pith. sign in

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
0
0 comments X

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

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

63 extracted references · 30 canonical work pages

  1. [1]

    Kistler, Richard Naud, and Liam Paninski.Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition

    Wulfram Gerstner, Werner M. Kistler, Richard Naud, and Liam Paninski.Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition. Cambridge University Press, 2014. ISBN 978-1- 107-06083-8

    2014

  2. [2]

    Peter Dayan and L. F. Abbott.Theoretical Neuroscience. MIT Press, Cambridge, 2001. ISBN 0-262- 04199-5

    2001

  3. [3]

    Rapid Neural Coding in the Retina with Relative Spike Latencies

    Tim Gollisch and Markus Meister. Rapid Neural Coding in the Retina with Relative Spike Latencies. Science, 319(5866):1108–1111, 2008. doi: 10.1126/science.1149639

    work page doi:10.1126/science.1149639 2008

  4. [4]

    Sparse and powerful cortical spikes.Current Opinion in Neurobiology, 20(3):306–312, 2010

    Jason Wolfe, Arthur R Houweling, and Michael Brecht. Sparse and powerful cortical spikes.Current Opinion in Neurobiology, 20(3):306–312, 2010. doi: 10.1016/j.conb.2010.03.006

    work page doi:10.1016/j.conb.2010.03.006 2010

  5. [5]

    Importance of spike timing in touch: an analogy with hearing?Current Opinion in Neurobiology, 40:142–149, 2016

    Hannes P Saal, Xiaoqin Wang, and Sliman J Bensmaia. Importance of spike timing in touch: an analogy with hearing?Current Opinion in Neurobiology, 40:142–149, 2016. doi: 10.1016/j.conb.2016.07.013

    work page doi:10.1016/j.conb.2016.07.013 2016

  6. [6]

    Sober, Simon Sponberg, Ilya Nemenman, and Lena H

    Samuel J. Sober, Simon Sponberg, Ilya Nemenman, and Lena H. Ting. Millisecond Spike Timing Codes for Motor Control.Trends in Neurosciences, 41(10):644–648, 2018. doi: 10.1016/j.tins.2018.08.010

    work page doi:10.1016/j.tins.2018.08.010 2018

  7. [7]

    Wittig, Julio I

    Weizhen Xie, John H. Wittig, Julio I. Chapeton, Mostafa El-Kalliny, Samantha N. Jackson, Sara K. Inati, and Kareem A. Zaghloul. Neuronal sequences in population bursts encode information in human cortex. Nature, 635(8040):935–942, 2024. doi: 10.1038/s41586-024-08075-8

    work page doi:10.1038/s41586-024-08075-8 2024

  8. [8]

    Mehonic and A

    A. Mehonic and A. J. Kenyon. Brain-inspired computing needs a master plan.Nature, 604(7905):255–260,

  9. [9]

    doi: 10.1038/s41586-021-04362-w

    work page doi:10.1038/s41586-021-04362-w

  10. [10]

    Vineyard, Tej Pandit, Cory Merkel, Rajkumar Kubendran, James B

    Dhireesha Kudithipudi, Catherine Schuman, Craig M. Vineyard, Tej Pandit, Cory Merkel, Rajkumar Kubendran, James B. Aimone, Garrick Orchard, Christian Mayr, Ryad Benosman, Joe Hays, Cliff Young, Chiara Bartolozzi, Amitava Majumdar, Suma George Cardwell, Melika Payvand, Sonia Buckley, Shruti Kulkarni, Hector A. Gonzalez, Gert Cauwenberghs, Chetan Singh Thak...

    2025

  11. [11]

    MIT Press, 2016

    Ian Goodfellow, Yoshua Bengio, and Aaron Courville.Deep Learning. MIT Press, 2016

    2016

  12. [12]

    Bishop and Hugh Bishop.Deep Learning - Foundations and Concepts

    Christopher M. Bishop and Hugh Bishop.Deep Learning - Foundations and Concepts. Springer, 2024. ISBN 978-3-031-45467-7

    2024

  13. [13]

    Visualizing the Loss Landscape of Neural Nets

    Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. Visualizing the Loss Landscape of Neural Nets. InAdvances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018

    2018

  14. [14]

    How Does Batch Normalization Help Optimization? InAdvances in Neural Information Processing Systems, volume 31, pages 2488–2498

