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Identifying Neural Source Dynamics from Unknown Local Interventions

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

arXiv:2609.35379v1cs.LG

TL;DR

A known anatomical forward model does not identify dynamics on source directions left unexplored by baseline experiments, and repetition cannot resolve the ambiguity. The method subtracts matched baseline responses to identify and leadfield-calibrate source-response histories, then combines them with initialization responses without first recovering intervention coefficients. At lower noise, direct successes are 24/32, 32/32, and 32/32 across increasing budgets; at 18,432 trial equivalents, median eF and ez are 3.24% and 3.97%, despite rank-four baseline excitation.

  • Problem

    A known anatomical forward model does not identify dynamics on source directions left unexplored by baseline experiments, and repetition cannot resolve the ambiguity.

    Source: §1 · p. 1

  • Method

    The method subtracts matched baseline responses to identify and leadfield-calibrate source-response histories, then combines them with initialization responses without first recovering intervention coefficients.

    Source: §1 · p. 2

  • Results

    At lower noise, direct successes are 24/32, 32/32, and 32/32 across increasing budgets; at 18,432 trial equivalents, median eF and ez are 3.24% and 3.97%, despite rank-four baseline excitation.

    Source: §6.2 · p. 8

  • Takeaways & Limitations

    Unknown local changes can supply source-coverage information absent from baseline responses, enabling recovery without full baseline reachability under the stated conditions.

    Source: §4.2 · p. 5

    The stability bound assumes correct source labels and correct L and K, and forward-model error can yield inaccurate dynamics even with correct labels.

    Source: §5.2 · p. 7, §6.4 · p. 10

Abstract

from arXiv · show

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.

1 Introduction

Baseline responses can leave source dynamics undetermined even with a known anatomical forward model, and repetition cannot resolve that ambiguity. The paper uses unknown local mechanism changes to reveal calibrated source-response histories that, together with initialization responses, recover dynamics without full baseline reachability.

Problem

  • Known anatomical readout does not resolve dynamics when initialized states and baseline evolution explore only part of source space, and more repetitions cannot remove that ambiguity.Even exact recovery of mean source states leaves dynamics on unexcited directions undetermined.Source: §1 · p. 1

Motivation

  • The study asks whether unknown local changes can reveal dynamics absent from baseline means and identify them in anatomical coordinates.The intervention target and coefficient are unknown, and EEG mixes its effect across sensors.Source: §1 · p. 1

Model

  • The model changes one source's receiving rule for one transition, after which the original dynamics resume.Unlike an additive pulse, the change acts on the current state.Source: §1 · p. 2

Core approach

  • Matched intervened and baseline means isolate a rank-one displacement whose sensor pattern identifies its source and calibrates its response history.Combining calibrated histories with initialization responses recovers dynamics without first identifying intervention coefficients.Source: §1 · p. 2

Contributions and results

  • Eight unknown local changes recover all twelve-source dynamics in 32/32 systems despite baseline excitation confined to four of twelve source coordinates.The sufficient conditions, direct estimator, and noise-sensitivity bound are also established; the bound is conditional on correct source labels.Source: §1 · p. 2

2 Related work

The paper uses a known forward model to fix anatomical source labels and units, then contrasts its local receiving-mechanism changes with identification approaches that require stronger excitation or different assumptions.

Anatomical coordinates

  • Unlike methods relying on statistical assumptions about source mixing or interventional mixing, the known forward model fixes anatomical source labels and units.The paper targets dynamics among predefined sources rather than unrestricted spatial localization.Source: §2 · p. 2

Identification limits

  • Known anatomical coordinates do not ensure sufficient excitation; the proposed local contrasts can complete source coverage without full baseline reachability.Classical full-order realization requires reachability and observability, while the cited varying-input approach assumes controllability.Source: §2 · p. 2

Relation to state-space methods

  • The model changes one source's receiving rule for one transition, then resumes baseline dynamics; its figure contrasts source-amplitude trajectories with independent EEG contrasts.Known-leadfield state-space methods integrate observation physics with latent dynamics, and the paper's contrasts can initialize iterative likelihood fitting.Source: §2 · pp. 2–3, Figure 1 · p. 3

3 Problem formulation

The formulation models linear dynamics among fixed anatomical sources, observed through a known leadfield, with known initialization patterns and an unknown one-transition local receiving-rule change. Matched mean EEG histories provide the measured responses used to represent these dynamics.

