Spatially Masked Regression (SMR)

• Spatially Masked Regression (SMR) is a reconstruction framework that uses progressive spatial masking to isolate local redundancy from broader network information in neural data.
• The method applies a linear elastic-net model with L1 loss, reconstructing electrode signals from non-masked channels and controlling spatial locality via configurable exclusion masks.
• Empirical findings show that even with full masking, residual predictability remains, supporting potential applications in channel selection, artifact localization, and network mapping in EEG/iEEG studies.

Spatially Masked Regression (SMR) is a reconstruction-based framework that reconstructs a target electrode’s time series from the rest of the array while excluding a configurable spatial neighborhood around the target. By progressively increasing the exclusion mask, the method turns spatial locality into an explicit experimental control for quantifying how much predictive information remains after local neighbors are withheld, thereby separating local redundancy—driven by spatial smoothness and volume conduction—from broader distributed structure in the network. In the formulation introduced for electrophysiological recordings, SMR was applied to intracranial EEG (iEEG) with heterogeneous electrode coverage and to scalp EEG with standardized montages over sensorimotor cortex, and evaluated with distance correlation between original and reconstructed signals (Memar et al., 9 Jun 2026).

1. Conceptual basis and problem setting

SMR addresses a basic interpretive question in multichannel electrophysiology: to what extent does an electrode’s signal reflect local versus distributed information in the underlying system. The motivating observation is that neural recordings are often interpreted as local measurements, yet the signal at any one sensor can also reflect structured activity distributed across the broader network. In this setting, SMR reconstructs each electrode’s time series from the remaining electrodes while excluding a configurable neighborhood around the target, so that increasing the mask quantifies how much predictive information survives after nearby channels are withheld (Memar et al., 9 Jun 2026).

The framework is grounded in two facts stated explicitly in the source study. First, field potentials are spatially mixed through conductive tissues and exhibit frequency- and distance-dependent attenuation, so nearby sensors inherit strong, smooth correlations. Second, neural activity organizes into oscillatory and task-locked patterns that span larger-scale networks, so any sensor can jointly express local and network-level dependencies. Masking therefore functions as a controlled perturbation of the predictor set rather than as an arbitrary ablation. Nearby electrodes contribute strongly to reconstruction, but the design asks whether they account for all predictability.

In this sense, SMR is not primarily a source-separation method. It is an interpretable reconstructability assay whose main output is the change in held-out reconstruction quality as local predictors are progressively removed. This suggests that SMR is best understood as a diagnostic of the balance between local redundancy and broader distributed organization rather than as a direct estimator of anatomical connectivity.

2. Formal definition and optimization

For subject $s$ with $N_s$ channels and $T$ time points, let $X^{(s)} \in \mathbb{R}^{N_s \times T}$ denote multichannel data, and let $x_i^{(s)}(t)$ be the target time series. The instantaneous SMR model predicts channel $i$ at time $t$ from non-masked channels at the same time point through a linear map with elastic-net regularization and $L_1$ reconstruction loss (Memar et al., 9 Jun 2026):

$$\widehat{x}_i^{(s)}(t) = \sum_{j=1}^{N_s} (1 - K_{ij}^{(s)})\, w_{ij}^{(s)}\, x_j^{(s)}(t).$$

The optimization problem is

$$\min_{w_i^{(s)},\, b_i^{(s)}} \frac{1}{T}\sum_{t=1}^{T}\Bigg|x_i^{(s)}(t) - \sum_{j=1}^{N_s} (1 - K_{ij}^{(s)})\, w_{ij}^{(s)}\, x_j^{(s)}(t) - b_i^{(s)}\Bigg| + \lambda_1 \|\theta_i^{(s)}\|_1 + \lambda_2 \|\theta_i^{(s)}\|_2^2,$$

subject to $N_s$0 for all $N_s$1, where $N_s$2. Adam is used for mini-batch stochastic optimization with early stopping. The $N_s$3 term promotes sparsity, and the $N_s$4 term discourages excessively large coefficients.

The neighborhood $N_s$5 is defined as the set of $N_s$6 nearest electrodes by Euclidean distance in projected coordinates. For EEG, the paper uses in-plane scalp coordinates $N_s$7 under a standardized montage; for iEEG, it uses MNI coordinates projected to $N_s$8. The electrode itself is excluded. The binary mask is

$N_s$9

Progressive masking is implemented with mask intensity $T$0, where $T$1 channels are randomly masked from the neighborhood, plus the target itself.

A lagged variant extends the design to delayed predictors with $T$2 ms:

$T$3

where delays are converted to sample offsets by $T$4.

