pith. sign in

arxiv: 2512.09502 · v2 · pith:VZYGZRRAnew · submitted 2025-12-10 · 💻 cs.DC · cs.NE· physics.comp-ph· q-bio.NC

Scalable Construction of Spiking Neural Networks using up to thousands of GPUs

Pith reviewed 2026-05-21 17:55 UTC · model grok-4.3

classification 💻 cs.DC cs.NEphysics.comp-phq-bio.NC
keywords spiking neural networksmulti-GPU clustersMPInetwork constructionscalable simulationcortical modelsspike exchangeexascale systems
0
0 comments X

The pith

Each MPI process builds only its local part of a spiking neural network to support efficient spike exchange on large GPU clusters.

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

The work describes a construction approach for spiking neural networks on clusters with many GPUs that uses the Message Passing Interface. In this method, every process is responsible for constructing the connections that are local to the neurons it manages and for setting up the data needed to exchange spikes with processes on other GPUs. This is shown to scale well for two different models of the cerebral cortex, one relying on direct communication between pairs of processes and the other on collective operations. The goal is to make it possible to run simulations of networks the size of the human brain on future supercomputers.

Core claim

A novel method for building spiking neural networks on multi-GPU systems allows each process to construct its local connectivity and prepare data structures for efficient spike exchange during simulation, achieving good scaling performance on two cortical models with point-to-point and collective communication.

What carries the argument

The per-process local connectivity construction and preparation of spike exchange data structures using MPI.

If this is right

  • Large cortical models with billions of neurons can be simulated without a central construction bottleneck.
  • Both point-to-point and collective MPI communication patterns support efficient scaling.
  • Memory and communication overheads are managed locally per process for sparse networks.
  • The technique is suitable for exascale systems with thousands of GPUs.

Where Pith is reading between the lines

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

  • Local construction may allow simulations to start faster by avoiding a global network assembly step.
  • This approach could extend to other types of large-scale network simulations in physics or biology.
  • Load balancing during construction might need additional techniques for models with uneven connectivity.

Load-bearing premise

That constructing connectivity locally per process will result in communication patterns that stay efficient at large scales without causing bottlenecks or uneven workloads in realistic brain models.

What would settle it

If tests on larger GPU counts or more detailed cortical models show that spike exchange time dominates and scaling efficiency falls below linear, the scalability claim would be falsified.

read the original abstract

Diverse scientific and engineering research areas deal with discrete, time-stamped changes in large systems of interacting delay differential equations. Simulating such complex systems at scale on high-performance computing clusters demands efficient management of communication and memory. Inspired by the human cerebral cortex -- a sparsely connected network of $\mathcal{O}(10^{10})$ neurons, each forming $\mathcal{O}(10^{3})$--$\mathcal{O}(10^{4})$ synapses and communicating via short electrical pulses called spikes -- we study the simulation of large-scale spiking neural networks for computational neuroscience research. This work presents a novel network construction method for multi-GPU clusters and upcoming exascale supercomputers using the Message Passing Interface (MPI), where each process builds its local connectivity and prepares the data structures for efficient spike exchange across the cluster during state propagation. We demonstrate scaling performance of two cortical models using point-to-point and collective communication, respectively.

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 / 2 minor

Summary. The paper presents a novel MPI-based network construction method for large-scale spiking neural networks on multi-GPU clusters and exascale systems. Each process independently builds its local connectivity and prepares data structures for spike exchange during simulation. Scaling performance is demonstrated for two cortical models, one using point-to-point communication and the other collective communication.

Significance. If the scaling claims hold with detailed verification, the work could enable efficient simulation of cortical-scale networks (O(10^10) neurons) on thousands of GPUs, addressing key challenges in communication and memory for discrete-event systems in computational neuroscience.

major comments (2)
  1. [Abstract] Abstract: The claim that scaling performance was demonstrated for two cortical models provides no quantitative metrics (e.g., speedup, communication volume, or wall-clock times), error analysis, or description of the performance measurement methodology. This is load-bearing for the central scalability claim up to thousands of GPUs.
  2. [Methods] Network construction description: The method states that each process builds local connectivity independently, but does not specify the partitioning of global adjacency information or whether any collective MPI operations occur during the build phase itself. This detail is required to confirm absence of hidden global coordination costs or load imbalance for structured, distance-dependent cortical topologies.
minor comments (2)
  1. [Results] Figure captions and axis labels in the scaling results could more explicitly state the model sizes, number of processes, and exact communication primitives used to improve reproducibility.
  2. The abstract mentions O(10^10) neurons and O(10^3)--O(10^4) synapses per neuron; a brief comparison table to the two specific cortical models tested would clarify how the demonstration relates to these scales.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment in detail below and have made revisions to improve clarity and completeness.

