statistic derived through local linearization of VAR model coefficients [1,2]. These p-values will be sorted in ascending order, and we will reject null hypotheses where $p_{(k)} \leq \frac{k}{m} \cdot q$, with $m$ representing the total number of tests. This approach ensures effective control of false positives in high-dimensional neuroimaging data.

Complementarily, for null model testing, we will evaluate the null hypothesis ($A_i[j,k] = 0$) by comparing the full VAR model, which includes all potential causal coefficients, against a restricted model with specific coefficients set to zero. We will employ a Likelihood Ratio Test (LRT): $LRT = -2(\mathcal{L}{\text{restricted}} - \mathcal{L}{\text{full}})$, with the de-biased test statistic appropriately adjusted for high-dimensionality and sparsity to ensure robust inference [1,2].

We will validate these methods via cross-validation and bootstrap resampling to ensure rigorous statistical treatment, thereby addressing the reviewer's concerns regarding multiple comparisons and null model testing. The detailed implementation of these statistical procedures will be provided in the Appendix.

References:

[1] Roebroeck et al. Mapping directed influence over the brain using Granger causality and fMRI. Neuroimage, 25(1), 230-242, 2005.

[2] Soleimani et al. NLGC: Network localized Granger causality with application to MEG directional functional connectivity analysis. NeuroImage, 260, 119496, 2022.

3. Reply to W3 & Q3 regarding signal leakage and volume conduction mitigation

We thank the reviewer for raising this important point. While our current work discusses signal leakage and ghost interactions as a limitation, we acknowledge the need for more systematic evaluation, which has been an open and urgent problem in the field of EEG/MEG source imaging. In the current numerical simulations, evaluation metrics such as accuracy will implicitly reflect errors caused by ghost interactions. Since ghost interactions are fundamentally false positive connections in brain network analysis, we have conducted calculation of the false positive rate of the proposed model and the result is provided below:

Table: False positive rate of different methods under varying observation noise levels. (Brain network estimation using Granger causality.)

The result above showed our framework is better at handling false positive rates than all the benchmark algorithms. We provide the following rationales that make this framework have an advantage in reducing ghost interactions:

1. Integrated estimation: Unlike traditional two-step methods that propagate source localization errors into connectivity estimation, our joint optimization of source activity and connectivity reduces error accumulation.

2. Multimodal constraints: The incorporation of high-fidelity iEEG observations ($z_t = Cx_t + \epsilon$) provides ground-truth-like constraints that are less affected by volume conduction, as demonstrated in our results, where traditional two-step methods showed "highly dense brain networks" due to over-diffused source estimation.

3. Sparsity-induced regularization: We have already implemented L1 regularization on the state transition matrix (as shown in Eq. 4) to penalize weak or artificial connections that may arise from volume conduction effects, thereby promoting sparse and biologically plausible connectivity patterns.

Building upon our existing framework, additional strategies might be helpful and will be conducted in future work. For example, we will implement DTI-derived and/or fMRI-derived structural-functional connectivity as a prior condition to enhance the estimation accuracy.

References:

[1] Yektaeian et al. Accelerated algorithms for source orientation detection and spatiotemporal LCMV beamforming in EEG source localization. Frontiers in Neuroscience, 2025.

[2] Wan et al. Electrophysiological brain source imaging via combinatorial search with provable optimality. AAAI, 2023

[3] Liu et al. Imaging brain extended sources from EEG/MEG based on variation sparsity using automatic relevance determination. Neurocomputing, 2020.

4. Reply to W4 & Q4 regarding validation ground truth and envision of translation pathways

We sincerely appreciate the reviewer's insightful comments regarding localization error as a comparative metric. However, we would like to clarify that localization error is primarily suited for traditional Electrophysiological Source Imaging (ESI) methods, which focus on identifying the spatial positions of a few sparse, active brain sources, typically applied in clinical contexts such as epileptogenic zone localization or brain function mapping under evoked response potential (ERP) paradigms.

Our proposed framework represents a paradigmatic shift from sparse source localization to large-scale electrophysiological network characterization, enabling high-temporal-resolution analysis of brain network dynamics. This approach assumes distributed activation across all brain regions (supported by intracranial EEG evidence) rather than focal sparse sources. Given the theoretical constraints of the Restricted Isometry Property (RIP) and empirical limitations in accurately estimating numerous simultaneously active sources for connectivity analysis, traditional localization metrics become less relevant for whole-brain network estimation. Therefore, localization error does not align with our network-focused methodology. We will clarify these distinct application scenarios and the rationale for our methodological approach in the revised manuscript.

