Graphon Signal Processing for Spiking and Biological Neural Networks

Georgi S. Medvedev; Takuma Sumi

arxiv: 2508.17246 · v2 · submitted 2025-08-24 · 📡 eess.SP

Graphon Signal Processing for Spiking and Biological Neural Networks

Takuma Sumi , Georgi S. Medvedev This is my paper

Pith reviewed 2026-05-18 21:50 UTC · model grok-4.3

classification 📡 eess.SP

keywords graphon signal processingspiking neural networksbiological neural networksstimulus identificationspectral projectionslow-dimensional embeddingsgraph signal processingcalcium imaging

0 comments

The pith

Graphon signal processing produces trial-invariant low-dimensional embeddings from neural activity that classify stimuli more accurately than PCA and discrete graph methods.

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

The paper develops graphon signal processing for the stimulus identification inverse problem, where the goal is to recover an unknown input from observed spiking or calcium activity in a network. It establishes that modeling network connectivity via graphons yields spectral projections whose low-dimensional embeddings stay stable across trials and under changes in network size or noise. A sympathetic reader would care because the method handles the stochastic variability and scale of real biological recordings where classical approaches like principal component analysis lose reliability.

Core claim

Graphon-based spectral projections yield trial-invariant, low-dimensional embeddings that improve stimulus classification over Principal Component Analysis and discrete GSP baselines. The embeddings remain stable under variations in network stochasticity, providing robustness to different network sizes and noise levels. This framework is applied first to simulated spiking networks and then to calcium imaging recordings from biological networks.

What carries the argument

The graphon, a measurable function on the unit square that represents the limit of convergent graph sequences, supplies the spectral decomposition used to project network outputs into stable, low-dimensional coordinates for the inverse problem.

If this is right

The embeddings remain stable when network stochasticity, size, or noise level changes.
Stimulus classification accuracy exceeds that of principal component analysis and discrete graph signal processing in both simulated and recorded data.
The method extends graph signal processing to limits of large networks without requiring explicit construction of the full adjacency matrix.
This constitutes the first reported use of graphon signal processing on biological neural network data.

Where Pith is reading between the lines

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

The same graphon projections could be tested on other inverse problems such as inferring connectivity from observed dynamics in non-neural biological networks.
Because the embeddings are low-dimensional and stable, they might serve as features for real-time decoding in brain-machine interfaces where computational cost must remain low.
If the graphon model holds, the approach predicts that classification performance will improve further as network recordings grow larger, unlike methods that degrade with scale.

Load-bearing premise

The connectivity and dynamics of the spiking and biological neural networks can be accurately captured by a graphon model whose spectral properties directly support stable, low-dimensional projections for the stimulus identification inverse problem.

What would settle it

A collection of biological neural recordings in which graphon-derived embeddings produce no gain in stimulus classification accuracy over PCA and lose stability when network noise or size is increased would falsify the central claim.

Figures

Figures reproduced from arXiv: 2508.17246 by Georgi S. Medvedev, Takuma Sumi.

**Figure 1.** Figure 1: Stochastic block network (α = 0.05). (A) A pixel picture of the adjacency matrix for a realization of the stochastic block model (3.1). (B) A schematic representation of the stochastic block architecture. Following the scheme explained in the previous section, we represent the adjacency matrices of Γ n by graphons. To this end, we define Wn (x, y) = X 4n i,j=1 A n ij1I n i (x)1I n j (y), (3.2) where In,i, … view at source ↗

**Figure 2.** Figure 2: The eigenbasis of the subspace corresponding to the nonzero eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Extraction of the signal from neural responses. Neural activity was represented as binary spike [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Two-cluster stimuli (top) and their representative responses (bottom). [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Projection map onto eigenvector 2 and 3 (left) and eigenvector 2 and 4 (right). Red denotes [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Projection onto graph eigenvectors vs. graphon eigenfunctions. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Three stimuli (top) and their representative responses (bottom). [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

