Beyond Gaussian Assumptions: Distribution-Aware Channel Capacity for Effective Connectivity
Authors: Jianan Jian, Jacob Kang, Nurahmed Multezem, Benjamin Li, Nan Xu
Organizations: Fischell Department of Bioengineering University of Maryland, College Park, MD, USA · Department of Electrical and Computer Engineering University of Maryland, College Park, MD, USA · Department of Computer Science University of Maryland, College Park, MD, USA · Brain and Behavior Institute University of Maryland, College Park, MD, USA · Artificial Intelligence Interdisciplinary Institute at Maryland University of Maryland, College Park, MD, USA
Effective-connectivity estimation from brain signals often relies on Gaussian residual modeling, which enables tractable estimation but can discard informative distributional structure and distort inferred directed interactions when empirical residuals are non-Gaussian. We show across multiple modalities, species, and experimental conditions that both brain signals and fitted channel residuals frequently deviate from Gaussianity. We therefore introduce a distribution-aware, information-theoretic measure of effective connectivity based on channel capacity under general residual distributions. To estimate the resulting capacity from empirical, potentially non-Gaussian residuals, we develop a dual-flow min-max estimator based on normalizing flows, in which a generator searches over admissible input distributions under a power constraint while an observer estimates output entropy. We provide a theoretical characterization of the estimator, showing that the observer objective recovers differential entropy up to a KL approximation term, that the formulation reduces to classical Gaussian capacity as a special case, and that residual entropy can alter achievable information rates beyond variance; game-gap and error analyses further characterize optimization and approximation sources. In brain-like simulations with known directed connectivity, Dual-flow achieves the highest AUROC and AUPRC across ten conditions spanning diverse network topologies, hidden drivers, feedback, and heterogeneous hemodynamics, compared with Gaussian capacity, Granger causality, VAR-LiNGAM, and GIMME. Applied to multimodal brain signals, the method reveals time- and condition-resolved directed interactions consistent with known neurobiological circuitry. Together, these results establish a principled distribution-aware framework for effective-connectivity estimation beyond Gaussian residual modeling.
Figures & tables
Figure 1: Proposed distribution-aware channel-capacity framework for EC. Steps 1 & 2: Select the source and target ROIs and retrieve their signals. Step 3: Perform a linear regression to obtain the linear channel model and a residual pool. Steps 4 & 5: Compute the channel capacity using a dual-flow minimax game. Step 6: Repeat for all directed ROI pairs to obtain the EC graph.
Figure 2: Dual-flow estimator and flow architecture. (a) The generator searches over admissible input distributions under a power constraint, while the observer estimates channel-output entropy. (b) A 1D Real NVP coupling layer used in the generator and observer.
Figure 3: Gaussianity diagnostics across neural signal modalities and datasets. Anderson–Darling (A–D) statistics quantify deviations from matched Gaussian distributions. A , Rat resting LFP–BOLD signals show frequency-dependent signal non-Gaussianity ( i ); sliding-window FIR residuals show time-varying Gaussianity, with the red curve indicating the fraction of subjects rejecting residual normality ( ii ). B , HCP motor fMRI shows ROI- and task-dependent Gaussianity variation ( i ) with representative task-ROI distributions and matched Gaussian fits ( ii ). C , Concurrent EEG–fMRI DMN signals show modest finite A–D statistics for fMRI BOLD ( i ) but widespread infinite A–D statistics for broadband EEG; infinite values are summarized by prevalence ( ii ), excluded from finite-value boxplots ( iii ), and illustrated by a representative distribution ( iv ). Dataset descriptions, preprocessing, and Gaussianity-analysis details are provided in Appendix D.1 .
Figure 4: Directed-edge recovery across brain-like simulations. Bars show mean AUROC ( A ) and AUPRC ( B ) across 50 realizations per condition; error bars denote +1 SD (upper half shown for clarity). Dual-flow achieved the highest mean AUROC and AUPRC in every condition. Full results are reported in Appendix E.2 .
Figure 5: Task-related directed connections of Dual-flow (left) and GCap (right) visualized in BrainNet Viewer ( Xia et al., 2013 ) . Result with all four EC metrics, as well as their respective connectivity matrices in the task state and rest states, are reported in Fig. 8 .
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Method
Distributional structure
EC estimation principle
Dual-flow
Residual distribution approximated by empirical samples.
Incorporates empirical residual shape into channel-capacity estimation while optimizing the input distribution.
GCap
Gaussian residuals; variance estimated from empirical data.
Computes channel capacity using a matched-variance Gaussian residual model.
GC
Gaussian residuals for conventional parametric inference.
Compares residual variances of reduced and full autoregressive models to quantify directed predictability.
VAR-LiNGAM
Independent non-Gaussian structural errors; distributional form unspecified.
Uses non-Gaussianity and independence to identify directed connections.
GIMME
Gaussian signal model under conventional SEM estimation.
