Identifying Neural Source Dynamics from Unknown Local Interventions

Ayana Mussabayeva; Jiaqi Sun; Anuar Aimoldin; Olivier Oullier; Kun Zhang

cs.LGSep 28, 2026

Identifying Neural Source Dynamics from Unknown Local Interventions

Authors: Ayana Mussabayeva, Jiaqi Sun, Anuar Aimoldin, Olivier Oullier, Kun Zhang

Organizations: Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE · Carnegie Mellon University (CMU), Pittsburgh, PA, USA

Abstract

Electroencephalography (EEG) records mixtures of brain-source activity. Even with a known anatomical forward model, experiments that excite only part of the source-state space leave the dynamics unidentified, and repetition cannot resolve the ambiguity. We show that unknown local mechanism changes can supply the missing information. We consider linear dynamics among fixed anatomical sources with known source-state initialization patterns. Changing one source's update rule for one transition leaves a rank-one, source-specific signature in subsequent EEG: subtracting matched baseline responses isolates it, and the forward model identifies the source and calibrates its response history. Combining these histories with initialization responses recovers source interactions without baseline reachability and without first identifying the intervention coefficients. We establish sufficient recovery conditions, a direct estimator, and a noise-sensitivity bound conditional on correct source labels. Simulated EEG on anatomy derived from magnetic resonance imaging confirms the information gain: with baseline excitation confined to four of twelve source coordinates, eight unknown changes recover all dynamics in 32/32 systems, whereas baseline realization, baseline regression through an invertible forward model, and changes that leave the tested states unexposed all fail, and explicitly constructed alternative dynamics reproduce every baseline mean. Where baseline information suffices, direct reconstruction is also more reliable than a matched-information spectral estimator. Nonlocal changes and forward-model error limit accuracy even when source labels are correct.

Figures & tables

Figure 1: Source dynamics and EEG. (A) One receiving row of $F$ changes; $L$ stays fixed. (B) The first transition uses $F+D_{e_{0}}$ ; $F$ then resumes without resetting the state. Nodes in (B) are vectors of source amplitudes $z_{\tau}$ or independent EEG contrasts $x_{\tau}$ . Intervention superscripts are omitted from labels; colors and entries are schematic; noise is omitted.

Figure 2: Recovery from simulated EEG. (A–B) Direct with 4/12 baseline-reachable sources (32 anatomy–seed cases): joint recovery and median $e_{F}$ at sensor/process noise 0.01/0.002 or fourfold larger; paired Gaussian draws make the higher-noise 73,728 and lower-noise 4,608 cells coincide ( $4/\sqrt{192}=1/\sqrt{12}$ ). (C–D) Separate full-reachability comparison (160 cases): Direct versus Spectral-local under matched information. Panel D pairs $e_{z}$ at 98,304 trial equivalents; points above the diagonal favor Direct; seven invalid Spectral-local fits occupy a nonmetric upper strip. Joint recovery requires $e_{F},e_{z}\leq 10\%$ .

Direct, all eight modes exposed
Sensor/process SD	$N$	Success	med. $e_{F}$	med. $e_{z}$	p90 $e_{z}$
0.01/0.002	4,608	24/32	6.33	8.08	11.85
0.01/0.002	18,432	32/32	3.24	3.97	6.02
0.01/0.002	73,728	32/32	1.66	1.99	3.04
0.04/0.008	4,608	0/32	26.50	29.52	40.88
0.04/0.008	18,432	3/32	12.79	15.34	22.50

Table 1: Recovery beyond baseline reachability. Twelve-source systems whose baseline means reach four coordinates at every horizon; 32 anatomy–seed cases per row; errors in percent;

N

counts trial equivalents. Top: Direct with all eight modes exposed (noise rows are paired acquisitions). Bottom: controls at 18,432 trial equivalents, lower noise; “coverage” is

\rank[K,E_{J}]

over targets with nonzero population contrasts. Square-leadfield rows use fresh systems from the same generator with

m=q=12

(Appendix D ).

	Joint success $\uparrow$		p90 $e_{z}$ (%) $\downarrow$
Intervention law	Direct	Spectral-local	Direct	Spectral-local
Row suppression (primary)	155/160	141/160	4.28	9.20
General row change (control)	157/160	139/160	4.16	9.99

Table 2: Matched-information recovery under full baseline reachability. Methods share simulated responses within rows; 160 anatomy–seed cases per row at 98,304 trial equivalents. p90: empirical 90th percentile of

e_{z}

, including invalid fits as

\infty

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Dimension	Meaning
$z_{\tau},x_{\tau},u$	$q,m,r$	Source state, sensor observation, controlled input
$T,s$	Positive integers	Output-history and insertion-time horizons
$t,\tau_{\mathrm{int}},\tau$	Nonnegative integers	Output lag $t=0,\ldots,T-1$ ; insertion time $\tau_{\mathrm{int}}=0,\ldots,s-1$ ; absolute episode time $\tau$
$F$	$q\times q$	Baseline transition matrix
$L,K$	$m\times q,q\times r$	Leadfield and initialization map
$D_{e}=e_{j_{e}}v_{e}^{\top}$	$q\times q$	One-transition change to receiving row $j_{e}$

Appendix

Table 3: Core notation.

|J|

is the number of distinct exposed targets.