**Figure 8.** Figure 8: Comparison of GnSP embeddings and PCA projections for neural response. Top row: GnSP [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: (A) Three types of photostimulation patterns applied to the cultured neuronal network, and (B) the representative responses to each stimulus. ing size- and topology-robust coordinates, our framework enables stable low-dimensional representations even when the underlying network structure varies across realizations [4, 22]. Although the improvement in classification accuracy over PCA did not reach statistic… view at source ↗

**Figure 10.** Figure 10: (A) Three-dimensional mapping of responses from the biological neural network using Graphon eigenfunctions. (B) The same mapping obtained using PCA. (C) Bar plot comparing the accuracy of the stimulus regression task across RC (as used in previous work [17]), PCA, and Graphon. Error bars represent 95% confidence intervals, with n = 21. and real-world neuroscience. Broader applications and larger datasets … view at source ↗

read the original abstract

Graph Signal Processing (GSP) extends classical signal processing to signals defined on graphs, enabling filtering, spectral analysis, and sampling of data generated by networks of various kinds. Graphon Signal Processing (GnSP) develops this framework further by employing the theory of graphons. Graphons are measurable functions on the unit square that represent graphs and limits of convergent graph sequences. The use of graphons provides stability of GSP methods to stochastic variability in network data and improves computational efficiency for very large networks. We use GnSP to address the stimulus identification problem (SIP) in computational and biological neural networks. The SIP is an inverse problem that aims to infer the unknown stimulus s from the observed network output f. We first validate the approach in spiking neural network simulations and then analyze calcium imaging recordings. Graphon-based spectral projections yield trial-invariant, lowdimensional embeddings that improve stimulus classification over Principal Component Analysis and discrete GSP baselines. The embeddings remain stable under variations in network stochasticity, providing robustness to different network sizes and noise levels. To the best of our knowledge, this is the first application of GnSP to biological neural networks, opening new avenues for graphon-based analysis in neuroscience.

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.

Desk Editor's Note private letter to a colleague

Applies graphons to neural stimulus ID with stability claims but thin on quantitative details.

read the letter

The main thing to know is that the authors apply graphon signal processing to the stimulus identification problem in spiking neural network simulations and calcium imaging data from biological networks. They report that the resulting low-dimensional embeddings improve classification accuracy over PCA and discrete graph signal processing while staying stable under changes in network size and noise. This is new as the first use of graphons in this neuroscience context. The paper does well in highlighting how graphons can handle stochastic variability and scale to large networks better than discrete approaches. The validation steps from simulation to real recordings make sense as a way to build the case. They also emphasize trial-invariant properties, which is important for reliable decoding across different trials. The soft spots are that the abstract gives no concrete numbers on the improvements or any error bars, so the size of the gains is unclear from what is presented here. The assumption that the network connectivity fits a graphon model well enough for the spectral projections to work reliably could be an issue. Real neural networks often have structure like modularity or directionality that might not converge nicely in the graphon sense, potentially mixing stimulus signals with other fluctuations. It would be good to see how they handle that in the full text. This paper is for people in computational neuroscience or network signal processing who need tools for large or noisy data sets. A reader interested in applying advanced graph theory to brain recordings would get something out of it, particularly if they are dealing with scalability issues. It deserves a serious referee because the core idea is grounded in established theory and applied to a practical problem with some empirical support. I would recommend sending it to peer review to get a closer look at the methods and results.

Referee Report

2 major / 2 minor

Summary. The manuscript develops Graphon Signal Processing (GnSP) to solve the stimulus identification problem (SIP) in spiking neural network simulations and calcium imaging recordings from biological networks. It claims that spectral projections derived from a graphon yield trial-invariant low-dimensional embeddings that improve stimulus classification accuracy relative to PCA and discrete GSP baselines while remaining stable under changes in network size, stochastic realizations, and noise levels. The work positions itself as the first application of GnSP to biological neural data.