read point-by-point responses
  1. Referee: [Abstract] Abstract: The claim that scaling performance was demonstrated for two cortical models provides no quantitative metrics (e.g., speedup, communication volume, or wall-clock times), error analysis, or description of the performance measurement methodology. This is load-bearing for the central scalability claim up to thousands of GPUs.

    Authors: We agree that the abstract would be strengthened by including quantitative indicators of the demonstrated scaling. In the revised version we will add specific metrics (e.g., wall-clock times and speedup factors on up to thousands of GPUs for both models) together with a concise reference to the performance-measurement approach used in the experiments. revision: yes

  2. Referee: [Methods] Network construction description: The method states that each process builds local connectivity independently, but does not specify the partitioning of global adjacency information or whether any collective MPI operations occur during the build phase itself. This detail is required to confirm absence of hidden global coordination costs or load imbalance for structured, distance-dependent cortical topologies.

    Authors: We thank the referee for noting this omission. The global network is partitioned by a spatial decomposition that assigns neurons to MPI processes according to their cortical coordinates; each process then generates its local outgoing connections from the distance-dependent probability rules without ever materializing the full global adjacency matrix. No collective MPI operations are invoked during construction—all inter-process communication is confined to the subsequent simulation phase. We will insert an explicit paragraph describing this partitioning strategy and confirming the absence of collectives in the build phase. revision: yes

Circularity Check

0 steps flagged

No circularity: methods paper with independent algorithmic description and scaling results

full rationale

The paper presents a novel MPI-based method for constructing spiking neural networks on multi-GPU clusters, with each process building local connectivity and preparing spike-exchange data structures, followed by empirical scaling demonstrations on two cortical models. No mathematical derivations, equations, fitted parameters, or self-referential claims are indicated in the provided abstract or description. The work is self-contained as a methods and performance report; claims rest on the described construction algorithm and observed scaling behavior rather than reducing to inputs by construction or via load-bearing self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard MPI primitives and existing SNN simulation frameworks; the main added element is the distributed construction procedure whose correctness is assumed to follow from local data preparation.

axioms (1)
  • domain assumption The chosen cortical models are representative and their connectivity patterns can be partitioned without loss of essential dynamics.
    Invoked when claiming that scaling results on these models generalize to the target use case of large-scale brain simulation.

pith-pipeline@v0.9.0 · 5745 in / 1151 out tokens · 40916 ms · 2026-05-21T17:55:08.874118+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

47 extracted references · 47 canonical work pages · 1 internal anchor

  1. [1]

    arXiv, 2505–21185 (2025) https://doi.org/10.48550/arXiv.2505.21185

    Senk, J., Kurth, A.C., Furber, S., Gemmeke, T., Golosio, B., Heittmann, A., Knight, J.C., M¨ uller, E., Noll, T., Nowotny, T., Coppola, G.P., Peres, L., Rhodes, O., Rowley, A., Schemmel, J., Stadtmann, T., Tetzlaff, T., Tiddia, G., Albada, S.J., Villamar, J., Diesmann, M.: Constructive community race: full-density spik- ing neural network model drives neu...

  2. [2]

    Frontiers in Neuroinformatics 17(2023) https://doi.org/10.3389/fninf.2023.1157418

    Aimone, J.B., Awile, O., Diesmann, M., Knight, J.C., Nowotny, T., Sch¨ urmann, F.: Editorial: Neuroscience, computing, performance, and benchmarks: Why it matters to neuroscience how fast we can compute. Frontiers in Neuroinformatics 17(2023) https://doi.org/10.3389/fninf.2023.1157418

  3. [3]

    Journal of large-scale research facilities JLSRF9(1) (2024) https://doi.org/10.17815/jlsrf-8-186

    Turisini, M., Cestari, M., Amati, G.: Leonardo: A pan-european pre-exascale supercomputer for hpc and ai applications. Journal of large-scale research facilities JLSRF9(1) (2024) https://doi.org/10.17815/jlsrf-8-186

  4. [4]

    Technical report, Knoxville, Tennessee (2009)