Method	Valid	Success	Median $e_{F}$	Median $e_{z}$	p90 $e_{z}$
Direct	160	155	2.925	2.203	4.279
Spectral-local	153	141	3.252	2.361	9.204
Target-oracle direct	160	155	2.925	2.203	4.279
Calibrated full- $D$ commutator	160	124	3.771	3.116	15.754
Diagonal-only approximation	160	0	250.988	47.935	54.743

Appendix

Table 4: Main confirmation at 98,304 trial equivalents. All rows contain 160 anatomy–seed cases. Errors are percentages; p90 uses the nearest observed order statistic and includes invalid outcomes.

$q/m$	$n$	$N_{\rm tot}$	Success	Valid	Median $e_{z}$	p90 $e_{z}$
12/8	256	98,304	63/57	73/69	2.69/3.48	32.85/ $\infty$
24/16	64	98,304	42/14	80/73	7.35/13.08	12.60/34.16
48/32	16	98,304	0/0	77/0	26.02/ $\infty$	38.02/ $\infty$
24/16	256	393,216	77/65	80/80	3.62/5.33	6.42/10.77
48/32	256	1,572,864	60/19	80/75	6.32/8.94	10.80/39.60

Appendix

Table 5: Full fresh-cohort grid. Each entry compares direct/spectral-local on 80 cases.

N_{\rm tot}

counts trial equivalents. Errors are percentages; medians and p90 are inverse empirical cumulative-distribution quantiles including infinite invalid outcomes. The shared

q=12

row appears once.

Sensor/process SD	$n$	$N_{\rm tot}$	Success	$e_{F}$ (%)		$e_{z}$ (%)
				med.	p90	med.	p90
0.01/0.002	12	4,608	24/32	6.33	10.29	8.08	11.85
0.01/0.002	48	18,432	32/32	3.24	5.24	3.97	6.02
0.01/0.002	192	73,728	32/32	1.66	2.64	1.99	3.04
0.04/0.008	12	4,608	0/32	26.50	37.15	29.52	40.88
0.04/0.008	48	18,432	3/32	12.79	19.70	15.34	22.50

Appendix

Table 6: All-exposed direct contrast reconstruction. Error columns are percentages;

N_{\rm tot}

counts trial equivalents. Each row contains 32 cases; noise/budget rows are paired. SD denotes standard deviation; med. and p90 denote median and the specified 90th percentile.

Condition	Coverage rank	Gate	Finite direct	Finite gated	Success
All exposed	12	32/32	32/32	32/32	Table 6
One unexposed	11	3/32	1/32	1/32	0/32
All unexposed	4	0/32	0/32	0/32	0/32

Appendix

Table 7: Exposure controls, with 32 cases per paired cell. “Gate” means all eight modes pass; “Finite” counts valid numerical estimates, not successful recovery. Success counts are the same for direct and gated-direct. The three false passes are the same systems across grid cells.

	Partial reachability			Full reachability
Method	$e_{F}$	$e_{z}$	Success	$e_{F}$	$e_{z}$	Success
Direct	1.62	1.78	32/32	2.45	3.06	32/32
Inverse + contrasts, OLS	18.69	74.40	0/32	23.34	92.29	0/32
Inverse + contrasts, GCV	18.63	75.40	0/32	23.43	95.94	0/32
Baseline half, OLS	704.53	363.71	0/32	27.10	42.62	3/32
Baseline half, GCV	81.91	195.09	0/32	24.06	31.74	3/32

Appendix

Table 8: Square leadfield (

m=q=12

), lower noise, 18,432 trial equivalents. Errors are medians in percent; success requires both errors at most 10%. Each input regime contains the same 32 anatomy–seed cases. OLS and GCV have the meanings defined above. “Half” means shared baseline only; “all” denotes the separately acquired equal-total-budget control.

Figure 3: All seven methods across nested sensor counts at lower noise and 18,432 trial equivalents. A) Partial baseline reachability (rank four). B) Full baseline reachability (rank twelve). Points are median dynamics errors over 32 cases; the vertical scale is logarithmic. Inverse-based values at $m=8$ are noninjective minimum-norm diagnostics. The horizontal 10% threshold concerns $e_{F}$ only; joint success also requires $e_{z}\leq 10\%$ . All underlying responses are simulated on MRI-derived anatomy.