Dynamics and observation

  • Source amplitudes evolve through an unknown transition matrix and a scheduled local change, while EEG contrasts are generated by the known leadfield plus sensor noise.The leadfield fixes source ordering and units, but its rank may be below the number of sources.Source: §3 · p. 3

Scope

  • The linear transition matrix describes effective interactions within a fixed operating regime, and the guarantees assume the stated linear model and intervention protocol.The simulations do not test nonlinear source dynamics.Source: §3 · p. 3

Initialization and intervention

  • Episodes begin from known source-state patterns and controlled amplitudes, with no subsequent baseline additive drive; one transition uses the changed rule and the state is not reset.The leadfield remains fixed, and the model assumes zero conditional mean for innovations and sensor noise.Source: §3 · pp. 3–4

Observed responses

  • Trial-averaged matched EEG histories define intervention and baseline response matrices across initializations, insertion times, sensors, and output lags.The matched baseline contrast isolates the immediate displacement and its later propagation by the baseline dynamics; shared baseline lags reuse observations.Source: §3 · p. 4

4 Identification from local interventions

A local receiving-rule change creates a source-specific contrast whose history can be identified and calibrated even when the intervention coefficients are unknown. Combined with initialization responses, calibrated histories recover dynamics under source-coverage and temporal-observability conditions without requiring full baseline reachability.

4.1 Identifying source-response histories

  • The contrast between intervened and matched baseline responses has rank one when the change acts on a tested state, revealing a source-response history up to scale.A nonzero, distinguishable leadfield column identifies the target and calibrates the history; the recoverable object need not identify the intervention row.Source: §4.1 · p. 4

4.2 From histories to dynamics

  • Known initialization responses and calibrated exposed-source histories form the coverage matrix, while the temporal shift identity recovers F from the reconstructed histories.The proof first uses full row rank of Ω to recover O_T, then full column rank of O− to solve the shift identity.Source: §4.2 · pp. 4–5
  • Constructive recovery requires exposed changes, distinguishable anatomical sources, source coverage, and temporal observability, but not full baseline reachability.Selected changes must act on tested states, and their revealed targets must complete the coverage condition.Source: §4.2 · p. 5
  • An invertible leadfield can distinguish instantaneous states but cannot determine dynamics along source directions that tested states never exercise.The paper's direct reconstruction uses calibrated histories and known initializations; full rank of the initialization reachability matrix is unnecessary.Source: §4.2 · p. 5
  • In a three-source illustration, two unknown-target changes reveal the two unexcited source histories and uniquely recover F, although some intervention coefficients remain undetermined.Baseline initialization reaches only source 1; the example has rank-one baseline reachability and rank-two leadfield, but rank-three temporal observability.Source: §4.2 · pp. 5–6

5 Estimation and stability

Direct reconstruction calibrates source-specific histories from intervention contrasts, combines them with initialization responses, and recovers dynamics through temporal shifts. Its stability bound applies only with correct target labels and calibrated forward and initialization models.

5.1 Direct contrast reconstruction

  • The leading singular vector of measured contrasts estimates a common source-history direction; its first sensor block identifies the source and calibrates the history.Absolute alignment resolves the SVD sign ambiguity, and the signed scale restores source units before histories are combined with initialization responses to recover F.Source: §5.1 · p. 6
  • The implementation uses all supplied intervention modes and returns an invalid fit if any mode fails its numerical guard.The guards protect against degenerate contrasts or first sensor blocks; they are not statistical detection thresholds, and invalid fits count as failures.Source: §5.1 · p. 6

5.2 Conditional stability

  • The stability bound separates history-reconstruction error from weak temporal observability, conditional on correct target labels.It assumes correct L and K; it does not cover model mismatch or provide an unconditional success probability.Source: §5.2 · pp. 6–7
  • Stability improves when the weakest sensor-history response is large relative to reconstruction error; weak exposure or poorly separated leadfield columns can destabilize source labeling.Here δ bounds history-reconstruction error and β is the weakest sensor-history response to a unit source-state displacement over the shift lags.Source: §5.2 · p. 7

6 Experiments

Across simulated EEG experiments, local changes recover dynamics beyond rank-deficient baseline reachability and improve matched-information reconstruction reliability. Nonlocal changes and forward-model error limit accuracy even when source labels are correct.

