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arxiv: 2606.03935 · v1 · pith:TPAAJR2Nnew · submitted 2026-06-02 · 💻 cs.NE · cs.LG

Quadratic integrate-and-fire neurons exhibit less fragmented loss landscapes and outperform leaky integrate-and-fire neurons in spike-based gradient descent

Carlo Wenig , Raoul-Martin Memmesheimer , Christian Klos This is my paper

Pith reviewed 2026-06-28 07:19 UTC · model grok-4.3

classification 💻 cs.NE cs.LG
keywords spiking neural networksquadratic integrate-and-fire neuronsleaky integrate-and-fire neuronsgradient descentloss landscapesSpiking Heidelberg Digitsneuromorphic computing
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The pith

QIF neurons produce less fragmented loss landscapes and outperform LIF neurons in spike-based gradient descent on the Spiking Heidelberg Digits dataset.

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

The paper shows that quadratic integrate-and-fire neurons maintain continuous spiking dynamics that prevent sudden spike appearances or disappearances when parameters change slightly. In contrast, leaky integrate-and-fire neurons suffer from these discontinuities, which fragment their loss landscapes and produce erratic gradients during training. After a thorough hyperparameter search on the Spiking Heidelberg Digits dataset, QIF networks achieve better performance, and visualizations confirm the smoother landscapes for QIF. A reader would care because reliable gradient-based training of spiking networks is crucial for neuromorphic computing and brain modeling. The central mechanism is the preservation of spike temporal order without disruptive changes.

Core claim

QIF neurons exhibit less fragmented loss landscapes and outperform LIF neurons in spike-based gradient descent because their continuous spiking dynamics avoid the discontinuities that cause spike (dis)appearances and unstable representations in LIF neurons, as demonstrated by superior accuracy and smoother visualized landscapes on the Spiking Heidelberg Digits dataset after hyperparameter optimization.

What carries the argument

The continuity of spiking dynamics in the QIF neuron model, which eliminates parameter-induced discontinuities in spike timing that fragment loss landscapes in the LIF model.

If this is right

  • QIF networks achieve higher classification accuracy than LIF networks on the Spiking Heidelberg Digits dataset.
  • Loss landscapes for LIF appear more fragmented due to changes in the temporal order of spikes.
  • Gradients in QIF training are less erratic than in LIF training.
  • Replacing LIF with QIF or similar continuous models is advocated for stable gradient descent training of spiking networks.

Where Pith is reading between the lines

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

  • Continuous dynamics might benefit other spike-based learning methods beyond exact gradient descent.
  • QIF models could lead to more reliable training on neuromorphic hardware where on-chip learning is desired.
  • Similar advantages may appear on other datasets or with different network architectures.

Load-bearing premise

The hyperparameter search was equally exhaustive and unbiased for both neuron models, with any performance gap due to spiking continuity rather than optimization differences.

What would settle it

Finding that an equally exhaustive hyperparameter search allows LIF networks to match or exceed QIF accuracy on the same dataset would falsify the performance advantage.

Figures

Figures reproduced from arXiv: 2606.03935 by Carlo Wenig, Christian Klos, Raoul-Martin Memmesheimer.

Figure 1
Figure 1. Figure 1: Loss and gradient landscapes of LIF networks are more fragmented than those of QIF [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the three selected models, the best-performing QIF model, the best [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Loss and gradient landscapes for three selected models. (A) Sample loss landscapes for an [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: D). By examining the accompanying change in network dynamics, we find that it is the result of the disappearance of a spike of a hidden-layer neuron at one point in time and the appearance of a new spike of that neuron at an earlier point in time (Figure 1F). This (dis)appearance event is caused by a small change in the input weights, which results in the membrane potential reaching the threshold directly … view at source ↗
Figure 5
Figure 5. Figure 5: (A) Loss change along linear path between parameters [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results of rough hyperparameter search with QIF neurons. For each configuration, we [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Results of rough hyperparameter search with LIF neurons. For each configuration, we show [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results of fine hyperparameter search with QIF neurons. For each configuration, we show [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Results of fine hyperparameter search with LIF neurons and low [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Results of fine hyperparameter search with LIF neurons and high [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Final test accuracy plotted against distinctness for hyperparameter configurations on the [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Change of the voltage dynamics of the hidden neuron that results in the gradient disconti [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example of a region of a QIF sample landscape exhibiting a sharp loss transition. (A) [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Additional landscapes. (A) Gradient amplitude landscapes of the same sample shown in [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Same landscapes as in Figure 14 but zoomed in by a factor of 20. Individual training steps [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Actual loss differences plotted against expected ones for all training steps. Corresponding [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
read the original abstract