Performance is quantified by distance correlation (DistCorr) between original and reconstructed signals. For vectors $T$5, pairwise distances are computed as $T$6 and $T$7, double-centered, and combined into $T$8, $T$9, and $X^{(s)} \in \mathbb{R}^{N_s \times T}$0, yielding

$X^{(s)} \in \mathbb{R}^{N_s \times T}$1

The use of DistCorr rather than a purely linear metric is significant because it scores dependence between original and reconstructed traces without requiring the dependence to be only Pearson-linear.

3. Data, preprocessing, and evaluation protocol

The electrophysiological study evaluated SMR on two modalities. The iEEG data came from the AJILE12 dataset, comprising 12 subjects with single hemisphere coverage per subject and heterogeneous electrode positions and counts; trials were 2-second segments time-locked to upper-limb movements. The EEG data came from the Upper Limb Movements EEG dataset of Ofner et al. (2017), with 15 healthy subjects and 61 electrodes covering frontal, central, parietal, and temporal regions, including right-arm movements and motor imagery under standardized montages over sensorimotor cortex (Memar et al., 9 Jun 2026).

Preprocessing differed by modality. For iEEG, the pipeline included DC removal and linear detrending; 0.5–150 Hz Butterworth band-pass; 60 Hz notch; 120 Hz low-pass for residual high-frequency noise; artifact rejection by replacing samples with $X^{(s)} \in \mathbb{R}^{N_s \times T}$2 per channel with the channel median; Common Average Reference; per-channel z-scoring across time; and segmentation into 2-second trials centered on upper-limb movement events. For EEG, the paper reports band-pass filtering, notch at line frequency, per-channel z-scoring, and trial segmentation according to movement or imagery events.

The evaluation protocol separates intra-subject reconstruction from cross-subject transfer. Intra-subject analyses z-score data across trials, then use a 64/16/20% train/validation/test split with nested 80/20 random splits and early stopping based on validation loss. DistCorr is reported on held-out test trials and averaged across electrodes as mean ± SD. Cross-subject transfer uses a 50/50 validation/test split in the target subject. For EEG, the learned inter-electrode matrix from a reference subject is transferred directly because electrode positions are standardized. For iEEG, cross-subject comparisons use correlation-based electrode mapping (CBEM) with a rectangular assignment solved by the Hungarian algorithm. The paper defines the cross-subject correlation matrix as

$X^{(s)} \in \mathbb{R}^{N_s \times T}$3

followed by a binary one-to-one assignment matrix $X^{(s)} \in \mathbb{R}^{N_s \times T}$4 maximizing total matched correlation, and transfer of the learned matrix into the target electrode space.

This design is central to the interpretation of modality differences. EEG offers standardized positions and broader spatial mixing; iEEG offers more focal measurements but heterogeneous coverage, so portability of learned inter-electrode structure is inherently more constrained.

4. Main empirical findings

The principal quantitative findings concern within-subject reconstruction, cross-subject transfer, masking curves, and surrogate controls. Within subjects, SMR achieved mean DistCorr $X^{(s)} \in \mathbb{R}^{N_s \times T}$5 for EEG and mean DistCorr $X^{(s)} \in \mathbb{R}^{N_s \times T}$6 for iEEG. Cross-subject transfer was markedly stronger in EEG than in iEEG: with standardized montage and full local-neighborhood masking, EEG reached $X^{(s)} \in \mathbb{R}^{N_s \times T}$7, whereas iEEG with CBEM alignment, donor selection, and trial-averaged signals reached $X^{(s)} \in \mathbb{R}^{N_s \times T}$8 (Memar et al., 9 Jun 2026).

Lagged predictors changed little relative to the instantaneous model. EEG yielded $X^{(s)} \in \mathbb{R}^{N_s \times T}$9 versus $x_i^{(s)}(t)$0, and iEEG yielded $x_i^{(s)}(t)$1 versus $x_i^{(s)}(t)$2. Progressive masking produced a monotonic decrease in DistCorr as mask intensity increased from $x_i^{(s)}(t)$3 to $x_i^{(s)}(t)$4 in both modalities. However, performance remained above zero at full masking, demonstrating residual distributed predictability beyond the local neighborhood.

Setting	EEG	iEEG
Within-subject reconstruction	$x_i^{(s)}(t)$5	$x_i^{(s)}(t)$6
Cross-subject transfer	$x_i^{(s)}(t)$7	$x_i^{(s)}(t)$8
Lagged SMR	$x_i^{(s)}(t)$9	$i$0
Instantaneous SMR	$i$1	$i$2

The local-versus-non-local comparison sharpened this interpretation. Best performance was obtained with all electrodes, meaning local plus non-local predictors. Local-only outperformed non-local-only, indicating concentrated predictive structure in the immediate neighborhood, yet non-local inputs added complementary information when combined.