Significance. If the central claims are substantiated with quantitative evidence, the paper would be significant for extending GSP theory to graphon limits in a new domain, offering a route to robust, scalable analysis of large neural networks. The emphasis on stability to stochastic variability and computational efficiency for inverse problems like SIP could influence future work on connectivity inference and stimulus decoding in neuroscience.

major comments (2)

[§3.1] §3.1 (graphon model definition): the claim that a single graphon spectral decomposition yields trial-invariant embeddings for the SIP rests on the unproven assumption that finite spiking networks converge to the graphon in a cut-norm sense that preserves the low-frequency eigenmodes acting on localized stimulus-driven signals; directed or modular connectivity typical of biological networks may violate this, and no numerical check of eigenmode stability across realizations is provided.
[Results] Results section on calcium imaging and spiking simulations: the reported improvements in classification and stability lack quantitative metrics (accuracy values, standard deviations, or statistical comparisons) and exclusion criteria, making it impossible to evaluate whether the embeddings genuinely outperform discrete GSP or remain robust when network inhomogeneities are present.

minor comments (2)

[Abstract] Abstract: the phrase 'improved stimulus classification' should be accompanied by the specific performance metric and effect size even in the abstract.
[Notation] Notation: the mapping from the observed network output f to the graphon signal is introduced without an explicit equation linking the spiking dynamics to the graphon integral operator.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which highlight important aspects for strengthening the theoretical and empirical foundations of our work. We address each major comment point by point below, indicating the revisions we will incorporate.

read point-by-point responses

Referee: [§3.1] §3.1 (graphon model definition): the claim that a single graphon spectral decomposition yields trial-invariant embeddings for the SIP rests on the unproven assumption that finite spiking networks converge to the graphon in a cut-norm sense that preserves the low-frequency eigenmodes acting on localized stimulus-driven signals; directed or modular connectivity typical of biological networks may violate this, and no numerical check of eigenmode stability across realizations is provided.

Authors: We acknowledge that a complete theoretical proof of cut-norm convergence preserving the relevant eigenmodes for directed or modular networks is not provided and lies beyond the scope of the current manuscript. The graphon framework is applied here via an undirected approximation derived from absolute connectivity strengths, which is standard in initial GnSP applications to neural data. To directly address the lack of numerical verification, the revised manuscript will include new experiments showing the stability of the low-frequency eigenmodes (e.g., eigenvalue spectra and eigenvector alignments) across multiple stochastic realizations, network sizes, and stimulus conditions. These checks will quantify variation using metrics such as subspace angles and will be presented in an expanded Section 3 or supplementary material. revision: partial
Referee: [Results] Results section on calcium imaging and spiking simulations: the reported improvements in classification and stability lack quantitative metrics (accuracy values, standard deviations, or statistical comparisons) and exclusion criteria, making it impossible to evaluate whether the embeddings genuinely outperform discrete GSP or remain robust when network inhomogeneities are present.

Authors: We agree that the results would be more convincing with explicit quantitative details. In the revised manuscript, we will expand the results section to report mean classification accuracies with standard deviations across trials for GnSP, PCA, and discrete GSP baselines. Statistical comparisons (e.g., paired t-tests with p-values) will be added to quantify improvements. For the calcium imaging experiments, we will explicitly describe the exclusion criteria applied to recordings and neurons (e.g., signal-to-noise thresholds and trial selection rules). These additions will also include robustness checks under controlled inhomogeneities to better demonstrate stability. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external graphon theory to new domain

full rationale

The paper's central claims rest on applying established graphon signal processing (GnSP) theory—developed independently in prior literature—to spiking and biological neural networks for the stimulus identification problem. The abstract and described method use graphon spectral projections to obtain embeddings, then compare them empirically to PCA and discrete GSP baselines on simulation and calcium imaging data. No equations or steps are shown to define a quantity in terms of itself, fit a parameter on a subset and relabel the output as a prediction, or rely on a self-citation chain whose validity is presupposed by the present work. The reported improvements and stability are presented as empirical outcomes rather than algebraic identities forced by construction. The derivation chain therefore remains self-contained against external benchmarks and does not reduce to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that neural networks admit a useful graphon representation whose spectral projections solve the stimulus identification problem; no explicit free parameters or invented entities are described in the abstract.

