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REVIEW 3 major objections 5 minor 41 references

Three factor delay learning rules for spiking neural networks

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proposes three-factor learning rules that let LIF spiking networks learn synaptic and axonal delays online, achieving accuracy close to offline backpropagation on speech recognition tasks.

desk verdict A genuinely useful online delay-learning rule for SNNs, but the printed delay gradient has a sign/notation error that must be corrected. read the letter →

arxiv 2601.00668 v2 pith:PGOKCA7P submitted 2026-01-02 cs.NE cs.LG

Luke Vassallo , Nima Taherinejad This is my paper
classification cs.NEcs.LG
keywords spikingneuralnetworksdelaylearningthree-factorrulesonlineeligibilitytracesGaussiansurrogatekeywordspottingLIFneurons
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

The paper tries to show that temporal delays in spiking neural networks can be treated as learnable parameters and trained online, without backpropagation through time. It derives three-factor learning rules — an eligibility trace capturing how a delay shift changes a neuron's spike, multiplied by a top-down error signal — and uses a smooth Gaussian surrogate to make the non-differentiable spike-delay relationship differentiable. On speech keyword-spotting tasks, this online method reaches accuracy within 0.15% of an offline backpropagation baseline on the SHD dataset, improves accuracy by up to 20% over weights-only networks, and yields up to 14% higher accuracy at comparable parameter counts. If correct, it lets neuromorphic chips learn temporal features on-device with far less memory and latency than offline training.

What carries the argument

The load-bearing object is the three-factor update in Equation 8: the delay gradient for a synapse decomposes into a top-down error signal and a local eligibility trace that recursively tracks the sensitivity of the neuron's output to the delay. Because the delayed spike is not differentiable, the paper replaces the Dirac spike train in Equation 10 with a Gaussian kernel in Equation 11 and uses its derivative in Equation 12; this surrogate supplies the missing gradient and is only used during learning, not inference. The recursive eligibility trace allows the computation to happen online with a ring buffer, despite the kernel being nominally non-causal.

What would settle it

Run the same training pipeline with a causal exponential or truncated causal Gaussian kernel of matched width replacing Eq. 11; if delays still learn and accuracy gains survive, the method does not depend on non-causal smoothing as the paper argues. Conversely, verify Eq. 12 by finite differences on Eq. 11; if the sign or normalization is wrong, the reported gradient is not the one being implemented.

Watch

Extended reading notes

Core claim

The central discovery is that delay parameters — both synaptic delays at individual connections and axonal delays at virtual input neurons — can be updated online using a three-factor rule whose eligibility trace is computed with a Gaussian surrogate for the spike train. Although shifting a spike by a delay is discontinuous in the delay parameter, representing the delayed spike as a Gaussian kernel makes the derivative well-defined; the product of this derivative with the surrogate spiking nonlinearity and the error signal gives a gradient-equivalent update. In experiments, this local online rule closely matches offline backpropagation for fully connected feedforward and recurrent networks,

Load-bearing premise

The Gaussian surrogate in Eq. 11 and its derivative in Eq. 12 correctly capture the sensitivity of a postsynaptic neuron to a shift in spike timing, even though the true spike-delay relationship is a discontinuous shift and the surrogate kernel is non-causal; causal exponential kernels did not support learning.

Editorial extensions

If this is right

  • On the SHD speech dataset, online delay learning reaches 92.64% test accuracy, within 0.15 percentage points of the 92.79% offline backpropagation baseline under the same network.
  • Adding learnable delays improves accuracy by up to 20% over weights-only networks; jointly learning weights and delays adds up to 14% at comparable parameter counts, with the largest gains in small and sparse models.
  • Because delay updates are computed from recursively maintained eligibility traces rather than unrolled sequences, learning can proceed sample-by-sample in real time, removing update locking.
  • A sparse model with learnable delays can match a denser weights-only model's accuracy with a 6.6x smaller model and 67% lower inference latency in the comparison reported.
  • On-chip memory planning changes: axonal delays add about 8.3% memory overhead, synaptic delays about 57.7%, but the delays' accuracy gains pay off once the network is sparse enough.

Reading between the lines

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

  • The consistent failure of causal kernels reported in the paper suggests a testable design principle: non-causal smoothing may be what makes delay gradients informative, so hardware that wants causal operation would need to approximate or accept a learning-accuracy penalty.
  • If axonal delay updates accumulate error because each axonal parameter pools contributions from many postsynaptic neurons, then switching to time-to-first-spike coding, where delays attach to hidden neurons rather than input neurons, could make those updates local and more accurate — a natural extension.
  • Across setups, accuracy on SHD saturates around 92.5% even when SRNNs double parameters; this hints that delay learning is supplying the temporal feature extraction that additional recurrent capacity would otherwise provide, and that scaling to harder tasks would likely need more layers rather than more parameters per layer.
  • The memory-overhead figures imply a concrete design rule: delay learning is memory-favorable only in sparse regimes; testing at which sparsity each delay type breaks even (43% for axonal, 74% for synaptic per the paper) turns the result into an engineering guideline.
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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

