Identifiable Signed Causal Learning via Latent Stochastic Differential Equations for Inferring Brain Dynamic Effective Connectivity

Yiding Wang; Chen Qiao

cs.LGNov 2, 2026

Identifiable Signed Causal Learning via Latent Stochastic Differential Equations for Inferring Brain Dynamic Effective Connectivity

Authors: Yiding Wang, Chen Qiao

Organizations: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049 P.R. China.

Abstract

Understanding how brain regions interact to support cognitive tasks requires characterizing directed causal influences among them, known as effective connectivity (EC). As inter-regional coupling changes over time, a dynamic EC (dEC) estimate is required to track this temporal variation. Existing dEC learning methods typically lack a causal rationality guarantee or an identifiability analysis, and output unsigned connectivity that cannot distinguish excitatory from inhibitory influences, which limits further study of brain mechanisms. We propose ISCL, a latent stochastic differential equation framework that derives dEC from latent dynamics through a causal influence operator defined in observation space. As established by a rationality theorem, the operator captures a directed interventional response at each brain state, and under invariance assumptions the operator is identifiable from data, with its signed output separating excitatory from inhibitory influences. On synthetic benchmarks, \model recovers the causal structure on Lorenz96 and achieves the lowest mean structural Hamming distance among compared methods on NetSim, while preserving signed influence structures that distinguish excitatory from inhibitory interactions. Applied to HCP Social Cognition task-fMRI, ISCL reveals a signed architecture whose stability across seeds supports the identifiability guarantee, identifies dominant source and target networks consistent with social-cognition systems, and shows that condition-specific training strengthens causal outflow from the default-mode and frontoparietal networks during theory-of-mind processing. In this task, the recovered signed structure reveals network-level excitatory and inhibitory organization, with the dorsal-attention network as the only consistently positive out-strength source.

Figures & tables

Fig. 1: Overview of the ISCL framework. High-dimensional observations

\mathbf{x}_{t}

are mapped into a low-dimensional latent representation

\hat{\mathbf{z}}_{t}

via latent inference with encoder

\mathbf{m}_{\phi}

. Temporal evolution is modeled by latent dynamics with drift

\boldsymbol{\mu}_{\psi}

, producing dynamically consistent latent trajectories. Observation mapping via decoder

\mathbf{f}_{\theta}

reconstructs observations and predicts future states from the latent dynamics. The Jacobian

\nabla\mathbf{f}_{\theta}

propagates latent drift to the observation space via the chain rule, and the Hessian

\nabla^{2}\mathbf{f}_{\theta}

provides a second-order correction, yielding an identifiable dynamic EC operator

\mathbf{C}_{\tilde{\mathbf{x}}}\in\mathbb{R}^{D_{x}\times D_{x}}

. Training jointly optimizes reconstruction accuracy, latent dynamic consistency, and future-state prediction.

Fig. 2: Loss ablation and $\mathbf{C}$ -matrix comparison on the Lorenz96 setting. (a) Estimated $\hat{\mathbf{C}}$ from the base configuration. (b) Ground-truth $\mathbf{C}_{\mathrm{true}}$ with colorbar. (c) Estimated $\hat{\mathbf{C}}$ with $\mathcal{L}_{\mathrm{edge}}$ . (d) $\mathrm{AUROC}$ and $\mathrm{WCS}$ for the base, base augmented with $\mathcal{L}_{\mathrm{edge}}$ , and ablations that remove $\mathcal{L}_{\mathrm{dyn}}$ or $\mathcal{L}_{\mathrm{pred}}$ .

Fig. 3: Dynamic causal coupling tracking on the non-identity Lorenz96 setting. (a) Ground-truth $\mathbf{C}_{\mathrm{true}}$ ; estimated $\hat{\mathbf{C}}$ with model $D_{z}=20$ (b) and $D_{z}=15$ (c). (d–i) For $D_{z}=20$ and $D_{z}=15$ , per-edge scatter of estimated versus true coupling and the time series of the top and mid-ranked edge.

$D_{z}$	Dropout	AUROC(%) $\uparrow$	SHD $\downarrow$	WCS(%) $\uparrow$
10	$20\%$	$71.25\pm 1.39$	$66.4\pm 4.4$	$41.21\pm 4.69$
	$40\%$	$70.81\pm 7.35$	$66.4\pm 14.0$	$42.42\pm 6.60$
	$60\%$	$78.97\pm 7.41$	$60.4\pm 15.4$	$45.80\pm 7.98$
15	$20\%$	$95.68\pm 1.24$	$22.4\pm 4.8$	$70.39\pm 2.43$
	$40\%$	$95.33\pm 1.94$	$22.4\pm 4.1$	$65.78\pm 2.76$
	$60\%$	$89.86\pm 3.64$	$36.0\pm 9.0$	$60.08\pm 1.68$

TABLE I: Latent dimension and dropout grid on the 20-node identity Lorenz96 setting across

5

seeds. Mean

\pm

95% CI computed by Student’s

t

-distribution.

