Direction as a Parameter, Hierarchy as a Graded Asymmetry: What Generative Effective Connectivity Actually Delivers for Cortico-Subcortical Circuits

Question: How do generative effective connectivity models of resting-state fMRI estimate directionality and hierarchy in cortico-subcortical circuits, and how do their inferences compare with dynamic causal modeling and Granger-based approaches?

Answer

Across the retrieved literature, generative effective connectivity (EC) models estimate directionality by inverting an explicit data-generating model in which the directed coupling is a free parameter, and they estimate hierarchy by reading a monotonic ordering off the resulting asymmetry matrix — but the papers that demonstrate reliable recovery do so only after importing external constraints (structural connectivity priors, structural valency, gene-expression gradients, or tracer-derived directionality), not from BOLD data alone. The comparison with Granger causality is less a contest than an algebraic identity: generative EC and Granger causality rest on the same linear stochastic assumptions and are related by an approximately quadratic mapping once unequal noise variances are rescaled, a relation that only becomes observable at group level with large samples. Neither family currently supplies cortico-subcortical hierarchy estimates that are validated against subcortical ground truth; the subcortical directionality claims in these preprints come from structural, tracer-transfer, and transcriptomic evidence, while the generative-inversion demonstrations are cortical. Gradient-based hierarchy findings from generative models should therefore be read as anatomically constrained inferences rather than as data-driven discoveries about thalamus, basal ganglia, or limbic subcortex.

Generative inversion makes direction a parameter, but reliable recovery depends on priors imported from anatomy or spectra

The core generative claim is that EC is obtained by inverting a model of the measurements, so that the quality of the estimated network depends heavily on the modelling assumptions — in particular on how the endogenous fluctuations driving neural activity are represented [2]. The strongest current demonstrations of recovery are those that constrain the inversion with structure: a hierarchical empirical Bayes model built on a well-established dynamic causal model integrates structural connectivity into resting-state EC via priors, recovers ground-truth EC in silico, compares favourably with a popular alternative, and improves model evidence on empirical data [1]. A chromatic dynamic causal model instead makes the spectral colour of each region's endogenous fluctuations a function of structural valency, recovers ground-truth parameters across network sizes and noise conditions, and outperforms standard spectral DCM [11]. The convergent implication is that in the spectral/generative setting, direction is only weakly identified by the data and is effectively set by the prior structure the modeller supplies.

Hierarchy in generative models is a graded asymmetry that reproduces known laminar ordering — but the demonstrations are cortical

Structural priors yield a hierarchy claim that is testable and passes: the coupling between structural and effective connectivity differs across networks in a way that recapitulates a well-known unimodal–transmodal hierarchy, described as the first evidence of network-dependent modulation of the structure–EC relationship in humans [1]. Likewise, the valency-to-spectral-exponent mapping varies across a cortical hierarchy and is conserved across a homologous network in humans, macaques, marmosets, and mice [11]. Independent directed-connectivity frameworks reach the same geometric conclusion from functional data: an integrated effective connectivity measure recovered connectome directionality in simulation and empirical validation, and its derived hierarchy increased monotonically from sensorimotor through association to paralimbic cortex, flattening under externally oriented conditions and steepening under internally focused ones [5]. These are hierarchy estimates of the cortical sheet; none of these papers establishes that the same inversion recovers a thalamic or striatal position in the ordering.

Granger and generative EC are re-parameterizations of one another, not independent arbiters

Exploiting a mapping between multivariate Ornstein–Uhlenbeck and multivariate autoregressive processes, one study derived an approximately quadratic relation between EC and Granger causality after rescaling GC to compensate for unequal noise variances of source and target, showed in simulation that this relation is observable in finite time series only when a large amount of data is available, and verified at group level in a large Human Connectome Project dataset [8]. This substantially deflates the expectation that the two approaches will cross-validate one another on single subjects: agreement is a large-sample, group-level phenomenon, and the residual differences are attributable to identifiable biases rather than to different causal concepts [8]. Frequency-domain Granger variants add a separate finite-length problem: with short series, generalized partial directed coherence and frequency-domain GC tend to produce more false positives while an imaginary-coherency measure becomes insensitive to true connectivity [7].

Acyclic, Gaussian-noise estimators fail precisely where cortico-subcortical loops live

Classical causal models including Granger causality and structural equation modelling are largely restricted to acyclic interactions and struggle to represent cyclic and higher-order dynamics; a Hodge-theoretic decomposition of resting-state fMRI network flow revealed robust cyclic causal patterns that conventional causal models do not detect [6]. The same limitation is framed as a deficit for intervention: regression-based Granger causality, structural equation modelling, and dynamic causal modelling are characterized as descriptive and acyclic, leaving open what the causal organization would become under pathway disruption or modulation [9]. Generative DCM is explicitly noted to face the computational difficulty of estimating large cyclic causal models, motivating multiscale parcellation with Bayesian model reduction that scales to hundreds or thousands of regions [20]. Because cortico-basal-ganglia-thalamo-cortical circuits are canonically recurrent, this is exactly the regime in which the acyclic Granger family is weakest.