    Shibani Santurkar, Dimitris Tsipras, Andrew Ilyas, and Aleksander Madry. How Does Batch Normalization Help Optimization? InAdvances in Neural Information Processing Systems, volume 31, pages 2488–2498. Curran Associates, Inc., 2018

    2018

  15. [15]

    On the difficulty of training recurrent neural networks

    Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. On the difficulty of training recurrent neural networks. InProceedings of the 30th International Conference on Machine Learning, volume 28, pages 1310–1318. PMLR, 2013

    2013

  16. [16]

    Smooth Exact Gradient Descent Learning in Spiking Neural Networks.Physical Review Letters, 134(2):027301, 2025

    Christian Klos and Raoul-Martin Memmesheimer. Smooth Exact Gradient Descent Learning in Spiking Neural Networks.Physical Review Letters, 134(2):027301, 2025. doi: 10.1103/PhysRevLett.134.027301

    work page doi:10.1103/physrevlett.134.027301 2025

  17. [17]

    Bohte, Joost N

    Sander M. Bohte, Joost N. Kok, and Han La Poutré. Error-backpropagation in temporally encoded networks of spiking neurons.Neurocomputing, 48(1):17–37, 2002. doi: 10.1016/S0925-2312(01)00658-0. 10

    work page doi:10.1016/s0925-2312(01)00658-0 2002

  18. [18]

    Eshraghian, Max Ward, Emre O

    Jason K. Eshraghian, Max Ward, Emre O. Neftci, Xinxin Wang, Gregor Lenz, Girish Dwivedi, Mohammed Bennamoun, Doo Seok Jeong, and Wei D. Lu. Training Spiking Neural Networks Using Lessons From Deep Learning.Proceedings of the IEEE, 111(9):1016–1054, 2023. doi: 10.1109/JPROC.2023.3308088

    work page doi:10.1109/jproc.2023.3308088 2023

  19. [19]

    Specaugment on large scale datasets

    Iulia M. Comsa, Krzysztof Potempa, Luca Versari, Thomas Fischbacher, Andrea Gesmundo, and Jyrki Alakuijala. Temporal Coding in Spiking Neural Networks with Alpha Synaptic Function: Learning with Backpropagation. InICASSP 2020 – 2020 IEEE International Conference on Acoustics, Speech and Signal Processing, pages 8529–8533. IEEE, 2020. doi: 10.1109/ICASSP40...

    work page doi:10.1109/icassp40776.2020.9053856 2020

  20. [20]

    Göltz, L

    J. Göltz, L. Kriener, A. Baumbach, S. Billaudelle, O. Breitwieser, B. Cramer, D. Dold, A. F. Kungl, W. Senn, J. Schemmel, K. Meier, and M. A. Petrovici. Fast and energy-efficient neuromorphic deep learning with first-spike times.Nature Machine Intelligence, 3(9):823–835, 2021. doi: 10.1038/s42256-021-00388-x

    work page doi:10.1038/s42256-021-00388-x 2021

  21. [21]

    Supervised Learning Based on Temporal Coding in Spiking Neural Networks.IEEE Transactions on Neural Networks and Learning Systems, pages 1–9, 2017

    Hesham Mostafa. Supervised Learning Based on Temporal Coding in Spiking Neural Networks.IEEE Transactions on Neural Networks and Learning Systems, pages 1–9, 2017. doi: 10.1109/TNNLS.2017. 2726060

    work page doi:10.1109/tnnls.2017 2017

  22. [22]

    Loss shaping enhances exact gradient learning with Eventprop in spiking neural networks.Neuromorphic Computing and Engineering, 5(1):014001,

    Thomas Nowotny, James P Turner, and James C Knight. Loss shaping enhances exact gradient learning with Eventprop in spiking neural networks.Neuromorphic Computing and Engineering, 5(1):014001,

  23. [23]

    doi: 10.1088/2634-4386/ada852

    work page doi:10.1088/2634-4386/ada852

  24. [24]

    S4NN: temporal backpropagation for spiking neural networks with one spike per neuron.International Journal of Neural Systems, 30(06):2050027, 2020

    Saeed Reza Kheradpisheh and Timothée Masquelier. S4NN: temporal backpropagation for spiking neural networks with one spike per neuron.International Journal of Neural Systems, 30(06):2050027, 2020. doi: 10.1142/S0129065720500276

    work page doi:10.1142/s0129065720500276 2020

  25. [25]