While direct ground truth validation presents inherent challenges in real clinical datasets due to the lack of definitive "gold standard" references, our results demonstrate strong convergent validity with previously published studies using the same dataset. For future translation pathways, we envision validation through: (1) prospective studies with expert clinical consensus as surrogate ground truth; (2) correlation analysis with established clinical biomarkers; (3) integration with existing clinical decision-support systems. These approaches would provide a pathway toward clinical translation while addressing the inherent challenges of ground truth validation in real-world clinical scenarios.

5. Reply to W5 concerning clarity

We thank the reviewer for the valuable feedback regarding the clarity. We acknowledge that the current presentation emphasizes mathematical rigor at the expense of conceptual clarity. To address the issue, we will move detailed mathematical derivations (Eqs. 5-12 for the EM algorithm and Eqs. 20-22 for the FIS method) to the Appendix, while prioritizing essential problem formulations and fundamental explanations in the main text. A summary algorithm is presented in Appendix A1.

6. Reply to Q2 about hyperparameter selection

We were using a grid search method on a simulated validation set for hyperparameter selection.

Thank you for the valuable feedback. We hope our clarifications have addressed the concerns and demonstrated the value of our contributions.

Dear Reviewer xCAQ,

We sincerely appreciate for your thoughtful and constructive feedback. We have carefully responded to your questions and suggestions. Please see our detailed point-by-point responses below.

1. W1: While the overall motivation of integrating EEG and iEEG is compelling, the narrative in the introduction would benefit from a clearer and more coherent progression. This paper initially positions fMRI’s high cost, limited temporal resolution, and non-portability as key limitations, suggesting the need for more accessible and temporally precise modalities like EEG or MEG. However, the subsequent introduction of iEEG, which is inherently more invasive, expensive, and clinically constrained than fMRI, seems to conflict with this premise. It would strengthen the manuscript if the authors could better contextualize the use of iEEG, clarifying its role not as a scalable alternative but as a complementary modality that enables partial ground-truth observation for improved modeling accuracy.

We thank the reviewer for the insightful observation about the narrative coherence issue in our introduction. We completely agree that our introduction needs to have a better explanation of iEEG's role and we agree with the reviewer that iEEG is a more expensive modality than fMRI. We will remove high-cost limitations of fMRI and only focus on its limited temporal resolution and as the secondary (metabolic) measurement modality for brain activities. We will emphasize that iEEG is not a scalable alternative but as a complementary modality for both EEG and fMRI.

Traditionally, regularization has to be implemented to ensure a unique solution to solve the source localization problem. However, when there are multiple active sources (which is usually true under the resting state or non-ERP settings), the Electrophysiological Source Imaging (ESI) accuracy is limited due to the Restricted Isometry Property (RIP) constraint.

The definition of RIP is that for a lead field matrix $\Phi \in \mathbb{R}^{m \times n}$ and sparsity level $s$, the RIP is mathematically defined as:

$(1 - \delta_s) ||x||_2^2 \leq ||\Phi x||_2^2 \leq (1 + \delta_s) ||x||_2^2$

where $\delta_s \in (0,1)$ is the Restricted Isometry Constant (RIC) and the inequality holds for all $s$-sparse vectors $x$. The RIP can be equivalently expressed using operator norms:

$\delta_s = \max_{|S| \leq s} ||\Phi_S^T \Phi_S - I||_{2 ->2}$

where $\Phi_S$ denotes the submatrix of $\Phi$ with columns indexed by set $S$, and $I$ is the identity matrix. In terms of eigenvalues, RIP ensures: $1 - \delta_s \leq \lambda_{\min}(\Phi_S^T \Phi_S) \leq \lambda_{\max}(\Phi_S^T \Phi_S) \leq 1 + \delta_s$

Cai et al.[1] rigorously proved that as the sparsity level $s$ increases, even optimal measurement matrices cannot maintain sufficiently small $\delta_s$, making accurate reconstruction fundamentally unreliable beyond a certain number of active sources. The realistic EEG leadfields often exhibit poor RIP behavior for moderate-to-high sparsity, which theoretically limits the reconstruction accuracy under multiple active source conditions, thus making whole brain level source localization also theoretically infeasible.

In addition to the RIP condition, recent discoveries in brain sciences highlight a network analytical perspective to view the brain dynamics. However, most of the whole brain network analysis is derived from fMRI or structural brain connectivity, the whole brain network analysis with high temporal resolution can be valuable to uncover the brain network dynamics for clinical applications and neuroscience studies. The integration of iEEG recording is to make the partial brain regions observable, thus improving the accuracy of estimation of whole brain electrophysiological networks, addressing both the scientific needs of WBEN analysis and technical challenges.

References:

[1] Cai *et al.*New bounds for restricted isometry constants. IEEE Transactions on Information Theory, 56(9), 4388-4394, 2010.

