Dynamics of random recurrent neural networks: colored noise and sublinear activations
Abstract
We study the dynamics of recurrent neural networks with random couplings, sublinear odd activation functions, and temporally correlated noise. Using large N methods, we determine stationary correlations and distinguish the loss of local linear stability from the onset of chaos, characterized by a positive largest Lyapunov exponent. These thresholds delimit an intermediate regime in which local instability coexists with asymptotic stability under a common noise history. Their separation depends on both the noise correlation time and the asymptotic growth of the activation function. We show that internal noise with any finite correlation time cuts off the divergence of the network's intrinsic correlation time tau_corr at the edge of chaos; it exhibits a critical scaling tau_{corr,max} propto 1/sqrt{D_0}, where D_0 is the integrated noise strength. Static random input instead produces coincident thresholds and a divergent correlation time. We also calculate the frequency-dependent participation dimension from a four-point correlation function. Its zero-frequency value is inversely proportional to the fourth power of the network correlation time, predicting a smooth minimum at the weak-noise chaos threshold. Numerical simulations support the large-network predictions and reveal appreciable finite-size effects.
Related
- Local connectivity balance shapes population dynamics in random recurrent networks
- Attractor dynamics underlie sequence working memory in data-constrained recurrent neural networks
- Bio-Inspired Auditory Processing: Coupling Parallel Resonate-and-Fire Dynamics with Ternary Recurrent Networks for Speech Recognition
- Shaping Slow Manifolds in Equivariant Recurrent Neural Networks
- Reusable modular architecture enables flexible cognitive operations in the mouse brain and artificial recurrent networks
BibTeX
@misc{mato2026dynamics,
title = {{Dynamics of random recurrent neural networks: colored noise and sublinear activations}},
author = {German Mato and Facundo Rigatuso and Gonzalo Torroba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31575},
eprint = {2610.31575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31575},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dynamics of random recurrent neural networks: colored noise and sublinear activations
AU - German Mato
AU - Facundo Rigatuso
AU - Gonzalo Torroba
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31575
UR - https://arxiv.org/abs/2610.31575
AB - We study the dynamics of recurrent neural networks with random couplings, sublinear odd activation functions, and temporally correlated noise. Using large N methods, we determine stationary correlations and distinguish the loss of local linear stability from the onset of chaos, characterized by a positive largest Lyapunov exponent. These thresholds delimit an intermediate regime in which local instability coexists with asymptotic stability under a common noise history. Their separation depends on both the noise correlation time and the asymptotic growth of the activation function. We show that internal noise with any finite correlation time cuts off the divergence of the network's intrinsic correlation time tau_corr at the edge of chaos; it exhibits a critical scaling tau_{corr,max} propto 1/sqrt{D_0}, where D_0 is the integrated noise strength. Static random input instead produces coincident thresholds and a divergent correlation time. We also calculate the frequency-dependent participation dimension from a four-point correlation function. Its zero-frequency value is inversely proportional to the fourth power of the network correlation time, predicting a smooth minimum at the weak-noise chaos threshold. Numerical simulations support the large-network predictions and reveal appreciable finite-size effects.
ER -