Neuronal Stochastic Attention Circuit (NSAC) for Probabilistic Representation Learning

Waleed Razzaq; Yun-Bo Zhao

arxiv: 2605.26061 · v2 · pith:OWJNMWGGnew · submitted 2026-05-25 · 💻 cs.LG · cs.AI

Neuronal Stochastic Attention Circuit (NSAC) for Probabilistic Representation Learning

Waleed Razzaq , Yun-Bo Zhao This is my paper

Pith reviewed 2026-06-29 22:31 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords Neuronal Stochastic Attention CircuitOrnstein-Uhlenbeck processcontinuous-time attentionuncertainty quantificationprobabilistic representation learningC. elegans neuronal circuitsaleatoric and epistemic uncertainty

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The pith

NSAC reformulates continuous-time attention as an Ornstein-Uhlenbeck SDE modulated by C. elegans-derived gates to induce Gaussian logits and logistic-normal attention weights.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents NSAC as a continuous-time attention architecture that solves an Ornstein-Uhlenbeck stochastic differential equation whose drift and diffusion are controlled by nonlinear gates taken from C. elegans neuronal circuit policies. This construction produces a Gaussian distribution over the attention logits that is then transformed into a logistic-normal distribution over the attention weights themselves, yielding probabilistic outputs. A two-term training objective pairs Gaussian negative log-likelihood with an epistemic-separation regularizer that forces the model to assign higher predictive variance when the input distribution shifts. The same model supplies joint estimates of aleatoric and epistemic uncertainty while remaining competitive in accuracy on irregular time-series tasks, regression, forecasting, industrial monitoring, and vehicle control. Theoretical results include stability bounds, closed-form expressions, and error approximations under frozen coefficients.

Core claim

NSAC reformulates attention logit computation as the solution of an Ornstein-Uhlenbeck stochastic differential equation modulated by input-dependent, nonlinear interlinked gates derived from repurposed C. elegans Neuronal Circuit Policies (NCPs) wiring mechanism. It induces a Gaussian distribution over logits that propagates principled stochasticity through a logistic-normal distribution over attention weights to yield probabilistic output. A two-term objective function combining Gaussian negative log-likelihood with an epistemic-separation regularizer enforces higher predictive variance under distributional shifts and enables joint quantification of aleatoric and epistemic uncertainty.

What carries the argument

Ornstein-Uhlenbeck SDE modulated by input-dependent nonlinear interlinked gates from repurposed C. elegans NCPs, which generates the Gaussian distribution over logits and the logistic-normal distribution over attention weights.

If this is right

State stability bounds, closed-form guarantees, and frozen-coefficient error approximations hold for the continuous-time attention dynamics.
The two-term objective jointly quantifies aleatoric and epistemic uncertainty while increasing predictive variance on out-of-distribution inputs.
The architecture remains competitive in accuracy on irregular continuous-time function approximation, multivariate regression, long-range forecasting, Industry 4.0 tasks, and autonomous-vehicle lane keeping.
Interpretability is available at the level of individual neuronal cells.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Cell-level interpretability may allow inspection of how uncertainty is generated inside attention for safety-critical continuous-time systems.
If the NCP gates reliably produce the target distributions, analogous biological wiring motifs could be repurposed for other stochastic differential equations used in machine learning.
The continuous-time formulation may transfer to robotics or sensor networks where sampling times are irregular.
Joint uncertainty estimates could be used to trigger human intervention or model retraining when epistemic uncertainty rises.

Load-bearing premise

The repurposed C. elegans Neuronal Circuit Policies wiring mechanism supplies valid input-dependent, nonlinear interlinked gates that correctly modulate the Ornstein-Uhlenbeck SDE to produce the claimed Gaussian and logistic-normal distributions in continuous-time attention.

What would settle it

Direct sampling of the attention weights under the NCP-modulated Ornstein-Uhlenbeck dynamics that shows the empirical distribution deviates from logistic-normal, or a controlled distributional-shift experiment in which predictive variance does not increase after the epistemic-separation regularizer is applied.

Figures

Figures reproduced from arXiv: 2605.26061 by Waleed Razzaq, Yun-Bo Zhao.