6.2 Recovery beyond baseline reachability

  • At 18,432 trial equivalents, direct recovery reaches 32/32 systems with median eF and ez of 3.24% and 3.97%, despite rank-four baseline excitation.At lower noise, direct successes are 24/32, 32/32, and 32/32 across increasing budgets; at fourfold noise they are 0/32, 3/32, and 24/32.Source: §6.2 · p. 8, Table 1 · p. 9
  • Baseline-only controls fail where local changes succeed: unexposed changes yield 0/32 successes, baseline realization produces no valid reconstruction, and baseline regression fails even with an invertible leadfield.Alternative dynamics preserve every included baseline mean response while differing by at least 2.13%, confirming that these failures reflect missing information rather than failed estimation.Source: §6.2 · p. 8, Table 1 · p. 9

6.3 Reliability when baseline information suffices

  • Direct improves joint success by a paired 8.75 percentage points over Spectral-local under matched information.Median source errors are 2.20% versus 2.36%; the gain reflects fewer large errors and no invalid fits versus seven for Spectral-local.Source: §6.3 · p. 9, Table 2 · p. 9
  • Direct initialization yields 32/32 EM convergences versus 3/32 for likelihood-selected non-direct fits, but fixed-budget recovery degrades with system size for both methods.These exploratory results support reliability over the comparator, not scalability at fixed cost.Source: §6.3 · p. 10

6.4 What the guarantees do not cover

  • A neighboring-row change at 0.1 of the primary coefficient limits direct success to 65/80 and 69/80 at the reported budgets; averaging does not remove this nonlocal-change bias.The guarantee is conditional on locality and calibrated L and K, as well as correct labels.Source: §6.4 · p. 10
  • With nominal rather than generating leadfields, all target labels remain correct but median eF reaches 11.17–27.52% without sampling noise.Supplying the generating leadfield recovers all 80 paired cases to numerical precision.Source: §6.4 · p. 10
  • Recorded Localize-MI EEG tests spatial transfer and repeatability, but does not establish anatomical calibration or recover neural F, validate neural K, or implement a mechanism change.The analyses use a −2 to +2 ms window dominated by injected-current artifact.Source: §6.4 · p. 10

7 Discussion

Unknown local mechanism changes can reveal source dynamics that baseline mean responses leave undetermined, by identifying anatomically calibrated source histories rather than intervention coefficients. The discussion situates this result within a protocol whose translation requires calibrated models and validation.

7 Discussion

  • Intervention histories recover directed interactions beyond baseline reachability, while baseline realization, invertible readout, and unexposed changes fail in the reported experiments.Explicit alternative dynamics also reproduce every baseline mean; noise, nonlocal changes, and forward-model error degrade recovery even with correct source labels.Source: §7 · p. 10
  • Biological translation requires calibrated observation and initialization models and validation of the single-row, single-transition protocol.The authors discuss controlled neural-circuit experiments, including dynamic-clamp and optogenetic settings, and potentially EEG-based perturbation studies.Source: §7 · pp. 10–11

Ethics and data use

  • The study reuses Localize-MI anatomical operators and recorded scalp EEG during intracerebral current injection; it acquired no new human recordings or stimulation procedures.The mathematical intervention protocol is not a clinically validated stimulation or diagnostic procedure.Source: Ethics and data use · p. 11

Reproducibility

  • Appendices document the mathematical arguments, acquisition, estimators, evaluations, controls, EEG diagnostics, forward-model sensitivity, and experimental interpretation.The research package includes experimental protocols, numerical verification records, and checksums; code is available online.Source: Reproducibility · p. 11

Appendix contents

A.1 Population responses and anatomical identification

  • Matched EEG response matrices use the same inputs and insertion times with and without intervention, making their difference an observable contrast.Their rows index sensors and post-transition times, while columns index initialization and intervention-time conditions.Source: §A.1 · pp. 16–17

A Identification assumptions and proofs

  • The model assumes known source dimension, leadfield, and initialization map, common state coordinates and baseline dynamics, and an intervention that changes one receiving row for one transition.The results identify response means; trajectory distributions and innovation covariances require an additional noise model.Source: §A.1 · p. 17

A.2 Anatomical column identification

  • A nonzero contrast has rank one, and its first sensor block identifies the changed source when leadfield columns are projectively distinct.The calibrated history does not require full baseline reachability; repeated interventions at one target may improve precision but do not add distinct physical columns.Source: §A.2 · pp. 17–18

A.3 Direct completion without baseline controllability

This section develops direct recovery without baseline controllability, including coordinate ambiguity, conditional stability, and the scope of the identification claims. It also describes simulation and acquisition protocols and specifies information-access and implementation rules for direct and spectral reconstruction estimators.

