Inference of Whole Brain Electrophysiological Networks Through Multimodal Integration of Simultaneous Scalp and Intracranial EEG
Authors: Shihao Yang
Feng Liu
Venue: NeurIPS 2025 poster ()
Keywords: Electrophysiological Brain Networks
State-Space Models
Multimodal Neuroimaging
Brain Connectome
Working Memory
Primary area: neuroscience_and_cognitive_science
TL;DR: First framework integrates scalp and intracranial EEG to estimate whole-brain networks via state-space models and EM algorithm, outperforming traditional methods and showing cortical-subcortical flows in working memory.

Abstract: Brain imaging research has transitioned over the past decades from identifying isolated regions of task-evoked activation to characterizing the spatiotemporal dynamics of large-scale brain networks. Electrophysiological signals are the direct manifestation of brain activity; thus, characterizing whole-brain electrophysiological networks (WBEN) can serve as a fundamental tool for neuroscience studies and clinical applications. In this work, we introduce a framework for integrating scalp EEG and intracranial EEG (iEEG) for WBEN estimation through a principled state-space modeling approach, where an Expectation-Maximization (EM) algorithm is designed to infer the state variables and brain connectivity simultaneously. We validated the proposed method on synthetic data, and the results revealed improved performance compared to traditional two-step methods using scalp EEG only, demonstrating the importance of including iEEG signals for WBEN estimation. For real data with simultaneous EEG and iEEG, we applied the developed framework to understand the information flows during encoding and maintenance phases of a working memory task. The information flows between subcortical and cortical regions are delineated, highlighting more significant information flows from cortical to subcortical regions during encoding than during maintenance. The results are consistent with previous research findings, but from a whole-brain perspective, which underscores the unique utility of the proposed framework.

Inference of Whole Brain Electrophysiological Networks Through Multimodal Integration of Simultaneous Scalp and Intracranial EEG
Shihao Yang
Feng Liu
Department of Systems Engineering Stevens Institute of Technology
Department of Systems Engineering Stevens Institute of Technology Hoboken, NJ 07030 fliu220stevens.. edu
Hoboken, NJ 07030 syang570stevvnns... edu
Abstract
Brain imaging research has transitioned over the past decades from identifying isolated regions task-evoked activation to characterizing the spatiotemporal dynamics of large-scale brain networks. Electrophysiological signals are the direct manifestation of brain activity; thus, characterizing whole-brain electrophysiolog ical networks (WBEN) serve fundamental tool for neuroscience studies and clinical applications. In this work, we introduce framework for integrating scalp EEG and intracranial EEG (iEEG) for WBEN estimation through a princi- pled state-space modeling approach, where an Expectation Mitiuttionn-aiizzzationn (EM) algorithm is designed infer the state riables and brain connectivity simultane- ously. We validated the proposed method on synthetic data, and the results revealed improved performance compared traditional two-step methods using scalp EEG only, demonstrating the importance of including iEEG signals for WBEN estima- tion. For real data with simultaneous EEG and iEEG, applied the developed framework to understand the information flows during encoding and maintenance phases of working memory task. The information flows between subcortical and cortical regions are delineated. highlighting more significant information flows from cortical to subcortical regions during encoding than during maintenance. The results are consistent with previous research findings, but from whole-brain perspective, which underscores the unique utility of the proposed framework.
Introduction
Brain networks represent the intricate and dynamic connectivity of neurons that facilitates communi- cation across different brain regions. These networks are essential for supporting cognitive functions, from basic sensory processing to complex Decision-making 21 Existing studies have suggested that accurately inferred brain connectivity patterns can help gain insights into the coordination and interactions between different brain regions [3] reveal the brain network underpinnings of cognitive processes, and uncover the mechanisms and biomarkers neuropsychiatric diseases [5] 6) Over the past decades, functional Magnetic Resonance Imaging (fMRI) has been the most widely used brain imaging modality for functional brain network modeling and analysis [7]. By analyzing fMRI data, researchers can explore the connectomes of the human brain across different cognitive tasks ( 9 consciousness states [10), degenerative diseases (II) and mental disorders (12). However, fMRI has limitations, including non-portability and low temporal resolution. which restrict its use in applica- tions that require characterization of instantaneous brain activity at sub-millisecond timescales [3]
CCoresponning author: Dr. Feng Liu, Email: fliu22 @ stevens.edu
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Ventoonaa network antiptx
peniolly cbsevoblee rodes
EGG TEAAANNTTNE
EIG meBuImmen
Figure 1: The overall pipeline of integration of EEG and iEEG for brain network reconstruction
Neuroimaging techniques with high temporal resolution, such as Magnetoencephallgrapyy (MEG) and Electroeneephalographyy (EEG), can be used measure electrical and magnetic brain signals, offering a direct measurement of brain activity rather than metabolic signals. To establish the electrophysiological connectome of the human brain, inferring whole-brain electrophysiolggcaa networks (WBEN) is important since it provides direct network-level delineation of brain connec- tivity. Existing studies that use MEG/EEG to reconstruct brain clectrophysiological networks have typically adopted two-step procedure, with EEG/MEG source imaging (ESI; Step 1) followed by brain connectivity measures (Step 2), such as amplltude coupling [4]. coherence [1], phase synchronization [1]. and Granger causality [17].
Existing ESI frameworks based on brain source localization suffer from low accuracy in estimating whole-brain activity across regions due to the ill-posedness of the inverse problem and are theoreti- cally limited by the Restricted Isometry Property (RIP, the definition provided in Appendix A..] [88 While a number of recent works have sought to alleviate these limitations through refined spatial priors and filtering strategics [19H21], the fundamental challenges remain significant. To address the limitations of two-step approaches, Yang et al. proposed one-step state-space model that jointly estimates source localization and dynamic connectivity by modeling ROI mean activities with time-varying autoregression (22]. Pirondini et al. developed computationally efficient algorithms combining spatial covariance estimation, linear state-space dynamics, and sparsity constraints, achiev- ing improved source localization performance with significant reduction in computation time through steady-state Kalman filtering [23]. However, estimating whole-brain networks poses additional chal- lenges for the ESI problem, further relaxes sparsity constraints
(24). More recently, Soleimani al. demonstrated that Granger causal links can be directly inferred from MEG measurements without an intermediate source localization step, achieving superior performance with low false alarm rates through integrated parameter estimation and statistical analysis (25). Additionally. Sanchez- -orroooooo al. introduced multiple penalized state-space models with novel algorithms based on backprop- agation, gradient descent, and alternating least squares, enabling simultancous solution of source localization and functional connectivity problems for thousands of cortical sources using data-driven regularization [26].
In addition non-invasive modalities, invasive neuroimaging technologies such intracranial Elec- troencephalography (iEEG), including Fllectrocorticographh (ECoG) and strrepelectrvenceehaaoggee phy (sEEG), which place subdural electrodes on the brain surface (ECoG) or penetrating electrodes in subcortical regions (sHEG), can achieve more accurate connectivity mapping among different regions of interest with high temporal resolution 27 28 Recent studies have leveraged simultancous scalp EEG and iEEG recordings to improve the reliability of elcctrophysiologicaa source imaging. For example, Jiao et al. proposed an explainable deep learning framework (XDL-ESI) that unrolls optimization algorithms into neural networks and achieves accurate, interpretable source localization validated on simultaneous EEG-iEEG data [29]. Despite these advances, iEEG recordings remain limited in spatial coverage due to their invasive nature, making the brain only partially observable through iEEG measurement. As iEEG electrodes can only cover part of the brain, important neural activity or connectivity patterns might be undetected in other regions. For example, in the analysis of seizure onset zones epilepsy, Shu el al. used MEG/EEG source imaging and identified interictal spikes that are missed by iEEG [30], highlighting the value of synergy between scalp EEG and inva-

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sive iEEG. The partial observability of iEEG and the challenges of localizing networked brain sources from EEG motivate the integration scalp EEG and iEEG as complementary modalities. To address the above issues, we propose to integrate scalp EEG and iEEG to provide accurate delineation of electrophysoologica activation and connectivity at the whole-brain scale. Integrating iEEG and scalp EEG can yield faithful reconstruction of WBEN, however, a principled multimodal integration modeling and inference framework has not been explored before. Early work leveraging state-space models solving the ESI and brain networks demonstrated significant potential [22 25 31. In this paper, we propose a new inference framework based on state-space dynamical systems and Bayesian inference that leverages multimodal fusion of scalp EEG and iEEG for WBEN estimation. This work represents a new and unified computational paradigm that integrates scalp EEG and intracranial recordings for whole-brain network inference, treating both modalities complementary observa- tions of shared underlying neural dynamics within rigorous Bayesian state-space paradigm. This framework enables comprehensive characterization of whole-brain electrophysioloiiaal networks when simultaneous recordings are available. The pipeline of the proposed approach is illustrated in Fig. D
2
Method
2.1 Basic problem definition
The linear discrete dynamic system of brain sources, as well as the linear model of EEG and iEEG observations, can be defined as:
PuTI-R I1,, to 1 T, E E
11 Lx, + We: V1 E R.
(I)
2, Ca, + ct 2 E R.
where N, M, and O are the number of the source regions, EEG electrodes, and iEEG electrodes, respectively. 4s E R NxN is the state transition matrix that delineates the impact the source state at time - k to n. RN, T4 (00, Q) is the noise in source state space which is assumned to be multivariate Gaussian distribution with mean O and diagonal covariance matrix Q. L E R MMN is the E RM w. (0, P) is the measurement noise in EEG observation which is
lead field matrix. also assumed to be a multivariate Gaussian distribution with mean 0 and a covariance matrix that assumed be known by measuring transformation matrix that selects the source signal where its region can be observed by iEEG directly. E R e (O, S) is the iEEG observational noise which is assumed to follow multivariate Gaussian distribution with mean 0 and covariance matrix that is also can be measured in a similar manner as P. Thus, according the model definition, the parameters that need be estimated are $ and Q. Define the unknown parameters as 0 [Φ, Q}. Then, the log-lik likeliiodooo can be written in the form:
realistic head model. And C E R is full-row rank
log (
The log-likelihood of the model can be defined as
logp(21.7, #1.7 (0), 0 == ( Q).
Since the number of observations substantially exceeds the number of sources, the inverse estimation problem is highly ill-posed. To alleviate this phenomenon and simplify the problem, a regularization term was added based on the assumption that the connection from all other regions to a given region is sparse. In this case, the regularized maximum ligeliiddoood of parameters for the model can be defined as
N
8" arg max logp(y1:r. 31r
A
Σ
1 P..l1
where Tn. is the th row of the state transition matrix. À is a regularization weight for model estimation that can be decided manually according to the experience by grid search [25).
3

