First and second order firing statistics of large neuronal networks with community structure

Abstract

We study a discrete-time stochastic model for the spiking activity of a network of n interacting neurons partitioned into k communities. The interaction network is a random directed weighted graph generated by a Stochastic Block Model with k+1 edge types (k intra-community, $1$ inter-community). Each edge is present independently with a probability that depends only on its type. When present, an edge carries a type-dependent deterministic weight scaled by n⁻¹, which is excitatory or inhibitory depending on an independent random sign. Given the interaction network, the spiking process is a stationary Markov chain. Our results describe how the network's community structure manifests itself in the first- and second-order statistics of this chain, as n → ∞ with k fixed. Representing these statistics as explicit functions of the random interaction network, we prove that, with high probability: (i) the vectors of mean firing activity and firing variances are approximated in supremum norm by deterministic vectors that are constant within each community, with an error of order √(log(n)/n); (ii) the off-diagonal part of the $0$-lagged firing covariance matrix is approximated in max norm, with an error of order √(log(n)/n³), by a deterministic community-block matrix (constant on each block defined by a pair of communities), with block values determined by the unique solution to a k × k Stein matrix equation; (iii) in contrast, the t-lagged covariance matrix, for every lag t ≥ 1, is approximated in max norm, again with an error of order √(log(n)/n³), by a random matrix built from two components: a random matrix capturing the interaction network and a deterministic community-block matrix. Finally, we discuss conditions on the model parameters under which these limiting statistics enable the identifiability of the community structure.

Identifiability of a linear dynamical reaction-diffusion system with directed interactions from the second-order statistics of a single snapshot
Pirmin Schlicke
arXiv5 Oct 2026PreprintCC BY 4.0Full textBioinformatics & computational biology ProbabilityarXiv
Stochastic Dynamics of Large-Scale Motif-Embedded Spiking Neuronal Networks
Gurpreet Jagdev, Richard Bertram, Na Yu
arXiv1 Oct 2026PreprintarXiv licenceFull textNeuroscience Dynamical systemsarXiv
Stochastic dynamics and synchronization in motif-based neuronal networks
Gurpreet Jagdev, Yifei Lu, Richard Bertram +1
arXiv1 Oct 2026PreprintarXiv licenceFull textNeuroscience Dynamical systemsarXiv
Structure-Aware Online Optimization of the Age of Information with Unknown System Statistics
Yoshiaki Inoue, Michel Mandjes, Liron Ravner
arXiv9 Oct 2026PreprintCC BY 4.0Full textOptimisation & controlarXiv
Robust second-order time integration of tree tensor networks
Jonas Kusch, Dominik Sulz
arXiv8 Oct 2026PreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv

Cite

BibTeX RIS

BibTeX

@misc{coelho2026first,
  title = {{First and second order firing statistics of large neuronal networks with community structure}},
  author = {Thiago H. N. Coelho and Daniel Figueiredo and Giulio Iacobelli and Guilherme Ost and Conrado C. Pinto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41706},
  eprint = {2610.41706},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41706},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - First and second order firing statistics of large neuronal networks with community structure
AU  - Thiago H. N. Coelho
AU  - Daniel Figueiredo
AU  - Giulio Iacobelli
AU  - Guilherme Ost
AU  - Conrado C. Pinto
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41706
UR  - https://arxiv.org/abs/2610.41706
AB  - We study a discrete-time stochastic model for the spiking activity of a network of n interacting neurons partitioned into k communities. The interaction network is a random directed weighted graph generated by a Stochastic Block Model with k+1 edge types (k intra-community, $1$ inter-community). Each edge is present independently with a probability that depends only on its type. When present, an edge carries a type-dependent deterministic weight scaled by n^−1, which is excitatory or inhibitory depending on an independent random sign. Given the interaction network, the spiking process is a stationary Markov chain. Our results describe how the network's community structure manifests itself in the first- and second-order statistics of this chain, as n → ∞ with k fixed. Representing these statistics as explicit functions of the random interaction network, we prove that, with high probability: (i) the vectors of mean firing activity and firing variances are approximated in supremum norm by deterministic vectors that are constant within each community, with an error of order √(log(n)/n); (ii) the off-diagonal part of the $0$-lagged firing covariance matrix is approximated in max norm, with an error of order √(log(n)/n^3), by a deterministic community-block matrix (constant on each block defined by a pair of communities), with block values determined by the unique solution to a k × k Stein matrix equation; (iii) in contrast, the t-lagged covariance matrix, for every lag t ≥ 1, is approximated in max norm, again with an error of order √(log(n)/n^3), by a random matrix built from two components: a random matrix capturing the interaction network and a deterministic community-block matrix. Finally, we discuss conditions on the model parameters under which these limiting statistics enable the identifiability of the community structure.
ER  -

Abstract

Related

Cite