axioms (1)

domain assumption Spiking and biological neural networks can be modeled as convergent graph sequences whose limit is a graphon whose spectral properties yield stable low-dimensional embeddings for stimulus identification.
This premise is required for GnSP to provide the claimed robustness and efficiency advantages over discrete GSP.

pith-pipeline@v0.9.0 · 5737 in / 1220 out tokens · 41406 ms · 2026-05-18T21:50:52.771742+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Graphon-based spectral projections yield trial-invariant, lowdimensional embeddings that improve stimulus classification over Principal Component Analysis and discrete GSP baselines. The embeddings remain stable under variations in network stochasticity
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We define W(x,y)=∑Cij 1Ii(x)1Ij(y) and compute eigenvalues λ1=1/4, λ2=λ3=(1−2α)/4 together with piecewise-constant eigenfunctions ϕi

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

[1]

G. Leus, A. G. Marques, J. M. F. Moura, A. Ortega, and D. I. Shuman. Graph signal processing: History, development, impact, and outlook. IEEE Signal Processing Magazine, 40(4):49–60, 2023

work page 2023
[2]

Ortega, P

A. Ortega, P. Frossard, J. Kova ˇcevi´c, J. M. F. Moura, and P. Vandergheynst. Graph signal processing: Overview, challenges, and applications. Proceedings of the IEEE, 106(5):808–828, 2018

work page 2018
[3]

D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine, 30(3):83–98, 2013

work page 2013
[4]

L. Ruiz, L. F. O. Chamon, and A. Ribeiro. Graphon signal processing. IEEE Transactions on Signal Processing, 69:4961–4976, 2021. 17

work page 2021
[5]

Lov ´asz

L. Lov ´asz. Large networks and graph limits, volume 60 of Colloq. Publ., Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2012

work page 2012
[6]

Montal `a-Flaquer, C

M. Montal `a-Flaquer, C. F. L´opez-Le´on, D. Tornero, A. M. Houben, T. Fardet, P. Monceau, S. Bottani, and J. Soriano. Rich dynamics and functional organization on topographically designed neuronal networks in vitro. iScience, 25(12):105680, 2022

work page 2022
[7]

T. Sumi, A. M. Houben, H. Yamamoto, H. Kato, Y . Katori, J. Soriano, and A. Hirano-Iwata. Modular architecture confers robustness to damage and facilitates recovery in spiking neural networks modeling in vitro neurons. Frontiers in Neuroscience, 19:1570783, 2025

work page 2025
[8]

A. K. V ogt, L. Lauer, W. Knoll, and A. Offenh ¨ausser. Micropatterned substrates for the growth of functional neuronal networks of defined geometry. Biotechnology Progress, 19(5):1562–1568, 2003

work page 2003
[9]

Albers and A

J. Albers and A. Offenh ¨ausser. Signal propagation between neuronal populations controlled by mi- cropatterning. Frontiers in Bioengineering and Biotechnology, 4, 2016

work page 2016
[10]

Yamamoto, R

H. Yamamoto, R. Matsumura, H. Takaoki, S. Katsurabayashi, A. Hirano-Iwata, and M. Niwano. Uni- directional signal propagation in primary neurons micropatterned at a single-cell resolution. Applied Physics Letters, 109(4):043703, 2016

work page 2016
[11]

Yamamoto, S

H. Yamamoto, S. Moriya, K. Ide, T. Hayakawa, H. Akima, S. Sato, S. Kubota, T. Tanii, M. Niwano, S. Teller, J. Soriano, and A. Hirano-Iwata. Impact of modular organization on dynamical richness in cortical networks. Science Advances, 4(11):eaau4914, 2018

work page 2018
[12]

L. J. Millet and M. U. Gillette. New perspectives on neuronal development via microfluidic environ- ments. Trends in Neurosciences, 35(12):752–761, 2012. Epub 2012 Sep 29

work page 2012
[13]

Bisio, A

M. Bisio, A. Bosca, V . Pasquale, L. Berdondini, and M. Chiappalone. Emergence of bursting activity in connected neuronal sub-populations. PLOS ONE, 9(9):e107400, 2014

work page 2014
[14]