3 major / 5 minor

Summary. The manuscript proposes three-factor learning rules for simultaneously learning synaptic and axonal delays in single-hidden-layer LIF-based SNNs and SRNNs. Delays are treated as parameters of a Gaussian surrogate spike kernel; the eligibility trace is computed recursively from the derivative of this kernel with respect to the delay, and a top-down error signal from the readout layer provides the third factor. The method is evaluated on the SHD and SSC keyword-spotting datasets, with comparisons to a DCLS/BPTT offline baseline, weights-only baselines, fixed-delay controls, and sparse-network variants. The paper reports that delays improve accuracy by up to 20%, that joint weight-delay learning yields up to 14% higher accuracy at comparable parameter counts, and that the online method matches offline BPTT to within 0.15% on SHD.

Significance. If the central derivation is correct and reproducible, this is a timely and practically relevant contribution to online, on-device learning in neuromorphic systems. The experimental design has several strengths: results are reported with 95% confidence intervals from repeated runs; the online method is compared directly to an offline DCLS/BPTT baseline; fixed-delay controls isolate the contribution of delay learning; and the memory-footprint analysis for on-chip implementation is useful. However, the central delay-gradient equation contains a sign/notation error, and a key kernel hyperparameter is not reported. These issues must be resolved before the claimed results can be independently reproduced from the text.

major comments (3)
  1. [Eq. (12), Sec. III-D] Equation (12) is stated as the derivative of the Gaussian surrogate with respect to the delay parameters, but as printed the left-hand side is d x(t-D_ji)/dt and the right-hand side is negative of the time derivative of the Gaussian. Since x(t-D_ji) = G(t - t_k - D_ji), the correct derivative with respect to D_ji is + (t - t_k - D_ji)/(sqrt(2 pi) sigma^3) * exp(-(t - t_k - D_ji)^2/(2 sigma^2)), which is the negative of the printed right-hand side. This derivative enters the eligibility trace in Eq. (9) and hence the update rule in Eq. (8). A reader implementing Eq. (12) exactly as written would update delays in the opposite direction of the intended gradient. The reported experimental results must therefore have been obtained with a corrected derivative. This is a load-bearing reproducibility defect in the central learning rule; please correct Eq. (12) and explicitly confirm the sign use
  2. [Sec. III-D / IV-B] The Gaussian width sigma in Eq. (11) is never specified. The magnitude and direction of all delay gradients depend directly on sigma, and the paper reports that causal kernels 'did not support effective learning', so the choice of sigma is not a neutral detail. Without a reported value for sigma (or a sigma schedule, if used), the experiments in Table I cannot be reproduced from the text. Please add the sigma value used for each experiment and, ideally, a short sensitivity analysis.
  3. [Sec. VI / Table I] The paper claims a 6.6x model size reduction and 67% lower inference latency relative to the DCLS baseline, but Table I shows the same FC-128 configuration with 32.3k parameters for both the BP baseline (row 1) and the online method (row 4) in the synaptic-delay condition. The basis for the 6.6x reduction is therefore unclear from the reported data. Please specify the exact baseline architecture and parameter count used for this claim, or revise the claim so that it is directly supported by Table I or a companion table.
minor comments (5)
  1. [Eq. (9)] The notation in Eq. (9) is inconsistent: the eligibility trace is written as e_t_ji = partial z_t_j / partial v_t'_j, but the prime on t' is not defined and the subsequent bracket includes partial v_t_j / partial D_ji. Please rewrite with consistent time indices, e.g., partial z_t_j / partial v_t_j times the recursive trace.
  2. [Eq. (10)-(11)] Equation (10) writes x^{t-D_ji}_i as a sum of Dirac deltas, but in the discrete-time dynamics of Eq. (3) the input is a sequence indexed by time. Please clarify the relationship between the continuous-time kernel notation and the discrete-time update used in the implementation.
  3. [Table I / Sec. V] The text states that the weights-only SRNN baseline is 85.72%, while Table I row 9 reports 85.77% +/- 0.94%. Please correct the inconsistency.
  4. [Sec. VI-A] There is a typo, 'Section Section VI', and the discussion of ring-buffer sizes would benefit from a more precise explanation of when the buffer depth is 'kernel length' versus 'maximum delay'.
  5. [Sec. IV-B] The paper attributes the large weights-only performance gap between online and DCLS/BPTT to analytical LIF dynamics versus Euler integration, but no direct comparison or citation is given for that claim in this context. Please add a reference or a small control experiment to support this attribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the delay-learning result is validated against external offline BPTT baselines and fixed-delay controls.