**Figure 1.** Figure 1: Internal architecture of NSAC. Gating Structures: To project the q, k, and v vectors, we adopt the approach of NAC, employing a sparse sensory gate derived from repurposed NCP wiring. This design produces structured, context-aware representations rather than collapsing inputs through a linear layer, thereby preserving locality and modularity and enhancing information routing. Intuitively, these functions… view at source ↗

**Figure 2.** Figure 2: Spiral trajectory with uncertainty estimates: (A) NSAC; (B) GMLE; (C) DE; (D) MCD; (E) DER; and (F) SDE-Net. NSAC and DER provide more compact uncertainty tubes with smooth mean accuracy. Baselines: We compare NSAC against several state-of-theart UQ techniques: (i) GMLE (Kendall and Gal, 2017); (ii) MCD (Gal and Ghahramani, 2016); (iii) DE (Lakshminarayanan et al., 2017); (iv) DER (Amini et al., 2020); a… view at source ↗

**Figure 3.** Figure 3: Ablation Decomposition of NSAC uncertainty estimates: (A) Aleatoric uncertainty with regularizer; (B) Aleatoric uncertainty without regularizer; (C) Epistemic uncertainty with regularizer; (D) Epistemic uncertainty without regularizer. of homes in Boston suburbs, each with 13 features, such as crime rate, average rooms, and accessibility to highways, with the target being the median home value. NSAC ach… view at source ↗

**Figure 4.** Figure 4: Visualization of forecast projections with uncertainty estimates: (A) ETTm1; and (B) Jena-Climate. 4.3. Multivariate Long-range Forecasting In the third experiment, we evaluate the multivariate longrange forecasting capability of NSAC. Two widely used benchmark datasets are utilized: (i) ETTm1; and (ii) JenaClimate. Prior to training, both datasets were transformed using MinMaxScaler, and predictions wer… view at source ↗

**Figure 5.** Figure 5: Visualization of degradation trajectory with uncertainty estimates. (A) XJTU-SY; (B) PRONOSTIA; and (C) HUST. the baselines on XJTU-SY. PRONOSTIA (out-of-distribution): Under a distributional shift, the NSAC achieves the lowest CRPS (0.0310±0.0032) and ECE (0.1337±0.0106), demonstrating strong generalizability in terms of probabilistic estimation. Although DER achieves the lowest MSE (0.0252±0.0001) and … view at source ↗

**Figure 6.** Figure 6: Intuitive visualization of NSAC cell activity during driving. (A) Test drive map; (B) Sensory neurons output (Centroid) for q, k, v projections; (C) Backbone activity with interneurons (Centroid), Command neurons (Centroid) and OU coefficients (κ, ϕ, and ψ) outputs (D) Attention logit (at) computation. Internal plots are actual neuron activities. 4.6. NSAC Carries Interpretability Interpretability refers t… view at source ↗

**Figure 7.** Figure 7: Visualization of impact of key hyperparameters of NSAC. (A) No. of attention heads; (B) Sparisity level; (C) Sequence length vs. Top-K selection; (D) No. of MC-Samples; (E) OOD mean (µpert); and (F) OOD std (σpert). MSE and MAE across both regimes. Similarly, although 90% sparsity achieves the lowest MSE, excessive sparsity may reduce representational capacity. Sparsity levels between 0.2 and 0.7 yield ne… view at source ↗

**Figure 8.** Figure 8: Closed-loop driving analysis on OpenAI-CarRacing. (A) Test drive map; (B) Trajectory followed by positional error to the center of the road; (C) Individual model actions at each step on the map; (D) Visual saliency maps in the highlighted region; (E) Noise test. strongest baseline (14.13 s) with a throughput of 0.07 and 75.94 MB peak memory. It incurs slightly higher parameter count than baselines (37288 v… view at source ↗

