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Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A biologically plausible learning rule, e-prop, can match backpropagation through time in how closely trained recurrent networks reproduce recorded neural activity, once task accuracy is matched.

desk verdict A solid empirical existence result—e-prop can match BPTT in neural similarity at matched accuracy—that is slightly overstated in the abstract, but the core finding survives the main caveat. read the letter →

arxiv 2506.06904 v1 pith:TBO7Z2RH submitted 2025-06-07 cs.NE cs.AIq-bio.NC

Yuhan Helena Liu , Guangyu Robert Yang , Christopher J. Cueva This is my paper
classification cs.NEcs.AIq-bio.NC
keywords biologicalplausibilitye-proptemporalcreditassignmentneuraldatasimilarityProcrustesdistancerecurrentnetworksbackpropagationthroughtimerepresentationalalignment
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether a biologically plausible temporal credit-assignment rule can produce recurrent-network activity that looks as much like real brain recordings as activity produced by backpropagation through time (BPTT), the standard but biologically questionable training method. On the Sussillo 2015 and Mante 2013 primate datasets, and using Procrustes distance at matched task accuracy, it argues that the answer is yes: e-prop, a local three-factor learning rule built on gradient truncation, achieves neural similarity comparable to BPTT. The paper also reports that architecture and initial weight settings can shift neural similarity more than the choice of learning rule does, and that BPTT and e-prop end up with similar post-training weights and dynamics. If the claim holds, biological plausibility at individual synapses does not have to come at the cost of brain-like population activity.

What carries the argument

The load-bearing machinery has two parts. On the learning side, e-prop truncates the recurrent gradient so each weight update depends only on locally available pre- and postsynaptic activity and a third top-down instructive signal, implemented as an eligibility trace times an error signal; this is the biologically plausible temporal credit-assignment rule under test. On the evaluation side, Procrustes distance, the residual angle between two neural representations after optimal orthogonal alignment, quantifies how close RNN hidden states are to recorded neural activity, and matching task accuracy before comparing distances isolates the learning rule's effect. A comparison noise floor built from data-to-data distances between neuron subsets gives the scale for interpreting model-data distances.

What would settle it

Record neural activity while animals perform a third temporal task, train e-prop and BPTT networks to matched accuracy, and compare both Procrustes distance and a measure that retains single-trial variability such as dynamical similarity analysis on un-averaged trials; the claim would fail if e-prop sat consistently farther from the data than BPTT on either measure across seeds and initializations.

Watch

Extended reading notes

Core claim

The central discovery is an existence result: there is at least one biologically plausible learning rule, e-prop, whose trained recurrent networks match BPTT-trained networks in how closely their trial-averaged hidden-state geometry aligns with electrode recordings, once task accuracy is matched. On the Sussillo 2015 motor-cortex task and the Mante 2013 prefrontal context-dependent task, Procrustes distances from model activity to neural data are statistically indistinguishable for BPTT and e-prop, and additional measures (CCA, CKA, dynamical similarity analysis) and qualitative dPCA comparisons point the same way. The authors further report that initial weight gain influences model-data distance more than the learning-rule choice, that e-prop behaves like BPTT with a lower learning rate plus a gradient approximation error, and that a one-dimensional linear RNN toy model shows e-prop can converge to or diverge from the BPTT solution depending on initialization.

Load-bearing premise

The paper's argument assumes, as it acknowledges in its Discussion, that Procrustes distance between trial-averaged RNN activity and trial-averaged recordings is a trustworthy measure of brain-likeness; if that scalar misses functionally important structure, matched distances would not establish that e-prop is as brain-like as BPTT.

