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REVIEW 3 major objections 4 minor 56 references

An analog-electronic implementation of a harmonic oscillator recurrent neural network

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read An analog circuit can implement a harmonic-oscillator recurrent network and, once the readout is retrained, match a digital twin's classification performance.

desk verdict First real analog-electronic HORN with a fair SVM comparison, but the precision explanation is confounded by per-sample input scaling. read the letter →

arxiv 2509.04064 v1 pith:ZOMQ4RIG submitted 2025-09-04 q-bio.NC physics.app-phphysics.comp-ph

Pedro Carvalho , Bernd Ulmann , Wolf Singer , Felix Effenberger This is my paper
classification q-bio.NCphysics.app-phphysics.comp-ph
keywords analogcomputingharmonicoscillatorrecurrentnetworkreservoirsequentialMNISTdigitaltwintransientdynamicsphysicalneuralnetworksneuromorphichardware
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 proof-of-concept paper asks whether oscillatory recurrent neural networks, specifically the Harmonic Oscillator Recurrent Network (HORN), can run on analog electronic hardware without losing computational performance. The authors train a four-node HORN in silico on sequential MNIST, transfer its parameters to an analog circuit, and measure how faithfully the circuit reproduces the digital dynamics. They find that the analog dynamics match the digital twin in most test cases, but the original digital readout transfers poorly (28.39% label agreement) because parts of the decision space require more precision than analog hardware provides. When the analog system is instead treated as a reservoir with a retrained linear SVM readout, its accuracy reaches 75.50% versus the digital twin's 73.75%, indicating the task-relevant information survives in the analog dynamics. The paper concludes that analog electronic circuits can effectively implement oscillatory neural networks, and that adapting the readout, not fixing the internal dynamics, is the key engineering step.

What carries the argument

The central mechanism is the damped harmonic oscillator (DHO) unit, whose second-order dynamics are given by ẍ_i + 2γ_i ẋ_i + ω_i² x_i = F(x, ẋ, t), with the full HORN forcing F = α tanh(V x + W ẋ + I s). For the analog implementation, the paper removes the tanh nonlinearity and self-feedback, constrains coupling strengths to [0,1], and discretizes the system with symplectic Euler integration. On the analog computer, each DHO is realized as a circuit of integrators, summers, and programmable coefficients that solve the same ODE continuously. The parameter transfer is carried by a scaling relation (Eq. 5) that maps digital discrete-time parameters to experimental continuous-time machine units

What would settle it

A decisive check would be to take the 10,000 test digits, apply exactly the same per-sample input scaling from Appendix B in the digital simulation, and run the original digital readout on the resulting states. If the agreement between scaled-digital and unscaled-digital predictions falls to near the 28.39% level, analog precision is not the main cause of the readout mismatch; if it remains high, the precision explanation is confirmed. A second observation would be to test an SVM readout trained on analog data against the small-volume digit classes 0, 3, 7, and 9 to verify the volume-precision

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a HORN—normally simulated with floating-point arithmetic—can be physically instantiated on an analog computer whose integrators solve the oscillator differential equations continuously, and that the transient dynamics of the physical circuit carry enough information to reproduce the digital network's classification once the readout accounts for analog precision. After training a homogeneous four-node HORN with BPTT on sequential MNIST, the authors map the digital parameters to machine units via an explicit scaling relation and a per-sample input rescaling loop that keeps node amplitudes inside the analog range. Across 10,000 test samples, custo

Load-bearing premise

The conclusion that the 28.39% readout mismatch is caused by analog precision assumes that the per-sample input scaling described in Appendix B does not itself change what the trained digital readout would predict; because the readout is affine, rescaling the node states changes its outputs, and the paper does not report the digital readout's predictions on a digitally simulated version of the scaled system.