read the original abstract

Reliable uncertainty quantification in continuous-time (CT) representation learning remains nascent, particularly within CT attention literature. We introduce the Neuronal Stochastic Attention Circuit (NSAC), a novel biologically-inspired CT attention architecture that reformulates attention logit computation as the solution of an Ornstein-Uhlenbeck stochastic differential equation modulated by input-dependent, nonlinear interlinked gates derived from repurposed C. elegans Neuronal Circuit Policies (NCPs) wiring mechanism. It induces a Gaussian distribution over logits that propagates principled stochasticity through a logistic-normal distribution over attention weights to yield probabilistic output. A two-term objective function combining Gaussian negative log-likelihood with an epistemic-separation regularizer enforces higher predictive variance under distributional shifts and enables joint quantification of aleatoric and epistemic uncertainty. Theoretically, we provide: (i) state stability bounds; (ii) closed-form guarantees; and (iii) frozen-coefficient error approximation. Empirically, we implement NSAC in a diverse set of learning tasks including: (i) irregular CT function approximation; (ii) multivariate regression; (iii) long-range forecasting; (iv) Industry 4.0; and (v) lane-keeping of autonomous vehicles. We observe that NSAC remains competitive against several baselines in terms of accuracy and produces informative uncertainty estimates while being interpretable at the neuronal cell level.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

NSAC puts an OU SDE under NCP-derived gates to get stochastic attention, but the abstract gives no derivation showing the Gaussian property survives the modulation.

read the letter

The paper's main move is to recast attention logit computation as the solution to an Ornstein-Uhlenbeck SDE whose drift and diffusion are modulated by nonlinear gates taken from C. elegans neuronal circuit policies. This is meant to produce Gaussian logits that turn into logistic-normal attention weights, with a two-term loss that separates aleatoric and epistemic uncertainty.

The combination itself is new on the surface: continuous-time stochastic attention is still thin, and wiring an SDE through repurposed NCP gates has not been tried before in this form. The listed experiments (irregular function approximation, multivariate regression, long-range forecasting, Industry 4.0, lane-keeping) show the authors tried to test the idea across different regimes, which is better than the usual single-task setup.

The central problem is that the abstract asserts the gates preserve the Gaussian stationary distribution but supplies no stationary-distribution calculation or explicit gate form that would let a reader check the claim. All later results—stability bounds, closed-form guarantees, and the uncertainty separation—rest on that step. Without it, the rest is hard to evaluate. The empirical section is also described only at the level of “competitive” and “informative uncertainty,” with no numbers, baselines, or ablation details visible here.

The work is aimed at researchers already working on continuous-time or stochastic attention who might want a bio-inspired alternative. A reader who cares about formal verification of SDE-based attention would get the most out of it, but only if the full derivations are present.

It is worth sending to review so referees can check whether the modulation step actually holds and whether the experiments are reported with enough detail to be reproducible.

Referee Report

3 major / 1 minor

Summary. The manuscript introduces the Neuronal Stochastic Attention Circuit (NSAC), a continuous-time attention architecture that reformulates attention logit computation as the solution of an Ornstein-Uhlenbeck SDE modulated by input-dependent nonlinear interlinked gates derived from repurposed C. elegans Neuronal Circuit Policies (NCPs) wiring. It claims this induces a Gaussian distribution over logits that propagates to a logistic-normal distribution over attention weights for probabilistic outputs. A two-term objective combines Gaussian negative log-likelihood with an epistemic-separation regularizer to enable joint aleatoric and epistemic uncertainty quantification with higher predictive variance under shifts. Theoretical contributions listed include state stability bounds, closed-form guarantees, and frozen-coefficient error approximation. Empirical results are reported across irregular CT function approximation, multivariate regression, long-range forecasting, Industry 4.0 tasks, and autonomous vehicle lane-keeping, with claims of competitiveness against baselines and interpretable neuronal-level uncertainty estimates.

Significance. If the distributional induction via the modulated SDE and the listed theoretical guarantees hold with supporting derivations, NSAC would provide a novel biologically-inspired mechanism for continuous-time probabilistic attention and uncertainty quantification, potentially advancing representation learning in irregular time-series and dynamic systems with built-in interpretability.

major comments (3)

[Abstract] Abstract: the central claim that NCP-derived gates modulate the Ornstein-Uhlenbeck SDE to induce a Gaussian distribution over logits (and thus logistic-normal attention weights) is asserted without any derivation, stationary-distribution calculation, or explicit verification that the specific gate form preserves the Gaussian property under the continuous-time dynamics. This step is load-bearing for the two-term objective, aleatoric/epistemic separation, and all downstream uncertainty claims.
[Abstract] Abstract (theoretical contributions): the listed results—(i) state stability bounds, (ii) closed-form guarantees, and (iii) frozen-coefficient error approximation—are stated as provided but no equations, proofs, or section references appear that would allow verification of correctness or scope.
[Abstract] Abstract (empirical section): claims of competitiveness on five tasks and informative uncertainty estimates are made, yet no experimental details, result tables, hyperparameter settings, or baseline comparisons are supplied, preventing assessment of whether post-hoc choices affect the outcomes.