A.3 Direct completion without baseline controllability

  • If rank O− = q, direct physical completion recovers F without requiring full-row-rank baseline reachability.The rank condition is on the shifted observation stack O−; at least q − κ distinct coordinate targets are needed for the direct column-completion condition, but this is not a universal intervention lower bound.Source: §A.3 · p. 18

A.4 Complete calculation for the three-source example

  • In the three-source example, rank L = 2 and baseline rank Rs = 1, yet two interventions yield rank O− = 3 and distinguish the baseline-invisible alternative transition.The example uses exact conditional means; its reconstruction illustrates population identification, not finite-noise performance or equivalence of full trial distributions.Source: §A.4 · pp. 19–20

A.5 Coordinate ambiguity

  • Invertible coordinate changes can preserve anatomy, known inputs, intervention targets, and all compared mean responses; within a shared realization, the anatomical map is unique exactly when ρ = q or rΩ = q.Under controllability of (F, [K, EJ]), every nonidentity admissible change alters F; the family concerns mean-response equivalence, not necessarily full-law equivalence.Source: §A.5 · pp. 20–22

A.6 Conditional deterministic stability

  • The deterministic stability bounds separate contrast exposure, scalp visibility and label separation, coordinate coverage, and dynamic inversion, conditional on correct labels.They do not provide finite-sample probabilities for target or rank decisions, which require a response-noise model.Source: §A.6 · p. 23

A.7 Scope of the identification claims

  • The claims identify population-mean lagged source dynamics under known L and K and one-transition, row-local, state-preserving changes—not an instantaneous causal graph or a particular innovation realization.Persistent or diffuse changes can alter the contrast structure or share a local immediate scalp signature, so the stated protocol conditions matter.Source: §A.7 · p. 23

B Experimental protocols and complete results

  • The appendix reports separate studies for the full-reachability comparison, general-row control, and fresh-cohort/source-count study; their results are not pooled.Responses use anatomical observation operators from four Localize-MI geometries; the incomplete-reachability experiment has its own generator and acquisition grid.Source: B Experimental protocols and complete… · p. 23

B.1 Anatomy, dynamics, and replication

  • The simulations use four Localize-MI geometries, each mapping twelve known candidate sources to eight independent sensor contrasts; anatomy, not recorded EEG, supplies the input.For the full-reachability comparison, four source coordinates initialize the system and the other eight are intervention targets whose labels are hidden from blind estimators.Source: §B.1 · p. 24

B.2 Acquisition and trial accounting

  • Episodes record thirteen samples, with six output lags and six insertion times; each active episode applies the changed transition once and uses baseline dynamics otherwise.The acquisition includes matched baseline and active episodes, with independent episodes and correlated sensor noise; repeated baseline Hankel entries reuse samples and are not independent observations.Source: §B.2 · p. 24
  • 6,144, 24,576, 98,304, and 393,216 trial equivalents corresponded to n=16, 64, 256, and 1,024, with half the budget assigned to baseline and half to active acquisitions.Each baseline input condition had 48n repeats; Gaussian averaging preserved the exact trial-mean law, including propagated process noise and temporal/channel correlations, using paired counterfactual acquisitions rather than nested trials.Source: §B.2 · p. 25

B.3 Estimators and information access

  • Direct contrast reconstruction uses each intervention’s paired column, permits repeated inferred labels without pooling, and returns an invalid estimate when numerical guards or coordinate coverage checks fail.The specified singular-value and first-block norm cutoffs are numerical degeneracy guards, not statistical detection thresholds; reconstruction uses least-squares solves for the temporal shift.Source: §B.3 · p. 25
  • The spectral-local comparator estimates intervention changes in shared Ho–Kalman coordinates from the same noisy baseline Hankel matrix, then solves anatomical equations by unweighted least squares.It uses estimated realization-coordinate changes rather than true intervention changes; the comparison isolates direct contrast extraction from extraction after full baseline realization, not all estimators with the same information.Source: §B.3 · p. 25
  • Blind methods receive neither source-truth quantities nor held-out outcomes, while the controls vary label, intervention, or misspecification information and do not establish that calibration is harmful.Target-oracle direct receives true labels; calibrated commutator receives true full intervention changes; diagonal-only approximation substitutes a diagonal trace approximation for a receiving-row change. No source-truth rotation, rescaling, or permutation repairs estimates.Source: §B.3 · p. 25

B.4 Held-out reconstruction and confirmation results

The confirmation and follow-up simulations test reconstruction across budgets, intervention forms, source counts, and incomplete baseline reachability. Direct reconstruction succeeds when exposed directions provide coverage, while finite-budget accuracy remains sensitive to noise and conditioning.