Editorial extensions

If this is right

  • If the central claim is correct, analog oscillator networks can be trained in silico and transferred to physical circuits, with only the readout needing to be retrained on the physical states.
  • A 16-parameter analog network reaching roughly 75% accuracy on sequential MNIST as a reservoir suggests that useful classification is feasible under severe hardware constraints, relevant for edge and sensor applications.
  • The comparable SVM accuracy of analog and digital reservoirs implies the physical dynamics do not lose task-relevant information, so scaling up analog HORNs can focus engineering effort on readout design.
  • The velocity-coupled network variant shows even better analog-digital label agreement (58.9%), suggesting that choosing which state variable to couple can ease hardware transfer.
  • The decision-space volume analysis gives a design heuristic: analog precision (machine-unit resolution around ±0.03) sets a lower bound on how small a decision-region can be for a direct readout, so training should favor representations with larger, well-separated label volumes.
  • The paper's discussion concludes that readout mechanisms may matter more than internal dynamics for achieving target performance in analog networks, pointing toward reservoir computing as a particularly suitable paradigm for analog oscillatory hardware.

Reading between the lines

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

  • Going beyond the paper: a clean control—applying the same per-sample input scaling in a purely digital simulation and then re-running the original digital readout—would separate hardware precision from the paper's own normalization. The paper does not report this control, so the precision explanation, while plausible, is not fully isolated.
  • The paper motivates energy efficiency but reports no power measurements; a direct energy-per-inference comparison between the analog circuit and its digital twin would turn the feasibility claim into a quantitative advantage claim.
  • The decision-space volume analysis yields a testable prediction: on any classification task mapped to analog hardware, digit classes with small state-space volumes at readout time will be the ones the original digital readout fails on, and training with an objective that enlarges or separates these volumes should recover accuracy.
  • The success of Hilbert-decoder and SVM readouts on the velocity-coupled network suggests that phase information may be a more robustly transferable substrate than raw amplitude at a single time point, pointing toward temporal and phase-based decoders for future analog implementations.
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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 / 4 minor

Summary. The paper reports a proof-of-concept implementation of a four-node Harmonic Oscillator Recurrent Network (HORN) on an analog electronic computer. A digital twin is trained on sequential MNIST, and the trained weights are transferred to the analog hardware. The analog dynamics are compared with the digital simulation using custom error metrics; using the original digital affine readout on analog states gives only 28.39% agreement with the digital model's predictions. The authors attribute this mismatch to limited analog precision and show that retraining a linear SVM readout on the analog states restores classification accuracy to roughly the digital baseline (75.50% vs. 73.75%). They conclude that analog circuits can effectively implement oscillatory RNNs and preserve task-relevant information.

Significance. If the central comparison is valid, this is a useful proof-of-concept for analog physical neural networks based on transient oscillatory dynamics. The paper reports real hardware measurements and makes a concrete attempt to benchmark the analog system against a digital twin, which is a strength. The contribution is modest in scale: the network has only four nodes, no nonlinearity, no feedback, and a low baseline accuracy (59.24% with the trained readout), so the broader claims about energy-efficient brain-inspired computing are not yet demonstrated. The main significance hangs on the reservoir-computing result, which is exactly the part potentially confounded by the per-sample input scaling described in Appendix B.