Setting	Method	AUROC(%) $\uparrow$	SHD $\downarrow$	WCS(%) $\uparrow$	Time (min)
sim2	NGM	$85.13\pm 1.43$	$17.2\pm 3.8$	$-65.09\pm 0.45$	86.6
	cMLP	$86.81\pm 0.36$	$12.0\pm 0.0$	$-23.85\pm 0.06$	85.3
	CUTS	$97.46\pm 0.34$	$9.6\pm 2.1$	$-28.67\pm 0.05$	48.6
	CUTS+	$91.22\pm 0.57$	$16.4\pm 1.1$	$-36.63\pm 0.14$	70.7
	ISCL	$\mathbf{98.52\pm 0.66}$	$\mathbf{6.8\pm 2.8}$	$\mathbf{94.64\pm 0.25}$	$\mathbf{1.0}$
sim4	NGM	$84.57\pm 0.43$	$108.8\pm 5.4$	$-55.96\pm 0.20$	89.0

TABLE II: NetSim benchmark. Mean

\pm

95% CI over

5

seeds computed by Student’s

t

-distribution.

Fig. 4: NetSim results. (a) Cross-seed pairwise cosine similarity of $\mathbf{C}$ matrices across 5 independent seeds, upper triangle for sim2 and lower triangle for sim4 . (b)–(c) Scatter of estimated versus true edge weights.

Metric	$D_{z}{=}15$	$D_{z}{=}20$
RSN $\|\tilde{\mathbf{C}}\|$ cosine	$0.986\pm 0.006$	$0.989\pm 0.005$
ROI $\|\tilde{\mathbf{C}}\|$ cosine	$0.950\pm 0.015$	$0.932\pm 0.015$
RSN signed $\tilde{\mathbf{C}}$ cosine	$0.935\pm 0.017$	$0.915\pm 0.030$
ROI signed $\tilde{\mathbf{C}}$ cosine	$0.914\pm 0.019$	$0.858\pm 0.041$

TABLE III: Cross-seed consistency of ISCL on HCP Social Cognition. Mean

\pm

95% CI over

\binom{5}{2}=10

pairwise seed comparisons, computed by bootstrap resampling of the pairwise cosine.

Fig. 5: Signed dEC architecture. (a) Per-subject normalized $200\times 200$ ROI-level $\tilde{\mathbf{C}}$ matrix at $D_{z}=15$ . (b) Corresponding $7\times 7$ RSN-aggregated signed matrix with marginal bar charts showing per-RSN out-strength and in-strength; error bars are cross-seed standard deviation.

Fig. 6: Condition-specific dEC at $D_{z}=15$ . (a) Mentalizing condition circos of the top-50 strongest ROI-level edges; (b) random condition circos. Edge width encodes connection magnitude. (c) Dorsal view of the 50 strongest mentalizing-strengthened directed edges ranked by $|\Delta|\tilde{\mathbf{C}}||$ ; (d) dorsal view of the 50 strongest mentalizing-weakened directed edges.

Fig. 7: Condition-specific effective connectivity difference. (a) $\Delta|\tilde{\mathbf{C}}|$ at the ROI level, subject-average across five seeds. (b) RSN-level $\Delta|\tilde{\mathbf{C}}|$ with marginal bars showing per-RSN out-strength and in-strength; error bars are cross-seed standard deviation. (c-d) Per-subject distribution (n=339) of directed RSN-level and ROI-level $\Delta|\tilde{\mathbf{C}}|$ for the top-4 strongest RSN pairs and the top-4 hub ROIs.

Metric	Value
Cross-condition cosine ( $\|\tilde{\mathbf{C}}_{\mathrm{mental}}\|$ , $\|\tilde{\mathbf{C}}_{\mathrm{random}}\|$ )	$0.929\pm 0.024$
Cross-seed cosine, mentalizing	$0.948\pm 0.015$
Cross-seed cosine, random	$0.939\pm 0.025$
Top-20 RSN pair by $\|\Delta\|\tilde{\mathbf{C}}\|\|$ , 5/5 sign agreement	16 / 20
strengthened / weakened	8 / 12
Top-400 ROI pair by $\|\Delta\|\tilde{\mathbf{C}}\|\|$ , 5/5 sign agreement	349 / 400

TABLE IV: Condition-specific EC stability and difference metrics.

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Timothy Doyeon Kim, Ulises Pereira-Obilinovic, Yiliu Wang +2

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Query Identifiability Recurrent Neural Networks Functional Connectivity

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We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.

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1LIACS, Leiden University, Netherlands

Stochastic Differential Equations Structural Causal Models Causal Identifiability

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We introduce Causal Computational Asymmetry (CCA), a principle for causal direction identification based on optimization dynamics in which one neural network is trained to predict

Y

from

X

and another to predict

X

from

Y

, and the direction that converges faster is inferred to be causal. Under the additive noise model

Y = f(X) + \varepsilon

with

\varepsilon \perp X

and

f

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Independent Researcher

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