The ability to train spiking neural networks is essential for modeling biological neural networks as well as for neuromorphic computing. However, for the extensively used leaky integrate-and-fire (LIF) neurons, arbitrarily small parameter changes can induce spike (dis)appearances that disrupt subsequent activity, leading to unstable neural representations and permanently silent neurons during exact spike-based gradient descent. Recent work shows that a class of neuron models, which includes the quadratic integrate-and-fire (QIF) neuron, avoids these discontinuities and enables continuous and even smooth spike-based gradient descent. However, it remains unclear whether these advantages translate into practice. Here, we demonstrate that they do so via a controlled comparison between networks of LIF and QIF neurons on the popular Spiking Heidelberg Digits dataset. Specifically, in a first step, we perform a thorough hyperparameter search to optimize both models, revealing a clear performance advantage of QIF neurons. In a second step, we visualize the loss and gradient landscapes. Consistent with their inferior performance, we find that the loss landscapes of LIF neurons, which are discontinuous, appear more fragmented and the related gradients more erratic. An analysis of the landscapes of single samples indicates that these features arise from changes in the temporal order of spikes, which often cause disruptive spike (dis)appearances. Overall, our results advocate replacing LIF neurons with neuron models exhibiting continuous spiking dynamics, such as QIF neurons, for gradient descent training.

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.

Referee Report

1 major / 0 minor

Summary. The paper claims that quadratic integrate-and-fire (QIF) neurons produce less fragmented loss landscapes and outperform leaky integrate-and-fire (LIF) neurons when training spiking networks via exact spike-based gradient descent on the Spiking Heidelberg Digits dataset, attributing the advantage to the continuity of QIF spiking dynamics that avoids disruptive spike (dis)appearances.

Significance. If the attribution holds, the result would provide empirical support for preferring neuron models with continuous spiking dynamics over standard LIF models in gradient-based training of spiking networks, with potential benefits for stability in neuromorphic hardware and biological modeling. The controlled comparison on a public benchmark together with loss-landscape visualizations constitutes a direct test of the theoretical continuity advantage.

major comments (1)
  1. [Abstract] Abstract (and the corresponding methods description): the central performance claim rests on a 'thorough hyperparameter search' having been performed equally for both models, yet no quantitative information is supplied on search ranges, number of trials, adaptation of ranges between models, or statistical comparison of the resulting optima. Without these details the reported accuracy gap cannot be confidently attributed to spiking continuity rather than differences in optimization effort or implementation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for highlighting the need for greater transparency in our hyperparameter search procedure. We agree that this information is essential for attributing performance differences to the neuron model rather than optimization effort, and we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the corresponding methods description): the central performance claim rests on a 'thorough hyperparameter search' having been performed equally for both models, yet no quantitative information is supplied on search ranges, number of trials, adaptation of ranges between models, or statistical comparison of the resulting optima. Without these details the reported accuracy gap cannot be confidently attributed to spiking continuity rather than differences in optimization effort or implementation.

    Authors: We acknowledge that the current manuscript lacks sufficient quantitative details on the hyperparameter optimization. In the revised version, we will expand the methods section (and update the abstract if space permits) to specify the search ranges used for each model, the total number of trials or evaluations performed, whether ranges were adapted differently between LIF and QIF, and any statistical comparisons (e.g., mean and variance of top-performing configurations). This will enable readers to evaluate the fairness of the comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical results on external benchmark.

full rationale

The paper's central claims rest on a controlled empirical comparison: hyperparameter search followed by training and loss-landscape visualization on the public Spiking Heidelberg Digits dataset. No equations, fitted parameters, or predictions are shown to reduce by construction to inputs inside the paper. The continuity property of QIF is attributed to prior work (not self-cited as load-bearing uniqueness theorem here), while the performance and landscape differences are measured directly. This is a standard experimental design against external benchmarks; no self-definitional, fitted-input, or self-citation-chain reductions occur.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the empirical outcomes of a hyperparameter search and landscape analysis performed on one public dataset. No additional free parameters, mathematical axioms, or invented entities are introduced beyond standard machine-learning practice.

pith-pipeline@v0.9.1-grok · 5798 in / 1146 out tokens · 26155 ms · 2026-06-28T07:19:15.773382+00:00 · methodology

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