Beyond Gaussian Assumptions: Distribution-Aware Channel Capacity for Effective Connectivity

Jianan Jian; Jacob Kang; Nurahmed Multezem; Benjamin Li; Nan Xu

cs.LGSep 26, 2026

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 ·

Abstract

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.

Panel of Figure 3 — 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. $\Delta$
Gaussian	.128 $\pm$ .006	.129 $\pm$ .008	.131 $\pm$ .007	.132 $\pm$ .011	.134 $\pm$ .011	2.25%
Uniform	.236 $\pm$ .011	.238 $\pm$ .010	.237 $\pm$ .012	.239 $\pm$ .012	.238 $\pm$ .011	1.11%
Student’s $t$	.184 $\pm$ .011	.181 $\pm$ .010	.184 $\pm$ .006	.185 $\pm$ .013	.190 $\pm$ .017	3.23%
Exponential	.337 $\pm$ .030	.346 $\pm$ .034	.332 $\pm$ .038	.344 $\pm$ .029	.341 $\pm$ .037	4.30%

Appendix

Table 2: Capacity estimates across training sequence lengths. Values are mean

\pm

SD across 50 random initializations. Max.

\Delta

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

\pm

s.e.m. where applicable.

Panel of Figure 7 — 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
Sender ROI	Receiver ROI	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

\to

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	$1\!\to\!2$ , $1\!\to\!3$ ; weights 0.4/0.4	Equal-coupling common driver	Xu et al. (2017)
CD-u	3/3	2	$1\!\to\!2$ , $1\!\to\!3$ ; weights 0.8/0.4	Unequal-coupling common driver	Adapted from Xu et al. (2017)
Diamond	5/5	5	$1\!\to\!2$ , $1\!\to\!3$ , $2\!\to\!4$ , $3\!\to\!4$ , $4\!\to\!5$ ; 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	Hidden common driver of ROIs 2 and 3	Present-study extension
Chain	5/5	5	$1\!\to\!2\!\to\!3\!\to\!4\!\to\!5$ and $1\!\to\!5$ ; weights 0.4	Indirect path with direct shortcut	Smith S5 ( Smith et al., 2011 )
Chain (HRF)*	5/5	5	Same as Chain	Regional HRF delays $[-0.60,0.70]$ s; transit multipliers $[0.85,1.18]$	Smith S5 with HRF manipulation

Appendix

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 $\pm$ .108	.857 $\pm$ .132	.887 $\pm$ .206	.620 $\pm$ .216	.070 $\pm$ .134
	GCap	.700 $\pm$ .129	.623 $\pm$ .155	.472 $\pm$ .069	.960 $\pm$ .137	.545 $\pm$ .120
	VAR-LiNGAM	.725 $\pm$ .315	.739 $\pm$ .273	.618 $\pm$ .404	.570 $\pm$ .378	.200 $\pm$ .226
	GIMME	.498 $\pm$ .018	.333 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000	.005 $\pm$ .035
	GC	.580 $\pm$ .221	.572 $\pm$ .201	.332 $\pm$ .206	.550 $\pm$ .323	.555 $\pm$ .249
CD-u	Dual-flow	.858 $\pm$ .104	.830 $\pm$ .112	.890 $\pm$ .273	.470 $\pm$ .120	.040 $\pm$ .093

Appendix

Table 5: Directed-edge recovery in brain-like simulations. Entries are mean

\pm

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 ) .

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