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2.2 An Expeciatoon-Maximizttonn estimation framework
In the domain of statistical methodology, the Expectation Expectaton-Maxxiizztionn (EM) algorithm serves as an iterative computational approach employed to determine the maximum likelihood, whether it pertains to local optima or the maximum of posterior estimations (MAP) of parameters in the context of statistical model. These models, in particular, depend latent variables hidden from direct observations, underscoring the importance of this sophisticated approach. The EM is characterized by iterative execution, with pivotal steps: firstly, the Expectation (E) step, where Q-function is calculated to encapsulate the expected value of the log-likelihood Secondly, the Maximization (M) step, during which optimal parameters are derived to maximize the anticipated log- -iglliielihood obtained during the E step. Consequently, these derived parameters play an important role in illustrating the distributional characteristics of the latent variables prepare for the subsequent E step within the iterative loop. Since the data distribution in the source domain is unknown, the Expectation- Maximization framework takes the source state as the latent variable, which can be applied to find the optimal estimation of the logelikel and then obtain the approximated state in the source domain with the problem defined above.
We first illustrate the E-step by starting from 2 and using the facts that the observations of EEG and iEEG conditional independent the source state The log- lellikelihood can be rewritten and derived by introducing source state 11.7 as
logp(yı::: F1.T F1.TT 10) log Dogp(yiT: #1.T/ 0) + p(11((1)))) plogg(yy:r/pggg M1: 0) + log (( 0) + log p(r:((1))))))
(5)
The first and second terms of the right- hand side of the second row can be easily obtained based on the Gaussian noise assumption while source na is given which are
T
log Dog((vir/arg FI.T, 0) log plu. 0) _ log (2
Σ
1
Laallp-11
(6)
t=1
and
logp(sn.)
T (ogp(e = - (2g((
| Cxalls-1, Σ
where i the matrix determinant, and ||||] w xf W v is the quadratic form in the exponential term of multivariate Gaussian distribution. The last term can be obtained based on the linear dynamical model from (T). and utilize the presumption that Q is a diagonal covariance matrix with the items {o, i 1, l.....N N} on its diagonal, then we have
M 1
logp(a1110) logp(z.10) T [og(22
Σ
1x. o,
XO.
(8)
2 2=1
where X; = [ti 21
X [ I1 1, [ Fi,
IN.T!
EN N. I NN. x. "", and di [ k = 1,... K.j I,..., N. By substituting Eqs. (6- (8) into Eq. (). the formula can be reformulated, and then take the expectation to get the Q-
function for EM
T (0) = 10g PI)
1
log p(y 21:11,,,777.... Z1:7
2
1lve Laz 1
2
T log(22
Σ
||z
Cai
2
2
t=1
T (log(2m)
T N
N
Σ
log(
|زا
x6.11
(9)
2
2
2
i=1
i=1
T N
N
C.
Σ
log(o?)
Σ
|
x
2
2
0
i=1
1=1

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2(0; ) = p(1i., R1,T, 21.r (0),, 21.2, 8609
N
Cacn
T
log(o!019)
1
Σ
-x x1) 11)
(10)
2 1=1
2
1=1
Cocs 2
T
N
log(o, (1)
Σ
(E|x, 60²
x; (0)
IVI: 2IT, U
2
G)
1=1
-2 (1) x07x(10
(1)
V1.T S1., 16
(1)
(3)T
Eix(N 2 VI:T, F15T 0 (0)
+ oi
T
Cous
2 12T (P)
_ lIve La llp-1
(1I)
T 1/2 (ogg(2
1
|| Σ t=1
T
Cal
T lls - (2g()
2
where the bracket superscript represents the jth iteration in EM, and Cgus is constant term when O is given at jth iteration. We can find that the p(11:r #1:T, 0) is also Gaussian due to the Gaussian property on v, y: and = given 0 32 To find the Q-function, we can permute the equation and notice that the first and third expectation terms in the last row of the Eq. (T) consist of the (M, second-order moment of the density P(r1TT//::T 21:T, 0) while the second expectation term can be
expressed by the first-order moment of 1 VI.T, Z1:T, 0) whose mean well the covariance matrix can be estimated via Fixed Interval Smoothing (FIS), and the details of it will be listed in the next section. In the M-step, the optimal A { Q) that maximizes the Q-function defined above should be found. Since the number of observations is far less than that of the source regions, the inverse problem is ill ill-posed. The 4 regularization on + is introduced based the premise that the functional connection among a given region and others possesses sparse properties to reduce problem-solving difficulty. Thus, the equation of the maximization can be described with the form
)
(+++1 arg max (00 ( +
4 pl1,,
(12)
which can be efficiently addressed through the implementation of the Fast Adaptive Shrinkage/ Threesh olding Algorithm (FASTA). At this juncture, the EM)framework for source estimation has been established, providing a statistically rigorous foundation for the subsequent analytical procedures.
2.3
Source density estimation with FIS
Fixed interval smoothing is a statistical technique used in time series analysis and signal processing. It involves retrospectively estimating and improving the values of time series over fixed time intervals, considering both past and future observations. This method is particularly useful for reducing noise or uncertainty historical data and obtaining more accurate, smoothed estimates of the underlying trends or states within the time series. The principle of FIS consists of two parts: forward filtering and backward smoothing. The forward filter is executed to derive posterior estimates and covariances up to the given time t. Subsequently. the backward filter is applied to yield prior estimates and covariances, effectively extending the timeline backward to time or providing prior perspective in reverse chronology, in other words. Finally, the estimates and covariances derived from both forward and backward filtering at time are integrated to produce the ultimate estimation of the state and covariance matrix. Recall the main problem in Eq. (T). Considering computing performance factors, the fusion observation can be described by combining y and 2. The idea is that EEG and iEEG can be viewed as components of a unified electrophysiological system. Still, there is no correlation between the measuring noise of the two modalities, and the conditional density of merged observation is also Gaussian. Thus, observation is redefined as
Vr = Ltt + w.
(13)
5

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where
L+
L
L (a(+))))N
C
V1
vi E RM
(14)
2t
w
Wt
.w ~ N(O,, w/ ER(M+0)
E1
O P e E
p
((xxM)
S
logplvi:r, 01r = log peuplliinn poggiilrr + logglo11:11
N
C2 2
log(o
Σ
|ix; x6.
(15)
0,
1=1
C = 2 (ogg(2((*1))
1
Σ
llyt
L x0
p+ log(2
Then the log-likelihood problem can be transformed to Eq. (15), and the same transformation can be applied to Q-function. The estimation of the mean and the covariance matrix of p(z1.r//1177 F1.7, 0)
can be redefined as that of the p(I1.r / 0). And one can easily find that p(T1:r yyt 0) is
Gaussian, since for two jointly Gaussians, the conditional distribution is also Gaussian (32. Then,
the FIS can be applied to find the mean and covariance matrix for ((x1r1l9i 0). The forward Kalman filtering can give the estimation on p(z, lutte 0), Vt, while the backward Kalman smoothing can calculate the p(z, l 0), Vt. The merging of two estimates can generate the final estimate on p(z1 vit.m. 0), Vt. Next, referring to the estimation framework [25], we start with Vector Auto Regressor ( (VAR) generate the initial value for estimation. Since the source state time depends on the state of the former K time points, it is necessary redefine the augmented source state as
x, [zt; 12 Ft K+ 1 X, E (KN) and the augmented dynamic model as (22) transform VAR(K) problem into a VAR(1) one.
X, #
= -x,,_ + n,, Vt,
(16)
where
+ M)
$2
+
(K
K
0
0
0
R
KNNKN
0
((xxM)
(17)
0
11.
D. E
R KN
n. ~ (
Q
the augmented state transition matrix and disturbance in source, respectively. And Q is the covariance matrix for the augmented disturbance, which is also diagonal, whose diagonal values are the same variance Q and elsewhere. Then, the augmented version of merged observations is
y = x, + w Vt,
(18)
where
(L 0) L+ e R MXKN
(19)
In this way, the p(x, W 0) can be caleulated via FIS if the observations well as are known. Next, according to (33] one can define

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St E l8itzh
Eig1 = Cou(x, Wliea),
B / if 4 l
(20)
B ta Cov(X,, 3t, w(rr
Enjleg if 12
K2, t1 t2
Ky [t 4 2ll
as the mean, covariance matrix as well as the cross-covariance matrix for any given & and tg. Then, the model can be fitted with the FIS framework. We started with the initial value obtained via VAR in the forward filtering step. Then, for t 0,1,2,...
can have
/////1 1 X/jt
Eet1/t = + x + Q,
K/41 E411 7 [L E +
(21)
M441/+41 Kttjit + K44 (Vit1 L' X+t1jh
Eet1j++1 = E+11: - KK41 (L E t + +P*)
Taking the results from the filtering step, we can further do backward smoothing for = - 1,T 2,..., 0 as
K Edje 4 Σ 241jes
1 M:j: K (X++ITT Mr+1je);
(22)
Σg Enee K 11+ Σ,+1 K)
Then, the cross-covariance matrix can be obtained according to (0 and finally, simply extract the first N rows of X and the 4 order submatrix of the upper left corner of the matrix B +1,+ for V41, l2 1,,2,... T, which are exactly the first- and second-order moment of p(reltier 0), Vt, to finalize the E-step. The algorithmic pipeline is described in Appendix A.2
3 Results
To validate the added value of the integration of scalp EEG and iEEG in estimating the WBEN, we first conducted experiments with simulated data, and then we tested the proposed inference framework on the Sternberg verbal working memory task to explore the connectivity maps between cortical and subcortical regions during the encoding and maintenance phases (34
3.1 Numerical experiment on synthetic data
Realistic simulated data are generated and several baseline methods are used for comparison. Detailed experimental configurations are provided the Appendix A.3
Firstly, the impact of iEEG coverages of the partially observable brain regions (state variables) on the estimation of brain connectivity was evaluated. In the experiment, the source space state variables are partial observable with preseribed portion rendered by the iEEG electrodes. The coverage ratio was set to range from 0% to 50% with stepsize of 10%. The SNR levels for EEG and iEEG observations were set to be -ddB and 30dB, respectively. The network complexity was set with the number of activations as 10 and in-degree as 2. In the full factorial design of experiments, 10 repetitive simulations were conducted.
Validation on the added value using simultaneous scalp EEG and iEEG:
We use Acc and Sen (definitions provided in Appendix A.3 to evaluate the network estimation performance. The impact of the percentages of observable variables given in Fig. [ accompanied with the numerical results given in Table [] Not surprisingly, as the proportion of observable state space variables increases, both Sen and Acc score are increasing. It is worth noting that when the percentage of observable state space variables reaches 30%, there is significant improvement in both Ace and Sen, as shown in Fig. 3 The result shows that the performance of WBEN estimation can be significantly improved when appropriate amount of activated areas are observable from iEEG
7