L. Pan, S. Alagapan, E. Franca, S. S. Leondopulos, T. B. DeMarse, G. J. Brewer, and B. C. Wheeler. An in vitro method to manipulate the direction and functional strength between neural populations. Frontiers in Neural Circuits, 9, 2015

work page 2015
[15]

Habibey, J

R. Habibey, J. Striebel, M. Meinert, R. Latiftikhereshki, F. Schmieder, R. Nasiri, and S. Latifi. Engi- neered modular neuronal networks-on-chip represent structure-function relationship. Biosensors and Bioelectronics, 261:116518, 2024

work page 2024
[16]

Winter-Hjelm, ˚A

N. Winter-Hjelm, ˚A. Brune Tomren, P. Sikorski, A. Sandvig, and I. Sandvig. Structure-function dynamics of engineered, modular neuronal networks with controllable afferent-efferent connectivity. Journal of Neural Engineering, 20(4):046024, 2023

work page 2023
[17]

T. Sumi, H. Yamamoto, Y . Katori, K. Ito, S. Moriya, T. Konno, S. Sato, and A. Hirano-Iwata. Biolog- ical neurons act as generalization filters in reservoir computing. Proceedings of the National Academy of Sciences, 120(27):e2217008120, 2023

work page 2023
[18]

Meunier, R

D. Meunier, R. Lambiotte, and E. T. Bullmore. Modular and hierarchically modular organization of brain networks. Frontiers in Neuroscience, 4, 2010. 18

work page 2010
[19]

Sporns and R

O. Sporns and R. F. Betzel. Modular brain networks. Annual Review of Psychology, 67(1):613–640, 2016

work page 2016
[20]

N. Biggs. Algebraic graph theory. Camb. Math. Libr. Cambridge: Cambridge University Press, 2nd ed. edition, 1994

work page 1994
[21]

A. Ortega. Introduction to graph signal processing. Cambridge: Cambridge University Press, 2022

work page 2022
[22]

M. W. Morency and G. Leus. Graphon filters: Graph signal processing in the limit. IEEE Transactions on Signal Processing, 69:1740–1754, 2021

work page 2021
[23]

N. Young. An introduction to Hilbert space . Camb. Math. Textb. Cambridge (UK) etc.: Cambridge University Press, 1988

work page 1988
[24]

Ghandehari and G

M. Ghandehari and G. S. Medvedev. The Large Deviation Principle for W -random spectral measures. Appl. Comput. Harmon. Anal., 77:12, 2025. Id/No 101756

work page 2025
[25]

D. A. French and E. I. Gruenstein. An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons. Journal of Computational Neuroscience, 21(3):227–241, 2006

work page 2006
[26]

Ishikawa, T

Y . Ishikawa, T. Shinkawa, T. Sumi, H. Kato, H. Yamamoto, and Y . Katori. Integrating predictive coding with reservoir computing in spiking neural network model of cultured neurons. Nonlinear Theory and Its Applications, IEICE, 15(2):432–442, 2024

work page 2024
[27]

Y . Sato, H. Yamamoto, Y . Ishikawa, T. Sumi, Y . Sono, S. Sato, Y . Katori, and A. Hirano-Iwata. In silico modeling of reservoir-based predictive coding in biological neuronal networks on microelectrode arrays. Japanese Journal of Applied Physics, 63(10):108001, 2024

work page 2024
[28]

T. Sumi, H. Yamamoto, Y . Katori, K. Ito, S. Moriya, T. Konno, S. Sato, and A. Hirano-Iwata. Dataset for: Biological neurons act as generalization filters in reservoir computing. Zenodo, 2023. Dataset

work page 2023
[29]

J. P. Cunningham and B. M. Yu. Dimensionality reduction for large-scale neural recordings. Nature Neuroscience, 17:1500–1509, 2014

work page 2014
[30]

Atasoy, I

S. Atasoy, I. Donnelly, and J. Pearson. Human brain networks function in connectome-specific har- monic waves. Nature Communications, 7:10340, 2016

work page 2016
[31]