Uses signal covariance structure to select group- and individual-level directed paths.
Appendix
Table 1: Distributional structure and EC estimation principles of the proposed method and evaluated baselines.
Distribution
L=256
L=512
L=1024
L=2048
L=4096
Max. Δ
Gaussian
.128 ± .006
.129 ± .008
.131 ± .007
.132 ± .011
.134 ± .011
2.25%
Uniform
.236 ± .011
.238 ± .010
.237 ± .012
.239 ± .012
.238 ± .011
1.11%
Student’s t
.184 ± .011
.181 ± .010
.184 ± .006
.185 ± .013
.190 ± .017
3.23%
Exponential
.337 ± .030
.346 ± .034
.332 ± .038
.344 ± .029
.341 ± .037
4.30%
Appendix
Table 2: Capacity estimates across training sequence lengths. Values are mean ± SD across 50 random initializations. Max. Δ is the largest relative deviation from the L=1024 value.
Figure 6: Controlled validation of Dual-flow. A , Validation in analytically tractable settings. (i) Entropy-estimation bias relative to ground truth across known noise distributions over 60 runs. (ii) Capacity-estimation bias relative to analytic Gaussian FIR capacity across representative channels over 60 runs. B , Matched-variance non-Gaussian residuals in a fixed FIR channel. Bars show the actual lower bound, Gaussian surrogate capacity, Dual-flow estimate, and actual upper bound for residual distributions with matched variance. Error bars denote mean ± s.e.m. where applicable.
Figure 7: Sliding-window analysis of the LFP–BOLD dataset. (a) Time-resolved noise histograms in a representative BOLD recording reveal time-variability of the distribution and the deviations from Gaussianity. (b) The difference between GCap and Dual-flow estimates correlates with the Anderson–Darling (AD) statistic of the windowed time series (Pearson r=0.56 ). Each dot represents the results computed for a window from a BOLD or an LFP band-limited power timeseries.
Sender ROI
Receiver ROI
Dual-flow
GCap
GC
VAR-LiNGAM
pFDR
CV
pFDR
CV
pFDR
CV
pFDR
CV
VI.R
VI.L
.1101
3.6144
.0915
3.7060
.9966
3.9374
.2715
3.7033
VI.R
BA3b,OP4.R
.1413
4.1552
.0767
3.1041
.9966
4.5693
.9912
2.6882
VI.R
BA3b,OP4.L
.0959
2.6447
.0438
2.6062
.9966
4.0495
.9880
8.2120
VI.L
VI.R
.1537
5.1530
.1963
7.1669
.9966
3.5852
.9880
10.9397
VI.L
BA3b,OP4.R
.1570
5.8323
.1963
6.7362
.9966
2.9687
.9880
7.0685
Appendix
Table 3: Student’s t -test positive false discovery rate (pFDR) and coefficient of variation (CV) comparing Dual-flow, GCap, GC, and VAR-LiNGAM between task and rest conditions for each directed ROI pair in the HCP tongue task dataset. A directed connection is considered task-related if pFDR <0.1 and CV < 30% percentile; values meeting each threshold are shown in bold.
Figure 8: Connectivity matrices of Dual-flow (upper left), GCap (upper right), GC (lower left), and VAR-LiNGAM (lower right) during tongue motion and rest. Significant task-related directed interactions are marked with stars and visualized in BrainNet Viewer ( Xia et al., 2013 ) . Since no edges met the task-sensitivity criteria for GC or VAR-LiNGAM, the four nonsignificant edges with the lowest FDR-adjusted p -values are visualized for each method. GIMME was excluded because the task segments were shorter than its documented minimum timeseries length.
Figure 9: Cross-modal correspondence of time-resolved EC estimates across six LFP frequency bands. Each panel shows one estimator. Violins represent correlations between BLP- and BOLD-derived EC estimates (left S1 → right S1), pooled across windows, over 10,000 bootstrap resamples of the 22 scans. Black markers and vertical bars indicate bootstrap means and 95% percentile intervals. Values below the violins are one-sided bootstrap q values after Benjamini–Hochberg correction across bands within each method; significant values ( q<0.05 ) appear in bold red.
Condition
ROIs
Edges
Ground-truth topology
Manipulation tested
Topology source
CD-e
3/3
2
1→2 , 1→3 ; weights 0.4/0.4
Equal-coupling common driver
Xu et al. (2017)
CD-u
3/3
2
1→2 , 1→3 ; weights 0.8/0.4
Unequal-coupling common driver
Adapted from Xu et al. (2017)
Diamond
5/5
5
1→2 , 1→3 , 2→4 , 3→4 , 4→5 ; weights 0.4
Parallel indirect paths and convergence
Present study
Diamond (H)
4/5
5 total; 3 evaluated
Same diamond; ROI 1 simulated but omitted from the analysis
Table 4: Simulation topologies and manipulations. ROIs are given as observed/total, so that conditions with a hidden node have fewer observed than total ROIs. Asterisks mark conditions in which hemodynamic response functions varied across realizations.