2. W2: Several derivations introduce redundancy and detract from the clarity of the proposed framework. These are encouraged to move standard or duplicated derivations to the appendix and instead provide more intuitive and conceptual explanations of how the proposed framework integrates scalp EEG and iEEG to infer whole-brain networks.

We are grateful to the reviewer for this valuable feedback regarding the clarity and presentation of our proposed framework. We agree that the current presentation prioritizes mathematical rigor over conceptual clarity. We will move the detailed mathematical formulations from the EM algorithm (e.g., derivations in Eqs. 5-12) and FIS method (e.g., derivations in Eqs. 20-22) to the appendix, thereby highlighting the most essential problem definitions and explanations.

Theoretically, we will add explanations as we mentioned in response to W1 about the RIP constraints and highlight our motivation to develop WBEN leveraging state-space models and pricinpled inference framework based on EM algorithm. Intuitively, we add hypothesis that by making some brain regions directly observable from iEEG electrodes, the whole brain electrophysiological network estimation is made easier and possible.

3. W3: The author mentions that the ground truth connectivity is based on a ‘generated’ state transition matrix Φ, but does not clearly explain how this matrix was constructed. Without an explicit description of how the state transition matrix Φ was generated, it’s difficult to assess if the experimental setup fairly evaluates the proposed method.

We thank the reviewer for pointing out the insufficiency of clarity regarding the construction of the connectivity matrix. The transition matrix in the simulation of this work was generated based on different parameters such as the number of sources and the complexity of the networks. We set the absolute value of the eigenvalues to be less than 1 to ensure system stability. This constraint guarantees that the system's state trajectories remain bounded over time and do not diverge to infinity, thereby generating realistic and physically plausible synthetic signals. The stability condition also ensures the existence of a stationary distribution for the state variables, which is essential for meaningful statistical analysis and reliable estimation of system properties. Then, we followed the Berlin Brain Connectivity Benchmark [1] methodology to generate synthetic causal signals with Gaussian noise.

For statistical analysis, we repeated the experiments with randomly generated connectivity matrices to ensure the validity and robustness of the proposed method. In addition to the randomly generated connectivity pattern, we will add more experiments with other derived brain connectivity patterns used in literature, either derived from fMRI or DTI (add reference), in the Appendix material.

References:

[1] Haufe, S., & Ewald, A. A simulation framework for benchmarking EEG-based brain connectivity estimation methodologies. Brain topography, 32(4), 625-642, 2019.

4. Q1: The authors compare their method with classical two-step ESI methods; however, the evaluation is limited to accuracy and sensitivity metrics only. In this work, is it not possible to include localization error as a comparative metric in the simulation experiments?

We appreciate the reviewer's suggestion regarding the inclusion of localization error as a comparative metric. While localization error is indeed a valuable metric for traditional ESI methods, it is used to characterize the distance of activated brain regions to the algorithm-reconstructed activation regions. The traditional ESI methods have great applications to clinical diagnosis and neuroscience studies when there are only a few sparse brain sources activated, primarily used for epileptogenic zone localization and brain function mapping under the evoked response potential (ERP) experimental setting.

However, we would like to highlight that the proposed framework represents a shift of paradigm from localizing active sparse brain sources to characterizing a large-scale of electrophysiological brain networks, which provides the delineation of brain network dynamics in high-temporal resolution. Due to the theoretical constraints of RIP and earlier practical evidence, estimating a large number of activating brain sources (step 1) and conducting connectivity measurement (step 2) has very limited accuracy to estimate the electrophysiological networks at the whole brain scale.

Given our emphasis on electrophysiological brain network estimation (assuming all brain sources are active, evidenced by intracranial EEG recordings at resting state), rather than measuring the ESI localization discrepancy of a few sparsely activated brain sources, we did not use the localization error as a measurement metric. Thanks to the reviewer, we will provide better clarity of the application context of this proposed framework.

Again, we sincerely appreciate your insightful comments and suggestions on the clarity of our presentation (W1, W2), details on experiments (W3), and suggestions on additional metrics (Q1), which can significantly improve the readability of this paper.

We appreciate your valuable comments. Given that it is the first principled framework integrating invasive EEG recording and non-invasive recording, and its unique value in characterizing the whole-brain electrophysiology of high temporal resolution with state-of-the-art accuracy, as well as our exploration of neuroscience investigation on the cortical-subcortical information flow, we request your kind reevaluation of our contribution.

Dear Reviewer vdqF,

We sincerely appreciate your time evaluating our paper, and highlighting our contribution as "particularly promising for clinical applications", and "use of EM to enable the "one-step" optimization is also an innovative method".