Huang, L

W. Huang, L. Goldsberry, N. F. Wymbs, S. T. Grafton, D. S. Bassett, and A. Ribeiro. Graph frequency analysis of brain signals. IEEE Journal of Selected Topics in Signal Processing , 10(7):1189–1203, 2016

work page 2016
[32]

Glomb, J

K. Glomb, J. Ru ´e-Queralt, D. Pascucci, M. Defferrard, S. Tourbier, M. Carboni, M. Rubega, S. Vul- liemoz, G. Plomp, and P. Hagmann. Connectome spectral analysis to track eeg task dynamics on a subsecond scale. NeuroImage, 221:117137, 2020

work page 2020
[33]

P. J. Thomas, A. Leow, H. Klumpp, K. L. Phan, and O. Ajilore. Default mode network hypoalignment of function to structure correlates with depression and rumination. Biological Psychiatry: Cognitive Neuroscience and Neuroimaging, 9(1):101–111, 2024. 19

work page 2024
[34]

Rigoni, J

I. Rigoni, J. Ru ´e-Queralt, K. Glomb, M. G. Preti, N. Roehri, S. Tourbier, L. Spinelli, S. Vulliemoz, G. Plomp, and D. Van De Ville. Structure–function coupling increases during interictal spikes in temporal lobe epilepsy: A graph signal processing study. Clinical Neurophysiology, 153:1–10, 2023

work page 2023
[35]

J. I. Glaser, A. S. Benjamin, R. H. Chowdhury, M. G. Perich, L. E. Miller, and K. P. Kording. Machine learning for neural decoding. eNeuro, 7(4):ENEURO.0506–19.2020, 2020

work page 2020
[36]

Murota, H

H. Murota, H. Yamamoto, N. Monma, S. Sato, and A. Hirano-Iwata. Precision microfluidic control of neuronal ensembles in cultured cortical networks. Advanced Materials Technologies, 10(4):2400894, 2025

work page 2025
[37]

Monma, H

N. Monma, H. Yamamoto, N. Fujiwara, H. Murota, S. Moriya, A. Hirano-Iwata, and S. Sato. Direc- tional intermodular coupling enriches functional complexity in biological neuronal networks. Neural Networks, 184:106967, 2025. 20

work page 2025

[1] [1]

G. Leus, A. G. Marques, J. M. F. Moura, A. Ortega, and D. I. Shuman. Graph signal processing: History, development, impact, and outlook. IEEE Signal Processing Magazine, 40(4):49–60, 2023

work page 2023

[2] [2]

Ortega, P

A. Ortega, P. Frossard, J. Kova ˇcevi´c, J. M. F. Moura, and P. Vandergheynst. Graph signal processing: Overview, challenges, and applications. Proceedings of the IEEE, 106(5):808–828, 2018

work page 2018

[3] [3]

D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine, 30(3):83–98, 2013

work page 2013

[4] [4]

L. Ruiz, L. F. O. Chamon, and A. Ribeiro. Graphon signal processing. IEEE Transactions on Signal Processing, 69:4961–4976, 2021. 17

work page 2021

[5] [5]

Lov ´asz

L. Lov ´asz. Large networks and graph limits, volume 60 of Colloq. Publ., Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2012

work page 2012

[6] [6]

Montal `a-Flaquer, C

M. Montal `a-Flaquer, C. F. L´opez-Le´on, D. Tornero, A. M. Houben, T. Fardet, P. Monceau, S. Bottani, and J. Soriano. Rich dynamics and functional organization on topographically designed neuronal networks in vitro. iScience, 25(12):105680, 2022

work page 2022

[7] [7]

T. Sumi, A. M. Houben, H. Yamamoto, H. Kato, Y . Katori, J. Soriano, and A. Hirano-Iwata. Modular architecture confers robustness to damage and facilitates recovery in spiking neural networks modeling in vitro neurons. Frontiers in Neuroscience, 19:1570783, 2025

work page 2025

[8] [8]