Causal-ordering identification requires assumptions Granger does not carry, and subcortical hierarchies have been recovered by adding them

Methods explicitly premised on Granger's perspective are argued to be unable to identify causal hierarchy in modular brain networks because of spurious correlation and the indistinguishability of causal direction under Gaussian observational noise in linear systems; a Jacobian-informed VAR-LiNGAM recovered causal ordering in synthetic data and identified causally hierarchical structures in mouse and human neural oscillatory signals [15]. A nonparametric structural-causal-model framework with lagged conditional independence tests proved consistent, outperformed baselines on linear, nonlinear, and recurrent-network simulations, and applied to mouse visual cortex, thalamus, and hippocampus uncovered stimulus-specific causal pathways and inter-regional hierarchies aligned with known anatomy [16]. The subcortical hierarchy results therefore arrive attached to non-Gaussianity or nonparametric conditional-independence assumptions, not to standard linear GC.

Subcortical directionality is currently supplied by anatomy and transcriptomics, not by generative inversion of fMRI

Noninvasive neuroimaging is described as blind to the directionality of neural wiring, and estimating edge-level directionality from function is characterized as ill-posed and underdetermined; the proposed resolution recovers ground-truth directionality from gene co-expression gradients with r = 0.70 in C. elegans, 0.57 in mouse, and 0.46 in macaque, and yields subject-specific directed connectomes in 770 humans that reproduce established corticothalamic asymmetries [4]. Transferring macaque tracer projection patterns to humans produces a directed structural connectome with a directionality-based hierarchy, and adding directionality reshapes the propagation and persistence of sensory inputs and best captures empirical spatiotemporal stimulus-evoked dynamics [22]. Diffusion tractography itself supplies two corticothalamic motifs consistent with hierarchy — focal posterolateral thalamic connections for sensorimotor cortex associated with fast feed-forward flow, diffuse anteromedial connections for associative cortex associated with slow feedback, replicated in macaques [14]. Structurally, the whole-brain connectome is modular and hierarchical with mixed cortical–subcortical modules and a subcortically dominated rich club built from hubs of all subcortical structures rather than the thalamus alone [3], while functional convergence onto thalamic, basal ganglia, and subcortical limbic communities is topographically organized and removing subcortical regions degrades information-transmission efficiency [19]. All of these are directed or hierarchical claims derived without inverting a generative model of resting BOLD in subcortical nodes.

Conflicts and gaps

The papers disagree on what carries the information about direction. One line treats endogenous-noise/spectral modelling as the crucial assumption [2] and shows a spectral re-parameterization beats standard spectral DCM [11], while another treats structural priors as the decisive ingredient and reports improved model evidence from them [1]; these prescriptions are not adjudicated against each other. There is a direct tension between the claim that generative EC and GC are near-equivalent at group level [8] and the claim that cyclic components invisible to conventional causal models are robustly present in resting fMRI [6]. Most importantly, no retrieved paper applies a DCM-family generative model to basal ganglia or thalamic nodes with a validated ground truth: the DCM applications reported here target cortical and cortico-limbic systems — a conductance-based canonical microcircuit model resolving hierarchical DLPFC-to-sgACC effects under theta-burst stimulation [13], and two-state DCM of the default-mode network with spectroscopic priors selected by Bayesian model comparison [12] — while a representative Granger application to subcortical regions has n = 10 per group [10], and an alternative generative scheme (probabilistic Boolean networks) has been used only for cortical-to-medial-temporal-lobe influence [23]. Whether the valency-exponent mapping or the structural-prior variance mapping holds for thalamus and striatum is untested.

What would change the answer

Three concrete tests would settle it. First, an in-silico benchmark with a known directed cortico-subcortical ground truth — a generative model whose thalamic and striatal edges are specified and then inverted by spectral DCM, chromatic DCM, structural-prior DCM, iEC, and GC on identical simulated BOLD — would establish whether any estimator recovers subcortical hierarchy without anatomical priors, extending the recovery designs already used for cortical EC [1] [11] [5]. Second, invasive causal ground truth is available: repeated single-pulse electrical stimulation across 5,794 intracranial electrodes including multiple thalamic nuclei showed that presence of a structural connection predicted causal electrophysiological connectivity with probability ~0.95 and absence predicted its lack with probability ~0.8, with thalamic involvement in indirect pathways increasing latency beyond 200 ms [18]; scoring generative and Granger estimates against these stimulation-defined directed edges would convert the hierarchy question into a prediction problem. Third, out-of-sample prediction of the held-out region's time series or of perturbation responses — as done by fitting closed-form cross-spectral densities and predicting unseen TMS-evoked potentials from stimulation-site-specific EC [17], and by modelling dynamic hierarchy alignment in source-resolved MEG [21] — would discriminate among models that fit equally well in-sample, since the acyclicity and noise-model disagreements above are otherwise not identifiable from resting covariance alone.