Editorial extensions

If this is right

  • BPTT-trained RNNs are not uniquely privileged in reproducing brain-like representations on this measure; local learning rules can match them.
  • Neural-similarity comparisons between learning rules should control architecture and initialization, since those factors can dominate the effect of the rule itself.
  • e-prop-trained networks can serve as biologically plausible alternatives in brain modelling without sacrificing neural alignment, at least on the tasks tested.
  • The e-prop/BPTT gap shrinks at lower BPTT learning rates, suggesting the practical difference between the rules is partly a learning-rate effect.
  • Because e-prop's convergence depends on initialization even in a one-dimensional linear case, matched similarity is not guaranteed for all initial conditions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the results suggest that any 'locality penalty' may be invisible in representation geometry; the dominant drivers of brain-likeness could be task, architecture, and initialization rather than the credit-assignment algorithm.
  • A testable extension would compare e-prop and BPTT on single-trial dynamics or on similarity measures that penalize trial-to-trial variability, since trial averaging could in principle mask differences the Procrustes metric does not see.
  • The paper's noise-floor analysis implies that future model-brain comparisons should report distances relative to a data-data baseline; otherwise 'close to neural data' has no absolute meaning.
  • One could also test whether other local rules, such as ModProp or node perturbation, show the same matched-similarity pattern across more tasks, which would generalize the existence claim from one rule to a class.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper asks whether biologically plausible temporal credit assignment rules, specifically e-prop, can match BPTT in neural data similarity when task accuracy is matched. RNNs are trained on two primate neuroscience datasets (Sussillo 2015 and Mante 2013) using BPTT, e-prop, truncated BPTT, and (in the appendix) ModProp, node perturbation, and evolutionary strategies. Similarity to neural recordings is quantified primarily with Procrustes distance, with additional CCA, CKA, dPCA, and DSA analyses. The authors report that e-prop and BPTT achieve comparable Procrustes distances at matched accuracies, that initial weight gain and architecture affect similarity more than the learning rule, that lower BPTT learning rates increase similarity to e-prop, and that the two rules produce similar weight eigenspectra and dynamical properties. A toy 1D linear RNN result (Proposition 4.1 / B.1) illustrates that e-prop can converge to or diverge from a BPTT-found solution depending on initialization. The central empirical claim is that a biologically plausible learning rule can match BPTT in neural data similarity under matched performance.

Significance. If the claim holds, it is a meaningful contribution: it suggests that synaptic-level locality constraints do not inherently prevent a learning rule from producing brain-like representations, and it provides a useful benchmark for future comparisons of learning rules. The paper's strengths include the use of multiple similarity measures (Procrustes, CCA, CKA, DSA), several control conditions (noise removal, Dale's law, feedback alignment, node perturbation, evolutionary strategies), and publicly available code. The formal toy proposition in Appendix B is clearly stated and proved, though very limited in scope. The main empirical findings are internally consistent, but the headline existence claim is currently qualified by a confound in the primary Sussillo 2015 result (the sparsity mask) and by the small number of seeds/datasets.

major comments (2)
  1. [Appendix A.3 and Figure 1B] The headline Sussillo 2015 result in Figure 1B is computed under a 25% recurrent-weight sparsity mask (only 25% of recurrent weights are nonzero and trainable), as stated in Appendix A.3. This mask is not mentioned in the abstract, is not applied to the Mante 2013 panel, and is not part of the settings described in Table 2 or Figure 5. Since the paper's own Figure 2 and Table 1 show that architectural and initialization choices (e.g., gain) can affect Procrustes distance more than the learning-rule choice, the sparsity mask is a plausible driver of the e-prop/BPTT similarity in Figure 1B. The abstract's unqualified existence claim therefore overreaches until the comparison is repeated without the mask, or the claim is explicitly restricted to sparse e-prop.
  2. [Table 1 and Table 2] The equivalence between e-prop and BPTT is supported mainly by non-significant differences (p > 0.05) in uncorrected t-tests across four random seeds per dataset. With n = 4, these tests have low power, and non-significance is not evidence of equivalence. The paper should report effect sizes and confidence intervals, or use an equivalence testing framework (e.g., TOST), to make the 'comparable similarity' claim quantitative. The current presentation, while suggestive, is weaker than the stated conclusion.
minor comments (6)
  1. [Appendix A.3, Table 4 caption] The text refers to 'Table A.3' but the table is labeled 'Table 4'; please fix the cross-reference.
  2. [Introduction, Contribution 1] The phrase 'demonstrate thereexists' contains a typo; it should read 'demonstrate there exists.'
  3. [Figure 3 caption] The caption reads 'E-prop — has been viewed as BPTT with a reduced learning rate...' which is grammatically awkward; please rephrase.
  4. [Appendix A.2] The text says 'we explore devolutionary strategies' — this is a typo for 'evolutionary strategies.'
  5. [Equation (4)] The use of φ in the Procrustes distance formula is not defined; presumably it is a whitening transform, but it should be stated explicitly for reproducibility.
  6. [Table 1] The row labeled 'Noise ceiling' is described as untrained models; 'noise ceiling' conventionally refers to an upper bound on achievable similarity given data variability, so the label is misleading and should be changed to 'untrained models' or explained.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the e-prop/BPTT neural-similarity comparison is an independent empirical benchmark against external recordings; the only self-cited decomposition (Liu et al. 2022a) is a non-load-bearing explanatory aside.