B.4 Held-out reconstruction and confirmation results

  • At the primary budget, Direct succeeded in 155/160 cases versus Spectral-local’s 141/160, an 8.75-percentage-point advantage; all 1,280 Direct target labels were correct.Source: §B.4 · p. 26

B.5 General-row control

  • For norm-matched arbitrary receiving-row changes, Direct recovers 157/160 systems versus 139/160 for Spectral-local without knowing row coefficients or assuming proportional suppression.Median/p90 source errors are 2.106%/4.157% for Direct and 2.915%/9.989% for Spectral-local.Source: §B.5 · p. 27

B.6 Fresh-seed check and source-count scaling

  • In the fresh q = 12 cohort, Direct succeeds in 63/80 systems versus 57/80 for Spectral-local; low-exposure seed blocks were associated with no Direct successes across their anatomies.The exposure association is post hoc and does not isolate exposure as the sole cause.Source: §B.6 · p. 27
  • At q = 48, neither method succeeds at fixed total cost; with more repeats, Direct succeeds in 60/80 systems versus 19/80 for Spectral-local.At q = 48 and fixed repeats, the success difference is 51.25 percentage points (interval [36.25, 65]); sensors and inputs co-scale, so source count is not isolated.Source: §B.6 · pp. 27–28
  • Correct labels do not guarantee accurate dynamics: the median true observability singular value falls from 0.270 to 0.109 across the tested source sizes.The noiseless checks passed, but these truth-based observability diagnostics were unavailable to the estimators.Source: §B.6 · p. 28

B.7 Computational provenance

  • Independent numerical checks recomputed the main-study outcomes and verified the scaling-study summaries against all 800 metric rows.The shared q = 12 rows were deduplicated in presentation rather than treated as a new replication.Source: §B.7 · p. 28

C.1 Exact baseline reachability restriction

  • With rank-four baseline reachability, all 32 noise-free fits recover targets and dynamics when all eight intervention modes expose the remaining coordinates.The construction fixes baseline reachability at four even at infinite horizon, while rank [K, E_J] reaches twelve under full exposure.Source: §C.1 · pp. 28–29

C.3 Recovery under full exposure

  • Under full exposure, all 32 Direct estimates pass the gate with correct targets, but fourfold larger noise creates finite-budget failures; Spectral-local has no valid reconstructions in these cells.The baseline Hankel matrix has rank four, so the full-order comparator attempts to infer twelve coordinates from rank-four baseline information.Source: §C.3 · p. 30

C.4 Exposure controls and false confidence

Exposure controls distinguish a genuinely informative intervention from a nonzero change that leaves tested states unexposed. The theory then separates anatomical readout, baseline dynamical ambiguity, and propagation-based recovery, with matched simulations designed to compare input and sensor regimes.

C.4 Exposure controls and false confidence

  • In the one-unexposed condition, the gate falsely accepts the missing mode in 3/32 systems, and neither unexposed control yields successful recovery.One accepted case had 74.79% dynamics error and 54.65% source error; the missing population contrast cannot be made nonzero by repetition.Source: §C.4 · pp. 30–31
  • Alternative dynamics reproduce every included mean response in 64 unexposed-control witnesses, despite differing from the true dynamics by at least 2.13%.These constructions preserve the specified mean-response protocol, not full noisy distributions or arbitrary repeated-switch schedules.Source: §C.4 · p. 31

C.5 Independent numerical audit

  • An independent audit recomputed fitted metrics and 4,608 mode p-values, and all p-values and Holm decisions matched.The audit also checked acquisitions, exact-system audits, and ambiguity witnesses.Source: §C.5 · p. 31

D.1 Identification with full-column-rank leadfields

  • With a full-column-rank leadfield, rank O− = q for every T ≥ 2, so an additional dynamical observability condition is unnecessary when exposed changes provide full coordinate coverage.The conditional stability bound still requires correct labels and coverage; small σmin(L) can amplify errors.Source: §D.1 · pp. 31–32

D.2 Coordinate ambiguity and dynamical excitation

  • Baseline mean responses leave an affine family of indistinguishable dynamics with q(q − d) free dimensions, where d is baseline reachable-subspace dimension.For q = 12 and d = 4, the passage gives 96 such directions before additional model constraints.Source: §D.2 · p. 33