    Wunderlich and Christian Pehle

    Timo C. Wunderlich and Christian Pehle. Event-based backpropagation can compute exact gradients for spiking neural networks.Scientific Reports, 11(1):12829, 2021. doi: 10.1038/s41598-021-91786-z

    work page doi:10.1038/s41598-021-91786-z 2021

  26. [26]

    jaxsnn: Event-driven Gradient Estimation for Analog Neuromorphic Hardware, 2024

    Eric Müller, Moritz Althaus, Elias Arnold, Philipp Spilger, Christian Pehle, and Johannes Schemmel. jaxsnn: Event-driven Gradient Estimation for Analog Neuromorphic Hardware, 2024

    2024

  27. [27]

    Training event-based neural networks with exact gradients via Differentiable ODE Solving in JAX, 2026

    Lukas König, Manuel Kuhn, David Kappel, and Anand Subramoney. Training event-based neural networks with exact gradients via Differentiable ODE Solving in JAX, 2026

    2026

  28. [28]

    SparseProp: Efficient Event-Based Simulation and Training of Sparse Recurrent Spiking Neural Networks

    Rainer Engelken. SparseProp: Efficient Event-Based Simulation and Training of Sparse Recurrent Spiking Neural Networks. InAdvances in Neural Information Processing Systems, volume 36, 2023

    2023

  29. [29]

    From LIF to QIF: Toward Differentiable Spiking Neurons for Scientific Machine Learning, 2025

    Ruyin Wan, George Em Karniadakis, and Panos Stinis. From LIF to QIF: Toward Differentiable Spiking Neurons for Scientific Machine Learning, 2025

    2025

  30. [30]

    Neftci, Hesham Mostafa, and Friedemann Zenke

    Emre O. Neftci, Hesham Mostafa, and Friedemann Zenke. Surrogate Gradient Learning in Spiking Neural Networks: Bringing the Power of Gradient-Based Optimization to Spiking Neural Networks.IEEE Signal Processing Magazine, 36(6):51–63, 2019. doi: 10.1109/MSP.2019.2931595

    work page doi:10.1109/msp.2019.2931595 2019

  31. [31]

    Elucidating the theoretical underpinnings of surrogate gradient learning in spiking neural networks.Neural Computation, 37(5):886–925, 2025

    Julia Gygax and Friedemann Zenke. Elucidating the theoretical underpinnings of surrogate gradient learning in spiking neural networks.Neural Computation, 37(5):886–925, 2025. doi: 10.1162/neco_a_01752

    work page doi:10.1162/neco_a_01752 2025

  32. [32]

    David Balduzzi, Marcus Frean, Lennox Leary, J. P. Lewis, Kurt Wan-Duo Ma, and Brian McWilliams. The shattered gradients problem: if resnets are the answer, then what is the question? InProceedings of the 34th International Conference on Machine Learning, volume 70, pages 342–350. PMLR, 2017

    2017

  33. [33]

    On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima

    Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima. InInternational Conference on Learning Representations, 2017

    2017

  34. [34]

    Flat minima.Neural Computation, 9(1):1–42, 1997

    Sepp Hochreiter and Jürgen Schmidhuber. Flat minima.Neural Computation, 9(1):1–42, 1997. doi: 10.1162/neco.1997.9.1.1

    work page doi:10.1162/neco.1997.9.1.1 1997

  35. [35]

    Sharpness-Aware Minimization for Efficiently Improving Generalization

    Pierre Foret, Ariel Kleiner, Hossein Mobahi, and Behnam Neyshabur. Sharpness-Aware Minimization for Efficiently Improving Generalization. InInternational Conference on Learning Representations, 2021

    2021

  36. [36]

    Loss surfaces, mode connectivity, and fast ensembling of DNNs

    Timur Garipov, Pavel Izmailov, Dmitrii Podoprikhin, Dmitry Vetrov, and Andrew Gordon Wilson. Loss surfaces, mode connectivity, and fast ensembling of DNNs. InAdvances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018

    2018

  37. [37]

    Hamprecht

    Felix Draxler, Kambis Veschgini, Manfred Salmhofer, and Fred A. Hamprecht. Essentially no barriers in neural network energy landscape. InProceedings of the 35th International Conference on Machine Learning, volume 80, pages 1309–1318. PMLR, 2018. 11

    2018

  38. [38]

    Stable irregular dynamics in complex neural networks.Physical Review Letters, 100(4):048102, 2008

    Sven Jahnke, Raoul-Martin Memmesheimer, and Marc Timme. Stable irregular dynamics in complex neural networks.Physical Review Letters, 100(4):048102, 2008. doi: 10.1103/PhysRevLett.100.048102

    work page doi:10.1103/physrevlett.100.048102 2008

  39. [39]