major comments (3)
  1. [Section IV, Fig. 4B; Appendix B] The reservoir comparison is confounded by the per-sample input scaling. Appendix B states that before every analog run the input matrix I is multiplied by a sample-specific scalar s so that max(|x|) lies in [0.1, 0.6]. Since the trained readout is affine (Mx+b), and since s changes the state magnitudes, the analog states are not directly comparable to the original digital model's states. The manuscript does not state whether the digital SVM baseline was trained on states obtained with the same per-sample scaling. If it was trained on the unscaled digital model, the similarity 75.50% vs. 73.75% could be produced by the scaling transformation alone. The authors should report the SVM accuracy of the digitally simulated HORN when the same Appendix B scaling is applied, and also the agreement of the original digital affine readout when applied to scaled digital states.
  2. [Section IV, 'Volume occupancy' and Fig. 4E; Figs. 4D and 5] The precision-mismatch explanation is not uniquely identified. The 28.39% agreement is measured between the analog scaled system and the digital model's original readout, which was trained on unscaled states. A perfect analog implementation of the scaled system would not necessarily reproduce that readout. The volume-occupancy analysis over the original decision space therefore cannot separate precision effects from the scaling-induced shift. The authors should include a control: apply the original affine readout to digitally simulated states after the same per-sample scaling. If agreement with the original digital predictions is already low, the precision hypothesis is not supported. Similarly, the error metrics in Figs. 4D and 5 should specify whether the digital reference traces are generated with the scaled or unscaled input matrix; otherwise they measure fidelity to a different dyna
  3. [Section V and Abstract] The central claim that 'analog electronic circuits can effectively implement oscillatory neural networks for computation' is stronger than what the current experiment establishes. The experiment shows that an SVM can read out the analog dynamics with accuracy comparable to some digital baseline. Given the scaling confound, the claim should be narrowed to 'the analog hardware adds little distortion beyond the scaled digital model' until the missing control is provided.
minor comments (4)
  1. [Section III B] Typo: 'dampled harmonic oscillator' should be 'damped harmonic oscillator'.
  2. [Section IV] Typo: 'These fidings indicate' should be 'These findings indicate'.
  3. [Equation (5)] The line for I_E is typeset ambiguously ('IE = 1/c IM ωM'). Please clarify whether the factor is ω_M or 1/ω_M, and define all symbols (ΔS, k0, c) in one place.
  4. [References] References [40] and [52] appear to be the same Baronig et al. preprint; consolidate.

Circularity Check

1 steps flagged · score 6.0 of 10

The SVM-reservoir 'recovery' is described as a trained readout and then reported as classification performance on the same analog-generated data, making the central evidence an in-sample fit rather than an independent prediction.

  1. fitted input called prediction [Section IV (Results), Fig. 4B; Appendix D; Discussion]
    "We then performed inference on the 10, 000 sMNIST test samples using the analog twin ... we discarded the affine readout layer trained as part of the digital model and replaced it with a linear Support Vector Machine (SVM) trained on the data generated by the analog implementation. ... For comparison and to avoid creating an unfair advantage for the analog twin, the same procedure was applied to its digital counterpart."

    The SVM is a fitted readout. The only analog-generated dataset described in the paper is the 10,000-sample test set used for inference. The SVM is 'trained on the data generated by the analog implementation' and the resulting accuracy (75.50% vs 73.75%) is then reported as recovered classification performance. As written, training and evaluation use the same samples, so the accuracy is an in-sample training score, not an independent prediction. The conclusion that the analog dynamics 'preserves sufficient information for accurate decoding' is therefore supported by the readout's fit to the very data it decodes, not by a held-out test. This is the fitted-input-called-prediction pattern: the 'prediction' is the SVM's own training accuracy.

full rationale

The core hardware demonstration is otherwise empirical and self-contained: the analog–digital dynamics comparison, the custom error metrics, the 28.39% readout-agreement observation, and the trace-level comparisons are direct measurements, not derivations from the model's assumptions. The self-citations to the authors' prior HORN work [19] are not load-bearing for the analog claim; the model definition and parameter choices are inputs, not the result, and no uniqueness theorem is imported. The Appendix B per-sample input scaling is a potential confound for the digital-vs-analog SVM comparison, but that is a missing-control/correctness issue, not circularity. The one circular step is the reservoir SVM evaluation: as reported, the SVM readout is trained on the analog-generated test data and then that same dataset is used to claim recovery of classification accuracy, so the quantitative support for 'preserved information content' reduces to an in-sample fit rather than a predictive test.