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Figure 2: Evaluation of the Sen and Acc score varied with the percentage of observable brain regions on the WBEN estimation. The curve plots show the mean value Acc and Sen.
electrodes, which highlights the value of using simultancous recordings of scalp EEG and iEEG to characterize the whole brain network dynamics. An example with 30% partially observable brain nodes is shown in Fig. When using two-step methods, due to inaccuracy caused by classical ESI approaches, the over-diffused source estimation results in highly dense brain networks. When using one-step approach with 0% of observable brain regions (without iEEG electrodes) [25]. the performance 0.461 for sensitivity and 0.189 for accuracy respectively, which a significant amount of false positive predictions.
D-3 -00335
Figure 3: The significance difference on accuracy and sensitivity between 0% and 30% source observation using group t-test.
Table 1: Evaluation of the percentage of observable brain regions on the WBEN estimation.
Metrics
0%
10%
20%
30%
40%
50%
60%
Sen
0.46
0.517 = 0.042
0.356 + 0.070
0.639
0.690
0.147
0.131
Acc
0.189
0.176
0.179 = 0.161
0.180 = 0.174 0.545
= 0.224
0.657
0.194
0.687
0.184
0.735
0.201
Impact of SNR on WBEN estimation: The sealp EEG SNR is key factor for WBEN estimation. Since the SNR in the iEEG signal is known to be significant greater than that the EEG signal [35). we mainly evaluate the impact of scalp EEG SNR on the WBEN estimation. In the experiment, the SNR of iEEG signal was set 30dB with different levels of SNR for the scalp EEG signal. It is unknown how the performance of WBEN estimation will be improved when integrating the iEEG into the WBEN inference framework with a high level of noise presence in scalp EEG recordings. The SNR for the EEG signal was set from - 11OBB to 10dB 35 meanwhile, the percentage of observable brain regions from iEEG was set to be 30%, and the network complexity configuration is the same as in Sec 3.1 Table 2 shows the statistical result for the experiment. The pairwise comparison of 0% and 30% observable state variables is illustrated
Fig. 3 The group level t-test shows the significant difference when making 30% of state variables (brain regions) directly observable from iEEG electrodes, with a pronounced improvement in Ace score by reducing the false positive rates. From Table [ the classical two-step methods do not achieve satisfactory results even under high SNR conditions.
contrast, when employing the one-step state space modeling approach, per- formance remains comparable with either 0% or 30% iEEG observations given high SNR scalp FEG, highlighting the advantages of the one-step approach with state-space modeling paradigm for WBEN estimation. As SNR decreases with increased noise, Acc and Sen metrics deteriorate

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Figure 4: Visualization for network estimation in different methods, where the activated patch centers are highlighted with yellow color, and the iEEG observed patch centered are highlighted with green color. For two-step methods, the eLORETA that has the best performance is selected for comparison.
across all methods. However, performance using the one-step state space dynamic model degrades significantly, producing numerous incorrectly predicted connections at lower SNR values, whereas integrating iEEG measurements yields relatively stable performance curve. This finding further confirms the value of incorporating iEEG for more accurate WBEN estimation, particularly when scalp EEG channels are contaminated with high noise levels.
Further analysis on the false positive rate, the different causal analysis baselines, and the impact of brain network complexity on the WBEN estimation are given in the Appendix A.6 A.5 and A.7
respectively.
3.2 Cortical-subeortiial network analysis of working memory task
In this section, we analyzed the networks between cortical and subcortical brain regions during the Working Memory (WM) tasks. WM commonly associated with learning, understanding. executive functioning, information processing, intelligence, and problem-solving in humans and various animals from infancy to old age [39]. Maintaining content in WM requires communication between an extensive network of brain regions [40.. In this section, the proposed method was applied to estimate and analyze the brain connectivity between cortical and subcortical regions at different WM phases, which are the encoding phase and maintenance phase, in verbal WM task conducted by patients with epilepsy where simultancous scalp EEG and iEEG were recorded. The data description and preprocessing
detailed in Appendix A.4
The experiment used second-order vector auto-regressor model for the estimation framework stated in section 2.2 The regional connectivity of brain activity during the encoding and maintenance phases was estimated separately for each human participant's tasks. Lastly, the final estimation was obtained by first taking the sum over the estimated state transition matrices for each task and then averaging all task-leveled state transition matrices according to different phases. To ensure the stability of the result, only the connections from region i to that contains signal power no less than 10% of the signal power from region j to itself were kept, i.e., (i, > 0.1 * $ (j.j). Moreover,
Table 2: Evaluation of performance with different levels of scalp EEG SNR.
Method
Metrics
SNR=5
SNR=0
SNR=-5
S1
Sen
0.328 :: 0.131
0.253 1 0.149
0.162 : 0.120
0.170
0.19
MNE
Acc
0.013 + 0.006
0.011:0.000
0.008 : 0.004
0.006 + 0.003
Sen
0.199
0.144
0.203
0.15210.134 0.144
0.149
dSPM
Acc
0.010 + 0.008
0.011 + 0.007
0.007 + 0.003
0.006 + 0.003
Sen
0.255 0.160
0.269
2166520.11
00.18
SLORETA
Acc
0.011 + 0.006
0.011 0.007
0.008 = 0.004
0.006 0.004
Sen
0.442 - 0.101
0.111
0.383 1 0.135
0.363 1.114
cLORETA
0.013
0.006
0.012
0.006
0.005
0.010
ALCMV 36
Sen
0.427
0.134
0.412 0.145
0.384
Acc
0.002 0.001
0.002 + 0.001
0.002 + 0.001
0.002 0.001
Sen
0.113 + 01011
0.107
+ 0.013
0.125 + 0.022
0.098 + 0.013
ASTAR 37
Acc
0.003
0.001
0.002 1 0.001
0.003 1: 0.001
0.003 0.002
Sen
0.448
0.117
0.421 0.128
0.361 = 0.124
0.338
0.1148
VSSI-ARD 38
0.003
0.002
0.003
0.001
0.003 0.003
0.003 0.001
BEG with 0%
Sen
0.707
0.194
0.592 1 0.169
0.461 1: 0.155
0.402 0.132
iEBG obs.
Acc
0.429
0.292
0.253 1 0.245
0.189 1 0.176
0.034 1 0.023
EEG with 30% of
Sen
0.780
+ 0.136
0.716 + 0.115
0.639 + 0.120
0.491
+ 0.103
iEBG abs.
Acc
0.404 0.270
0.501 4: 0.269
0.545 1: 0.224
0.595 1 0.167