    Message Passing Interface Forum: MPI: A message-passing interface stan- dard, version 2.2. Technical report, Knoxville, Tennessee (2009). http://www. mpi-forum.org/docs

  5. [5]

    Nature Computational Science4(12), 890–898 (2024) https://doi.org/10.1038/s43588-024-00731-3

    Lu, W., Du, X., Wang, J., Zeng, L., Ye, L., Xiang, S., Zheng, Q., Zhang, J., Xu, N., Feng, J., Bao, Y., Chen, B., Chen, S., Chen, Z., Dai, F., Ding, W., Du, X., Feng, J., Hou, Y., Ji, M., Ji, P., Li, C., Li, C., Li, X., Liu, Y., Lu, W., Lv, Z., Ma, H., Qi, Y., Rolls, E., Wang, H., Wang, H., Wang, S., Wang, Z., Xia, Y., Xie, C., Xue, X., Zeng, T., Zhang, C...

  6. [6]

    IEEE Transactions on Parallel and Distributed Systems35(6), 1056–1073 (2024) https://doi.org/10.1109/TPDS.2024.3387720

    Du, X., Wang, M., Lu, Z., Duan, Q., Liu, Y., Feng, J., Wang, H.: Hrcm: A hierarchical regularizing mechanism for sparse and imbalanced communication in whole human brain simulations. IEEE Transactions on Parallel and Distributed Systems35(6), 1056–1073 (2024) https://doi.org/10.1109/TPDS.2024.3387720

  7. [7]

    Scientific Reports6(1) (2016) https://doi.org/10

    Yavuz, E., Turner, J., Nowotny, T.: GeNN: a code generation framework for accelerated brain simulations. Scientific Reports6(1) (2016) https://doi.org/10. 25 1038/srep18854

  8. [8]

    Nature Computational Science1(2), 136–142 (2021) https: //doi.org/10.1038/s43588-020-00022-7

    Knight, J.C., Nowotny, T.: Larger GPU-accelerated brain simulations with pro- cedural connectivity. Nature Computational Science1(2), 136–142 (2021) https: //doi.org/10.1038/s43588-020-00022-7

  9. [9]

    Proceedings of the International Joint Confer- ence on Neural Networks (IJCNN), 1–8 (2015) https://doi.org/10.1109/IJCNN

    Beyeler, M., Carlson, K.D., Chou, T.S., Dutt, N.D., Krichmar, J.L.: Carlsim 3: A user-friendly and highly optimized library for the creation of neurobiologically detailed spiking neural networks. Proceedings of the International Joint Confer- ence on Neural Networks (IJCNN), 1–8 (2015) https://doi.org/10.1109/IJCNN. 2015.7280694

  10. [10]

    Cordone, B

    Niedermeier, L., Chen, K., Xing, J., Das, A., Kopsick, J., Scott, E., Sutton, N., Weber, K., Dutt, N., Krichmar, J.L.: Carlsim 6: An open source library for large-scale, biologically detailed spiking neural network simulation. In: 2022 International Joint Conference on Neural Networks (IJCNN), pp. 1–10 (2022). https://doi.org/10.1109/IJCNN55064.2022.9892644

  11. [11]

    Cambridge Uni- versity Press, ??? (2006)

    Carnevale, N.T., Hines, M.L.: The NEURON Book. Cambridge Uni- versity Press, ??? (2006). https://doi.org/10.1017/cbo9780511541612 . https://doi.org/10.1017/CBO9780511541612

  12. [12]

    Frontiers in Neuroinformatics13, 63 (2019) https://doi.org/10.3389/ fninf.2019.00063

    Kumbhar, P., Hines, M., Fouriaux, J., Ovcharenko, A., King, J., Delalondre, F., Sch¨ urmann, F.: Coreneuron: An optimized compute engine for the neuron simulator. Frontiers in Neuroinformatics13, 63 (2019) https://doi.org/10.3389/ fninf.2019.00063

  13. [13]

    Frontiers in Neuroinformatics16(2022) https://doi.org/10.3389/fninf.2022.884046

    Awile, O., Kumbhar, P., Cornu, N., Dura-Bernal, S., King, J.G., Lupton, O., Magkanaris, I., McDougal, R.A., Newton, A.J.H., Pereira, F., S˘ avulescu, A., Carnevale, N.T., Lytton, W.W., Hines, M.L., Sch¨ urmann, F.: Modernizing the NEURON simulator for sustainability, portability, and performance. Frontiers in Neuroinformatics16(2022) https://doi.org/10.33...