full rationale

The paper's central claim is an empirical existence result: e-prop and BPTT RNNs are trained independently under matched settings and then compared, via Procrustes distance, to external neural recordings (Sussillo 2015 and Mante 2013; Figure 1B, Table 2). Neither the e-prop update (Eq. 8) nor the Procrustes metric (Eq. 4) is defined in terms of the other rule or in terms of the neural-similarity outcome, so the comparison is not forced by construction. The self-citation to Liu et al. 2022a ('e-prop can be decomposed into a lower learning rate BPTT and an approximation error') appears only in the explanatory discussion around Figure 3 and is corroborated by the paper's own control in Figure 3B; it is not a premise of the main result. Proposition 4.1 is a toy theorem whose assumptions directly imply its conclusion, but it is explicitly presented as motivation rather than as evidence for the empirical claim. The Discussion's identifiability caveat and the Appendix A.3 note that the Sussillo 2015 panel applies a 25% recurrent-weight sparsity mask are legitimate generalizability and robustness limitations, not circularity. No step in the claimed derivation chain reduces to its own inputs by definition or by fit, so the only issue is the single non-load-bearing self-citation, which keeps the circularity score minimal.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. Its free parameters are standard training hyperparameters and initialization choices, though the analysis shows these choices matter more than the learning rule, which is itself a finding. The main assumptions are about the validity of the similarity metric and the trial averaged comparison, both acknowledged as limitations in the paper.

free parameters (6)
  • Initial weight gain g = 0.0, 0.5, 1.0, 1.5, 2.0 (scanned)
    The initial recurrent weight variance is N(0, g^2/N). Figure 2 and Table 1 show g has a large effect on Procrustes distance, so the similarity result depends on this choice. It is scanned, not fit, but it is a hand chosen parameter that strongly influences the outcome.
  • Learning rate = 1e-3 default; grid searched over {3e-3,1e-3,3e-4,1e-4}
    Learning rate is tuned by grid search and the paper shows e-prop aligns more closely with BPTT at lower learning rates. The reported similarity comparison depends on matched accuracy, which interacts with the learning rate.
  • Recurrent noise standard deviation = 0.1 (default), removed in controls
    Zero-mean Gaussian noise with std 0.1 is added to hidden activity. This is a modeling choice that could affect the geometry of representations and therefore Procrustes distances.
  • Sparsity constraint for Sussillo 2015 = 25% nonzero recurrent weights in Figure 1
    The main Sussillo plot applies a sparsity constraint limiting 25% of recurrent weights to be nonzero, while other figures do not. This inconsistency makes the comparison across figures harder to interpret and is a hand chosen condition.
  • ModProp hyperparameter mu = 0.25
    The ModProp filter tap strength mu is set to 0.25. Used only in Appendix Figure 7, so it does not affect the central claim.
  • DSA hyperparameters = n_delays in {5,10,15,20}, rank in {10,20,30,40}
    DSA trends are checked across these ranges; the paper reports consistent trends. Not central to the main claim.
assumptions (5)
  • domain assumption Procrustes distance is an appropriate and sufficient measure of neural similarity (Eq. 4).
    The paper relies on this metric for the main claim. The Discussion admits it focuses on specific geometric structures and may not capture the computational properties of interest.
  • domain assumption Trial averaged neural activity is the right substrate for comparing models to brains.
    Section A.3 explains why trial based subsampling was avoided due to trial to trial variability. The noise floor is computed with neuron splitting, which the authors note may be overly stringent due to neuron dependence.
  • domain assumption BPTT is the appropriate benchmark for brain-like models.
    The paper states BPTT trained RNNs are the predominant brain modeling method and the benchmark many bio plausible rules aim to approximate. The Discussion cites recent work questioning whether the brain uses BPTT at all, acknowledging the limitation of this benchmark.
  • standard math e-prop is equivalent to RFLO and its gradient truncation is a faithful local approximation.
    The paper cites Bellec et al. 2020 and Murray 2019 for the e-prop update, Eq. 8. This is background from prior literature, not derived in this paper.
  • domain assumption The 1D linear RNN toy model dynamics (Eqs. 12-14) capture the relevant behavior of e-prop convergence.
    The toy proof assumes a polynomial with a real root, a nondegenerate input, and constant sign conditions. The authors explicitly state the extension to higher dimensions is left for future work, so the proof is illustrative only.