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you
0.
Figure 5: Cortical-succortical connectivity estimation analyses across encoding and maintenance phases in verbal working memory. A depicts the distribution of absolute connectivity strength be- tween cortical and subcortical regions during both phases based on averaged results, with significant differences confirmed by group t-test (t 3.359, p < 0.001). B illustrates the distribution of con- nection frequencies across phases derived from event-level analyses: left panel represents outgoing information flows from cortical regions (t 5.487, p < 0.001), while right panel shows outgoing information flows from subcortical regions (1 = -13.336, p < 0.001).
the HO atlas regions in the cortical area were further merged into lobe-level granularity, while caudate, putamen, and pallidum were aggregated as basal ganglia to further pursue robust macro-scale results. The estimated dynamic networks (Appendix A.9 revealed significant phase-dependent differences. During encoding, predominant information flow occurred from cortical to subcortical regions, specifically from frontal, temporal (including auditory processing areas), and parietal lobes to thalamus and basal ganglia. Meanwhile, -eteortiallooiggnaatngg connections primarily targeted other subcortical structures, with directed pathways from basal ganglia and hippocampus to thalamus, and from amygdala to basal ganglia and thalamus. Conversely, maintenance phase exhibited a reversed directionality pattern, with prominent connections from thalamus and basal ganglia to frontal and parietal lobes, and from hippocampus to temporal regions (including auditory processing areas). Statistical analyses in Fig. connfirmed significant differences in both directional connectivity and distribution patterns between subcortical and cortical regions across phases. We further analyzed the connectivity strength and the direction of information flow between cortical and subcortical regions during the encoding and maintenance phases. As shown in Fig. 5 the overall connectivity strength during the encoding phase is stronger than that of the maintenance phase. Besides, the amount of connection that flows start from cortical regions is generally greater during the encoding phase than the maintenance phase, whereas brain connections during the maintenance phase are more active in the information flow that is directed from subcortical regions to elsewhere.
4 Conclusion
We proposed the first unified computational framework for integrating simultancously recorded scalp EEG and iEEG data to estimate the whole-brain clectrophysiologic. networks. This framework enables the delineation of neurophysiological networks at a whole brain scale and with millisccond- level temporal resolution. Results validate the complementary value both modalities, demonstrating that strategic multimodal scalp EEG and iEEG deployment reconstruction accuracy even with high volume of measurement noise. Numerical experiments confirm the robustness under broad experiment configurations, while application to Sternberg verbal working memory task yielded insightful findings consistent with previous studies 44) particularly the characterization of subcoricall information flows during encoding and maintenance phase (40), which is the first analysis of this type at the whole brain scale. The proposed framework can serve fundamental computational tool for clectropyysiolgiic. brain network analysis with applications to clinical studies and neuroscience research.
significantly improve network
Acknowledgment
Research reported in this publication was supported by the National Institute of Neurological Dis- orders and Stroke (NINDS) of the National Institutes of Health (NIH), United States under Award Number R2INS13548) (PI: Liu). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
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A Appendix
A.1 Definition of restricted isometry property
The definition of Restricted Isometry Property (RIP) is that for lead field matrix + R and sparsity level 8, the RIP is mathematically defined as:
5x)
| | (1 5,) 8,)
(23)
where 6, (0, 1) is the Restricted Isometry Constant (RIC) and the inequality holds for all s-sparse vectors æ.
The RIP can be equivalently expressed using operator norms:
8, max 05.05 1ll2-22
(24)
|s S
where s denotes the submatrix with columns indexed by set S, and I is the identity matrix. In terms of eigenvalues, RIP ensures:
1 6- 6, S Amin ( (D545) S Amax (4) S $1+ 8g.
(25)
Cai et al. proved that as sparsity 8 increases, even optimal measurement matrices cannot keep de sufficiently small, making recovery unreliable [48). Since EEG leadfields often violate RIP at moderate- high sparsity, ESI accuracy degrades under multi-source conditions, motivating the use of strong priors or data-driven constraints.
A.2
Algorithmic framework and procedure
Algorithm EM Framework for Parameter Estimation
Require:
EEG measurement {u) E iEEG measurement {=1)T= leadfield matrix for EEG L iEEG C, estimated noise covariance matrix of EEG P and iEEG S, other parameters for VAR model, regularization, and halting condition.
I:
Integrate the two modalities get augmented measurement (vi ) leadfield matrix L+, and noise covariance matrix P+ Initialize parameter ((0) according to any conventional source localization method,
e.g.. MNE.
2: while the convergence halting condition is not met do
3: E-step: Compute Q(0; ((3) based on the conditional density p(r1:r wit ((3) estimated via FIS.
4: M-step: Maximize ((++1) argmax Q(00 (65)) A DE [| [11) using FASTA
5: j=j+1
6: end while
Ensure: Optimal parameter 6"
A.3
Simulated experiment settings
To generate synthetic EEG data, we used realistic head model compute the leadfield matrix based on the TI-MRI images from FreeSurfer [49]. The brain tissue segmentation and tissue surface generation were conducted using FreeSurfer. 128-channel BioSemi EEG cap layout was used, and the EEG channels were co-registered with the scalp surface and further validated on the MNE-Python toolbox [50]. The source space was parcelled and resampled. Then a three- layer boundary element method (BEM) was built based on the reconstructed surfaces, resulting in leadfield matrix denoted as
Brain Forward Model:
Realistic data generation: To generate causal time series, the Berlin Brain Connectivity Benchmark (BBCB) [5I is used with randomly generated state transition matrices 4 with & E {1,-.. K). To ensure the convergence of source signals, each eigenvalue in & is further validated to be less equal to I. Then we add an independent random Gaussian noise to each source signal at every time step. I Lastly, acausal third-order Butterworth filter with zero phase delay was applied with band- pass frequency being HH,, 40Hz] [52]. With the generated source signal, the observation time series can be derived by multiplying the leadfield matrix with the source signal and adding the channel-wist
22

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correlated random Gaussian noise with a given signal-to -oonallto-oee to-noise ratio (SNR) level. The iEEG noise is generated separately with a relatively higher SNR level (35].
Evaluation Metrics: In the simulated experiment, the ground truth of the connectivity network is defined based on the generated state transition matrix $. If the j, element ((J)) is zero, there is no link from i to j. otherwise there is link from i to j. In this experiment, K' is set to be I so that the state at time is only dependent on the state at time t 1. The Sensitivity (Sen) and Accuracy (Acc) metrics for connectivity estimation are defined to evaluate the performance given as:
Nc
No
Sen
Acc = No + N
(26)
Nt
where N₂ denotes the number of correct predictions, and Noot is the total number of connectivities, and Nw is the number of wrong predictions (number of false positives).
Benchmark algorithms: The baseline methods include traditional two-step methods, which con- ducted brain source localization first with existing ESI algorithms, such as MNE [53], sLORETA (54), dSPM (55), eLORETA (56] etc., followed by applying the Granger causality analysis on the estimated source signals for source space connectivity estimation. The observation signals were set with sampling rate 100Hz and a group of 10s time series were generated. In order to analyze the impact of the integration of iEEG, several hyperparameters, i.e., the number of partially observ- able brain regions, different levels SNR, and complexity of brain networks (which includes the number of activated source regions and the maximum in-degree of each region), were evaluated comprehensively.
Computational resources: All experiments are run on Windows 11 pro desktop with 32G memory, an Intel (9-12900KS CPU and an A6000 GPU of 48G memory.
A.4 Real Data Description and Preprocessing
The dataset comprises intracranial recordings from 15 epilepsy patients undergoing clinical moni- toring for seizure localization while performing modified Stermberg verbal working memory task. This paradigm temporally segregated encoding, maintenance, and recall phases. The comprehensive dataset includes simultaneous scalp EEG recordings following the 10-20 system, depth electrode iEEG recordings, and the corresponding MNI coordinates with anatomical labels for all intracranial electrodes (34. Each participant completed multiple experimental sessions, with 50 distinct events per session. Each event epoch consisted of an 8-second recording (sampled at 200 Hz for EEG and 2000 Hz for iEEG), structured as follows: fixation (0-1s), working memory encoding (1-3s), maintenance (3-6s), and response (6-8s). Our analysis specifically targeted the encoding and mainte- nance phases. To account for potential temporal extension of the auditory encoding process beyond the visual stimulus presentation, we analyzed only the final 2 seconds of the maintenance phase, consistent with methodological considerations outlined in prior research 40.
All iEEG and EEG electrodes were co-registered to the standard 'fsaverage" template ward model was calculated using MNF--Python with source spacing set to oct-5, generating over 2000 patches per hemisphere. Forward solutions were computed independently for each subject according to the standard 10-20 montage of their specific electrode configurations. For regional connectivity analysis, source locations were aggregated into anatomical regions defined by the Harvard-Oxfort (HO) atlas (48 cortical and 21 subcortical areas) using nearest-neighhor mapping in MNI space, thereby reducing the high-dimensional source space to more tractable atlas-based representation. iEEG channels were mapped to HO atlas areas based on electrode positions, with signals averaged across channels assigned to identical atlas regions. The iEEG leadfield matrix was constructed as a binary mapping, with unit values representing measured atlas areas and zeros elsewhere. In the absence of empty room recordings, we modeled EEG noise as multivariate Gaussian distribution (mean=0, standard deviation=1) and iEEG noise as a multivariate Gaussian with reduced variance (mean=0, standard deviation=0.1), Both EEG and iEEG signals were subsequently downsampled to 50 Hz.
57
The for-
A.5 Assessment of model performance with respect to different causal analysis under difffernt SNR
We further use transfer entropy and partial directed coherence for two-stage causal analysis baselines. 3and
As can be seen from Table 3 and 4 The proposed method is consistently better than basclines.
23

