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arxiv: 2606.24588 · v1 · pith:72SNFFOSnew · submitted 2026-06-23 · 💻 cs.NE

Spatial Partial Functionalization of Neural Networks based on Noise Fields

Shuhei Ikemoto , Fabio DallaLibera This is my paper

Pith reviewed 2026-06-25 21:39 UTC · model grok-4.3

classification 💻 cs.NE
keywords noise fieldsneural networkscrossing activation functionpartial functionalizationmemory capacityspatial structurefunction approximation
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The pith

Spatially structured noise can assign different functions to overlapping subnetworks within one neural network.

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

The paper shows that noise, usually seen as interference, can instead define which parts of a network compute each function by varying its spatial pattern across the network. A crossing activation function combined with virtual noise fields lets the same weights support multiple independent tasks when each task is paired with its own noise location. In simple one-dimensional approximation experiments, the network holds more functions without interference when the noise locations are arranged to match how similar the functions are to each other. When the noise arrangement clashes with those similarities, effective capacity drops. This turns noise into a mechanism that selects functional subnetworks rather than merely adding randomness.

Core claim

By modulating a crossing activation function with virtual noise fields—an auxiliary continuous space that produces spatially varying noise—the network can store multiple functions such that each function is learned primarily by the subnetwork activated at its assigned noise location; memory capacity rises when the geometry of the noise fields mirrors the proximity relations among the target functions and falls when the geometries are mismatched.

What carries the argument

The virtual noise field, an auxiliary continuous space used to generate spatially structured noise that activates partially overlapping subnetworks via the crossing activation function.

If this is right

  • Multiple functions can be stored in a single set of weights by pairing each function with a distinct noise-field location.
  • Capacity increases when noise-field geometry reflects proximity among the functions being learned.
  • Mismatched noise-field geometry reduces the number of functions that can be stored independently.
  • Structured noise can act as a topology selector that chooses which subnetwork computes each function.

Where Pith is reading between the lines

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

  • The same principle might allow a network to switch among learned behaviors simply by changing the applied noise pattern at test time.
  • If extended to image or sequence data, noise fields could be shaped by the data’s own spatial or temporal structure rather than chosen by hand.
  • The approach offers a continuous alternative to discrete routing methods such as mixture-of-experts without requiring an explicit router network.

Load-bearing premise

The crossing activation function, in its sample, statistical, or analytical form, lets different noise fields drive largely independent learning of multiple functions with little cross-talk.

What would settle it

Run the one-dimensional approximation tasks with noise-field locations that match functional proximity yet observe no gain in the number of functions that can be stored without interference, or observe that mismatched locations produce no drop in capacity.

Figures

Figures reproduced from arXiv: 2606.24588 by Fabio DallaLibera, Shuhei Ikemoto.

Figure 1
Figure 1. Figure 1: Overview of Noise-modulated Neural Network. Left: the crossing activation [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Examples of different noise fields generated via the virtual noise field. Truncated [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Inference results (solid lines) compared against the target analytical functions [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Interpolation of noise fields by moving the center of the activation gaussian. [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Intermediate noise patterns generated during inference by linearly interpolating [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Network inference results observed during the continuous interpolation of the [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reconstruction loss for a unidimensional function space with phase variations [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reconstruction loss for a unidimensional function space with frequency variations [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Reconstruction loss for a bidimensional function space with both phase and [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The ratio of grid dimensions, m (2) v /m(1) v , plotted as a function of the total number of target functions Nf . Periodic spikes represent highly disproportionate grid aspect ratios caused by integer factorization constraints during grid construction. These extreme imbalances effectively collapse the dimensionality of the function space. these loss spikes are highly deterministic. Therefore, we believe … view at source ↗
read the original abstract

Noise in neural computation is typically regarded as a disturbance, but its spatial distribution may also actively regulate which parts of a network participate in computation. This paper investigates the spatial partial functionalization of Noise-modulated Neural Networks using noise fields. We first present an activation function suitable for this goal, the crossing activation function, using the sample-level, statistical-level, and analytical-level implementations, and examine parameter reuse across these implementations. We then introduce a virtual noise field, an auxiliary continuous space for generating spatially structured network noise fields that activate partially overlapping subnetworks. Using one-dimensional function approximation tasks, we evaluate how multiple functions can be stored in a single network when each function is assigned to a different noise-field location. The results show that memory capacity improves when the spatial arrangement of noise fields reflects the proximity relationships among the functions to be learned, whereas mismatches in noise field structure can reduce effective capacity. These findings suggest that structured noise can serve not only as a perturbation but also as a topology-defining factor for functional subnetwork selection.