Condition
Method
AUROC
AUPRC
Precision
Sensitivity
FPR
CD-e
Dual-flow
.888 ± .108
.857 ± .132
.887 ± .206
.620 ± .216
.070 ± .134
GCap
.700 ± .129
.623 ± .155
.472 ± .069
.960 ± .137
.545 ± .120
VAR-LiNGAM
.725 ± .315
.739 ± .273
.618 ± .404
.570 ± .378
.200 ± .226
GIMME
.498 ± .018
.333 ± .000
.000 ± .000
.000 ± .000
.005 ± .035
GC
.580 ± .221
.572 ± .201
.332 ± .206
.550 ± .323
.555 ± .249
CD-u
Dual-flow
.858 ± .104
.830 ± .112
.890 ± .273
.470 ± .120
.040 ± .093
Appendix
Table 5: Directed-edge recovery in brain-like simulations. Entries are mean ± SD across 50 realizations. Precision, sensitivity, and FPR use LOSO-MCC-calibrated thresholds. Bold marks the best mean in each metric within a condition (highest for AUROC, AUPRC, precision, and sensitivity; lowest for FPR). GIMME was not evaluated on Macq28 because its 28 ROIs exceed the recommended range of 3–20 ROIs ( Beltz and Gates, 2017 ) .
Forecasting time-varying functional connectivity from electroencephalography (EEG) requires modeling both history-dependent trends and structured variability across channels. Conditional flow matching provides a framework for distributional forecasting, yet it remains unclear whether graph-informed source distributions offer practical advantages over isotropic noise and strong deterministic predictors. We introduce a graph-structured residual flow framework that separates conditional mean prediction from stochastic residual transport. A history-only predictor estimates the future connectivity graph, while a graph Gaussian source encodes dependencies derived from past connectivity through a Laplacian-based covariance. A conditional velocity field transports source samples to future graph residuals, with transport time explicitly distinguished from physical EEG time. Our study identifies the conditions and controls needed to distinguish useful residual transport from improvements attributable to deterministic prediction, learned representations, and sampling effects.
Faculty of Information Science and Technology, Hokkaido University, Japan · SANKEN, The University of Osaka, Japan · Department of Computer Science, University of Illinois Urbana-Champaign, USA +1
Connectivity structure shapes neural computation, but inferring this structure from population recordings is degenerate: multiple connectivity structures can generate identical dynamics. Recent work uses low-rank recurrent neural networks (lrRNNs) to infer low-dimensional latent dynamics and connectivity from observed activity, enabling a mechanistic interpretation of the dynamics. However, standard approaches for training lrRNNs can recover spurious structures irrelevant to the underlying dynamics. We first characterize the identifiability of connectivity structures in lrRNNs and determine conditions under which a unique solution exists. To find such solutions, we develop an inference framework based on maximum entropy and continuous normalizing flows (CNFs), trained via flow matching. Instead of estimating a single connectivity matrix, our method learns a distribution over connection weights that is maximally unbiased over unidentifiable components while matching the observed dynamics. This approach captures complex yet necessary distributions such as heavy-tailed connectivity found in empirical data. We validate our method on synthetic datasets with connectivity structures that generate multistable attractors, limit cycles, and ring attractors, and demonstrate its applicability in recordings from rat frontal cortex during decision-making. Our framework shifts circuit inference from recovering connectivity to identifying which connectivity structures are computationally required, and which are artifacts of underconstrained inference.
Timothy Doyeon Kim, Ulises Pereira-Obilinovic, Yiliu Wang +2
Allen Institute, Seattle, WA, USA · University of Washington, Seattle, WA, USA
A growing number of techniques leverage the spatial structures that underlie many real-world datasets. Despite these advances, the complementary task of estimating spatial structures and understanding their role within these techniques has often been overlooked. In neurophysiological data analysis specifically, numerous methods exist to estimate brain connectivity, but most are not explicitly model-based, dynamic, multivariate, or directed. To address these limitations, we previously introduced noise-driven heat modelling on graphs for neurophysiological connectivity estimation. In this study, we extend this framework by relaxing earlier noise assumptions and adding regularisation to improve robustness. We also develop a simulation procedure to characterise and evaluate our technique in a controlled setting. Finally, we demonstrate that the technique is able to capture meaningful spatial structure across two experiments, each using two real-world datasets. The explicit model formulation of our connectivity estimator has the potential to improve the interpretability of graph-based techniques across a wide range of applications. The code implementing our method is available at https://github.com/sgoerttler/Heat_Connectivity.
Stephan Goerttler, Min Wu, Fei He
Centre for Computational Science and Mathematical Modelling, Coventry University, Coventry CV1 5FB, UK · Institute for Infocomm Research, A*STAR, Singapore