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Table 3: Evaluation of performance with different levels of scalp EEG SNR. The connectivity estimation for two-step methods is based on transfer entropy.
Method
Metrics
SNR=5
SNR=0
SNR==5
SNR=-10
Sen
0.255
0.093
0.255 1 0.093
0.255
0.093
0.255
0.099
MNE
Acc
0.002
0.001
0,002 1 0.001
0.002
0.001
0.002 0.001
Sen
0.000
01.000
0.000 + 0.000
0.008 + 0.024
0.008 + 0.024
DSPM
Acc
0.000 +: 0.000
0.000 4 0.000
0.033 1: 0.100
0.000 + 0.001
Sen
0.262 1 0.074
0.277 1: 0.043
0.292 1 0.046
0.296 1 0.053
SL.ORETA
Acc
0.002
0.000
0.002 1: 0.000
0.002 4 0.000
0,002 0.001
Sen
0.296
0.053
0.296 0.053
0.296 1: 0.053
0,291 0.058
ELORETA
Acc
0.004
0.001
0.003 + 0.001
0.003 + 0.000
0.002 + 0.000
Sen
0.140 + 0.104
0,166 + 0.095
0.212 + 0.113
0.232 + 0.103
AL.CMV 36
Acc
0.096 + 0.064
0.056 + 0.033
0.021 - 0.013
0.007 0.005
Sen
0.009 1 0.018
0.009 0.018
0.004 1 0.013
1.0013
ASTAR 37
Acc
0.014 0.031
0.014 + 0.031
0.005 0.014
0.004 0.011
Sen
0.087 0.065
0.067 0.058
0.066 0.053
0.062 0.056
VSSI-ARD 38
Acc
0.120 + 0.085
0.088 + 0.121
0.062 + 0.089
0.035 + 0.060
EFG with (%% of iEBG obs.
Sen
0.707 + 0.194
0.592 + 0.169
0.461 + 0.155
0,402 0.132
Acc
0.429
0.292
0.253 + 0.245
0.189 + 0.176
0.034 0.023
EEG with 30% of iEEG obs.
Sen
0.780 = 0.136
0.716
0.639 = 0.120
0.491 0.100
0.404 0.270
0.501 + 0.269
0.545 0.224
0.595 0.167
Table 4: Evaluation of performance with different levels of scalp EEG SNR. The connectivity estimation for two-step methods is based on partial directed coherence.
Method
Metrics
SNR=5
SNR=0
SNR==5
SNR=-10
Sen
0.306 0.151
0.217
0.150
0.110
MNE
Acc
0.011 + 0.009
0.011
0.007
0.012
0.011
0.017
DSPM
Sen
0.111 + 0.210
0.147
0.126 0.200
0.105
0.204
Acc
0.018
0.030
0.151 0.136
0.155 +: 0.122
0.129 0.118
Sen
0.270
0.194
0.184
0.175 0.191
0.221
SLORETA
Acc
0.007
0.006
0.006
0.003
0.007 + 0.004
0.005 + 0.006
Sen
0.467
0.170
0.431 0.143
0.366
0.163
0.417 0.168
ELORETA
Acc
0.019 0.011
0
0.007 0.002
0,006 1.0002
ALCMV 36
Sen
0.643 + 01.669
0.699
0.716+0.113
0.699
Acc
0.004 + 0.001
0.005
+ 0.000
0.005 + 0.001
0.005 + 0.001
Sen
0.078
0.042
0.068
1 0.060
0.055 1 0.039
0.064 1 0.048
ASTAR 37
Acc
0.038
0.023
0.034 1: 0.033
0.026 1 0.018
0.029 0.022
VSSI-ARD 38
Sen
0.648
01.255
0.658 + 0.22T
0.624 0.214
0.315 0.218
Avc
0.005
0.001
0.005 1 0.001
0.005 1: 0.001
0.004 1 0.001
EEG with 0% of iEEG obs.
Sen
0.707 1 0.194
0.592 1 0.169
0.461 1 0.155
0,402 0.132
Acc
0.429 + 0.292
0.253 1 0.245
0.189 + 0.176
0.034 0.023
EEG with 30% of iEEG obs.
Sen
0.780
0.136
0.716 1 0.115
0.639 1 0.120
0.491 0.100
Acc
0.404 0.270
0,501 1 0.269
0.545 1 0.224
0.595 0.167
A.6 Assessment of model performance using false positive rate
To address ghost interactions, which mainly represent false positive connections in brain network analysis, we calculated the false positive rate (FPR) of different methods. As shown in Table 5 the proposed method achieved significantly lower FPR compared other benchmark algorithms, demonstrating superior specificity in identifying true brain network connections while effectively suppressing spurious interactions caused by signal leakage.
Table 5: False positive rate of different methods under varying observation noise levels. The connectivity estimation for two-step methods is based on Granger causality.
Method
Metric
SNR,
SNR,..0
.N5..=-5
-NR...-110
MNE
FPR
0.154 + 0.273
0.156
+ 0.260
0.145
= 0.263
0.151
= 0.262
dSPM
FPR
0.129 1 0.276
0.137
1: 0.266
0.139 1 0.265
0.139
- 0.266
eLORETA
FPR
0.256 + 0.278
0.265
0.272
0.272
0.275
0.258
0.271
SLORETA
EPR
0.146 + 0.271
0.226
0.146
0.263
0.148
0.262
ALCMV [36]
FPR
0.643 1 0.225
0.700110.233
0.732
0.242
0.736
1 0.249
ASTAR [7]
FPR
0.172 + 0.071
0.228
0.188
0.085
0.165
0.092
VSSI-ARD [38
FPR
0.548 + 0.176
0.522:100117
0.197 0.468
0.371
0.194
BEG with 0% of
FPR
0.037 + 0.091
0.038 + 0.089
0.028
0.061
0.071 + 0.085
iEEG obs.
EEG with 30% of iEEG obs.
FPR
0.007 + 0.005
0.008 0.005
0.003 0.003
0.002 0.002
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A.7 Assessment of model performance with respect to network architectural complexity
Impact of brain network complexity on the WBEN estimation: The benefit of integrating iEEG and scalp EEG is further validated with different network complexity levels. The network complexity level is designed with varied number of activated source regions, and varied value of in-degree of node that effects the number of edges in the brain networks. The number of activated source regions set to 10, 15 and 20, while the in-degree was varied from I to with other parameters being the same as the previous experiments.
Table Evaluation of performance with different number of activated source regions
Method
Metrics
#Brain Regions=11
#Brain Regions=15
#Brain Regions=20
MNE
Sen
0.162 = 0.120
0.251 = 0.191
0.154 = 0.156
Acc
0,008 = 0.004
0.013 = 0.008
0.011 0.006
dSPM
Sen
0.152 1 0.134
0.254 0.195
0.208 0.211
Acc
0,007 = 0.003
0.012 0.009
0.010 0.005
SLORETA
Sen
0.165 0.123
0,272 = 0,189
0.206 0.214
Acc
0.008 = 0.004
0.013 = 0.008
0,010 = 0.005
eLORETA
Sen
0,383 = 0.135
0.390 + 0.169
0.389 + 0.146
Acc
0.010 + 0.005
0.009 + 0.005
0.011 + 0.005
EEG with 0% of iEEG observation
Scn
0.461 = 0.155
0,350 = 0.087
0.398 = 0.096
Acc
0.189 = 0.176
0.118 0.095
0.191 0.091
BBG with 30% of
Sen
0.639 = 0.120
0.518 0.110
0,547 0.093
iERG observation
Ace
0.545 + 0.224
0.500 0.172
0.552 0.182
Table 7: Evaluation of performance with different values of node in-degree
Method
Metries
In-degres=1
In-degree=2
In-degree=3
In-degree=4
MNE
Sen
0.117 0.043
0.162 1 0.120
0.230 1 0.197
0,304 0.259
Acc
0.005 0.002
0.008 1: 0.004
0.014 1 0.011
0.016 0.011
dSPM
Sen
0.121 = 0.037
0.152 0.134
0.423 1223000 0.294
0.504
Acc
0.005 + 0.002
0.007 + 0.003
0.011 = 0.010
0.011 0.009
SLORETA
Sen
0.131 + 0.058
0.165 4 0.123
0.428 : 0.292
0,511 0.229
Acc
0.005 0.002
0.008 + 0.004
0.010
CLORETA
Son
0.196
0.098
0.383 0.135
0.505 0.189
0.130
Acc
0.005
0.003
0.010 + 0.005
0.010 + 0.007
0.009 + 0.006
EEG with 0% of iEBG observation
Sen
0.570 0.194
0.461 1 0.155
0.401 = 0.198
0,422 0.2241
Acc
0.228
0.234
0.189 1: 0.176
0.149 1 0.201
0.109 0.166
EEG with 30% of
Sen
0.797 0.129
0.639 - 0.120
0.546 = 0.143
0.422
iEBG ohservation
0.697 0.248
0.545 + 0.224
0.498 0.159
0.410 +
Based on performance summarized in Table and and Table 7 the performance of the two-step methods is usually limited, with high false positive predictions. By integrating iEEG and scalp EEG in the WBEN estimation, both Acc and Sen showed improved performance compared with the comparison methods. The experiments further highlight the benefits of leveraging simultaneously recorded scalp EEG and iEEG for WBEN estimation.
A.8 Intra- and Inter-subject analysis
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.
The 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 working n-back memory tasks (each time different words are presented to the participants). 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 [58]. demonstrates the presence of shared functional networks underlying verbal working memory while preserving the individual neural diversity that characterizes brain function (59) 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.
25

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A.9
Visualization of connectivity estimation in real data experiment
A
B
Lacoding
Maintenance
Eaceding
Maintenance
C
Sucrtiill Connectrrty (Droding thone
Susoricil Commltirly (Rastonencs Mise
Figure 6: A visualization example of cortical-subcoriic connectivity estimation in working memory. Sub-figure shows the brain connectivity network in two WM phases according the HO atlas, while B is the network of aggregated lobe-leveled regions used for macroscopic expression. C is the circular brain connectivity graph in HO atlas parcellation for encoding and maintenance phases, respectively.
A.10 Approximated computational complexity
The computational complexity the proposed method is dominated by the FIS in the E-step. The forward Kalman filtering requires matrix operations each time step, including the covariance update and matrix inversion, both involving ( operations, where is the number source regions.
To align with established work brain network analysis using fMRI [60], we also adopted an basas- approach where we estimate
connectivity. In this work, we used the Harvard -Offord atlas regions (N 69). The computational cost for the atlas-based analysis will remain stable regardless of the different dimensions of the source space. Practically, the brain network estimation is computationally feasible (about 3 minutes wall clock for 1000 time points).
A.11
Limitations
Although this method one-step approach modeling source-space connectivity, false- cossitvve connections may still occur near true connections due to 'signal leakage' during source reconstruction
This problem occurs because estimating thousands sources from data collected by only hundred electrodes is inherently under-determined. As a result, residual signal leakage between locations will inevitably influence source data, causing false connectivities to appear as unintended artifacts of true interactions between source pairs. Morcover, the exact locations of these sources may
61
62
26

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be inaccurately determined. This issue, known as "ghost interaction," can be mitigated by novel method that organizes connections into hyperedges based on their adjaceney in signal mixing [63]
A.12 Safeguards
The code and result produced from our developed framework can give a delineation of electrophys- iological brain networks which can be used for surgical decision. However, epileptologists and neurosurgeons should use this as augmented tool for decision making and should double check the result with other brain imaging modalities.
27