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

2 major / 1 minor

Summary. The paper claims that a crossing activation function (implemented at sample-level, statistical-level, and analytical-level) combined with virtual noise fields enables spatial partial functionalization of neural networks. In 1D function approximation tasks, assigning functions to noise-field locations whose spatial arrangement matches the proximity relationships among the target functions improves memory capacity, while mismatches reduce effective capacity; structured noise thus acts as a topology-defining factor for subnetwork selection.

Significance. If the central empirical claim holds after verification, the work introduces two new concepts (crossing activation function and virtual noise field) and supplies initial evidence that noise can be harnessed to regulate functional subnetwork participation without explicit architectural partitioning. This perspective on noise as an active selector rather than pure perturbation could inform multi-task and continual-learning designs. The manuscript already supplies three distinct implementations of the crossing function and reports parameter-reuse comparisons, which are concrete strengths.

major comments (2)
  1. [1D function approximation tasks evaluation] The strongest claim—that capacity scales with spatial-proximity matching of noise fields—requires that the crossing activation function (across its three implementations) produces sufficiently independent subnetwork activation with negligible cross-interference. The 1D function-approximation evaluation provides no explicit quantification of cross-interference (e.g., performance drop when noise fields overlap versus when separated) and no ablation that removes the crossing mechanism, so the observed capacity difference could arise from task-specific regularization rather than the intended spatial partial functionalization.
  2. [Abstract and experimental description] The abstract and evaluation report positive results on capacity improvement yet supply no details on controls, baselines, error bars, or exact experimental conditions. This absence directly limits verification of the central empirical claim.
minor comments (1)
  1. [Abstract] The abstract states that parameter reuse across the sample-level, statistical-level, and analytical-level implementations is examined; the main text should include the concrete reuse metrics or tables that support this examination.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight areas where additional evidence and clarity will improve the manuscript. We address each major comment below and commit to revisions that directly target the concerns raised.

read point-by-point responses

  1. Referee: [1D function approximation tasks evaluation] The strongest claim—that capacity scales with spatial-proximity matching of noise fields—requires that the crossing activation function (across its three implementations) produces sufficiently independent subnetwork activation with negligible cross-interference. The 1D function-approximation evaluation provides no explicit quantification of cross-interference (e.g., performance drop when noise fields overlap versus when separated) and no ablation that removes the crossing mechanism, so the observed capacity difference could arise from task-specific regularization rather than the intended spatial partial functionalization.

    Authors: We agree that explicit quantification of cross-interference and an ablation of the crossing mechanism are needed to isolate the contribution of spatial partial functionalization. In the revision we will add (i) direct comparisons of performance for overlapping versus spatially separated noise fields and (ii) an ablation replacing the crossing activation with a standard activation while keeping all other elements fixed. These additions will test whether the observed capacity differences are attributable to the intended mechanism rather than incidental regularization. revision: yes

  2. Referee: [Abstract and experimental description] The abstract and evaluation report positive results on capacity improvement yet supply no details on controls, baselines, error bars, or exact experimental conditions. This absence directly limits verification of the central empirical claim.

