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

A more focused title/claim and why mainly one bio-plausible rule. We thank the reviewer for this thoughtful and detailed feedback. We agree that our study does not aim to characterize all biologically plausible learning rules; rather, our goal is to show that at least one such rule—e-prop—can match BPTT in neural similarity when trained to similar performance, thereby demonstrating existence rather than universality; as the reviewer noted, universality would be beyond the scope of a short paper. To clarify our focus, we will:

• Update the title to: “Can biologically plausible temporal credit assignment rules match BPTT for neural similarity? E-prop as an example”
• Modify lines 70–73 to end with: “— or can some such rules, under the right conditions, yield representations as brain-like as those learned by BPTT?”
• Revise the first contribution point to state explicitly that this is an existence result
• Emphasize in the Discussion that e-prop serves as a concrete example, not a general case

With this reframing, we believe that providing a single positive example is sufficient for the contribution we aim to make, though we do explore additional rules in appendices for curiosity.

Focusing on tasks with dynamics. While vision tasks are important, our study focuses on temporal credit assignment—a critical challenge in biological learning. This focus, aligned with prior work on dynamic tasks (e.g., Bellec et al., 2020), allows us to evaluate how well models capture evolving neural activity over time rather than static averages, making it ideal for representational similarity analyses. Our updated title now reflects this focus, and we will note that extension to vision tasks is an interesting direction for future work.

Coupling of learning rule and architecture. We agree that learning rules are often developed with specific architectures in mind, and we acknowledge this broader issue. In our study, we intentionally fix the task, architecture, and initialization to isolate the effect of the learning rule on neural similarity, following prior work (e.g., Liu et al., 2022). The architecture used is a standard RNN widely adopted in computational neuroscience (e.g., Yang & Wang, 2020). We will add a discussion point noting the importance of studying rule–architecture interactions more systematically in future work.

Additional similarity measures. Please see our response under “Alternative Metrics” to Reviewer AQPQ, including new results using additional metrics Table 3.

Theoretical claim. We agree the result may appear intuitive, but BPTT and e-prop do not generally share the same fixed points (e.g., $\hat{y}'(W) = 0$ yields a fixed point for BPTT but not generally for e-prop); even when fixed points overlap, the dynamics of convergence may differ due to distinct basins of attraction. Also, what's a bit more precise than saying they sometimes converge to different ones is the sign condition on initialization for e-prop, which determines whether it converges to W*. That said, we recognize the theoretical result is limited in scope and may distract from our main contributions. As noted in our response to Reviewer xVGX, we will revise the manuscript to de-emphasize the theoretical component and clarify that the theorem serves only as a simple illustrative example, not a general convergence result. Our aim is to highlight, in a concrete toy setting, e-prop’s sensitivity to initialization as a motivation for future theoretical work.

BPTT limitations. We agree. While BPTT-trained RNNs are widely used in brain modeling, recent studies (e.g., Pagan et al., 2024) show that BPTT tends to favor a limited subset of solutions. Zahorodnii et al. (bioRxiv, 2025) further demonstrate that handcrafted solutions can yield more brain-like dynamics. We will also cite Song et al. (2024) and others who question the usage of BPTT by the brain (see also significance of the e-prop–BPTT comparison in response to Reviewer 23UQ). Together, these points highlight the limitations of BPTT as a benchmark and motivate future work on alternative rules and rule–architecture co-design; we will update the manuscript accordingly.

Additional references. In the updated manuscript, we will discuss and cite all references the reviewer mentioned.

Is the question ill-posed? This is an excellent point. In under-constrained systems with unknown architectural variables, multiple learning rules may converge to similar solutions. Constraining the architecture using experimental data—or incorporating data during learning—could help narrow the search space, making this a promising direction for future work. We will update the manuscript with this point. That said, our goal is not to pinpoint the brain’s actual learning rule, but to test whether any existing biologically plausible rule (e.g., e-prop) can match BPTT in neural similarity, the de facto benchmark for brain-like models.

Theoretical claims. We appreciate the reviewer’s constructive feedback and acknowledge that the main text did not clearly convey the limited scope of our theoretical result. We agree that the theorem is restrictive in two key ways: (1) it applies only to a 1D setting, and (2) W* can be the solution of any algorithm Z, not specifically BPTT.

We recognize that our presentation may have unintentionally suggested a broader theoretical contribution than intended, potentially distracting from our main message. In response, we will make the following updates to our manuscript to (A) clarify the theorem’s assumptions and scope in the main text, and (B) re-center the reader’s attention on our core empirical focus and existence-based framing:

• Remove all phrases referring to the theoretical results in abstract and Contributions, including “to support this further, we provide theoretical results …”, to avoid implying that we seek to establish a general theoretical framework for e-prop convergence, which we do not aim to do.
• Clarify the restrictive assumptions in the main text’s informal theorem statement by replacing “linear RNN” with “1D linear RNN”, and “found via BPTT” with “found via an arbitrary algorithm (e.g., BPTT)”, to acknowledge that our result does not specifically hinge on BPTT alone.
• Update Appendix B heading to “Convergence and divergence of e-prop in a toy setting”
• Explain the theorem’s purpose in the surrounding text, clarifying that it serves as an existence demonstration in a toy setting to motivate future theoretical work: “We provide a 1D linear RNN example showing that e-prop can match BPTT under certain initializations, but diverge under others. While this illustrates key ideas—such as e-prop’s sensitivity to initialization—extending the analysis to higher-dimensional RNNs is a substantial effort (e.g., Schuessler et al., NeurIPS’20) and beyond the scope of this paper.”