Surrogate analyses tested whether SMR performance could be explained by marginal distributions or stationary spectra alone. In iEEG, the original data yielded $i$3, phase-shuffled surrogates $i$4, IAAFT surrogates $i$5, and block-shuffled surrogates $i$6. In EEG, the corresponding values were $i$7, $i$8, $i$9, and $t$0. These marked reductions support the paper’s conclusion that preserving marginal distributions and spectra is insufficient: SMR depends on structured temporal and cross-channel organization rather than on marginal statistics alone.

5. Interpretation, applications, and limitations

The study interprets SMR as evidence that individual channels reflect both local redundancy and broader distributed structure. The masking results show that nearby electrodes carry the strongest predictive information, consistent with spatial smoothness and volume conduction, yet substantial residual predictability persists when local neighbors are excluded. The modality contrast is equally explicit: EEG’s higher reconstructability and stronger cross-subject transfer reflect its broader spatial mixing and standardized montages, whereas iEEG is more focal and heterogeneous across subjects, limiting portability of learned inter-electrode structure (Memar et al., 9 Jun 2026).

Several applications are proposed directly in the source text. In clinical decoding and BCI, SMR can identify redundant sensors that are highly reconstructable from others versus uniquely informative sensors, thereby guiding montage optimization and channel selection. In artifact localization, channels that are overly predictable from nearby electrodes may reflect volume conduction or local artifacts, while masking helps quantify nonlocal contributions. In network mapping, SMR weights and masking curves can serve as interpretable signatures of local versus distributed organization across states such as sleep and wake or across pathologies.

The limitations are also explicit. SMR is linear with elastic net, so nonlinear dependencies may be under-represented. Spatial sampling is uneven in iEEG, while standardized EEG montages still interact with subject-specific anatomy. Potential confounds include volume conduction, referencing schemes, and filtering choices, all of which influence local correlations. Masking helps disentangle local versus distributed effects, but it does not eliminate biophysical mixing. The reported robustness derives from early stopping and regularization, while the surrogate analyses confirm reliance on structured dynamics.

Practical guidance in the paper is correspondingly concrete. Recommended defaults are neighborhood size $t$1, mask intensities $t$2, regularization parameters $t$3 and $t$4, and optional lags $t$5 ms. Adam is used with early stopping, and dropout is not used. Code is available at https://github.com/neurovium/SpatiallyMaskedRegression; the iEEG data are AJILE12 in DANDISET 000055, and the EEG data are available via the BNCI Horizon 2020 database.

6. Terminological scope and relation to other masked-regression frameworks

The expression “Spatially Masked Regression” is not unique to electrophysiology. In spatial statistics, Zhou, Dominici, and Louis used the term for fitting GLMs to spatially masked data produced by spatial smoothing, where masked exposures and outcomes are defined by row-stochastic weighting matrices, such as $t$6 and $t$7, and the central issue is the risk-utility tradeoff between statistical validity and disclosure protection (Zhou et al., 2010). In that literature, the form of masking is determined by the weight function $t$8, the degree of masking is controlled by bandwidth $t$9 or $L_1$0, and utility is evaluated through bias, MSE, and disclosure risk rather than channel reconstruction.

The electrophysiological SMR of (Memar et al., 9 Jun 2026) is therefore distinct in both objective and interpretation. It withholds spatially local predictors to quantify local versus distributed predictability in neural recordings; it does not smooth records to protect confidentiality. A plausible implication is that the shared label reflects a common structural idea—explicit manipulation of spatial locality through masking—while the inferential targets differ substantially.

Adjacent but separate theoretical work appears in masked self-supervised regression, where the objective is coordinate-wise ridge prediction with a zero-diagonal constraint, aggregated into a joint predictor $L_1$1 satisfying $L_1$2 (Zurich et al., 30 Jan 2026). That framework provides a random-matrix-theoretic analysis in the proportional regime and shows, for example, that in spiked covariance models the joint predictor undergoes a Baik–Ben Arous–Péché-type phase transition, whereas in AR(1)/Toeplitz settings masked regression can outperform PCA unless PCA uses a sufficiently large proportion of directions. This is not the same method as electrophysiological SMR, but it supplies a formal vocabulary for thinking about masked reconstruction as a structured predictor of one coordinate from the others.

A further neighboring usage appears in image reconstruction, where masked regularization builds a binary spatial mask from an edge map and regularizes only away from edges, using masked anisotropic TV-derived difference operators (Churchill et al., 2019). There the mask preserves edge information during inverse reconstruction rather than quantifying local versus distributed predictability. Taken together, these works show that “SMR” and closely related masked-regression constructions recur across fields, but their technical meaning is domain-specific. In electrophysiology, the defining feature is the use of progressive spatial exclusion as an experimental control on reconstructability (Memar et al., 9 Jun 2026).