  14. [14]

    Frontiers in Neuroinformatics15(2022) https://doi.org/10.3389/ fninf.2021.785068

    Pronold, J., Jordan, J., Wylie, B.J.N., Kitayama, I., Diesmann, M., Kunkel, S.: Routing brain traffic through the von neumann bottleneck: Parallel sorting and refactoring. Frontiers in Neuroinformatics15(2022) https://doi.org/10.3389/ fninf.2021.785068

  15. [15]

    Frontiers in Neuroinformatics16(2022) https://doi.org/10.3389/fninf.2022.883333

    Tiddia, G., Golosio, B., Albers, J., Senk, J., Simula, F., Pronold, J., Fanti, V., Pastorelli, E., Paolucci, P.S., Albada, S.J.: Fast simulation of a multi-area spiking network model of macaque cortex on an mpi-gpu cluster. Frontiers in Neuroinformatics16(2022) https://doi.org/10.3389/fninf.2022.883333

  16. [16]

    Scholarpedia 2(4), 1430 (2007) 26

    Gewaltig, M.-O., Diesmann, M.: NEST (NEural Simulation Tool). Scholarpedia 2(4), 1430 (2007) 26

  17. [17]

    http://www.openmp.org/mp-documents/spec30.pdf

    OpenMP Architecture Review Board: OpenMP Application Program Inter- face. http://www.openmp.org/mp-documents/spec30.pdf. Accessed: 2016-09-27 (2008)

  18. [18]

    Frontiers in Neuroinformatics8, 78 (2014) https://doi.org/10.3389/ fninf.2014.00078

    Kunkel, S., Eppler, J.M., Plesser, H.E., Pyka, A., Courcol, J.-D., Potjans, T.C., Diesmann, M., Morrison, A.,et al.: Spiking network simulation code for petascale computers. Frontiers in Neuroinformatics8, 78 (2014) https://doi.org/10.3389/ fninf.2014.00078

  19. [19]

    Frontiers in Neuroinformatics12(2018) https://doi.org/10.3389/fninf.2018.00002

    Jordan, J., Ippen, T., Helias, M., Kitayama, I., Sato, M., Igarashi, J., Diesmann, M., Kunkel, S.: Extremely scalable spiking neuronal network simulation code: From laptops to exascale computers. Frontiers in Neuroinformatics12(2018) https://doi.org/10.3389/fninf.2018.00002

  20. [20]

    PLOS Computational Biology14(10), 1006359 (2018) https://doi.org/10.1371/journal.pcbi.1006359

    Schmidt, M., Bakker, R., Shen, K., Bezgin, G., Diesmann, M., Albada, S.J.: A multi-scale layer-resolved spiking network model of resting-state dynamics in macaque visual cortical areas. PLOS Computational Biology14(10), 1006359 (2018) https://doi.org/10.1371/journal.pcbi.1006359

  21. [21]

    Cerebral Cortex34(10), 409 (2024) https://doi.org/10.1093/cercor/ bhae409

    Pronold, J., Meegen, A., Shimoura, R.O., Vollenbr¨ oker, H., Senden, M., Hilgetag, C.C., Bakker, R., Albada, S.J.: Multi-scale spiking network model of human cere- bral cortex. Cerebral Cortex34(10), 409 (2024) https://doi.org/10.1093/cercor/ bhae409

  22. [22]

    Cerebral Cortex34(10), 405 (2024) https://doi.org/10.1093/cercor/bhae405

    Senk, J., Hagen, E., Albada, S.J., Diesmann, M.: Reconciliation of weak pairwise spike–train correlations and highly coherent local field potentials across space. Cerebral Cortex34(10), 405 (2024) https://doi.org/10.1093/cercor/bhae405

  23. [23]

    Nature Computational Science3(3), 264–276 (2023) https://doi.org/10.1038/ s43588-023-00417-2

    Gandolfi, D., Mapelli, J., Solinas, S.M.G., Triebkorn, P., D’Angelo, E., Jirsa, V., Migliore, M.: Full-scale scaffold model of the human hippocampus ca1 area. Nature Computational Science3(3), 264–276 (2023) https://doi.org/10.1038/ s43588-023-00417-2

  24. [24]

    Nature637(8047), 801–812 (2025) https://doi.org/10.1038/s41586-024-08253-8

    Kudithipudi, D., Schuman, C., Vineyard, C.M., Pandit, T., Merkel, C., Kuben- dran, R., Aimone, J.B., Orchard, G., Mayr, C., Benosman, R., Hays, J., Young, C., Bartolozzi, C., Majumdar, A., Cardwell, S.G., Payvand, M., Buckley, S., Kulkarni, S., Gonzalez, H.A., Cauwenberghs, G., Thakur, C.S., Subramoney, A., Furber, S.: Neuromorphic computing at scale. Nat...