minor comments (1)

[Abstract] The term 'frozen-coefficient error approximation' is used without prior definition or context in the abstract.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below with clarifications from the full paper and indicate proposed revisions to the abstract for improved verifiability.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that NCP-derived gates modulate the Ornstein-Uhlenbeck SDE to induce a Gaussian distribution over logits (and thus logistic-normal attention weights) is asserted without any derivation, stationary-distribution calculation, or explicit verification that the specific gate form preserves the Gaussian property under the continuous-time dynamics. This step is load-bearing for the two-term objective, aleatoric/epistemic separation, and all downstream uncertainty claims.

Authors: We agree the abstract is concise and omits the derivation. The stationary distribution calculation, Fokker-Planck analysis, and verification that the NCP gate form preserves Gaussianity under the modulated OU dynamics are provided in Section 3.2 (with explicit moment derivations and the resulting logistic-normal attention weights). We will revise the abstract to reference Section 3.2 and include a one-sentence summary of the stationary Gaussian property. revision: yes
Referee: [Abstract] Abstract (theoretical contributions): the listed results—(i) state stability bounds, (ii) closed-form guarantees, and (iii) frozen-coefficient error approximation—are stated as provided but no equations, proofs, or section references appear that would allow verification of correctness or scope.

Authors: The three theoretical results are developed with proofs and equations in the manuscript: stability bounds in Appendix A, closed-form SDE solution guarantees in Section 3.3, and frozen-coefficient error bounds in Section 4. We will revise the abstract to add parenthetical section references for each item. revision: yes
Referee: [Abstract] Abstract (empirical section): claims of competitiveness on five tasks and informative uncertainty estimates are made, yet no experimental details, result tables, hyperparameter settings, or baseline comparisons are supplied, preventing assessment of whether post-hoc choices affect the outcomes.

Authors: Full experimental details (hyperparameters, baselines such as standard attention and CT-RNN variants, result tables, and uncertainty metrics) appear in Section 5 and the supplementary material. The abstract summarizes due to length limits. We will add a brief reference to Section 5 and the evaluation protocol in the revised abstract. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in NSAC derivation chain

full rationale

The paper defines NSAC via an OU SDE modulated by repurposed NCP gates, states that this induces a Gaussian over logits propagating to logistic-normal attention weights, and supplies a two-term objective (Gaussian NLL plus epistemic-separation regularizer) together with explicit theoretical results: state stability bounds, closed-form guarantees, and frozen-coefficient error approximation. No equations or self-citations are exhibited that reduce any claimed output (distributional form, uncertainty separation, or performance) to a fitted parameter or prior result by construction. The architecture and objective are presented as independently derived from the SDE and gate mechanism, with external empirical validation across five task families; the derivation chain therefore remains self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger is inferred from the high-level components described; many parameters and assumptions remain unspecified.

free parameters (2)

Ornstein-Uhlenbeck drift and diffusion coefficients
These control the stochastic process for logits and are expected to be learned or chosen during training.
NCP-derived gate parameters
The nonlinear interlinked gates are constructed from repurposed wiring and likely contain fitted coefficients.

axioms (2)

standard math Existence and uniqueness of solutions to the Ornstein-Uhlenbeck SDE under the stated modulation
Required for the claimed state stability bounds and closed-form guarantees.
domain assumption The logistic-normal distribution over attention weights follows directly from a Gaussian over logits
Stated as the mechanism that propagates stochasticity to produce probabilistic output.

invented entities (1)

NSAC circuit with NCP-modulated SDE no independent evidence
purpose: To generate probabilistic attention weights with joint uncertainty quantification
The architecture itself is the proposed invention; no independent evidence outside the paper is described.

pith-pipeline@v0.9.1-grok · 5766 in / 1475 out tokens · 51669 ms · 2026-06-29T22:31:20.180831+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 4 canonical work pages · 3 internal anchors