To offer more information about the last point, our rationale for focusing on a toy (1D) setup is twofold. First, to our knowledge, there is no existing theoretical framework for e-prop convergence that we could readily build upon, and developing such a framework (e.g., in the spirit of Schuessler et al.) would be a major effort in its own right and beyond the scope of this work. Second, the toy setting lets us illustrate the key existence story we wish to highlight: there are indeed cases where e-prop converges to the same solution as BPTT, and there are cases where it fails. This demonstration is not meant to be comprehensive, but rather to motivate future investigations (e.g., broader architectures, higher-dimensional RNNs, etc.).

By making these changes, we hope our updated manuscript can better convey that our theorem is meant primarily to provide an example—“e-prop can match BPTT under certain initial conditions, but not always”—rather than to establish a general theoretical foundation for e-prop, as well as motivating future work on theoretical work to examine insights for the simple toy model across more realistic settings. We hope these revisions clarify that the theoretical result plays a limited, illustrative role and is not essential to our main aims.

Amount of data/training required to achieve equal performance. We agree this is an important point. While we did not include these comparisons originally, they have been addressed in prior work evaluating the “(1) good neuroscience task performance” criterion (see Fig. 3, Bellec et al., 2020), which shows that e-prop typically requires more training iterations than BPTT to reach the same accuracy. Our findings are consistent with this, and we will include this information and citation in the revised manuscript.

Additional questions. Indeed, task performance is varied by changing the number of training iterations. At each iteration, we compute both task accuracy and neural similarity, and plot one against the other. As training progresses, models move from low accuracy/high distance (upper left) to high accuracy/low distance (bottom right) in Figures 1-3. The same procedure is applied to all learning rules. We will add this explanation to the caption for clarity.

Longer captions. In addition to the above explanation, we will ensure all figure captions include complete axis label explanations and refer to the relevant appendix sections. We acknowledge that short captions are more common in the appendix, and will revise those to match the clarity of the main text, including color legend descriptions (e.g., Fig. 8B).

Other comments. We thank the reviewer for catching the indexing error and will correct it in the updated version.

Alternative metrics. We thank this reviewer for their constructive comments and now add additional metrics to the manuscript:

Table 3: Additional measures.

These additional results again support the insignificant differences across the rules (p=0.834, 0.683, 0.373, for the respective measures).

While we provided initial justification for using Procrustes in the Discussion, we will expand on it in the revised manuscript. In addition to its interpretability and geometric grounding, a recent study (Cloos et al., ICLR 2025) found that optimizing Procrustes may better preserve task-relevant features than other metrics. This suggests it can capture more meaningful neural structure. They also observed that Procrustes is stricter than CKA: high Procrustes implies high CKA, but not vice versa (see also Harvey et al., UniReps 2024). We will add this discussion.

Novelty & contribution. We thank the reviewer and respectfully clarify that while we use existing tools, our contribution lies in addressing a fundamental, underexplored question: How do biologically plausible gradient approximations affect neural activity similarity? Using e-prop as an example, we show that despite gradient truncation, it is possible to match the similarity of BPTT (before the truncation). This is not obvious a priori—e-prop < BPTT or > BPTT were both plausible. The finding of matched similarity is thus nontrivial, and we further analyze why this occurs (Figs. 3 & Appx Fig. 8), as well as how confounds like architecture/initialization dominate (Fig. 2). These insights also help refine the precise future questions that can be systematically explored across learning rules, architectures, datasets, and metrics — all of which our flexible pipeline is designed to support. We will revise the Discussion to reflect these points and believe our insights meaningfully advance this emerging area, warranting consideration for ICML.

Why focus mainly on one bio-plausible rule. We thank the reviewer for raising this point. Our goal is not to claim universality — that all biologically plausible rules match BPTT in neural similarity — but to demonstrate existence, by showing that e-prop can do so under certain conditions. To support this existence claim, showing a single such example should suffice, which we believe is both non-trivial and novel, especially given how underexplored representational similarity is in this context.

That said, we sympathize with the reviewer and acknowledge that our current title may overstate the scope. To avoid this confusion, we plan to revise the title to better reflect our focus on a single illustrative case — e.g., “Can biologically plausible temporal credit assignment rules match BPTT for neural similarity? E-prop as an example”. Additional planned updates are outlined in our response to Reviewer buDs.