Reviews:
[{"review_id": "fIslxCVHvo", "reviewer": "Reviewer_xCAQ", "summary": "This paper proposes the novel principled estimation framework based on the state-space model for estimating whole-brain electrophysiological networks (WBEN) by integrating scalp EEG and iEEG. The proposed method infers latent source dynamics and connectivity structures using an Expectation-Maximization (EM) algorithm, incorporating smoothing and sparsity-regularized optimization. Experiments on synthetic and real data show that including partial iEEG observations improves network estimation, especially under low signal-to-noise ratio of scalp EEG. The proposed method enables millisecond- temporal resolution reconstruction of cortical-subcortical information flow during cognitive tasks, offering a principled tool for brain network modeling in research and clinical contexts.", "questions": "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?", "limitations": "Yes, as mentioned by the authors, given the nonlinear nature of EEG and iEEG, the proposed method may still be susceptible to ghost interactions.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\nThis paper proposes a novel framework that integrates scalp EEG and iEEG for WBEN. Unlike classical two-step ESI methods, the proposed joint inference approach simultaneously estimates source dynamics and connectivity, leading to improved structural accuracy and connectivity mapping. Furthermore, the framework is empirically validated on both synthetic and real datasets, showcasing its utility in resolving large-scale brain connectivity patterns, particularly in capturing directional information flow between cortical and subcortical regions.\n\n**Weaknesses:**\n\n1. 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.\n\n2. 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.\n\n3. The author mentions that the ground truth connectivity is based on a ‘generated’ state transition matrix $\\Phi$, but does not clearly explain how this matrix was constructed. Without an explicit description of how the state transition matrix $\\Phi$ was generated,  it’s difficult to assess if the experimental setup fairly evaluates the proposed method.", "was_revised": true, "final_justification": "The authors have provided clear explanations for the points raised, and I believe that incorporating their responses will enhance the overall contribution of the paper. Therefore, I have decided to raise the score. However, I still consider the novelty somewhat limited, as the use of EEG and iEEG alone may not be sufficiently innovative. As such, I have given a score of 4.", "rebuttal": "Dear Reviewer xCAQ,\n\nWe 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.\n\n### **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.**\n\nWe 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. \n\nTraditionally, 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.\n\nThe 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:\n\n$ (1 - \\delta_s) ||x||_2^2 \\leq ||\\Phi x||_2^2 \\leq (1 + \\delta_s) ||x||_2^2 $\n\nwhere $\\delta_s \\in (0,1)$ is the Restricted Isometry Constant (RIC) and the inequality holds for all $s$-sparse vectors $x$.\nThe RIP can be equivalently expressed using operator norms:\n\n$ \\delta_s = \\max_{|S| \\leq s} ||\\Phi_S^T \\Phi_S - I||_{2 ->2} $\n\nwhere $\\Phi_S$ denotes the submatrix of $\\Phi$ with columns indexed by set $S$, and $I$ is the identity matrix.\nIn terms of eigenvalues, RIP ensures:\n$ 1 - \\delta_s \\leq \\lambda_{\\min}(\\Phi_S^T \\Phi_S) \\leq \\lambda_{\\max}(\\Phi_S^T \\Phi_S) \\leq 1 + \\delta_s $\n\nCai 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. \n\nIn 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**. \n\n**References:**\n\n[1] Cai *et al.*New bounds for restricted isometry constants. IEEE Transactions on Information Theory, 56(9), 4388-4394, 2010.\n\n\n### **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.**\n\nWe 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. \n\n**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. \n\n\n\n### **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.**\n\nWe 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.  \n\nFor 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. \n\n\n**References:**\n\n[1] Haufe, S., & Ewald, A. A simulation framework for benchmarking EEG-based brain connectivity estimation methodologies. Brain topography, 32(4), 625-642, 2019.\n\n### **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?**\n\nWe 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. \n\nHowever, 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.  \n\nGiven 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**. \n\nAgain, 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. \n\nWe 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."}, {"review_id": "GyNR9OjuJ2", "reviewer": "Reviewer_vdqF", "summary": "This paper introduces an innovative computational framework that simultaneously analyzes noninvasive scalp EEG and invasive intracranial EEG (iEEG) to reconstruct whole‑brain electrophysiological networks. By modeling the brain's electrical dynamics using a state‑space formulation and leveraging an Expectation‑Maximization (EM) algorithm, the authors jointly estimate both the latent regional source activity and the underlying effective connectivity across the entire brain from the combined EEG+iEEG data. This unified, one‑step Bayesian inference approach significantly outperforms traditional two‑stage methods in simulations, demonstrating markedly higher accuracy in identifying true neural connections. Furthermore, when applied to real-world recordings, the inferred connectivity patterns align closely with established cognitive neuroscience findings.", "questions": "* In your simulation study, what is the number of nodes used to represent brain regions? Did you use a specific brain atlas?\n\n* Regarding the selection of iEEG coverage and the number of sources: were these assigned entirely at random? It may be more realistic to incorporate anatomical or montage constraints, such as arranging iEEG electrodes in strips or grids. You can also simulate some special cases where iEEG is covering one or two sources but missing others.", "limitations": "YES", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strength\nOne of the key contributions of this work lies in its joint modeling of scalp EEG and intracranial EEG (iEEG). Many prior studies tend to use iEEG as the \"ground truth\" for sources and connectivity analysis, which leads to mis-identification of important sources and connections. This work leverages simultaneous EEG+iEEG data, which bridges the spatial coverage of EEG and the focal precision of iEEG in a principled inference framework, an approach particularly promising for clinical applications such as epileptic network mapping. The use of EM to enable the \"one-step\" optimization is also an innovative method that in theory is suitable to solve the problem.\n\n\nOn the other hand, the paper could be improved in the following aspects:\n\n* False positives in connectivity estimation should be included for a more comprehensive assessment. Additionally, the study benchmarks the proposed method only against Granger causality. More diverse connectivity estimation methods should be considered.\n* The practical impact of the method in real-world scenarios remains underexplored. Including additional simulation case studies that illustrate how the framework can successfully identify known cognitive pathways or epileptogenic sources would strengthen the paper’s relevance and translational potential.", "was_revised": true, "final_justification": "The additional results further demonstrate the value of the work, I would recommend acceptance.", "rebuttal": "Dear Reviewer vdqF,\n\nWe 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**\".\n\n\n### **1. Reply to W1:**\n\nWe 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). \n\nAdditional 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. \n\n#### **Table: Evaluation of performance with different levels of scalp EEG SNR. (Brain network estimation using TE method.)** \n\n(Due to limited space, std in all tables is removed in the rebuttal, but will be added in the appendix.)\n\n| Method                    | Metrics | SNR=5 | SNR=0 | SNR=-5 | SNR=-10 |\n| ------------------------- | ------- | ----- | ----- | ------ | ------- |\n| MNE                       | Sen     | 0.255 | 0.255 | 0.255  | 0.255   |\n|                           | Acc     | 0.002 | 0.002 | 0.002  | 0.002   |\n| DSPM                      | Sen     | 0.000 | 0.000 | 0.008  | 0.008   |\n|                           | Acc     | 0.000 | 0.000 | 0.033  | 0.000   |\n| SLORETA                   | Sen     | 0.262 | 0.277 | 0.292  | 0.296   |\n|                           | Acc     | 0.002 | 0.002 | 0.002  | 0.002   |\n| ELORETA                   | Sen     | 0.296 | 0.296 | 0.296  | 0.291   |\n|                           | Acc     | 0.004 | 0.003 | 0.003  | 0.002   |\n| ALCMV [3]                 | Sen     | 0.140 | 0.166 | 0.212  | 0.232   |\n|                           | Acc     | 0.096 | 0.056 | 0.021  | 0.007   |\n| ASTAR [4]                 | Sen     | 0.009 | 0.009 | 0.004  | 0.004   |\n|                           | Acc     | 0.014 | 0.014 | 0.005  | 0.004   |\n| VSSI-ARD [5]              | Sen     | 0.087 | 0.067 | 0.066  | 0.062   |\n|                           | Acc     | 0.120 | 0.088 | 0.062  | 0.035   |\n| EEG with 0% of iEEG obs.  | Sen     | 0.707 | 0.592 | 0.461  | 0.402   |\n|                           | Acc     | 0.429 | 0.253 | 0.189  | 0.034   |\n| EEG with 30% of iEEG obs. | Sen     | 0.780 | 0.716 | 0.639  | 0.491   |\n|                           | Acc     | 0.404 | 0.501 | 0.545  | 0.595   |\n\n\n#### **Table: Evaluation of performance with different levels of scalp EEG SNR.  (Brain network estimation using PDC method.)**\n\n| Method                    | Metrics | SNR=5 | SNR=0 | SNR=-5 | SNR=-10 |\n| ------------------------- | ------- | ----- | ----- | ------ | ------- |\n| MNE                       | Sen     | 0.306 | 0.217 | 0.150  | 0.115   |\n|                           | Acc     | 0.011 | 0.011 | 0.012  | 0.011   |\n| DSPM                      | Sen     | 0.111 | 0.147 | 0.126  | 0.105   |\n|                           | Acc     | 0.018 | 0.151 | 0.155  | 0.129   |\n| SLORETA                   | Sen     | 0.270 | 0.214 | 0.175  | 0.121   |\n|                           | Acc     | 0.007 | 0.006 | 0.007  | 0.005   |\n| ELORETA                   | Sen     | 0.467 | 0.431 | 0.366  | 0.417   |\n|                           | Acc     | 0.019 | 0.013 | 0.007  | 0.006   |\n| ALCMV [3]                 | Sen     | 0.643 | 0.699 | 0.716  | 0.699   |\n|                           | Acc     | 0.004 | 0.005 | 0.005  | 0.005   |\n| ASTAR [4]                 | Sen     | 0.078 | 0.068 | 0.055  | 0.064   |\n|                           | Acc     | 0.038 | 0.034 | 0.026  | 0.029   |\n| VSSI-ARD [5]              | Sen     | 0.648 | 0.658 | 0.624  | 0.515   |\n|                           | Acc     | 0.005 | 0.005 | 0.005  | 0.004   |\n| EEG with 0% of iEEG obs.  | Sen     | 0.707 | 0.592 | 0.461  | 0.402   |\n|                           | Acc     | 0.429 | 0.253 | 0.189  | 0.034   |\n| EEG with 30% of iEEG obs. | Sen     | 0.780 | 0.716 | 0.639  | 0.491   |\n|                           | Acc     | 0.404 | 0.501 | 0.545  | 0.595   |\n\nFor 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. \n\n#### **Table: FPR of different methods under varying observation noise levels. (Brain network estimation using GC.)