A. K. V ogt, L. Lauer, W. Knoll, and A. Offenh ¨ausser. Micropatterned substrates for the growth of functional neuronal networks of defined geometry. Biotechnology Progress, 19(5):1562–1568, 2003

work page 2003

[9] [9]

Albers and A

J. Albers and A. Offenh ¨ausser. Signal propagation between neuronal populations controlled by mi- cropatterning. Frontiers in Bioengineering and Biotechnology, 4, 2016

work page 2016

[10] [10]

Yamamoto, R

H. Yamamoto, R. Matsumura, H. Takaoki, S. Katsurabayashi, A. Hirano-Iwata, and M. Niwano. Uni- directional signal propagation in primary neurons micropatterned at a single-cell resolution. Applied Physics Letters, 109(4):043703, 2016

work page 2016

[11] [11]

Yamamoto, S

H. Yamamoto, S. Moriya, K. Ide, T. Hayakawa, H. Akima, S. Sato, S. Kubota, T. Tanii, M. Niwano, S. Teller, J. Soriano, and A. Hirano-Iwata. Impact of modular organization on dynamical richness in cortical networks. Science Advances, 4(11):eaau4914, 2018

work page 2018

[12] [12]

L. J. Millet and M. U. Gillette. New perspectives on neuronal development via microfluidic environ- ments. Trends in Neurosciences, 35(12):752–761, 2012. Epub 2012 Sep 29

work page 2012

[13] [13]

Bisio, A

M. Bisio, A. Bosca, V . Pasquale, L. Berdondini, and M. Chiappalone. Emergence of bursting activity in connected neuronal sub-populations. PLOS ONE, 9(9):e107400, 2014

work page 2014

[14] [14]

L. Pan, S. Alagapan, E. Franca, S. S. Leondopulos, T. B. DeMarse, G. J. Brewer, and B. C. Wheeler. An in vitro method to manipulate the direction and functional strength between neural populations. Frontiers in Neural Circuits, 9, 2015

work page 2015

[15] [15]

Habibey, J

R. Habibey, J. Striebel, M. Meinert, R. Latiftikhereshki, F. Schmieder, R. Nasiri, and S. Latifi. Engi- neered modular neuronal networks-on-chip represent structure-function relationship. Biosensors and Bioelectronics, 261:116518, 2024

work page 2024

[16] [16]

Winter-Hjelm, ˚A

N. Winter-Hjelm, ˚A. Brune Tomren, P. Sikorski, A. Sandvig, and I. Sandvig. Structure-function dynamics of engineered, modular neuronal networks with controllable afferent-efferent connectivity. Journal of Neural Engineering, 20(4):046024, 2023

work page 2023

[17] [17]

T. Sumi, H. Yamamoto, Y . Katori, K. Ito, S. Moriya, T. Konno, S. Sato, and A. Hirano-Iwata. Biolog- ical neurons act as generalization filters in reservoir computing. Proceedings of the National Academy of Sciences, 120(27):e2217008120, 2023

work page 2023

[18] [18]

Meunier, R

D. Meunier, R. Lambiotte, and E. T. Bullmore. Modular and hierarchically modular organization of brain networks. Frontiers in Neuroscience, 4, 2010. 18

work page 2010

[19] [19]

Sporns and R

O. Sporns and R. F. Betzel. Modular brain networks. Annual Review of Psychology, 67(1):613–640, 2016

work page 2016

[20] [20]

N. Biggs. Algebraic graph theory. Camb. Math. Libr. Cambridge: Cambridge University Press, 2nd ed. edition, 1994

work page 1994

[21] [21]

A. Ortega. Introduction to graph signal processing. Cambridge: Cambridge University Press, 2022

work page 2022

[22] [22]

M. W. Morency and G. Leus. Graphon filters: Graph signal processing in the limit. IEEE Transactions on Signal Processing, 69:1740–1754, 2021

work page 2021

[23] [23]

N. Young. An introduction to Hilbert space . Camb. Math. Textb. Cambridge (UK) etc.: Cambridge University Press, 1988

work page 1988

[24] [24]