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

The central claim rests on the analog circuit faithfully solving the HORN ODE, on the equivalence between the discrete digital model and the continuous analog system under the stated scaling, and on the per-sample input scaling being information-preserving. The paper provides no direct verification of the first premise and no analysis of the third. The free parameters are the hand-chosen experimental settings and the per-sample gain.

free parameters (4)
  • per-sample input scaling factor s = per-test-sample, chosen so max|x| in [0.1, 0.6]
    Applied to I to maximize dynamic range of the analog computer; determined by iterative digital simulation (Appendix B). Not fitted to labels, but a per-input gain.
  • dynamic range thresholds floor/ceil = 0.1 / 0.6
    Chosen by hand to avoid clipping and precision issues (Appendix B).
  • experiment duration T and integration factor k0 = 6000 ms / 10
    Chosen to map omega* to 16 Hz and to avoid extreme parameter values (Section III D).
  • model hyperparameters omega_M, gamma_M = 0.22 rad/pixel, 0.01
    From prior HORN work [19]; homogeneous across nodes.
assumptions (4)
  • domain assumption The anabrid Model-1 analog computer implements the linear ODE dynamics of Eq. 1 with negligible unmodeled non-idealities beyond the stated +/-0.03 precision.
    The entire comparison of analog vs digital dynamics rests on the circuit faithfully solving the ODE; no characterization of integrator drift, crosstalk, or coefficient error is provided (Section III B).
  • domain assumption The discrete-time Euler model (Eq. 3) and the continuous-time analog system are equivalent under the scaling in Eq. 5.
    Parameter mapping assumes discretization and continuous time are interchangeable; the paper does not quantify integration error (Sections II, III E).
  • domain assumption The error metrics (area, phase, correlation, mismatch) are sufficient to infer 'successful replication' of dynamics.
    No threshold or baseline ties these metrics to task-relevant fidelity (Section IV, Appendix A).
  • ad hoc to paper Per-sample input scaling is information-preserving for the classification task.
    The scaling changes the magnitude of states and thus the affine readout outputs; its effect on the label predictions is not analyzed (Appendix B).

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Pith. "Pith review of An analog-electronic implementation of a harmonic oscillator recurrent neural network." pith.science (2026). https://pith.science/paper/ZOMQ4RIG

@misc{pith2026250904064,
  author       = {Pith},
  title        = {Pith review of: An analog-electronic implementation of a harmonic oscillator recurrent neural network},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOMQ4RIG}},
  note         = {Machine review of arXiv:2509.04064}
}
abstract

Oscillatory recurrent networks, such as the Harmonic Oscillator Recurrent Network (HORN) model, offer advantages in parameter efficiency, learning speed, and robustness relative to traditional non-oscillating architectures. Yet, while many implementations of physical neural networks exploiting attractor dynamics have been studied, implementations of oscillatory models in analog-electronic hardware that utilize the networks' transient dynamics so far are lacking. This study explores the feasibility of implementing HORNs in analog-electronic hardware while maintaining the computational performance of the digital counterpart. Using a digital twin approach, we trained a four-node HORN in silico for sequential MNIST classification and transferred the trained parameters to an analog electronic implementation. A set of custom error metrics indicated that the analog system is able to successfully replicate the dynamics of the digital model in most test cases. However, despite the overall well-matching dynamics, when using the readout layer of the digital model on the data generated by the analog system, we only observed $28.39\%$ agreement with the predictions of the digital model. An analysis shows that this mismatch is due to a precision difference between the analog hardware and the floating-point representation exploited by the digital model to perform classification tasks. When the analog system was utilized as a reservoir with a re-trained linear readout, its classification performance could be recovered to that of the digital twin, indicating preserved information content within the analog dynamics. This proof-of-concept establishes that analog electronic circuits can effectively implement oscillatory neural networks for computation, providing a demonstration of energy-efficient analog systems that exploit brain-inspired transient dynamics for computation.

Figures

Figures reproduced from arXiv: 2509.04064 by the authors.