    Authors: We accept that the current abstract and experimental section lack sufficient detail for independent verification. We will revise the abstract to summarize the evaluation protocol and expand the experimental section to report all controls, baselines (including standard multi-task and single-network baselines), error bars from repeated runs with different seeds, network architectures, training hyperparameters, and precise definitions of the three crossing-function implementations and virtual noise-field generation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical claims rest on independent experiments

full rationale

The paper defines a new crossing activation function (with sample-, statistical-, and analytical-level implementations) and virtual noise fields as auxiliary constructs, then reports empirical memory-capacity results from 1D function-approximation tasks. No load-bearing step equates a prediction to a fitted input, renames a known result, or reduces the central claim to a self-citation chain; the reported capacity differences are presented as experimental outcomes rather than algebraic identities or self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 2 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities beyond the newly introduced crossing activation function and virtual noise field are detailed. These two entities are postulated to enable the claimed functionality.

invented entities (2)
  • crossing activation function no independent evidence
    purpose: Enable partial network activation modulated by noise
    Introduced as suitable for spatial partial functionalization, with three implementation levels mentioned.
  • virtual noise field no independent evidence
    purpose: Generate spatially structured noise to activate partially overlapping subnetworks
    Auxiliary continuous space proposed to assign functions to different noise locations.

pith-pipeline@v0.9.1-grok · 5703 in / 1226 out tokens · 20580 ms · 2026-06-25T21:39:23.757637+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 36 canonical work pages

  1. [1]

    M. E. Raichle, The brain’s dark energy, Science 314 (5803) (2006) 1249–

    2006

  2. [2]

    URLhttps://www.science.org/doi/abs/10.1126/science.1134405

    arXiv:https://www.science.org/doi/pdf/10.1126/science.1134405, 31 doi:10.1126/science.1134405. URLhttps://www.science.org/doi/abs/10.1126/science.1134405

    work page doi:10.1126/science.1134405

  3. [3]

    N. K. Logothetis, What we can do and what we cannot do with fmri (6 2008). doi:10.1038/nature06976

    work page doi:10.1038/nature06976 2008

  4. [4]

    L. C. Katz, C. J. Shatz, Synaptic activity and the construc- tion of cortical circuits, Science 274 (5290) (1996) 1133–1138. doi:10.1126/science.274.5290.1133. URLhttps://www.science.org/doi/abs/10.1126/science.274.5290.1133

    work page doi:10.1126/science.274.5290.1133 1996

  5. [5]

    Architectures of neuronal circuits

    L. Luo, Architectures of neuronal circuits, Science 373 (6559) (2021) eabg7285. doi:10.1126/science.abg7285. URLhttps://www.science.org/doi/abs/10.1126/science.abg7285

    work page doi:10.1126/science.abg7285 2021

  6. [6]

    C. I. Bargmann, Beyond the connectome: How neuromodula- tors shape neural circuits, BioEssays 34 (6) (2012) 458–465. doi:https://doi.org/10.1002/bies.201100185. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1002/bies.201100185

    work page doi:10.1002/bies.201100185 2012

  7. [7]

    Marder, Neuromodulation of neuronal circuits: Back to the future, Neuron 76 (1) (2012) 1–11

    E. Marder, Neuromodulation of neuronal circuits: Back to the future, Neuron 76 (1) (2012) 1–11. doi:https://doi.org/10.1016/j.neuron.2012.09.010. URLhttps://www.sciencedirect.com/science/article/pii/S0896627312008173

    work page doi:10.1016/j.neuron.2012.09.010 2012

  8. [8]

    D. V. Buonomano, W. Maass, State-dependent computa- tions: Spatiotemporal processing in cortical networks (2 2009). doi:10.1038/nrn2558

    work page doi:10.1038/nrn2558 2009

  9. [9]

    A. S. Morcos, C. D. Harvey, History-dependent variability in population dynamics during evidence accumulation in cortex, Nature Neuroscience 19 (2016) 1672–1681. doi:10.1038/nn.4403

    work page doi:10.1038/nn.4403 2016

  10. [10]

    Destexhe, M

    A. Destexhe, M. Rudolph-Lilith, Neuronal Noise, 1st Edition, Springer Series in Computational Neuroscience, Springer US, 2012. doi:10.1007/978-0-387-79020-6

    work page doi:10.1007/978-0-387-79020-6 2012

  11. [11]

    R. M. Birn, The role of physiological noise in resting-state functional connectivity, NeuroImage 62 (2) (2012) 864–870. doi:https://doi.org/10.1016/j.neuroimage.2012.01.016. 32

    work page doi:10.1016/j.neuroimage.2012.01.016 2012

  12. [12]