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Pith. "Pith review of Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example." pith.science (2026). https://pith.science/paper/TBO7Z2RH

@misc{pith2026250606904,
  author       = {Pith},
  title        = {Pith review of: Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TBO7Z2RH}},
  note         = {Machine review of arXiv:2506.06904}
}
read the original abstract

Understanding how the brain learns may be informed by studying biologically plausible learning rules. These rules, often approximating gradient descent learning to respect biological constraints such as locality, must meet two critical criteria to be considered an appropriate brain model: (1) good neuroscience task performance and (2) alignment with neural recordings. While extensive research has assessed the first criterion, the second remains underexamined. Employing methods such as Procrustes analysis on well-known neuroscience datasets, this study demonstrates the existence of a biologically plausible learning rule -- namely e-prop, which is based on gradient truncation and has demonstrated versatility across a wide range of tasks -- that can achieve neural data similarity comparable to Backpropagation Through Time (BPTT) when matched for task accuracy. Our findings also reveal that model architecture and initial conditions can play a more significant role in determining neural similarity than the specific learning rule. Furthermore, we observe that BPTT-trained models and their biologically plausible counterparts exhibit similar dynamical properties at comparable accuracies. These results underscore the substantial progress made in developing biologically plausible learning rules, highlighting their potential to achieve both competitive task performance and neural data similarity.

Figures

Figures reproduced from arXiv: 2506.06904 by the authors.

Figure 1
Figure 1. (A) Setup overview: analysis of two neural datasets. We computed similarity scores between RNN activity and electrode recordings from (1) Mante et al. (2013) and (2) Sussillo et al. (2015). Schematics have been modified from those in the original papers. RNNs are trained on these respective tasks using various learning rules, including BPTT and bio-plausible alternatives. Subsequently, we evaluate the similarity bet… view at source ↗
Figure 2
Figure 2. Impact of Initial Weight Magnitude on Model-Data Distances Exceeds Variation from Learning Rules. Model-data distances versus normalized accuracy for various initial gain values (depicted by different colors) for (A) BPTT and (B) e-prop. Initial weight gain refers to the multiplier applied to the default initializations for recurrent and readout weights. The results shown are for the Sussillo 2015 task, with similar… view at source ↗
Figure 3
Figure 3. (A) Procrustes distances remain consistent across various learning rates when employing the same rule (BPTT). Different color shades represent different learning rates: 1e − 3, 3e − 4, and 1e − 4. These rates result in nearly indistinguishable Procrustes distances. The analysis in this figure is done using the Sussillo 2015 task. (B) E-prop — has been viewed as BPTT with a reduced learning rate plus some degree of g… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Demixed principal component analysis (dPCA) shows a qualitative match between model and data when projected onto the first time component and first condition component. The time component captures variance aligned with task time, while the condition component captures …
Figure 5
Figure 5. Figure 5: Data-to-data distance (noise floor) vs model-to-data distance (BPTT and e-prop before and after training). Left: Hatsopolous 2007; middle: Sussillo 2015; right: Mante 2013. The data-splitting procedure for obtaining the baseline (i.e. noise floor) is detailed in Append…
Figure 6
Figure 6. Figure 6: UMAP embedding and pairwise distance matrix heatmap for different models when (A) best e-prop accuracy, (B) 80%, (C) 60%, and (D) 40% accuracies are reached. Here, the Sussillo 2015 dataset is illustrated. Black: BPTT, blue: truncated BPTT, magenta: e-prop, orange: neu…
Figure 7
Figure 7. Figure 7: This figure replicates the core trend from [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: (A) presents the eigenvalues of the recurrent weight matrix post-training, with columns representing BPTT and e-prop respectively. Each row displays a different training setting: the base setting (referenced in [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Node perturbation (cyan) and evolutionary strategies (yellow) lead to higher Procrustes distances from the neural data compared to BPTT (black) and e-prop (magenta) when accuracies are equivalent. This figure presents the Procrustes distance versus accuracy plots, adhe…
Figure 10
Figure 10. Figure 10: The use of random feedback readout weights for gradient computation (red) (Lillicrap et al., 2016) resulted in distances comparable to those achieved using exact readout weights (black). Plotting conventions are consistent with previous figures: task performance varie…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.