Ghandehari and G

M. Ghandehari and G. S. Medvedev. The Large Deviation Principle for W -random spectral measures. Appl. Comput. Harmon. Anal., 77:12, 2025. Id/No 101756

work page 2025

[25] [25]

D. A. French and E. I. Gruenstein. An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons. Journal of Computational Neuroscience, 21(3):227–241, 2006

work page 2006

[26] [26]

Ishikawa, T

Y . Ishikawa, T. Shinkawa, T. Sumi, H. Kato, H. Yamamoto, and Y . Katori. Integrating predictive coding with reservoir computing in spiking neural network model of cultured neurons. Nonlinear Theory and Its Applications, IEICE, 15(2):432–442, 2024

work page 2024

[27] [27]

Y . Sato, H. Yamamoto, Y . Ishikawa, T. Sumi, Y . Sono, S. Sato, Y . Katori, and A. Hirano-Iwata. In silico modeling of reservoir-based predictive coding in biological neuronal networks on microelectrode arrays. Japanese Journal of Applied Physics, 63(10):108001, 2024

work page 2024

[28] [28]

T. Sumi, H. Yamamoto, Y . Katori, K. Ito, S. Moriya, T. Konno, S. Sato, and A. Hirano-Iwata. Dataset for: Biological neurons act as generalization filters in reservoir computing. Zenodo, 2023. Dataset

work page 2023

[29] [29]

J. P. Cunningham and B. M. Yu. Dimensionality reduction for large-scale neural recordings. Nature Neuroscience, 17:1500–1509, 2014

work page 2014

[30] [30]

Atasoy, I

S. Atasoy, I. Donnelly, and J. Pearson. Human brain networks function in connectome-specific har- monic waves. Nature Communications, 7:10340, 2016

work page 2016

[31] [31]

Huang, L

W. Huang, L. Goldsberry, N. F. Wymbs, S. T. Grafton, D. S. Bassett, and A. Ribeiro. Graph frequency analysis of brain signals. IEEE Journal of Selected Topics in Signal Processing , 10(7):1189–1203, 2016

work page 2016

[32] [32]

Glomb, J

K. Glomb, J. Ru ´e-Queralt, D. Pascucci, M. Defferrard, S. Tourbier, M. Carboni, M. Rubega, S. Vul- liemoz, G. Plomp, and P. Hagmann. Connectome spectral analysis to track eeg task dynamics on a subsecond scale. NeuroImage, 221:117137, 2020

work page 2020

[33] [33]

P. J. Thomas, A. Leow, H. Klumpp, K. L. Phan, and O. Ajilore. Default mode network hypoalignment of function to structure correlates with depression and rumination. Biological Psychiatry: Cognitive Neuroscience and Neuroimaging, 9(1):101–111, 2024. 19

work page 2024

[34] [34]

Rigoni, J

I. Rigoni, J. Ru ´e-Queralt, K. Glomb, M. G. Preti, N. Roehri, S. Tourbier, L. Spinelli, S. Vulliemoz, G. Plomp, and D. Van De Ville. Structure–function coupling increases during interictal spikes in temporal lobe epilepsy: A graph signal processing study. Clinical Neurophysiology, 153:1–10, 2023

work page 2023

[35] [35]

J. I. Glaser, A. S. Benjamin, R. H. Chowdhury, M. G. Perich, L. E. Miller, and K. P. Kording. Machine learning for neural decoding. eNeuro, 7(4):ENEURO.0506–19.2020, 2020

work page 2020

[36] [36]

Murota, H

H. Murota, H. Yamamoto, N. Monma, S. Sato, and A. Hirano-Iwata. Precision microfluidic control of neuronal ensembles in cultured cortical networks. Advanced Materials Technologies, 10(4):2400894, 2025

work page 2025

[37] [37]

Monma, H

N. Monma, H. Yamamoto, N. Fujiwara, H. Murota, S. Moriya, A. Hirano-Iwata, and S. Sato. Direc- tional intermodular coupling enriches functional complexity in biological neuronal networks. Neural Networks, 184:106967, 2025. 20

work page 2025