  25. [25]

    Frontiers in Neuroinformatics16(2022) https://doi.org/10.3389/fninf

    Albers, J., Pronold, J., Kurth, A.C., Vennemo, S.B., Mood, K.H., Patronis, A., Terhorst, D., Jordan, J., Kunkel, S., Tetzlaff, T., Diesmann, M., Senk, J.: A modular workflow for performance benchmarking of neuronal network simu- lations. Frontiers in Neuroinformatics16(2022) https://doi.org/10.3389/fninf. 2022.837549

  26. [26]

    Frontiers in Computational Neuroscience15(2021) https://doi.org/10.3389/fncom.2021

    Golosio, B., Tiddia, G., De Luca, C., Pastorelli, E., Simula, F., Paolucci, P.S.: 27 Fast simulations of highly-connected spiking cortical models using gpus. Frontiers in Computational Neuroscience15(2021) https://doi.org/10.3389/fncom.2021. 627620

  27. [27]

    Applied Sciences13(17), 9598 (2023) https://doi.org/10.3390/app13179598

    Golosio, B., Villamar, J., Tiddia, G., Pastorelli, E., Stapmanns, J., Fanti, V., Paolucci, P.S., Morrison, A., Senk, J.: Runtime construction of large-scale spiking neuronal network models on GPU devices. Applied Sciences13(17), 9598 (2023) https://doi.org/10.3390/app13179598

  28. [28]

    Hilfer fractional advection-diffusion equations with power-law initial condition; a Numerical study using variational iteration method

    Herten, A., Achilles, S., Alvarez, D., Badwaik, J., Behle, E., Bode, M., Breuer, T., Caviedes-Voulli` eme, D., Cherti, M., Dabah, A., Sayed, S.E., Frings, W., Gonzalez-Nicolas, A., Gregory, E.B., Mood, K.H., Hater, T., Jitsev, J., John, C.M., Meinke, J.H., Meyer, C.I., Mezentsev, P., Mirus, J.-O., Nassyr, S., Penke, C., R¨ ommer, M., Sinha, U., Vieth, B.v...

  29. [29]

    Brain Structure and Function223(3), 1409–1435 (2017) https://doi.org/10.1007/s00429-017-1554-4

    Schmidt, M., Bakker, R., Hilgetag, C.C., Diesmann, M., Albada, S.J.: Multi-scale account of the network structure of macaque visual cortex. Brain Structure and Function223(3), 1409–1435 (2017) https://doi.org/10.1007/s00429-017-1554-4

  30. [30]

    Journal of large-scale research facilities JLSRF 7(2021) https://doi.org/10.17815/jlsrf-7-179

    Vieth, B.V.S.: JUSUF: Modular tier-2 supercomputing and cloud infrastructure at j¨ ulich supercomputing centre. Journal of large-scale research facilities JLSRF 7(2021) https://doi.org/10.17815/jlsrf-7-179

  31. [31]

    Brunel, Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons, Journal of Computational Neuroscience, 8 (2000), pp

    Brunel, N.: Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience8(3), 183–208 (2000) https://doi.org/10.1023/a:1008925309027

  32. [32]

    Frontiers in Neurosciencevolume 5 - 2011(2011) https: //doi.org/10.3389/fnins.2011.00032

    Boucsein, C., Nawrot, M., Schnepel, P., Aertsen, A.: Beyond the cortical column: Abundance and physiology of horizontal connections imply a strong role for inputs from the surround. Frontiers in Neurosciencevolume 5 - 2011(2011) https: //doi.org/10.3389/fnins.2011.00032

  33. [33]

    PLOS Biology20(3), 3001575 (2022) https://doi.org/10.1371/journal.pbio.3001575

    Rosen, B.Q., Halgren, E.: An estimation of the absolute number of axons indicates that human cortical areas are sparsely connected. PLOS Biology20(3), 3001575 (2022) https://doi.org/10.1371/journal.pbio.3001575