Figure 1
Figure 1. FIG. 1. Experimental setup and performance comparison of digital and analog HORN implementations. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Analog circuit implementation of a damped harmonic [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of analog and digital HORN dynamics for representative sMNIST samples demonstrating different repro [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Analysis of the analog implementation of the HORN model. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Node-wise error metric distributions for prediction agreement analysis. Probability densities of error metrics (mismatch, [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Additional examples of the analog and digital dynamics comparison across different input samples. Left column [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Flowchart of the input matrix rescaling algorithm. This iterative procedure optimizes the dynamic range usage of the [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Prediction agreement analysis using Venn diagram representation. Overlap regions show the intersection between [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Digit-wise classification performance comparison across different evaluation scenarios. [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Classification performance comparison for velocity [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Circuit diagram for the oscillatory unit of the veloc [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. SVM readout performance analysis for velocity-coupled network. Temporal evolution of classification accuracy using [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Prediction agreement analysis for velocity-coupled [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Error metric distributions for velocity-coupled network. Comparison of error metrics between analog implementation [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Representative dynamics samples from runs of velocity-coupled network. Left column displays input in both sMNIST [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]

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Reference graph

Works this paper leans on

56 extracted references · 44 canonical work pages

  1. [1]

    L. G. Wright, T. Onodera, M. M. Stein, T. Wang, D. T. Schachter, Z. Hu, and P. L. McMahon, Deep physical neural networks trained with backpropagation, Nature 601, 549 (2022)

    2022

  2. [2]

    occupy significantly smaller volumes within the deci- sion space compared to other digits (Fig. 4 E). Access- ing these regions with small volumes necessitates high 9 numerical precision. This indicates that the digitally trained network exploits floating-point precision capabil- ities for making its predictions. However, the degree of precision of the di...

    work page

  3. [3]

    W. Yu, H. Guo, J. Xiao, and J. Shen, Physical neural networks with self-learning capabilities, Science China Physics, Mechanics & Astronomy 67, 287501 (2024)

    work page 2024

  4. [4]

    Markovi´ c, A

    D. Markovi´ c, A. Mizrahi, D. Querlioz, and J. Grollier, Physics for neuromorphic computing, Nature Reviews Physics 2, 499 (2020)

    2020

  5. [5]

    Dalgaty, F

    T. Dalgaty, F. Moro, Y. Demira˘ g, A. De Pra, G. Indi- veri, E. Vianello, and M. Payvand, Mosaic: in-memory computing and routing for small-world spike-based neu- romorphic systems, Nature Communications 15, 142 (2024)

    work page 2024

  6. [6]

    C. D. Schuman, S. R. Kulkarni, M. Parsa, J. P. Mitchell, P. Date, and B. Kay, Opportunities for neuromorphic computing algorithms and applications, Nature Compu- tational Science 2, 10 (2022)

    2022

  7. [7]

    P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cas- sidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, A million spiking- neuron integrated circuit with a scalable communication network and interface, Science...

    work page 2014

  8. [8]

    Aifer, Z

    M. Aifer, Z. Belateche, S. Bramhavar, K. Y. Cam- sari, P. J. Coles, G. Crooks, D. J. Durian, A. J. Liu, A. Marchenkova, A. J. Martinez, P. L. McMa- hon, F. Sbahi, B. Weiner, and L. G. Wright, Solving the compute crisis with physics-based ASICs (2025), arXiv:2507.10463 [cs]

    arXiv 2025

Show all 56 references
  1. [9]

    Left: L = 2 (dense grid near origin); Right: L = 20 (wide dynamic range)

    require higher precision than analog hardware provides. Left: L = 2 (dense grid near origin); Right: L = 20 (wide dynamic range). F. Error metrics visualization showing analog (green) vs digital (orange) node dynamics in the [5 , 789] pixel time interval. Displayed error metri...

  2. [10]

    Zhong, J

    Y. Zhong, J. Tang, X. Li, X. Liang, Z. Liu, Y. Li, Y. Xi, P. Yao, Z. Hao, B. Gao, H. Qian, and H. Wu, A memristor-based analogue reservoir computing system for real-time and power-efficient signal processing, Na- ture Electronics , 672 (2022)

    2022

  3. [11]

    Vandoorne, P

    K. Vandoorne, P. Mechet, T. Van Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, Experimental demonstration of reser- voir computing on a silicon photonics chip, Nature Com- munications 5, 3541 (2014)

    2014

  4. [12]

    Abreu, I

    S. Abreu, I. Boikov, M. Goldmann, T. Jonuzi, A. Lupo, S. Masaad, L. Nguyen, E. Picco, G. Pourcel, A. Skalli, L. Talandier, B. Vettelschoss, E. Vlieg, A. Argyris, P. Bienstman, D. Brunner, J. Dambre, L. Daudet, J. Domenech, I. Fischer, F. Horst, S. Massar, C. Mirasso, B. Offrei...