**\n\n| Method | Metric | SNR_obs=5 | SNR_obs=0 | SNR_obs=-5 | SNR_obs=-10 |\n|-------------------------|--------|-----------|-----------|------------|-------------|\n| MNE | FPR | 0.154 | 0.156 | 0.145 | 0.151 |\n| dSPM | FPR | 0.129 | 0.137 | 0.139 | 0.139 |\n| eLORETA | FPR | 0.256 | 0.265 | 0.272 | 0.258 |\n| sLORETA | FPR | 0.146 | 0.147 | 0.146 | 0.148 |\n| ALCMV | FPR | 0.643 | 0.700 | 0.732 | 0.736 |\n| ASTAR | FPR | 0.172 | 0.228 | 0.188 | 0.165 |\n| VSSI-ARD | FPR | 0.548 | 0.522 | 0.468 | 0.371 |\n| EEG with 0% of iEEG obs | FPR | 0.037 | 0.038 | 0.028 | 0.071 |\n| EEG with 30% of iEEG obs| FPR | 0.007 | 0.008 | 0.003 | 0.002 | \n\n**References:**\n\n[1] Schreiber, T. Measuring information transfer. Physical review letters, 85(2), 461. 2000\n\n[2] Baccalá *et al.* Partial directed coherence: a new concept in neural structure determination. Biological cybernetics, 84(6), 2001\n\n[3] Yektaeian *et al.* Accelerated algorithms for source orientation detection and spatiotemporal LCMV beamforming in EEG source localization. Frontiers in Neuroscience, 2025.\n\n[4] Wan *et al.* Electrophysiological brain source imaging via combinatorial search with provable optimality. AAAI, 2023\n\n[5] Liu *et al.* Imaging brain extended sources from EEG/MEG based on variation sparsity using automatic relevance determination. Neurocomputing, 2020.\n\n\n### **2. Reply to W2:** \n\nWe 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. \n\nFor a better explanation of the clinical indications of our finding, we have the following in the discussion section. \n\n“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].\"\n \n\n**References:**\n\n[1] Chang *et al.* Temporal dynamics of basal ganglia response and connectivity during verbal working memory. Neuroimage, 34(3), 1253-1269, 2007.\n\n[2] Moore *et al.* Bilateral basal ganglia activity in verbal working memory. Brain and language, 125(3), 316-323, 2013\n \n[3] Castellanos, & Proal. Large-scale brain systems in ADHD. Trends in Cognitive Sciences, 16(1), 17–26, 2012.\n\n[4] Minzenberg *et al.* Meta-analysis of executive function in schizophrenia. Archives of General Psychiatry, 66(8), 811–822, 2009\n\n\n### **3. Reply to Q1:** \n\nYes, we used the Harvard-Oxford atlas, which includes 48 cortical regions and 21 subcortical regions.\n\n### **4. Reply to Q2:**\n\nWe 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. \n\nFor 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.\n\nWe appreciate your insightful comments. We hope our clarifications have addressed your concerns and better conveyed the significance of our work."}, {"review_id": "NAO7bh9mi8", "reviewer": "Reviewer_BLF3", "summary": "The paper proposes a novel and principled framework for estimating Whole Brain Electrophysiological Networks (WBEN) by integrating simultaneous scalp EEG and intracranial EEG (iEEG) using a state-space model with an Expectation-Maximization (EM) algorithm. The method addresses the ill-posed nature of source localization and connectivity estimation with EEG alone by leveraging complementary information from iEEG. Validation includes extensive simulations and application to a verbal working memory task, revealing meaningful cortical-subcortical dynamics.", "questions": "How does computational cost scale with source dimensionality?\n\nHow are hyperparameters (e.g., regularization weights, VAR order) selected?\n\nWhat strategies are planned to address spurious connectivity due to volume conduction?\n\nCan the framework be validated against known clinical targets (e.g., seizure foci)? What are the envisioned pathways to translation?", "limitations": "The paper acknowledges ghost interactions and computational demands but lacks quantitative analysis or mitigation plans. Some ethical aspects, especially regarding clinical deployment, are noted but not comprehensively addressed.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths: The paper introduces the first principled framework for integrating scalp EEG and iEEG in whole-brain electrophysiological network (WBEN) inference, addressing a critical methodological gap with clear implications for neuroscience and potential clinical use. It demonstrates strong methodological rigor through a statistically grounded EM algorithm based on state-space modeling, validated across multiple configurations, including varying SNR levels, network complexities, and degrees of iEEG coverage. The real-data application, focusing on verbal working memory, uncovers phase-dependent cortical-subcortical dynamics consistent with existing literature. Altogether, the work has high relevance for computational neuroscience and shows promise for future applications in clinical settings, particularly in epilepsy and neuropsychiatric disorders.\n\nWeaknesses: The paper does not sufficiently address the computational complexity of the proposed method, leaving unresolved questions about its scalability to large-scale neural data. Real-data analyses lack rigorous statistical treatment, including corrections for multiple comparisons and the use of null models. While signal leakage and ghost interactions are acknowledged, they are not quantitatively evaluated, and proposed mitigation strategies remain underdeveloped. Clinical applications are suggested but not demonstrated through validation against ground truth (e.g., seizure localization). Finally, the clarity of technical exposition suffers, particularly in the dense mathematical derivations, which would benefit from more accessible formulations or algorithmic pseudocode.", "was_revised": true, "final_justification": "The authors successfully addressed my primary concerns through additional analyses and clarifications. The computational complexity analysis demonstrates practical feasibility for clinical deployment, while their commitment to rigorous statistical procedures appropriately addresses multiple comparison concerns. Their new false positive rate analysis provides compelling evidence that their integrated EEG-iEEG framework substantially outperforms traditional methods in reducing ghost interactions, achieving remarkably low false positive rates. While their clarification of the methodological paradigm shift from sparse source localization to whole-brain network characterization is well-reasoned, the validation approach remains focused on traditional metrics without incorporating distributed evaluation methods that could be relevant for distributed source estimation techniques. Nevertheless, the technical innovation represents a significant methodological advance and merits acceptance.", "rebuttal": "Dear Reviewer BLF3,\n\nWe 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**\".\n\n### **1. Reply to W1  & Q1 about computational complexity**\nWe 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:\n\n“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. \n\nTo 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). \n\n[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.\n\n### **2. Reply to W2 about lack of rigorous statistical treatment for real data analysis**\nWe thank the reviewer for the valuable feedback. We believe our method can address these issues through well-established statistical techniques, as detailed below.\n\nFirstly, 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.\n\nComplementarily, 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].\n\nWe 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.\n\n**References:**\n\n[1] Roebroeck *et al.* Mapping directed influence over the brain using Granger causality and fMRI. Neuroimage, 25(1), 230-242, 2005.\n\n[2] Soleimani *et al.* NLGC: Network localized Granger causality with application to MEG directional functional connectivity analysis. NeuroImage, 260, 119496, 2022.\n\n### **3. Reply to W3 & Q3 regarding signal leakage and volume conduction mitigation**\nWe 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:\n\n#### **Table: False positive rate of different methods under varying observation noise levels.  (Brain network estimation using Granger causality.)**\n| Method | SNR_obs=5 | SNR_obs=0 | SNR_obs=-5 | SNR_obs=-10 | \n|--------|------------|------------|------------|------------|\n| MNE | 0.154 ± 0.273 | 0.156 ± 0.260 | 0.145 ± 0.263 | 0.151 ± 0.262 | \n| dSPM | 0.129 ± 0.276 | 0.137 ± 0.266 | 0.139 ± 0.265 | 0.139 ± 0.263 | \n| eLORETA | 0.256 ± 0.278 | 0.265 ± 0.272 | 0.272 ± 0.275 | 0.258 ± 0.271 | \n| sLORETA | 0.146 ± 0.271 | 0.147 ± 0.263 | 0.146 ± 0.263 | 0.148 ± 0.262 | \n| ALCMV [1] | 0.643 ± 0.225 | 0.700 ± 0.236 | 0.732 ± 0.242 | 0.736 ± 0.249 | \n| ASTAR [2] | 0.172 ± 0.071 | 0.228 ± 0.101 | 0.188 ± 0.085 | 0.165 ± 0.092 | \n| VSSI-ARD [3] | 0.548 ± 0.176 | 0.522 ± 0.179 | 0.468 ± 0.197 | 0.371 ± 0.194 | \n| EEG with 0% of iEEG obs | 0.037 ± 0.091 | 0.038 ± 0.089 | 0.028 ± 0.061 | 0.071 ± 0.085 | \n| EEG with 30% of iEEG obs | 0.007 ± 0.005 | 0.008 ± 0.005 | 0.003 ± 0.003 | 0.002 ± 0.002 |\n\n\nThe 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:\n\n1. **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.\n\n2. **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.\n\n3. **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.\n \n\nBuilding 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.  \n\n**References:**\n\n[1] Yektaeian et al. Accelerated algorithms for source orientation detection and spatiotemporal LCMV beamforming in EEG source localization. Frontiers in Neuroscience, 2025.\n\n[2] Wan et al. Electrophysiological brain source imaging via combinatorial search with provable optimality. AAAI, 2023\n\n[3] Liu et al. Imaging brain extended sources from EEG/MEG based on variation sparsity using automatic relevance determination. Neurocomputing, 2020.\n\n### **4. Reply to W4 & Q4 regarding validation ground truth and envision of translation pathways**\nWe 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.\n\nOur 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.\nTherefore, 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.\n\nWhile 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.\nFor 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.\n\n### **5. Reply to W5 concerning clarity**\n\nWe 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.\n\n### **6. Reply to Q2 about hyperparameter selection**\nWe were using a grid search method on a simulated validation set for hyperparameter selection.\n\nThank you for the valuable feedback. We hope our clarifications have addressed the concerns and demonstrated the value of our contributions."}, {"review_id": "ZiBv1s6KpF", "reviewer": "Reviewer_r5Ha", "summary": "The paper presents a new method to estimate the whole-brain electrophysiological networks using EEG and iEEG recordings. By iteratively optimising the unknown parameters under the framework of the EM algorithm, the method estimates the state transition matrix and the variance of the noise in the source space. The proposed method was evaluated on simulated and experimental data. Results on simulated data show that the method outperforms traditional two-step methods in estimating the brain connection under different noise conditions and network complexities. The validation on the experimental data shows consistent findings with previous studies on the cortical-subcortical information flow during encoding and maintenance phase. The method can be used to better understand brain networks and thus contributes to neuroscience studies.", "questions": "1. Could the authors provide comparison results when only using EEG or iEEG signals? My concern is that the improvement in sensitivity and accuracy compared with two-step methods may come from 1) additional information from iEEG or 2) the one-step method itself. Providing ablation studies by removing one modality of inputs would be valuable.\n2. Can the authors discuss the inter-subject and intra-subject consistency on the derived brain connections? It would be interesting to see if the pattern is similar across subjects during the same phase.