    A. A. Faisal, L. P. J. Selen, D. M. Wolpert, Noise in the ner- vous system, Nature Reviews Neuroscience 9 (4) (2008) 292–303. doi:10.1038/nrn2258

    work page doi:10.1038/nrn2258 2008

  13. [13]

    A. R. McIntosh, N. Kovacevic, R. J. Itier, Increased brain signal variability accompanies lower behavioral variability in de- velopment, PLOS Computational Biology 4 (7) (2008) 1–9. doi:10.1371/journal.pcbi.1000106. URLhttps://doi.org/10.1371/journal.pcbi.1000106

    work page doi:10.1371/journal.pcbi.1000106 2008

  14. [14]

    D. D. Garrett, N. Kovacevic, A. R. McIntosh, C. L. Grady, The modulation of bold variability between cognitive states varies by age and processing speed, Cerebral Cortex 23 (3) (2012) 684–693. doi:10.1093/cercor/bhs055. URLhttps://doi.org/10.1093/cercor/bhs055

    work page doi:10.1093/cercor/bhs055 2012

  15. [15]

    J. S. Nomi, T. S. Bolt, C. C. Ezie, L. Q. Uddin, A. S. Heller, Moment-to- moment bold signal variability reflects regional changes in neural flexi- bility across the lifespan, Journal of Neuroscience 37 (22) (2017) 5539–

    2017

  16. [16]

    URLhttps://www.jneurosci.org/content/37/22/5539

    arXiv:https://www.jneurosci.org/content/37/22/5539.full.pdf, doi:10.1523/JNEUROSCI.3408-16.2017. URLhttps://www.jneurosci.org/content/37/22/5539

    work page doi:10.1523/jneurosci.3408-16.2017 2017

  17. [17]

    Gammaitoni, P

    L. Gammaitoni, P. H¨ anggi, P. Jung, F. Marchesoni, Stochas- tic resonance, Reviews of Modern Physics 70 (1998) 223–287. doi:10.1103/RevModPhys.70.223

    work page doi:10.1103/revmodphys.70.223 1998

  18. [18]

    Wiesenfeld, F

    K. Wiesenfeld, F. Moss, Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs, Nature 373 (6509) (1995) 33–36

    1995

  19. [19]

    Benzi, A

    R. Benzi, A. Sutera, A. Vulpiani, The mechanism of stochastic reso- nance, Journal of Physics A: Mathematical and general 14 (1981) 453– 457

    1981

  20. [20]

    Benzi, G

    R. Benzi, G. Parisi, A. Sutera, A. Vulpiani, Stochastic resonance in climatic change, Tellus 34 (1) (1982) 10–16. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.2153- 3490.1982.tb01787.x, doi:https://doi.org/10.1111/j.2153- 3490.1982.tb01787.x. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1111/j.2153-3490.1982.tb01787.x 33

    work page doi:10.1111/j.2153- 1982

  21. [21]

    Fauve, F

    S. Fauve, F. Heslot, Stochastic resonance in a bistable system, Physics Letters A 97 (1983) 5–7

    1983

  22. [22]

    McNamara, K

    B. McNamara, K. Wiesenfeld, R. Roy, Observation of stochastic resonance in a ring laser, Phys. Rev. Lett. 60 (1988) 2626–2629. doi:10.1103/PhysRevLett.60.2626. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.60.2626

    work page doi:10.1103/physrevlett.60.2626 1988

  23. [23]

    P. S. Landa, P. V. E. McClintock, Vibrational resonance, Jour- nal of Physics A: Mathematical and General 33 (45) (2000) L433. doi:10.1088/0305-4470/33/45/103. URLhttps://doi.org/10.1088/0305-4470/33/45/103

    work page doi:10.1088/0305-4470/33/45/103 2000

  24. [24]

    Guderian, G

    A. Guderian, G. Dechert, K.-P. Zeyer, F. W. Schneider, Stochas- tic resonance in chemistry. 1. the belousov-zhabotinsky reaction, The Journal of Physical Chemistry 100 (11) (1996) 4437–4441. arXiv:https://doi.org/10.1021/jp952243x, doi:10.1021/jp952243x. URLhttps://doi.org/10.1021/jp952243x

    work page doi:10.1021/jp952243x 1996

  25. [25]