    Dynamic Flux Tubes Form Reservoirs of Stability in Neuronal Circuits

    Michael Monteforte and Fred Wolf. Dynamic Flux Tubes Form Reservoirs of Stability in Neuronal Circuits. Physical Review X, 2(4):041007, 2012. doi: 10.1103/PhysRevX.2.041007

    work page doi:10.1103/physrevx.2.041007 2012

  40. [40]

    Sparse chaos in cortical circuits, 2024

    Rainer Engelken, Michael Monteforte, and Fred Wolf. Sparse chaos in cortical circuits, 2024

    2024

  41. [41]

    Causal pieces: analysing and improving spiking neural networks piece by piece, 2025

    Dominik Dold and Philipp Christian Petersen. Causal pieces: analysing and improving spiking neural networks piece by piece, 2025

    2025

  42. [42]

    Bower, Markus Diesmann, Abigail Morrison, Philip H

    Romain Brette, Michelle Rudolph, Ted Carnevale, Michael Hines, David Beeman, James M. Bower, Markus Diesmann, Abigail Morrison, Philip H. Goodman, Frederick C. Harris, Milind Zirpe, Thomas Natschläger, Dejan Pecevski, Bard Ermentrout, Mikael Djurfeldt, Anders Lansner, Olivier Rochel, Thierry Vieville, Eilif Muller, Andrew P. Davison, Sami El Boustani, and...

    work page doi:10.1007/s10827-007-0038-6 2007

  43. [43]

    The Heidelberg Spiking Data Sets for the Systematic Evaluation of Spiking Neural Networks.IEEE Transactions on Neural Networks and Learning Systems, 33(7):2744–2757, 2022

    Benjamin Cramer, Yannik Stradmann, Johannes Schemmel, and Friedemann Zenke. The Heidelberg Spiking Data Sets for the Systematic Evaluation of Spiking Neural Networks.IEEE Transactions on Neural Networks and Learning Systems, 33(7):2744–2757, 2022. doi: 10.1109/TNNLS.2020.3044364

    work page doi:10.1109/tnnls.2020.3044364 2022

  44. [44]

    Johansson and Ingvars Birznieks

    Roland S. Johansson and Ingvars Birznieks. First spikes in ensembles of human tactile afferents code complex spatial fingertip events.Nature Neuroscience, 7(2):170–177, 2004. doi: 10.1038/nn1177

    work page doi:10.1038/nn1177 2004

  45. [45]

    Spike-based strategies for rapid processing

    Simon Thorpe, Arnaud Delorme, and Rufin Van Rullen. Spike-based strategies for rapid processing. Neural Networks, 14(6):715–725, 2001. doi: 10.1016/S0893-6080(01)00083-1

    work page doi:10.1016/s0893-6080(01)00083-1 2001

  46. [46]

    & Panda, P

    Kaushik Roy, Akhilesh Jaiswal, and Priyadarshini Panda. Towards spike-based machine intelligence with neuromorphic computing.Nature, 575(7784):607–617, 2019. doi: 10.1038/s41586-019-1677-2

    work page doi:10.1038/s41586-019-1677-2 2019

  47. [47]

    Towards parameter-free attentional spiking neural networks.Neural Networks, 185:107154, 2025

    Pengfei Sun, Jibin Wu, Paul Devos, and Dick Botteldooren. Towards parameter-free attentional spiking neural networks.Neural Networks, 185:107154, 2025. doi: 10.1016/j.neunet.2025.107154

    work page doi:10.1016/j.neunet.2025.107154 2025

  48. [48]

    Scalable Event-by-event Processing of Neuromorphic Sensory Signals With Deep State-Space Models, 2024

    Mark Schöne, Neeraj Mohan Sushma, Jingyue Zhuge, Christian Mayr, Anand Subramoney, and David Kappel. Scalable Event-by-event Processing of Neuromorphic Sensory Signals With Deep State-Space Models, 2024

    2024

  49. [49]

    Advancing Spatio- Temporal Processing in Spiking Neural Networks through Adaptation, 2025

    Maximilian Baronig, Romain Ferrand, Silvester Sabathiel, and Robert Legenstein. Advancing Spatio- Temporal Processing in Spiking Neural Networks through Adaptation, 2025

    2025

  50. [50]

    2015, in 2015 IEEE International Conference on Computer Vision (ICCV), 1026–1034, doi: 10.1109/ICCV.2015.123

    Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification. In2015 IEEE International Conference on Computer Vision (ICCV). IEEE, 2015. doi: 10.1109/iccv.2015.123

    work page doi:10.1109/iccv.2015.123 2015

  51. [51]

    Alexandre Bittar and Philip N. Garner. A surrogate gradient spiking baseline for speech command recognition.Frontiers in Neuroscience, 16:865897, 2022. doi: 10.3389/fnins.2022.865897

    work page doi:10.3389/fnins.2022.865897 2022

  52. [52]

    JAX: composable transformations of Python+NumPy programs, 2018

    James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Yash Katariya, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. JAX: composable transformations of Python+NumPy programs, 2018

    2018

  53. [53]

    The DeepMind JAX Ecosystem, 2020

    DeepMind, Igor Babuschkin, Kate Baumli, Alison Bell, Surya Bhupatiraju, Jake Bruce, Peter Buchlovsky, David Budden, Trevor Cai, Aidan Clark, Ivo Danihelka, Antoine Dedieu, Claudio Fantacci, Jonathan Godwin, Chris Jones, Ross Hemsley, Tom Hennigan, Matteo Hessel, Shaobo Hou, Steven Kapturowski, Thomas Keck, Iurii Kemaev, Michael King, Markus Kunesch, Lena ...

    2020

  54. [54]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V . Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V . Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python.Journal of Machine Learning Research, 12:2825–2830, 2011

    2011

  55. [55]

    Data structures for statistical computing in python

    Wes McKinney. Data Structures for Statistical Computing in Python. InProceedings of the 9th Python in Science Conference, volume 445, pages 51–56, 2010. doi: 10.25080/Majora-92bf1922-00a

    work page doi:10.25080/majora-92bf1922-00a 2010

  56. [56]

    pandas-dev/pandas: Pandas, 2020

    The pandas development team. pandas-dev/pandas: Pandas, 2020

    2020

  57. [57]

    J. D. Hunter. Matplotlib: A 2D Graphics Environment.Computing in Science & Engineering, 9(3):90–95,

  58. [58]

    Computing in Science and Engineering , keywords =

    doi: 10.1109/MCSE.2007.55. 12 A Model details The model description was intentionally kept short in the main text to keep the focus on the main results. For completeness, we give more details about the models used in this work in this section. This is intended to be an extension of Section 2.1. A.1 Phase representation To simplify and unify the dynamics o...

    work page doi:10.1109/mcse.2007.55 2007

  59. [59]

    The resulting threshold phases are ϕLIF Θ =−τlog 1− VΘ I0 , ϕ QIF Θ = πτ a .(6) The inverse phase transformations are (ΦLIF)−1(ϕ) =I 0 1−e −ϕ/τ ,(Φ QIF)−1(ϕ) = 1 2 +atan aϕ τ − π 2 .(7) For infinitesimally short input currents, the membrane potential of neuron i jumps by wij when receiving an input spike from neuron j. The effect of this potential jump on...

  60. [60]

    This could either be an input spike or a spike in the layer that is simulated, depending on which appears earlier

    Determine the neuron index jk+1 and spike time tk+1 of the next spike. This could either be an input spike or a spike in the layer that is simulated, depending on which appears earlier. Because the phase grows linearly with time between spikes, the neuron that spikes next is always the one with the largest phase,j k+1 =argmax iϕik

  61. [61]

    This is trivial due to the linear dynamics between spikes

    Evolve neurons freely until shortly before the next spike at t− k+1. This is trivial due to the linear dynamics between spikes. We have ˙ϕi = 1 and, thus, ϕi(t− k+1) =ϕ ik + (tk+1 −t k)

  62. [62]

    Otherwise, if the spike occurs in the simulated layer itself, reset the phase of the spiking neuron, ϕj+1,k+1 = 0, and set ϕi,k+1 =ϕ i(t− k+1)for all other neurons

    If the next spike is an input spike, transfer it to the neurons in the simulated layer, using the phase transfer function ϕi,k+1 =H wijk+1 (ϕi(t− k+1)). Otherwise, if the spike occurs in the simulated layer itself, reset the phase of the spiking neuron, ϕj+1,k+1 = 0, and set ϕi,k+1 =ϕ i(t− k+1)for all other neurons. This process has to be repeated until t...

  63. [63]

    for principle component analysis, pandas [53, 54] for data management, and Matplotlib [55] for visualizations. 27