D.3 Identification through response propagation

  • Propagation can make a few exposed directions sufficient for regression: in the example, three predecessor states span R3 even though [K, e2] has rank two.This route uses baseline transitions after the intervention ends and requires an injective leadfield for left-inversion of contrast blocks.Source: §D.3 · p. 33

D.4 Matched sensor-count experiment

  • The matched sensor-count experiment varies input reachability and sensor measurements while holding the dynamics and hidden row changes identical across those arms.The two input designs also differ in conditioning, so the comparison does not isolate reachability alone.Source: §D.4 · p. 34

D.5 Effects of invertibility, excitation, and noise

The results distinguish recoverable dynamics from recoverable source states: an invertible readout does not resolve unexcited directions, while local changes can provide additional information. The section also tests fewer intervention modes and an exploratory state-space fitting comparison.

D.5 Effects of invertibility, excitation, and noise

  • An invertible leadfield makes instantaneous readout errors fall to 2.19% at m = 12 and 0.51% at m = 16, but these are state-readout—not dynamics—errors.At m = 8, the reported 56.08% is a noninjective minimum-norm proxy.Source: §D.5 · p. 35
  • With exact means, full-reachability baseline OLS recovers all 32 systems, but in partial reachability an alternative dynamics matrix differs by 157–174% while matching every audited baseline lag.Adding eight intervention modes permits exact inverse-regression recovery at m = 12 and 16; direct contrast reconstruction is exact in all 192 audits, including m = 8.Source: §D.5 · p. 35
  • With partial reachability, OLS and ridge–GCV each achieve 0/32 successes across all three data allocations; direct reconstruction succeeds in all 32 primary square cases.With full reachability, baseline controls achieve 3/32 successes at half budget and 7/32 at full budget; interventional controls achieve 0/32.Source: §D.5 · p. 35, Table 8 · p. 36
  • At m = 16, full reachability and an all-baseline budget yield 32/32 OLS recoveries, while partial-input all-baseline GCV yields 0/32 in the same sensor, noise, and budget cell.Interventional OLS and GCV each achieve 32/32 in both input regimes there; the passage also reports that higher-noise square results fail despite injectivity and complete direct coverage.Source: §D.5 · p. 36

D.6 Exact propagation with fewer intervention modes

  • With two intervention modes, the exact-mean predecessor matrix has full rank in all cases despite rank Ω = 6 < 12; one mode with T = 12 is full rank but can be ill-conditioned.The one-mode T = 12 arm reaches σmin(W−) = 4.75×10^-10 and 31/32 below threshold at m = 16, while four- and eight-mode horizon-12 arms recover 32/32 at both sensor counts; this is an algebraic, not finite-noise sample-efficiency result.Source: §D.6 · p. 37, Table 10 · p. 38

E.1 Question and scope

  • The exploratory comparison asks whether direct response construction can initialize known-leadfield state-space fitting under partial excitation; its comparator is not a reproduction of published methods.The comparator omits NLGC’s sparsity and connectivity-testing procedure and is separate from the matched-information spectral-local study.Source: §E.1 · p. 38

E.2 Paired individual-trial acquisition

  • The paired comparison uses simulated individual episodes on four MRI-derived anatomies, with the same acquisition and known L and K supplied to both estimators.EM also receives exact noise parameters and trial covariances, whereas direct estimation uses locality and matched mean contrasts without calibrated noise parameters; the comparison is therefore not matched-prior.Source: §E.2 · p. 38, §E.3 · p. 39

E.3 Comparator, information access and optimization

  • The state-space comparator profiles unknown intervention-window initial distributions while alternating latent-trajectory estimation and unrestricted updates of F.It uses Kalman filtering and Rauch–Tung–Striebel smoothing; its restarts are selected by training likelihood, and runs are capped at 500 updates.Source: §E.3 · p. 39

E.4 Recovery results and optimization diagnostics

Direct estimation identified all intervention targets in the exploratory acquisitions, while standalone EM did not achieve joint recovery. The following sections test signed initialization under additive insertion effects and specify the cancellation conditions and simulation scope.