Questions. We thank the reviewer for those meaningful questions and will add them to our manuscript:

1. This is a nuanced question that touches on how task timescale and complexity modulate the roles of both initialization and learning rules. Due to rebuttal word limits, we focus here on timescale. Prior work shows that performance gaps between BPTT and e-prop widen with longer task timescales (Liu et al., 2021). If neural similarity correlates with task performance—as supported by Yamins et al. (2014) and our results—we would expect a growing similarity gap as well (although we'd be violating the assumption of matched accuracy for comparison). Testing this directly would require datasets with varying temporal dependencies, which we view as a promising direction for future work.
2. Intuitively, if both metrics show similar results for e-prop and BPTT, a composite score (based on weighted or normalized average) would reflect the same conclusion. However, we prefer to report each metric separately to avoid arbitrary weighting and allow for clearer interpretation. We’ll add composite scores to the manuscript for completeness.
3. This is a profound question that touches on translational relevance: to what extent do findings from one species (e.g., primates) generalize to others (e.g., rodents or humans)? The widespread use of model organisms in neuroscience is based on shared circuit motifs and learning mechanisms. While we expect our conclusions may extend to other species or even SNNs with similar dynamics, systematic cross-species and model-type studies will be essential. We view this as an exciting direction for future work.

Other comments. We will cite and discuss BrainScale and others in Related Works. We didn’t use Tanh for type-1 firing for nonnegative firing rate; we will clarify this point in the manuscript.

Tabular results and statistical test for Fig 2. We appreciate this reviewer's concrete feedback and tabulate the results:

Table 1: Results from Fig2. *Noise ceiling corresponds to untrained models.

Non-overlapping error bars across rows suggest differences across gains (fixed rule), while overlapping bars within columns suggest similarity across rules (fixed gain). Values are mean ± standard deviation across seeds. Noise ceiling refers to untrained models. Gain=2.0 is excluded due to poor e-prop performance. Distances were measured at normalized accuracy ≈ 0.8 to ensure matched performance. To further support these trends, we find BPTT at gain=1.0 vs 0.0 differs significantly (p=2.07e-5), as does e-prop (p=1.3e-4). In contrast, BPTT vs e-prop yields no significant difference at fixed gain=1.0 (p=0.173) or gain=0.0 (p=0.985). We report these two gains for brevity (due to rebuttal char limits), but similar trends hold at other values and are available upon request.

Fig 5 updates. As suggested, we add tabulated bar values and statistical tests below; we apologize for initially omitting Mante13 (due to an earlier code we wrote for Hats07), but now include it and observe similar trends as Suss15 — trained BPTT/e-prop outperform the noise ceiling but still show a gap from the noise floor:

Table 2: Neural similarity scores (Fig5).

As noted, distances here use fewer neurons (to compute the noise floor), so values may differ slightly from earlier plots. Consistent with the bar plots, the gap from the noise floor is insignificant for Hats07 (p=0.311, BPTT-trained), but significant for Mante13 (p=1.29e-5) and Suss15 (p=5.02e-6). We will append these results to the updated manuscript. We will also add a noise ceiling to the main plots and visualization using normalized values to the manuscript.

Additional content in the main text. As suggested, we will add DSA and UMAP results to the main text. We’ve clarified the choice of Procrustes and added new results using CCA and CKA (see Table 3 for Reviewer AQPQ).

Other suggestions and questions. We will incorporate all these helpful suggestions to improve clarity. Fig2&5 comments are addressed above. For Fig5, the reviewer is correct: the noise floor was computed by comparing different neuron subsets matched across time and condition. Our baseline compares time-varying firing rates across conditions (not trial-to-trial variability), aligning with common practice in systems neuroscience. The variance in Fig. 1B (Mante) is largely due to the absence of learning rate decay; adding decay stabilizes training without altering the main trend — we will include this plot. We will also revise the main text to clarify the toy theorem’s assumptions, limited scope, and illustrative purpose, motivating comprehensive future work (see also Reviewer xVGX). We will cite and discuss all references the reviewer mentioned. Finally, we will update Fig1&7 to be on the same footing.

The significance of e-prop BPTT comparison. This is an important question. Beyond the motivation in lines 70–73, BPTT remains a widely used benchmark in brain modeling, especially in seminal works (Yamins, DiCarlo, Mante, Sussillo, …). Comparing it to a biologically plausible alternative like e-prop offers broadly relevant insight into neural similarity. While BPTT has known limitations (see response to Reviewer buDs), our pipeline aims to evaluate — and eventually surpass — it as a benchmark in future studies.

More broadly, our study addresses a fundamental and underexplored question: how do bio-plausible gradient approximations affect neural similarity? Using e-prop as a case study (demonstrating existence, not universality), we show that despite gradient truncation, it can match BPTT (before the truncation). This was not obvious a priori; both “e-prop < BPTT” and “e-prop > BPTT” were plausible outcomes (see further discussions on novelty&contribution with AQPQ). Our framework enables detecting future models that outperform BPTT. We will add these discussion points to the manuscript.