References

[1] Greaves, M. D., Novelli, L., Razi, A.. "Structurally informed resting-state effective connectivity recapitulates cortical hierarchy". bioRxiv. 2026-03-05T00:00:00.000Z. URL: https://doi.org/10.1101/2024.04.03.587831

[2] Giulia Prando, Mattia Zorzi, Alessandra Bertoldo, Alessandro Chiuso. "The role of noise modeling in the estimation of resting-state brain effective connectivity". arxiv. 2026-06-04T00:00:00.000Z. URL: https://arxiv.org/abs/1802.05533

[3] Julian Schulte, Mario Senden, Gustavo Deco, Xenia Kobeleva, Gorka Zamora-López. "The global communication pathways of the human brain transcend the cortical-subcortical-cerebellar division". arxiv. 2025-10-21T00:00:00.000Z. URL: https://arxiv.org/abs/2505.22893

[4] Sipes, B. S., Arab, F., Nagarajan, S., Raj, A.. "Gene Gradients Reveal Directed Structural Connectivity Across Species". bioRxiv. 2026-08-28T00:00:00.000Z. URL: https://doi.org/10.64898/2026.05.05.723068

[5] Oh, Y., Ann, Y., Lee, J., Ito, T., Froudist-Walsh, S., Paquola, C., Milham, M., Spreng, R. N., Margulies, D., Bernhardt, B., Woo, C.-W., Hong, S.-J.. "In vivo cartography of state-dependent signal flow hierarchy in the human cerebral cortex". bioRxiv. 2025-06-25T00:00:00.000Z. URL: https://doi.org/10.1101/2025.06.24.660962

[6] Moo K. Chung, D. Vijay Anand, Anass B El-Yaagoubi, Jae-Hun Jung, Anqi Qiu, Hernando Ombao. "Causality as a Minimum Energy Principle". arxiv. 2026-04-21T00:00:00.000Z. URL: https://arxiv.org/abs/2604.17151

[7] Sara Sommariva, Alberto Sorrentino, Michele Piana, Vittorio Pizzella, Laura Marzetti. "A comparative study of the robustness of frequency--domain connectivity measures to finite data length". arxiv. 2026-06-04T00:00:00.000Z. URL: https://arxiv.org/abs/1711.03332

[8] Allegra, M., Gilson, M., Brovelli, A.. "Directed neural interactions in fMRI: a comparison between Granger Causality and Effective Connectivity". bioRxiv. 2026-04-29T00:00:00.000Z. URL: https://doi.org/10.1101/2024.02.22.581068

[9] Moo K. Chung, Luigi Maccotta, Aaron Struck. "Counterfactual Analysis of Brain Network Dynamics". arxiv. 2026-04-01T00:00:00.000Z. URL: https://arxiv.org/abs/2603.29843

[10] Tahir, A., Abdullah, W., Azeem, W. u., Khalid, M. F., Abualait, T., Shakil, S., Ahmed, F., Bashir, S., Chaudhary, S. U.. "Causality Mapping Using Resting-state fMRI Reveals Hyperactivity and Hypoconnectivity in Schizophrenia Patients". bioRxiv. 2024-12-09T00:00:00.000Z. URL: https://doi.org/10.1101/2024.09.25.614909

[11] Greaves, M. D., Novelli, L., Pang, J. C., Fornito, A., Razi, A.. "Spectral imprint of structural embedding in effective connectivity". bioRxiv. 2025-06-25T00:00:00.000Z. URL: https://doi.org/10.1101/2025.06.19.660638

[12] Asadpour, A., Malaviya, A., Cook, J., George, A., Bhattacharya, P., Manitsa, I., Wilson, M. P., Dauvermann, M. R.. "Two-State Effective Connectivity Predicts the Effect of Anterior Cingulate Cortex Excitatory / Inhibitory Balance on Posterior Cingulate Cortex". bioRxiv. 2026-09-14T00:00:00.000Z. URL: https://doi.org/10.64898/2026.09.01.748001