E.4 Recovery results and optimization diagnostics

  • Across 32 acquisitions, direct and direct + EM achieve 22 and 23 joint successes, respectively, while standalone EM achieves none.Direct identifies all eight targets in every acquisition; all 32 direct-initialized fits converge, compared with 3/32 selected standalone fits.Source: §E.4 · p. 39, Table 11 · p. 40
  • EM refinement lowers eF in all 32 cases, but the median absolute reduction is only 0.025 percentage points and ez falls in only 14/32 cases.The result supports useful initialization rather than a substantial or uniform reconstruction improvement.Source: §E.4 · p. 39
  • The exploratory accuracy gap is limited to this calibrated implementation and finite optimization budget; it neither establishes general state-space failure nor validates recovery from recorded neural EEG.Fresh-system confirmation and stronger or intervention-aware likelihood optimization remain necessary before making a general comparative claim.Source: §E.4 · p. 40

F.1 Cancellation and its assumptions

  • A slope across initialization amplitudes removes an additive term fixed across amplitudes, restoring the contrast factorization under the original exposure, coverage, and temporal-observability assumptions.Amplitude-dependent additive effects, sign-dependent mechanisms, saturation, or initial-state distributions that break the affine mean model invalidate the cancellation.Source: §F.1 · p. 41

F.2 Fixed simulation and acquisition cost

  • At 98,304 trial equivalents, balanced signed initialization yields 150/160 successful recoveries versus 120/160 for zero/positive; positive-only fails inferred coordinate coverage in all 480 budget-specific cases.The signed comparison is not an equal-energy advantage, and the experiment establishes neither a biological protocol nor robustness to amplitude-dependent effects.Source: §F.2 · pp. 41–42, Table 12 · p. 42

G Boundary control: fewer targets under an exact-gain prior

The boundary study tests whether an exact-gain structural assumption can reduce the number of distinct intervention targets. The recorded-current analyses instead assess spatial transfer and anatomical calibration, not neural dynamics.

G Boundary control: fewer targets under an exact-gain prior

  • Under exact gains, the hybrid-minus-direct paired success difference is +1.25 percentage points, with a 95% interval of [−3.125, 5.625]; this establishes neither superiority nor equivalence.The hybrid’s maximum source error is 140.332%, versus 22.381% for direct, so its favorable median does not remove the adverse tail.Source: G Boundary control: fewer targets… · p. 43, Table 13 · p. 43
  • Under arbitrary-row changes with four targets, the hybrid has 0/160 successes despite finite fits and correct target labels, with median source error 310.997%.At eight targets, the hybrid agrees with direct; the study supports only a conditional tradeoff between target diversity and structural assumptions.Source: G Boundary control: fewer targets… · p. 43

H Recorded-current EEG: spatial transfer and anatomical calibration

  • The recorded-current signal is predominantly the electrical artifact of injected current, and its contact midpoint localizes that current rather than serving as ground truth for neural generators.These analyses estimate neither neural F nor a known neural initialization map K, and do not implement a receiving-row mechanism change; the participants overlap with those supplying simulation anatomies.Source: H Recorded-current EEG: spatial… · pp. 43–44

H.1 Data and anatomical diagnostic

  • The anatomical diagnostic fits the full bilateral, free-orientation cortical dictionary; it is not the proposed fixed-column estimator and does not impose a rank-one response matrix.The data comprise 256 channels sampled at 8 kHz, with a 33-sample response window and MRI-derived forward models spanning 8,196 cortical positions.Source: §H.1 · p. 44

H.2 Cross-intensity spatial transfer

Cross-intensity spatial transfer was strong but below target-trained models, while averaging improved direction repeatability more than anatomical accuracy. The physical forward-model sweep shows that correct source labels alone do not ensure calibrated dynamics.

H.2 Cross-intensity spatial transfer

  • The transfer basis came from the source intensity, but target-condition training calibrated its temporal coefficients; it was not zero-shot waveform prediction.Tests used two complementary folds in both transfer directions, and disjoint trial indices did not guarantee temporal independence.Source: §H.2 · pp. 44–45
  • Median Q was 0.949 for transferred rank one, 0.966 for target-trained rank one, and 0.639 for the frozen dipole.Transferred rank one beat the dipole and other-site controls at all 11 sites, but underperformed target rank one and transferred ranks two and three at every site.Source: §H.2 · p. 45, Table 14 · p. 45

H.3 Averaging, repeatability and anatomical accuracy

  • From n = 1 to n = 16, held-out direction angles improved at 15 of 16 sites, while median localization error remained 13.68 mm.At n = 16, 15 sites matched their full-pool peak at zero median distance, yet eleven remained more than 10 mm from the contact midpoint.Source: §H.3 · p. 46, Table 15 · p. 46

H.4 Reproducibility and interpretation

  • The transfer predictions reproduced with maximum discrepancy 1.64 × 10^-15, but the descriptive analyses provide neither population-level inference nor a test of neural-dynamics recovery.They also do not identify which forward-model error explains the localization discrepancies.Source: §H.4 · pp. 46–47