    M. D. McDonnell, L. M. Ward, The benefits of noise in neural systems: bridging theory and experiment, Nature Reviews Neuroscience 12 (7) (2011) 415–425. doi:10.1038/nrn3061

    work page doi:10.1038/nrn3061 2011

  26. [26]

    F. Moss, L. M. Ward, W. G. Sannita, Stochastic resonance and sensory information processing: a tutorial and review of application, Clinical Neurophysiology 115 (2) (2004) 267–281. doi:https://doi.org/10.1016/j.clinph.2003.09.014. URLhttps://www.sciencedirect.com/science/article/pii/S1388245703003304

    work page doi:10.1016/j.clinph.2003.09.014 2004

  27. [27]

    Douglass, L

    J. Douglass, L. Wilkens, E. Pantazelou, F. Moss, Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic reso- nance, Nature 365 (6444) (1993) 337–340

    1993

  28. [28]

    Levin, J

    J. Levin, J. Miller, Broadband neural encoding in the cricket cercal sensory system enhanced by stochastic resonance, Nature 380 (1996) 165–168. doi:10.1038/380165a0

    work page doi:10.1038/380165a0 1996

  29. [29]

    M. D. McDonnell, D. Abbott, What is stochastic resonance? definitions, misconceptions, debates, and its relevance to biology, PLoS Comput Biol 5 (5) (2009) e1000348. 34

    2009

  30. [30]

    Priplata, J

    A. Priplata, J. Niemi, M. Salen, J. Harry, L. A. Lipsitz, J. J. Collins, Noise-enhanced human balance control, Phys. Rev. Lett. 89 (2002) 238101. doi:10.1103/PhysRevLett.89.238101. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.89.238101

    work page doi:10.1103/physrevlett.89.238101 2002

  31. [31]

    Mart´ ınez, T

    L. Mart´ ınez, T. P´ erez, C. R. Mirasso, E. Manjarrez, Stochastic res- onance in the motor system: Effects of noise on the monosynap- tic reflex pathway of the cat spinal cord, Journal of Neurophysiology 97 (6) (2007) 4007–4016. arXiv:https://doi.org/10.1152/jn.01164.2006, doi:10.1152/jn.01164.2006. URLhttps://doi.org/10.1152/jn.01164.2006

    work page doi:10.1152/jn.01164.2006 2007

  32. [32]

    P. Jung, P. H¨ anggi, Amplification of small signals via stochastic resonance, Phys. Rev. A 44 (1991) 8032–8042. doi:10.1103/PhysRevA.44.8032. URLhttps://link.aps.org/doi/10.1103/PhysRevA.44.8032

    work page doi:10.1103/physreva.44.8032 1991

  33. [33]

    McNamara, K

    B. McNamara, K. Wiesenfeld, Theory of stochastic resonance, Phys. Rev. A 39 (1989) 4854–4869. doi:10.1103/PhysRevA.39.4854. URLhttps://link.aps.org/doi/10.1103/PhysRevA.39.4854

    work page doi:10.1103/physreva.39.4854 1989

  34. [34]

    Zozor, P.-O

    S. Zozor, P.-O. Amblard, Stochastic resonance in discrete time nonlinear ar(1) models, IEEE Transactions on Signal Processing 47 (1) (1999) 108–

    1999

  35. [35]

    doi:10.1109/78.738244

    work page doi:10.1109/78.738244

  36. [36]

    Maass, Networks of spiking neurons: The third generation of neural network models, Neural Networks 10 (9) (1997) 1659–1671

    W. Maass, Networks of spiking neurons: The third generation of neural network models, Neural Networks 10 (9) (1997) 1659–1671. doi:https://doi.org/10.1016/S0893-6080(97)00011-7. URLhttps://www.sciencedirect.com/science/article/pii/S0893608097000117

    work page doi:10.1016/s0893-6080(97)00011-7 1997

  37. [37]