[13] Toghi, A., Ghaffaripour Jahromi, G., Zarei, S., Aliyari, B., Moqtaderi, H., Motie Nasrabadi, A.. "Hierarchical neurobiological changes in dorsolateral prefrontal cortex and subgenual anterior cingulate cortex connectivity induced by theta-burst stimulation". medRxiv. 2025-05-24T00:00:00.000Z. URL: https://doi.org/10.1101/2025.05.23.25328212

[14] Howell, A. M., Warrington, S., Fonteneau, C., Cho, Y. T., Sotiropoulos, S., Murray, J. D., Anticevic, A.. "The spatial extent of anatomical connections within the thalamus varies across the cortical hierarchy in humans and macaques". bioRxiv. 2025-05-04T00:00:00.000Z. URL: https://doi.org/10.1101/2023.07.22.550168

[15] Yokoyama, H., Takeuchi, R., Shimizu, S.. "Causal Discovery of Synchronous Neural Oscillations based on Jacobian-informed VAR-LiNGAM". bioRxiv. 2026-05-01T00:00:00.000Z. URL: https://doi.org/10.64898/2026.04.28.721377

[16] Rahul Biswas, SuryaNarayana Sripada, Somabha Mukherjee, Reza Abbasi-Asl. "CITS: Nonparametric Statistical Causal Modeling for High-Resolution Neural Time Series". arxiv. 2026-03-10T00:00:00.000Z. URL: https://arxiv.org/abs/2508.0192

[17] Veronese, A., Momi, D., Sarasso, S., Corbetta, M., Allegra, M., Suweis, S.. "A Whole-Brain Dynamical Framework Linking Resting-State Activity to TMS-Evoked Responses". bioRxiv. 2026-07-16T00:00:00.000Z. URL: https://doi.org/10.64898/2026.07.10.737755

[18] Shailja, S., Lyu, D., Chau Loo Kung, G., Mortazavi, L., Dai, E., Zeineh, M. M., Buch, V. P., Deisseroth, K., Parvizi, J., McNab, J. A.. "Diffusion MRI Tractography Predicts Electrophysiological Connectivity and Explains Spectral Signatures of Evoked Potentials in the Human Brain". bioRxiv. 2026-09-21T00:00:00.000Z. URL: https://doi.org/10.64898/2026.09.15.751678

[19] Dang, S., Jallepalli, D.. "Cortico-subcortical converging organization at rest". bioRxiv. 2025-01-27T00:00:00.000Z. URL: https://doi.org/10.1101/2025.01.26.634847

[20] Zarghami, T. S.. "Multiscale Parcellation of Dynamic Causal Models of the Brain". bioRxiv. 2026-02-14T00:00:00.000Z. URL: https://doi.org/10.1101/2025.06.14.659698

[21] Liu, X., Wan, B., Liu, W., Lu, R., Wang, S., Zhang, X., Dong, L., Fan, T.. "Cortical Hierarchy Dynamically Organizes Large-Scale Neural Propagation". bioRxiv. 2026-09-03T00:00:00.000Z. URL: https://doi.org/10.64898/2026.08.29.747943

[22] Huang, N., Wang, H. E., Triebkorn, P., Gandini Wheeler-Kingshott, C. A. M., Jedyank, M., David, O., Destexhe, A., D'Angelo, E. U., Pedersen, N. P., Jirsa, V.. "Directed Human Structural Connectome Reveals Hierarchical Organization and Shapes Large-Scale Brain Dynamics". bioRxiv. 2026-06-17T00:00:00.000Z. URL: https://doi.org/10.64898/2026.06.16.732559

[23] Liu, Z., Zhang, L., Ching, W. K.. "Inferring Progressive Disconnection in Alzheimer's Disease with Probabilistic Boolean Networks". bioRxiv. 2025-10-17T00:00:00.000Z. URL: https://doi.org/10.1101/2025.07.10.664143

How this survey was made

This survey answers one question: “How do generative effective connectivity models of resting-state fMRI estimate directionality and hierarchy in cortico-subcortical circuits, and how do their inferences compare with dynamic causal modeling and Granger-based approaches?”

We keep our own collection of recent preprints and refresh it every day from bioRxiv, medRxiv, arXiv, PsyArXiv, SocArXiv and ChemRxiv. For each survey, we search that collection for recent work and keep the strongest matches: papers that use the same terms as the question, and papers on the same topic in different words.

The text was written from the titles and abstracts of those papers; the full texts were not read. Statements that draw on a particular paper are marked with a number in brackets. The reference list just above is the 23 papers whose work this text draws on, chosen from the 143 papers those searches returned; the rest of the shortlist is not printed. It is not a complete review of everything published on the question.

Treat this as a starting point and a reading list, not a peer-reviewed review. The searches behind this survey ran on October 21, 2026 and covered work from the previous two years.