	Partial reachability			Full reachability
Condition	$e_{F}$	$e_{z}$	Success	$e_{F}$	$e_{z}$	Success
$m=8$ , lower noise	3.27	3.90	31/32	5.14	6.68	30/32
$m=12$ , lower noise	1.62	1.78	32/32	2.45	3.06	32/32
$m=16$ , lower noise	1.22	1.33	32/32	2.03	2.22	32/32
4,608, higher noise	13.22	14.42	2/32	21.03	24.05	0/32
18,432, higher noise	6.56	7.16	28/32	9.82	12.48	9/32

Appendix

Table 9: Direct contrast reconstruction: sensor and noise sensitivities. All errors are median percentages over 32 cases. The upper block fixes lower noise and 18,432 trial equivalents; the lower block fixes

m=12

and higher noise and lists total budgets in trial equivalents. Rows are paired acquisitions, not independent replications.

$T=s$	Modes	$\rank\Omega$	$\rank W_{-}$	Recovered	Maximum $e_{F}$
6	1	5	9	0/32	$5.51\times 10^{-1}$
6	2	6	12	32/32	$1.51\times 10^{-11}$
6	4	8	12	32/32	$9.80\times 10^{-14}$
6	8	12	12	32/32	$1.12\times 10^{-14}$
12	1	5	12	31/32	$2.90\times 10^{-7}$
12	2	6	12	32/32	$1.25\times 10^{-12}$

Appendix

Table 10: Exact-mean fewer-mode boundary,

m=q=12

. “Recovered” means

e_{F}<10^{-8}

, a numerical threshold distinct from noisy joint success. Ranks are common to all 32 systems in each row. There is no added noise.

Noise	Episodes	Direct	Standalone EM	Direct + EM
Lower	18,432	2.99%; 8/8	53.78%; 0/8	2.95%; 8/8
Lower	73,728	1.57%; 8/8	53.66%; 0/8	1.57%; 8/8
Higher	18,432	12.19%; 1/8	46.22%; 0/8	11.96%; 1/8
Higher	73,728	6.40%; 5/8	47.62%; 0/8	6.36%; 6/8

Appendix

Table 11: Exploratory known-leadfield fitting under partial excitation. Each method cell gives median

e_{F}

(percent) and joint successes out of eight systems. The same systems recur in all four rows. Standalone EM is selected by training likelihood among three non-direct starts; direct + EM starts from our estimate. Selected standalone fits converge in 0/8, 0/8, 1/8 and 2/8 cases in row order, versus 8/8 in every direct + EM row. Capped fits remain included; these are not globally optimized likelihood comparisons.

$N_{\rm tot}$	Design	Valid	Success	Median $e_{F}$	Median $e_{z}$	p90 $e_{z}$
24,576	Positive-only	0	0	$\infty$	$\infty$	$\infty$
24,576	$0/+1$ slope	155	77	10.371	8.265	22.731
24,576	$-1/+1$ slope	158	120	5.165	4.144	10.695
98,304	Positive-only	0	0	$\infty$	$\infty$	$\infty$
98,304	$0/+1$ slope	158	120	5.165	4.144	10.695
98,304	$-1/+1$ slope	160	150	2.553	2.062	5.401

Appendix

Table 12: Complete signed-initialization confirmation. Each row contains 160 cases.

N_{\rm tot}

counts trial equivalents. Errors are percentages; medians and nearest-order-statistic p90 include invalid estimates as

\infty

. Designs and budgets are paired.

Row class	Estimator, targets	Valid	Success	Median $e_{z}$	p90 $e_{z}$
Exact gains	Hybrid, 4	160/160	144/160	1.965	6.704
Exact gains	Direct, 8	160/160	142/160	2.383	7.016
Arbitrary rows	Hybrid, 4	160/160	0/160	310.997	1267.994
Arbitrary rows	Direct, 8	160/160	157/160	2.106	4.157
Arbitrary rows	Spectral-local, 8	153/160	139/160	2.915	9.989

Appendix

Table 13: Selected comparisons from the boundary study, at 98,304 trial equivalents each. Joint success requires both

e_{F}\leq 0.1

and

e_{z}\leq 0.1

; source errors are percentages. The arbitrary-row direct and spectral-local rows are the same secondary control reported in Appendix B.5 .