\n3. Could the authors elaborate on why the sensitivity of two-step methods is quite low, even though there are more derived connections between the nodes, as shown in Figure 4.\n4. Could the authors explain more why the noise in EEG signals and iEEG signals are assumed to be independent? There are the same sources of noises, e.g., myoelectric signals and eye movement.\n5. Nit: Figure 2 shows the performance of the method with the coverage of brain regions between 0% and 60%; In section 3.1 line 196 it’s 0% to 50%.\n6. Nit: What’s the coverage of iEEG signals in the experimental data?", "limitations": "The authors have discussed limitations and safeguards in the supplementary.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. The method integrates scalp EEG signals and iEEG signals to estimate brain networks. Combining the two modalities covers the whole brain area and provides a more accurate recording in specific brain areas.\n2. The paper provides a detailed description of the proposed method, including the process of deriving mathematical formulas and algorithm implementation.\n3. The method is validated using both synthetic and experimental data, and was compared with multiple traditional two-step modelling methods. The authors generated synthetic data under different EEG SNR and brain network complexities, which confirms the reliability and practical applicability of the proposed method.\n\nWeaknesses\n1. The paper is packed with mathematical formulas. Moving the algorithmic framework at A.1 to the main and simplifying the derivations by moving some formulas to the appendix could make the paper more readable.\n2. The authors discussed the information flows between cortical and subcortical areas during the encoding and maintenance phases to interpret the estimated networks. Providing more details on how these findings align with or different from findings in previous studies would enhance the interpretability of the proposed method.\n3. The authors did not discuss if the performance of the proposed method will be affected by the amount of observation data. Providing the computational complexity of the method and discussing how the performance changes with the data size will be great.", "was_revised": true, "final_justification": "Thank you for your response. My questions have been well addressed. The analyses of connectivity pattern consistency within and across subjects are interesting. I suggest adding it to the appendix.", "rebuttal": "Dear Reviewer r5Ha,\n\nWe 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.\n\n### **1. Reply to W1:**\n \nThank 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. \n\n### **2. Reply to W2:**\n\nWe 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.. \n\n“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].\"\n\n**References:**\n\n[1] Chang *et al.* Temporal dynamics of basal ganglia response and connectivity during verbal working memory. Neuroimage, 34(3), 1253-1269, 2007.\n\n[2] Moore *et al.* Bilateral basal ganglia activity in verbal working memory. Brain and language, 125(3), 316-323, 2013.\n\n[3] Castellanos, & Proal. Large-scale brain systems in ADHD. Trends in Cognitive Sciences, 16(1), 17–26, 2012.\n\n[4] Minzenberg *et al.* Meta-analysis of executive function in schizophrenia. Archives of General Psychiatry, 66(8), 811–822, 2009.\n\n\n### **3. Reply to W3:**\n\nWe 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. \n\nAs 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. \n\nTo provide a better understanding of algorithm complexity, we will detail it in Appendix. \n\n“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.”\n\n\n### **4. Reply to Q1:**\n\nWe 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. \n\n### **5. Reply to Q2:**\nWe 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.\n\n**Intra-subject consistency:**\n\nEncoding phase: 0.539 ± 0.062\n\nMaintenance phase: 0.511 ± 0.043\n\n**Inter-subject consistency:**\n\nEncoding phase: 0.355 ± 0.037\n\nMaintenance phase: 0.353 ± 0.059\n\nThese 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.** \n\nThe 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.**\n\n**References:**\n\n[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.\n\n[2] Finn *et al.* Functional connectome fingerprinting: identifying individuals using patterns of brain connectivity. Nature neuroscience, 18(11), 1664-1671, 2015.\n\n### **6. Reply to Q3:**\nThank 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].\n\nIn 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).\n\n**References:**\n\n[1] Schoffelen, J. M., & Gross, J. Source connectivity analysis with MEG and EEG. Human brain mapping, 30(6), 1857-1865, 2009.\n\n[2] Colclough *et al.* A symmetric multivariate leakage correction for MEG connectomes. Neuroimage, 117, 439-448, 2015.\n\n### **7. Reply to Q4:**\n\nWe 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.\n\nWhile 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].\n\nAs 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.  \n\nTherefore, 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.\n\n**References:**\n\n[1] McMenamin *et al.* Electromyogenic artifacts and electroencephalographic inferences revisited. Neuroimage, 54(1), 4-9, 2011.\n\n[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.\n\n[3] Ball *et al.* Signal quality of simultaneously recorded invasive and non-invasive EEG. Neuroimage, 46(3), 708-716, 2009.\n\n[4]  Herff *et al.* The potential of stereotactic-EEG for brain-computer interfaces: current progress and future directions. Frontiers in neuroscience, 14, 123, 2020.\n\n### **8. Reply to Q5 & Q6:**\nWe 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.\n\nWe 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."}, {"review_id": "KCku42LGpT", "reviewer": "Reviewer_Dcww", "summary": "The paper introduces a novel framework for inferring whole brain electrophysiological networks (WBEN) by integrating simultaneous scalp EEG and intracranial EEG using a state-space model and an Expectation-Maximization (EM) algorithm. This approach estimates state variables and brain connectivity concurrently, addressing limitations of traditional two-step EEG/MEG source imaging methods, which suffer from low accuracy due to the ill-posed nature of source localization and are limited by Restricted Isometry Property (RIP) in theory.", "questions": "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.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "# Pros\n(1)\tThe paper clearly outlines the methodology, including the state-space model, EM algorithm, and Fixed Interval Smoothing (FIS), with mathematical formulations (e.g., Equations 2, 5, 9-10) and an algorithmic pipeline in Appendix A.1.\n(2)\tThe framework provides a tool for understanding brain networks in cognitive tasks (e.g., working memory) and identifying biomarkers for neuropsychiatric disorders, with potential applications in surgical decision-making for epilepsy (Appendix A.7).\n(3)\tComparative experimental results demonstrate strong performance, highlighting the effectiveness of the proposed EEG/iEEG integration framework. Ablation studies provide compelling evidence, further validating the contributions of individual components to the overall model’s efficacy.\n\n# Cons\n(1)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.\n\n\n(2) 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.", "was_revised": true, "final_justification": "After Rebuttal, I think the reviewer has solved some of my doubts, and I will raise the score", "rebuttal": "Dear Reviewer Dcww,\n\nWe are grateful for your insightful and valuable comments. We have carefully addressed every concern you raised. Please see our itemized responses below.\n\n### **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.**\n\nWe 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. \n\nThe distinguishing features of our method are:\n\n**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.\n\n**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.\n\n**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. \n\nWe 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. \n\n**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.\n\n### **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.**\n\nWe 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**.\n\n#### **Table: Evaluation of performance with different levels of scalp EEG SNR.**\n\n| Method                    | Metrics | SNR=5         | SNR=0         | SNR=-5        | SNR=-10       |\n| ------------------------- | ------- | ------------- | ------------- | ------------- | ------------- |\n| MNE                       | Sen     | 0.328 ± 0.131 | 0.253 ± 0.149 | 0.162 ± 0.120 | 0.170 ± 0.192 |\n|                           | Acc     | 0.013 ± 0.006 | 0.011 ± 0.005 | 0.008 ± 0.004 | 0.006 ± 0.003 |\n| dSPM                      | Sen     | 0.199 ± 0.144 | 0.203 ± 0.144 | 0.152 ± 0.134 | 0.149 ± 0.159 |\n|                           | Acc     | 0.010 ± 0.008 | 0.011 ± 0.007 | 0.007 ± 0.003 | 0.006 ± 0.003 |\n| sLORETA                   | Sen     | 0.255 ± 0.160 | 0.269 ± 0.159 | 0.165 ± 0.123 | 0.176 ± 0.189 |\n|                           | Acc     | 0.011 ± 0.006 | 0.011 ± 0.007 | 0.008 ± 0.004 | 0.006 ± 0.004 |\n| eLORETA                   | Sen     | 0.442 ± 0.101 | 0.412 ± 0.119 | 0.383 ± 0.135 | 0.363 ± 0.143 |\n|                           | Acc     | 0.013 ± 0.006 | 0.012 ± 0.006 | 0.010 ± 0.005 | 0.010 ± 0.005 |\n| ALCMV [1]                    | Sen     | 0.427 ± 0.134     | 0.390 ± 0.145     | 0.412 ± 0.161     | 0.384 ± 0.139     |\n|                           | Acc     | 0.002 ± 0.001     | 0.002 ± 0.001     | 0.002 ± 0.001     | 0.002 ± 0.001     |\n| ASTAR [2]                    | Sen     | 0.113 ± 0.011     | 0.107 ± 0.013     | 0.125 ± 0.022     | 0.095 ± 0.013     |\n|                           | Acc     | 0.003 ± 0.001     | 0.002 ± 0.001     | 0.003 ± 0.001     | 0.003 ± 0.002     |\n| VSSI-ARD [3]                      | Sen     | 0.448 ± 0.117     | 0.421 ± 0.128     | 0.361 ± 0.124     | 0.338 ± 0.148     |\n|                           | Acc     | 0.003 ± 0.002     | 0.003 ± 0.001     | 0.003 ± 0.003     | 0.003 ± 0.001     |\n| EEG with 0% of iEEG obs.  | Sen     | 0.707 ± 0.194 | 0.592 ± 0.169 | 0.461 ± 0.155 | 0.402 ± 0.132 |\n|                           | Acc     | 0.429 ± 0.292 | 0.253 ± 0.245 | 0.189 ± 0.176 | 0.034 ± 0.023 |\n| EEG with 30% of iEEG obs. | Sen     | 0.780 ± 0.136 | 0.716 ± 0.115 | 0.639 ± 0.120 | 0.491 ± 0.103 |\n|                           | Acc     | 0.404 ± 0.270 | 0.501 ± 0.269 | 0.545 ± 0.224 | 0.595 ± 0.167 |\n\n**References:**\n\n[1] Yektaeian & Makkiabadi, Accelerated algorithms for source orientation detection and spatiotemporal LCMV beamforming in EEG source localization. Frontiers in Neuroscience, 18, 1505017. 2025. \n\n[2] Wan *et al.*  Electrophysiological brain source imaging via combinatorial search with provable optimality. AAAI 2023.\n\n[3] Liu *et al.* Imaging brain extended sources from EEG/MEG based on variation sparsity using automatic relevance determination. Neurocomputing, 389, 132-145, 2020.\n\nThank 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."}]

Author rebuttal:

Meta-review: The work explores the combination of EEG and intracranial EEG for inferring statistical dependencies between brain regions (a.k.a. "brain connectivity" or "brain networks"). Paper covers a technical contribution based on state-space models combining the two modalities. Leveraging the improved spatial specificity of iEEG, authors demonstrate experimentally the improved performance of the method, which contributed to convincing all reviewers about the relevance of the work.
Decision: Accept (poster)