    Ponulak, A

    F. Ponulak, A. Kasi´ nski, Introduction to spiking neural networks: In- formation processing, learning and applications, Acta neurobiologiae ex- perimentalis 71 (2011) 409–33. doi:10.55782/ane-2011-1862

    work page doi:10.55782/ane-2011-1862 2011

  38. [38]

    Deep learning in spiking neural networks , volume=

    A. Tavanaei, M. Ghodrati, S. R. Kheradpisheh, T. Masquelier, A. Maida, Deep learning in spiking neural networks, Neural Networks 111 (2019) 47–63. doi:https://doi.org/10.1016/j.neunet.2018.12.002. URLhttps://www.sciencedirect.com/science/article/pii/S0893608018303332 35

    work page doi:10.1016/j.neunet.2018.12.002 2019

  39. [39]

    M. Ozer, M. Perc, M. Uzuntarla, Stochastic resonance on new- man–watts networks of hodgkin–huxley neurons with local pe- riodic driving, Physics Letters A 373 (10) (2009) 964–968. doi:https://doi.org/10.1016/j.physleta.2009.01.034. URLhttp://www.sciencedirect.com/science/article/pii/S0375960109000905

    work page doi:10.1016/j.physleta.2009.01.034 2009

  40. [40]

    M. Ozer, M. Perc, M. Uzuntarla, E. Koklukaya, Weak signal propagation through noisy feedforward neuronal networks, Neuroreport 21 (5) (2010) 338–343. doi:10.1097/wnr.0b013e328336ee62. URLhttp://europepmc.org/abstract/MED/20186108

    work page doi:10.1097/wnr.0b013e328336ee62 2010

  41. [41]

    Ikemoto, F

    S. Ikemoto, F. DallaLibera, K. Hosoda, Noise-modulated neural net- works as an application of stochastic resonance, Neurocomputing 277 (2018) 29 – 37. doi:https://doi.org/10.1016/j.neucom.2016.12.111

    work page doi:10.1016/j.neucom.2016.12.111 2018

  42. [42]

    Ikemoto, Noise-modulated neural networks for selectively functional- izing sub-networks by exploiting stochastic resonance, Neurocomputing 448 (2021) 1–9

    S. Ikemoto, Noise-modulated neural networks for selectively functional- izing sub-networks by exploiting stochastic resonance, Neurocomputing 448 (2021) 1–9. doi:https://doi.org/10.1016/j.neucom.2020.05.125. URLhttps://www.sciencedirect.com/science/article/pii/S0925231221004379

    work page doi:10.1016/j.neucom.2020.05.125 2021

  43. [43]

    Bengio, N

    Y. Bengio, N. L´ eonard, A. Courville, Estimating or propagating gra- dients through stochastic neurons for conditional computation (2013). arXiv:1308.3432. URLhttps://arxiv.org/abs/1308.3432

    Pith/arXiv arXiv 2013

  44. [44]

    Courbariaux, I

    M. Courbariaux, I. Hubara, D. Soudry, R. El-Yaniv, Y. Bengio, Bina- rized neural networks: Training deep neural networks with weights and activations constrained to +1 or -1 (2016). arXiv:1602.02830. URLhttps://arxiv.org/abs/1602.02830

    Pith/arXiv arXiv 2016

  45. [45]

    J. H. Lee, T. Delbruck, M. Pfeiffer, Training deep spiking neural networks using backpropagation, Frontiers in Neuroscience Volume 10 - 2016. doi:10.3389/fnins.2016.00508. URLhttps://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2016.00508

    work page doi:10.3389/fnins.2016.00508 2016

  46. [46]

    E. O. Neftci, H. Mostafa, F. Zenke, Surrogate gradient learning in spik- ing neural networks: Bringing the power of gradient-based optimization to spiking neural networks, IEEE Signal Processing Magazine 36 (6) (2019) 51–63. doi:10.1109/MSP.2019.2931595. 36

    work page doi:10.1109/msp.2019.2931595 2019

  47. [47]

    Rybicki, Y

    L. Rybicki, Y. Sugita, J. Tani, Reinforcement learning of multiple tasks using parametric bias, in: 2009 International Joint Conference on Neural Networks, IEEE, 2009, pp. 2732–2739. 37

    2009