    2024

  5. [13]

    Wetzstein, A

    G. Wetzstein, A. Ozcan, S. Gigan, S. Fan, D. Englund, M. Soljaˇ ci´ c, C. Denz, D. A. B. Miller, and D. Psaltis, Inference in artificial intelligence with deep optics and photonics, Nature 588, 39 (2020)

    2020

  6. [14]

    Romera, P

    M. Romera, P. Talatchian, S. Tsunegi, F. Abreu Araujo, V. Cros, P. Bortolotti, J. Trastoy, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa, M. Ernoult, D. Vo- denicarevic, T. Hirtzlin, N. Locatelli, D. Querlioz, and J. Grollier, Vowel recognition with four coupled spin- torque...

    2018

  7. [15]

    Y. Chen, M. Nazhamaiti, H. Xu, Y. Meng, T. Zhou, G. Li, J. Fan, Q. Wei, J. Wu, F. Qiao, L. Fang, and Q. Dai, All-analog photoelectronic chip for high-speed vision tasks, Nature 623, 48 (2023)

    2023

  8. [16]

    R. H. Lee, E. A. B. Mulder, and J. B. Hopkins, Mechan- ical neural networks: Architected materials that learn behaviors, Science Robotics 7, eabq7278 (2022)

    2022

  9. [17]

    Momeni, B

    A. Momeni, B. Rahmani, M. Mall´ ejac, P. del Hougne, and R. Fleury, Backpropagation-free training of deep physical neural networks, Science 382, 1297 (2023)

    2023

  10. [18]

    Stern, D

    M. Stern, D. Hexner, J. W. Rocks, and A. J. Liu, Su- pervised learning in physical networks: From machine learning to learning machines, Physical Review X 11, 10.1103/physrevx.11.021045 (2021)

    2021 doi

  11. [19]

    Momeni, B

    A. Momeni, B. Rahmani, B. Scellier, L. G. Wright, P. L. McMahon, C. C. Wanjura, Y. Li, A. Skalli, N. G. Berloff, T. Onodera, I. Oguz, F. Morichetti, P. del Hougne, M. L. Gallo, A. Sebastian, A. Mirhoseini, C. Zhang, D. Markovi´ c, D. Brunner, C. Moser, S. Gigan, F. Mar- quardt...

    2024 arXiv

  12. [20]

    T. K. Rusch and S. Mishra, Coupled Oscillatory Recur- rent Neural Network (coRNN): An accurate and (gra- dient) stable architecture for learning long time de- pendencies, arXiv:2010.00951 [cs, stat] (2021), arXiv: 2010.00951

    2010 arXiv

  13. [21]

    Effenberger, P

    F. Effenberger, P. Carvalho, I. Dubinin, and W. Singer, 11 The functional role of oscillatory dynamics in neocortical circuits: A computational perspective, Proceedings of the National Academy of Sciences 122, e2412830122 (2025), https://www.pnas.org/doi/pdf/10.1073/pnas.2412830122

    2025 doi

  14. [22]

    Fries, Rhythms for cognition: Communication through coherence, Neuron 88, 220 (2015)

    P. Fries, Rhythms for cognition: Communication through coherence, Neuron 88, 220 (2015)

    2015

  15. [23]

    Rosenblum, A

    M. Rosenblum, A. Pikovsky, J. Kurths, C. Sch¨ afer, and P. Tass, Chapter 9 phase synchronization: From the- ory to data analysis, in Handbook of Biological Physics, Vol. 4, edited by F. Moss and S. Gielen (North-Holland,

  16. [24]