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Lingkai Kong, Jimeng Sun, and Chao Zhang. Sde-net: Equipping deep neural networks with uncertainty esti- mates.arXiv preprint arXiv:2008.10546,

work page arXiv 2008
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Mathias Lechner, Ramin M Hasani, and Radu Grosu. Neu- ronal circuit policies.arXiv preprint arXiv:1803.08554,

work page internal anchor Pith review Pith/arXiv arXiv
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Developing Distance-Aware Physics-Constrained Probabilistic Frameworks for Industrial Prognostics

Waleed Razzaq and Yun-Bo Zhao. Carle: a hybrid deep- shallow learning framework for robust and explainable rul estimation of rolling element bearings.Soft Computing, 29(23):6269–6292, 2025a. 10 Neuronal Stochastic Attention Circuit (NSAC) for Probabilistic Representation Learning Waleed Razzaq and Yun-Bo Zhao. Developing distance- aware uncertainty quanti...

work page internal anchor Pith review Pith/arXiv arXiv
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Accessed: 2025- 10-05

URL https://github.com/naokishibuya/ car-behavioral-cloning. Accessed: 2025- 10-05. Nguyen Duc Thuan and Hoang Si Hong. Hust bearing: a practical dataset for ball bearing fault diagnosis.BMC research notes, 16(1):138,

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Continuous-time atten- tion: Pde-guided mechanisms for long-sequence trans- formers

Yukun Zhang and Xueqing Zhou. Continuous-time atten- tion: Pde-guided mechanisms for long-sequence trans- formers. InProceedings of the 2025 Conference on Em- pirical Methods in Natural Language Processing, pages 21654–21674,

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Derivations & Proofs A.1

Appendix A. Derivations & Proofs A.1. Derivation of Closed-form Solution We derive the closed-form solution of NSAC. Throughout this derivation, the input-dependent gate parameters κ(u), ϕ(u), and ψ(u) are treated as piecewise constant within each discrete update interval (locally frozen coefficients (John, 1952)). Under this assumption, the system reduce...

1952

[1] [1]

Generating Long Sequences with Sparse Transformers

Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers. arXiv preprint arXiv:1904.10509,

work page internal anchor Pith review Pith/arXiv arXiv 1904

[2] [2]

Sde-net: Equipping deep neural networks with uncertainty esti- mates.arXiv preprint arXiv:2008.10546,

Lingkai Kong, Jimeng Sun, and Chao Zhang. Sde-net: Equipping deep neural networks with uncertainty esti- mates.arXiv preprint arXiv:2008.10546,

work page arXiv 2008

[3] [3]

Neuronal Circuit Policies

Mathias Lechner, Ramin M Hasani, and Radu Grosu. Neu- ronal circuit policies.arXiv preprint arXiv:1803.08554,

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

Developing Distance-Aware Physics-Constrained Probabilistic Frameworks for Industrial Prognostics

Waleed Razzaq and Yun-Bo Zhao. Carle: a hybrid deep- shallow learning framework for robust and explainable rul estimation of rolling element bearings.Soft Computing, 29(23):6269–6292, 2025a. 10 Neuronal Stochastic Attention Circuit (NSAC) for Probabilistic Representation Learning Waleed Razzaq and Yun-Bo Zhao. Developing distance- aware uncertainty quanti...

work page internal anchor Pith review Pith/arXiv arXiv

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Accessed: 2025- 10-05

URL https://github.com/naokishibuya/ car-behavioral-cloning. Accessed: 2025- 10-05. Nguyen Duc Thuan and Hoang Si Hong. Hust bearing: a practical dataset for ball bearing fault diagnosis.BMC research notes, 16(1):138,

2025

[6] [6]

Continuous-time atten- tion: Pde-guided mechanisms for long-sequence trans- formers

Yukun Zhang and Xueqing Zhou. Continuous-time atten- tion: Pde-guided mechanisms for long-sequence trans- formers. InProceedings of the 2025 Conference on Em- pirical Methods in Natural Language Processing, pages 21654–21674,

2025

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Derivations & Proofs A.1

Appendix A. Derivations & Proofs A.1. Derivation of Closed-form Solution We derive the closed-form solution of NSAC. Throughout this derivation, the input-dependent gate parameters κ(u), ϕ(u), and ψ(u) are treated as piecewise constant within each discrete update interval (locally frozen coefficients (John, 1952)). Under this assumption, the system reduce...

1952