Participant	Site	Current (mA)	Trials	Transfer rank 1	Target	Other site
01	K13–14	1/5	38/36	.974	.977	.293
01	N2–3	1/5	38/36	.894	.966	.200
01	S1–2	1/5	27/34	.949	.953	.242
01	S5–6	1/5	38/32	.974	.976	.125
03	R ′ 2–3	.3/.5	35/42	.893	.896	.179
05	G ′ 8–9	.1/.3	44/43	.978	.980	.556

Appendix

Table 14: All 11 paired sites in the recorded-current transfer study. Current and trial counts are low/high intensity. Scores average both folds and directions; “Target” is target-trained rank one. A prime belongs to the dataset contact identifier. Sites reuse four participants.

$n$	Median error (mm)	Error p90 (mm)	Held-out angle	Peak agreement
1	13.68	33.24	$4.72^{\circ}$	66.88%
2	13.68	24.79	$4.77^{\circ}$	75.63%
4	13.68	24.28	$3.76^{\circ}$	83.13%
8	13.68	24.28	$3.30^{\circ}$	89.69%
16	13.68	24.28	$2.93^{\circ}$	92.50%

Appendix

Table 15: Averaging study across all 16 sites. Error and angle first take a median over 20 overlapping draws within site, then a median or 90th percentile across sites. Peak agreement is the mean within-site fraction matching the full-pool peak, not correct localization. These descriptive percentiles are not confidence intervals.

Skull conductivity (S/m)	Correct labels	Recovery	$e_{F}$ (%)	$e_{z}$ (%)
0.0030	100%	0/16	26.63	34.66
0.0045	100%	0/16	11.17	15.26
0.0060 (nominal)	100%	16/16	$<10^{-10}$	$<10^{-10}$
0.0090	100%	0/16	15.99	23.38
0.0120	100%	0/16	27.52	40.05

Appendix

Table 16: Exact conditional mean responses with the nominal leadfield supplied to the estimator. Each row contains the same sixteen paired anatomy–seed cases. Joint recovery requires

e_{F},e_{z}\leq 10\%

. Error columns are medians across the sixteen cases and are expressed in percent. Conductivity is an assumed physical parameter, not an estimated participant property.

	Nominal readout			True readout
$c$ (S/m)	Recovery	$e_{F}$ (%)	$e_{z}$ (%)	Recovery	$e_{F}$ (%)	$e_{z}$ (%)
0.0030	0/16	27.13	34.70	13/16	4.47	3.15
0.0045	0/16	12.61	15.76	13/16	3.64	2.45
0.0060	13/16	3.25	2.11	13/16	3.25	2.11
0.0090	0/16	16.61	23.27	14/16	2.92	1.79
0.0120	0/16	27.41	39.76	14/16	2.77	1.64

Appendix

Table 17: Noisy physical-readout sensitivity at 98,304 trial equivalents. The same sixteen systems are paired across all rows and both readout conditions. Errors are all-case medians in percent. “True” supplies the generating leadfield to the unchanged blind estimator; it does not supply targets, changed coefficients or held-out source states.

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NeuroDyn-EEG: An Interpretable Pre-trained Model for EEG Based on Neural Dynamics

Yi Cui, Tong Zhao, Jiaxin Lei, Chuyi Yang+4

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Electroencephalography Neural Dynamics

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Clinical scalp electroencephalography (EEG) offers a noninvasive window into neural dynamics of neuropsychiatric disorders. However, discriminative deep models often lack anatomically indexed physiological interpretability. We propose NeuroDyn-EEG, a pretraining framework integrating generative priors from neural dynamics. It couples an extended Jansen-Rit neural mass model, leadfield-based source projection, and simulation-based parameter inversion. Trained on synthetic parameter-EEG pairs within physiological ranges, NeuroDyn-EEG estimates 11 regional parameter families across 90 AAL regions plus one global parameter from standard 19-channel EEG, using only ~2.43M trainable parameters. We evaluate the framework across three levels. First, controlled simulations demonstrate robust parameter recovery under diverse noise conditions, while real resting-state EEG evaluations confirm spectral and phase consistency in an inverse-forward closed loop. Second, on four clinical benchmarks (AD65, PD31, Figshare MDD, and TUAB), NeuroDyn-EEG achieves competitive classification performance, securing the highest BACC, AUROC, and AUCPR on PD31 and MDD, and highest BACC on AD65. Third, post hoc regional analyses reveal disease-specific alterations: local synaptic connectivity C_1 involves the most altered regions in AD65, whereas the firing threshold theta ranks first in MDD, offering testable mechanistic hypotheses. Overall, NeuroDyn-EEG maps scalp EEG to anatomically indexed dynamical parameters, bridging representation learning and mechanistic neurophysiology. Code: https://github.com/Gnosis-Neurodynamics/NeuroDyn-EEG.

Abstract

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