    T. W. Hughes, I. A. D. Williamson, M. Minkov, and S. Fan, Wave physics as an analog recurrent neural net- work, Science Advances 5, eaay6946 (2019)

    2019

  17. [25]

    Buzs´ aki and X.-J

    G. Buzs´ aki and X.-J. Wang, Mechanisms of Gamma Os- cillations, Annual Review of Neuroscience35, 203 (2012)

    2012

  18. [26]

    A. V. Goltsev, M. A. Lopes, K.-E. Lee, and J. F. F. Mendes, Critical and resonance phenomena in neural networks, arXiv:1211.5686 [cond-mat, physics:physics, q- bio] , 28 (2013), arXiv: 1211.5686

    2013 arXiv

  19. [27]

    Dubinin and F

    I. Dubinin and F. Effenberger, Fading memory as induc- tive bias in residual recurrent networks, Neural Networks 173, 106179 (2024)

    2024

  20. [28]

    Ashwin and C

    P. Ashwin and C. Postlethwaite, Excitable networks for finite state computation with continuous time recurrent neural networks, Biological Cybernetics 115, 519 (2021)

    2021

  21. [29]

    Nikoli´ c, S

    D. Nikoli´ c, S. H¨ ausler, W. Singer, and W. Maass, Distributed Fading Memory for Stimulus Properties in the Primary Visual Cortex, PLoS Biology 7, e1000260 (2009)

    2009

  22. [30]

    K. B. Doelling and D. Poeppel, Cortical entrain- ment to music and its modulation by expertise, Pro- ceedings of the National Academy of Sciences 112, 10.1073/pnas.1508431112 (2015)

    2015 doi

  23. [31]

    Schmidt, J

    R. Schmidt, J. Rose, and V. Muralidharan, Transient os- cillations as computations for cognition: Analysis, mod- eling and function, Current Opinion in Neurobiology 83, 102796 (2023)

    2023

  24. [32]

    Palmigiano, T

    A. Palmigiano, T. Geisel, F. Wolf, and D. Battaglia, Flex- ible information routing by transient synchrony, Nature Neuroscience 20, 1014 (2017)

    2017

  25. [33]

    R. F. Helfrich, I. C. Fiebelkorn, S. M. Szczepanski, J. J. Lin, J. Parvizi, R. T. Knight, and S. Kastner, Neural mechanisms of sustained attention are rhythmic, Neuron 99, 854 (2018)

    2018

  26. [34]

    Csaba and W

    G. Csaba and W. Porod, Coupled oscillators for comput- ing: A review and perspective, Applied Physics Reviews 7, 011302 (2020)

    2020

  27. [35]

    Ulmann, anabrid Model-1 Ana- log Computer User Manual , http://analogparadigm.com/downloads/handbook.pdf

    B. Ulmann, anabrid Model-1 Ana- log Computer User Manual , http://analogparadigm.com/downloads/handbook.pdf

  28. [36]

    H. Chen, J. Yang, J. Chen, S. Wang, S. Wang, D. Wang, X. Tian, Y. Yu, X. Chen, Y. Lin, Q. Zhu, Y. He, X. Wu, Y. Li, X. Zhang, N. Lin, M. Xu, Y. Li, X. Zhang, X. Qi, Z. Wang, H. Wang, D. Shang, Q. Liu, K.-T. Cheng, and M. Liu, Continuous-time digital twin with analog memristive...

    2025 doi

  29. [37]

    LeCun, C

    Y. LeCun, C. Cortes, and C. Burges, MNIST hand- written digit database, ATT Labs [Online]. Available: http://yann.lecun.com/exdb/mnist 2 (2010)

    2010

  30. [38]

    Tanaka, T

    G. Tanaka, T. Yamane, J. B. H´ eroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, D. Nakano, and A. Hirose, Recent advances in physical reservoir comput- ing: A review, Neural Networks 115, 100 (2019)

    2019

  31. [39]

    C. Du, F. Cai, M. A. Zidan, W. Ma, S. H. Lee, and W. D. Lu, Reservoir computing using dynamic memris- tors for temporal information processing, Nature Com- munications 8, 10.1038/s41467-017-02337-y (2017)