full rationale

The paper's central claim is that the proposed three-factor rules allow online learning of synaptic and axonal delays and match offline backpropagation accuracy. The delay-gradient computation in Sec. III-D uses a Gaussian surrogate (Eq. 11) and its derivative (Eq. 12). This surrogate is an explicit modeling choice, justified by comparative preliminary experiments (Gaussian vs. causal exponential vs. double-exponential kernels), not a term defined in terms of the target result. The learned delays Dji are independently parameterized, clamped, and evaluated against (1) a weights-only baseline, (2) fixed-delay controls, and (3) an external DCLS/BPTT offline baseline from prior work; Table I's headline accuracies are therefore not fitted to the targets. No load-bearing step relies on the present authors' prior work: the reference list contains no self-citations by Vassallo or TaheriNejad, and e-prop is cited as an external algorithm (Ref. [28]) whose formulation is extended, not assumed. The only notable issue is that Eq. (12) is printed as a time derivative d x/dt with a negative sign while the surrounding text says it is the derivative with respect to delay Dji; that is a correctness/reproducibility concern (the correct delay derivative has the opposite sign), but it does not make the claimed result equivalent to its inputs by construction. The non-causal kernel and empirical hyperparameter selection are acknowledged limitations, not circularity. Accordingly, no circular step is identified.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a smooth surrogate for non-differentiable spike delays. The listed free parameters are hand-set or fitted for performance; the axioms are the modeling choices inherited from LIF/e-prop and the ad hoc Gaussian kernel. No new physical entities are introduced.

free parameters (5)
  • Gaussian kernel width σ = not reported
    Controls the surrogate spike width in Eq. (11); never specified in the paper, chosen by hand; learning and accuracy depend on it.
  • Delay learning rate = 1e-2
    Set in Sec. IV-B; chosen by hand, affects convergence of delay updates.
  • Weight learning rate = 1e-4
    Set in Sec. IV-B; standard hyperparameter, included for completeness.
  • Maximum delay Dmax = 25 timesteps (250 ms)
    Delay clamp bound in Sec. IV-B; chosen by hand and affects model capacity.
  • Surrogate gradient width γ_pd = 0.3
    Width of piecewise-linear surrogate in Eq. (7); chosen by hand.
assumptions (4)
  • domain assumption LIF membrane dynamics with reset-by-subtraction (Eq. 5) are a sufficient neuron model for delay learning.
    Sec. III-A; standard assumption in neuromorphic engineering.
  • ad hoc to paper Dirac-delta spike trains can be replaced by a Gaussian kernel to obtain delay gradients (Eq. 11).
    Sec. III-D; causal exponential kernels failed, so the Gaussian is a modeling choice validated only by performance, not derived.
  • domain assumption The eligibility-trace recursion in Eq. (9) yields gradients equivalent enough to BPTT for delay parameters.
    Sec. III-C; inherited from e-prop [28] for weights and assumed to extend to delays; empirically checked in Fig. 3c but not proven.
  • domain assumption Non-causal kernel information can be made available in an online setting via ring buffers.
    Sec. III-D/VI; the authors note non-causality is 'undesirable for real-time applications' and requires buffering, so strict online causality is not guaranteed.

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Cite this review

Pith. "Pith review of Three factor delay learning rules for spiking neural networks." pith.science (2026). https://pith.science/paper/PGOKCA7P

@misc{pith2026260100668,
  author       = {Pith},
  title        = {Pith review of: Three factor delay learning rules for spiking neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PGOKCA7P}},
  note         = {Machine review of arXiv:2601.00668}
}
read the original abstract

Spiking Neural Networks (SNNs) are dynamical systems that operate on spatiotemporal data, yet their learnable parameters are often limited to synaptic weights, contributing little to temporal pattern recognition. Learnable parameters that delay spike times can improve classification performance in temporal tasks, but existing methods rely on large networks and offline learning, making them unsuitable for real-time operation in resource-constrained environments. In this paper, we introduce synaptic and axonal delays to leaky integrate and fire (LIF)-based feedforward and recurrent SNNs, and propose three-factor learning rules to simultaneously learn delay parameters online. We employ a smooth Gaussian surrogate to approximate spike derivatives exclusively for the eligibility trace calculation, and together with a top-down error signal determine parameter updates. Our experiments show that incorporating delays improves accuracy by up to 20% over a weights-only baseline, and for networks with similar parameter counts, jointly learning weights and delays yields up to 14% higher accuracy. On the SHD speech recognition dataset, our method achieves similar accuracy to offline backpropagation-based approaches. Compared to state-of-the-art methods, it reduces model size by 6.6x and inference latency by 67%, with only a 2.4% drop in classification accuracy. Our findings benefit the design of power and area-constrained neuromorphic processors by enabling on-device learning and lowering memory requirements.

Figures

Figures reproduced from arXiv: 2601.00668 by the authors.

Figure 1
Figure 1. Illustration of coinciding pre-synaptic spikes leading to [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Comparison against offline methods on SHD and SSC [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Control experiments illustrating the efficacy of delay-learning. (a, b) Sweep model sparsity for axonal (AD) and synaptic [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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