1. Reply to W1:

We thank the reviewer for these constructive comments. Firstly, we acknowledge the reviewer's suggestion for broader comparisons. We selected Granger Causality (GC) as our primary baseline because it serves as the gold standard in temporal causality analysis. However, we recognize the value of broader comparisons and we expand our benchmark to include Transfer Entropy (TE) [1] and Partial Directed Coherence (PDC) [2] for connectivity estimation. In addition, we have added additional recent SOTA methods for a more comprehensive comparison (see the updated tables below).

Additional experimental results for other control variables (e.g., number of activations) are presented in the revised Appendix A4 and are not shown here due to the space limit. The empirical findings demonstrate that our proposed methods consistently achieve superior performance compared to SOTA and classical baseline approaches, validated using commonly used connectivity estimation methodologies, including GC, TE, and PDC.

Table: Evaluation of performance with different levels of scalp EEG SNR. (Brain network estimation using TE method.)

(Due to limited space, std in all tables is removed in the rebuttal, but will be added in the appendix.)

Table: Evaluation of performance with different levels of scalp EEG SNR. (Brain network estimation using PDC method.)

For False Positive Rate (FPR), we have incorporated this metric, which can be found in the following Table. It indicates our proposed framework has the lowest FPR.

Table: FPR of different methods under varying observation noise levels. (Brain network estimation using GC.)

References:

[1] Schreiber, T. Measuring information transfer. Physical review letters, 85(2), 461. 2000

[2] Baccalá et al. Partial directed coherence: a new concept in neural structure determination. Biological cybernetics, 84(6), 2001

[3] Yektaeian et al. Accelerated algorithms for source orientation detection and spatiotemporal LCMV beamforming in EEG source localization. Frontiers in Neuroscience, 2025.

[4] Wan et al. Electrophysiological brain source imaging via combinatorial search with provable optimality. AAAI, 2023

[5] Liu et al. Imaging brain extended sources from EEG/MEG based on variation sparsity using automatic relevance determination. Neurocomputing, 2020.

2. Reply to W2:

We appreciate the reviewer's suggestion about further exploration on the practical impact and translational potential of our approach. In the paper, we explored the real-data of the simultenously recorded iEEG and EEG during the working memory task, and uncovered the information flows between the cortical and subcortical brain structures during the maintanence and recall phase, which is consistent with and posit as an important extension of recent neuroscience studies, as this study provides a unique view by characterizing the dynamics of whole brain networks. We agree that epileptic network modeling can be a direct translational application validation; however, due to the scarce availability of simultaneous EEG and iEEG and lack of clinical annotations of epileptogenic zones and their propagation pathways, it can be our future plan to validate this approach on epilepsy dataset. We will work with our clinical collaborators for the validation on patients with epilepsy in the coming years.

For a better explanation of the clinical indications of our finding, we have the following in the discussion section.

“Given that working memory (WM) is a resource-limited system, the brain likely engages mechanisms to optimize attention, temporary storage, and manipulation of task-relevant information [1, 2]. Our findings support prior research showing that the basal ganglia, particularly the caudate nucleus, play a key role in verbal WM. The sustained activation of the caudate during encoding suggests involvement in goal maintenance and distraction suppression, functions often impaired in conditions such as ADHD, schizophrenia, and other neuropsychiatric disorders [3, 4]."

References:

[1] Chang et al. Temporal dynamics of basal ganglia response and connectivity during verbal working memory. Neuroimage, 34(3), 1253-1269, 2007.

[2] Moore et al. Bilateral basal ganglia activity in verbal working memory. Brain and language, 125(3), 316-323, 2013

[3] Castellanos, & Proal. Large-scale brain systems in ADHD. Trends in Cognitive Sciences, 16(1), 17–26, 2012.

[4] Minzenberg et al. Meta-analysis of executive function in schizophrenia. Archives of General Psychiatry, 66(8), 811–822, 2009

3. Reply to Q1:

Yes, we used the Harvard-Oxford atlas, which includes 48 cortical regions and 21 subcortical regions.

4. Reply to Q2:

We thank the reviewer for the very insightful question. For the numerical experiments, we used random selection of iEEG electrodes in the source space to validate the algorithms without relying on a specific configuration of (1) iEEG montages and manufactures, (2) choices of surface electrocorticography or stereoelectroencephalography or a combination of both, (3) specific locations of iEEG electrodes, to insure the robustness and generalization of this proposed framework. In the real-data experiments, the iEEG locations are derived from the specific iEEG montage of patients with epilepsy.

For the simulated study, our framework is validated using a comprehensive analysis varying iEEG configurations with 2520 numerical experiments and up to 2.5 million time points. Our analysis included coverage rates ranging from 0% to 60% (with random electrode selection at each level), scenarios with varying numbers of active sources, and a comprehensive statistical assessment of method performance across all these configurations, with detailed results presented in Table 1 and Appendix A4.

We appreciate your insightful comments. We hope our clarifications have addressed your concerns and better conveyed the significance of our work.