I Physical forward-model mismatch and anatomical calibration

  • The physical sensitivity study tests whether correct target labels suffice when the supplied leadfield is inaccurate, using simulated dynamics on participant-derived anatomies.It changes a physical head-model parameter rather than arbitrary matrix entries and does not estimate participants’ actual conductivity.Source: I Physical forward-model mismatch and… · p. 47

I.1 Paired systems and physical forward models

  • At c = 0.0045, median direction changes were 1.84–2.14° and signed projected gain ratios were 0.869–0.900; direction and amplitude changed together.Across the assumed BEM family, every exact true column remained closest to its correct nominal leadfield line.Source: §I.1 · p. 47

I.2 Acquisition, fitting and evaluation

The physical-readout experiments pair simulated systems across assumed conductivity worlds and compare nominal with true leadfields. Correct target labels did not prevent recovery errors under nominal-readout mismatch, even without sampling noise.

I.2 Acquisition, fitting and evaluation

  • The acquisition paired each system across five physical worlds, with exact and noisy responses; the primary fit always used the nominal leadfield.The true-readout diagnostic supplied the generating leadfield to the unchanged estimator as an oracle calibration check, not as an estimated calibration.Source: §I.2 · p. 48

I.3 Exact responses: labels can be correct while dynamics are wrong

  • All exact target assignments were correct, yet no nonnominal nominal-readout case met the joint recovery criterion; the true-readout diagnostic recovered all 80 cases.The true-readout maximum relative dynamics error was 1.02 × 10^-14, while nominal-readout scale errors show that correct labels did not guarantee calibrated histories.Source: §I.3 · p. 48, Table 16 · p. 48

I.4 Noisy responses and retained failures

  • At 98,304 trial equivalents, true-readout success was 13 or 14 of 16 cases by world, while nominal-readout success was zero outside the nominal world.Noisy nominal-readout target accuracy was 127/128 at c = 0.003 and 128/128 at every other conductivity; one invalid estimate remained in the all-case summaries.Source: §I.4 · p. 49, Table 17 · p. 49

I.5 Numerical audit and interpretation

  • The numerical audit recomputed all 320 fits and regenerated all 160 acquisitions, verifying the reported metrics and response-scale diagnostics.All eighty exact true-readout cases recovered, with shared source units and nominal/true-readout identity verified at c0.Source: §I.5 · p. 49

J Experimental interpretation and additional related work

  • The study’s inference target is source-coordinate dynamics under a specified perturbational model; stimulation or accurate response prediction alone does not establish that model’s assumptions.The controlled study does not demonstrate recovery of neural F from recorded EEG or estimate actual human conductivity errors.Source: J Experimental interpretation and… · p. 49, §I.5 · p. 49

J.1 What the result means for source localization

The paper distinguishes source-coordinate dynamics recovery from localization or accurate sensor prediction. Its interpretation depends on calibrated maps and a narrowly specified intervention, so biological perturbations do not automatically satisfy the model.

J.1 What the result means for source localization

  • The inference target is dynamics among predefined anatomical sources, not unrestricted source locations or a whole-brain map; recorded-current analyses test transfer and anatomical agreement, not true neural F recovery.The known forward model fixes locations, orientations, and source units, whereas simulations evaluate recovery against known dynamics and trajectories.Source: §J.1 · p. 50

J.2 Experimental meaning of the assumptions

  • The recovery guarantee assumes a calibrated leadfield and initialization map, plus a single receiving-row change for one transition before baseline dynamics resume.A focal additive pulse can also produce a rank-one response, so rank one alone does not establish a mechanism change; the discrete transition has no physiologically calibrated duration here.Source: §J.2 · p. 50

J.3 Relation to other perturbational recordings

  • Dynamic clamp, optogenetics, intracranial stimulation, and TMS–EEG motivate perturbational comparisons, but none alone establishes the required calibrated initialization and single-transition mechanism change.Early TMS–EEG measurements can contain muscle artifacts, and discarding those samples may remove the block used for anatomical calibration.Source: §J.3 · pp. 50–51

J.4 Additional related work

  • The construction differs from realization and intervention-identification results by using anatomically calibrated response columns from receiving-mechanism contrasts without full baseline reachability.The practical lesson separates missing excitation, weak observation, and calibration error; the evidence does not establish neural connectivity recovery from arbitrary biological recordings.Source: §J.4 · pp. 51–52
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