    2017 doi

  32. [40]

    F. K. Kneub¨ uhl,Oscillations and Waves(Springer Berlin Heidelberg, Berlin, Heidelberg, 1997)

    1997

  33. [41]

    Hairer, C

    E. Hairer, C. Lubich, and G. Wanner, Geometric numeri- cal integration illustrated by the St¨ ormer–Verlet method, Acta Numerica 12, 399 (2003)

    2003

  34. [42]

    Baronig, R

    M. Baronig, R. Ferrand, S. Sabathiel, and R. Legenstein, Advancing spatio-temporal processing in spiking neural networks through adaptation (2024), arXiv:2408.07517 [cs.NE]

    2024 arXiv

  35. [43]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Rai- son, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, PyTorch: An Imperative Style, High-P...

  36. [44]

    ADALM2000 Website, https://www.analog.com/ en/resources/evaluation-hardware-and-software/ evaluation-boards-kits/adalm2000.html

  37. [45]

    Ulmann and S

    B. Ulmann and S. K¨ oppel, TeensyLogger: Open-source analog data logger based on Teensy, https://github. com/anabrid/TeensyLogger (2025)

    2025

  38. [46]

    Ulmann, Analog Computing(De Gruyter Oldenbourg, Berlin, Boston, 2022)

    B. Ulmann, Analog Computing(De Gruyter Oldenbourg, Berlin, Boston, 2022)

    2022

  39. [47]

    Ulmann, Analog and Hybrid Computer Programming (De Gruyter Oldenbourg, Berlin, Boston, 2020)

    B. Ulmann, Analog and Hybrid Computer Programming (De Gruyter Oldenbourg, Berlin, Boston, 2020)

    2020

  40. [48]

    Ohkubo and M

    A. Ohkubo and M. Inubushi, Reservoir computing with generalized readout based on generalized synchroniza- tion, Scientific Reports 14, 30918 (2024)

    2024

  41. [49]

    Satyanarayanan, The emergence of edge computing, Computer 50, 30 (2017)

    M. Satyanarayanan, The emergence of edge computing, Computer 50, 30 (2017)

    2017

  42. [50]

    D. J. Amit, Modeling Brain Function: The World of At- tractor Neural Networks (Cambridge University Press, 1989)

    1989

  43. [51]

    Indiveri and S.-C

    G. Indiveri and S.-C. Liu, Memory and information pro- cessing in neuromorphic systems, Proceedings of the IEEE 103, 1379 (2015)

    2015

  44. [52]

    J. D. Kendall, R. D. Pantone, K. Manickavasagam, Y. Bengio, and B. Scellier, Training end-to-end analog neural networks with equilibrium propagation, CoRR abs/2006.01981 (2020), 2006.01981

    2006 arXiv

  45. [53]

    Kudithipudi, C

    D. Kudithipudi, C. Schuman, C. M. Vineyard, T. Pandit, C. Merkel, R. Kubendran, J. B. Aimone, G. Orchard, C. Mayr, R. Benosman, J. Hays, C. Young, C. Bartolozzi, A. Majumdar, S. G. Cardwell, M. Payvand, S. Buckley, S. Kulkarni, H. A. Gonzalez, G. Cauwenberghs, C. S. Thakur, A....

    2025

  46. [54]

    Baronig, R

    M. Baronig, R. Ferrand, S. Sabathiel, and R. Legenstein, Advancing Spatio-Temporal Processing in Spiking Neu- ral Networks through Adaptation (2024)

    2024

  47. [55]

    Higuchi, S

    S. Higuchi, S. Kairat, S. M. Boht´ e, and S. Otte, Balanced Resonate-and-Fire Neurons, in Proc. ICML 2024(2024)

    2024

  48. [56]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, 12 R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cour- napeau, M. Brucher, M. Perrot, and E. Duchesnay, Scikit-learn: Machine learning in Python, Journal of Ma- chine L...

    2011

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.