  34. [34]

    PLOS Computational Biology18(9), 1010086 (2022) https://doi.org/10.1371/journal.pcbi.1010086

    Senk, J., Kriener, B., Djurfeldt, M., Voges, N., Jiang, H.-J., Sch¨ uttler, L., Gramelsberger, G., Diesmann, M., Plesser, H.E., Albada, S.J.: Connectivity concepts in neuronal network modeling. PLOS Computational Biology18(9), 1010086 (2022) https://doi.org/10.1371/journal.pcbi.1010086

  35. [35]

    NVIDIA Corporation: CUDA Toolkit Documentation. (2024). Version 12.5. https: //developer.nvidia.com/cuda-toolkit 28

  36. [36]

    Cerebral Cortex 24(3), 785–806 (2014) https://doi.org/10.1093/cercor/bhs358

    Potjans, T.C., Diesmann, M.: The cell-type specific cortical microcircuit: Relating structure and activity in a full-scale spiking network model. Cerebral Cortex 24(3), 785–806 (2014) https://doi.org/10.1093/cercor/bhs358

  37. [37]

    Multi-Scale Modeling in Morphogenesis: A Critical Analysis of the Cellular Potts Model

    Schuecker, J., Schmidt, M., Albada, S.J., Diesmann, M., Helias, M.: Fundamen- tal activity constraints lead to specific interpretations of the connectome. PLOS Computational Biology13(2), 1005179 (2017) https://doi.org/10.1371/journal. pcbi.1005179

  38. [38]

    John Wiley & Sons, Inc., USA (1990)

    Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implemen- tations. John Wiley & Sons, Inc., USA (1990)

  39. [39]

    Frontiers in Neuroinformatics6(2012) https: //doi.org/10.3389/fninf.2012.00026

    Helias, M., Kunkel, S., Masumoto, G., Igarashi, J., Eppler, J.M., Ishii, S., Fukai, T., Morrison, A., Diesmann, M.: Supercomputers ready for use as dis- covery machines for neuroscience. Frontiers in Neuroinformatics6(2012) https: //doi.org/10.3389/fninf.2012.00026

  40. [40]

    Brain98(1), 81–90 (1975) https://doi.org/10.1093/ brain/98.1.81

    CRAGG, B.G.: The density of synapses and neurons in normal, mentally defective and ageing human brains. Brain98(1), 81–90 (1975) https://doi.org/10.1093/ brain/98.1.81

  41. [41]

    & Buehler, M

    Alonso-Nanclares, L., Gonzalez-Soriano, J., Rodriguez, J.R., DeFelipe, J.: Gen- der differences in human cortical synaptic density. Proceedings of the National Academy of Sciences105(38), 14615–14619 (2008) https://doi.org/10.1073/pnas. 0803652105

  42. [42]

    Frontiers in Neuroscience15(2021) https://doi.org/10.3389/fnins.2021.757790

    Dasbach, S., Tetzlaff, T., Diesmann, M., Senk, J.: Dynamical characteristics of recurrent neuronal networks are robust against low synaptic weight resolution. Frontiers in Neuroscience15(2021) https://doi.org/10.3389/fnins.2021.757790

  43. [43]

    [WKO19] WKO

    Waskom, M.L.: seaborn: statistical data visualization. Journal of Open Source Software6(60), 3021 (2021) https://doi.org/10.21105/joss.03021

  44. [44]

    The Annals of Mathematical Statistics27(3), 832–837 (1956) https://doi.org/10

    Rosenblatt, M.: Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics27(3), 832–837 (1956) https://doi.org/10. 1214/aoms/1177728190

  45. [45]

    Probability content of regions under spherical normal distributions, IV: The distribution of homogeneous and non-homogeneous quadratic functions of normal variables,

    Parzen, E.: On estimation of a probability density function and mode. The Annals of Mathematical Statistics33(3), 1065–1076 (1962) https://doi.org/10. 1214/aoms/1177704472

  46. [46]

    Chapman and Hall, London (1986)

    Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London (1986)

  47. [47]

    ! " " ! !

    Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Courna- peau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Millman, K.J., Mayorov, N., Nelson, A.R.J., Jones, 29 E., Kern, R., Larson, E., Carey, C.J., Polat, ˙I., Feng, Y., Moore, E.W., Vander- Plas, J., Laxalde, D., Perktold, J.,...