CoarseGraining

Code accompanying the preprint:

"Scaling and tuning to criticality in resting-state human magnetoencephalography" arXiv

Abstract

From 1/f noise to neuronal avalanches, evidence of scaling in brain activity has been increasingly linked to tuning to or near criticality. The concept of scaling is intimately related to the renormalization group (RG), in essence providing coarse-grained, simplified descriptions that generalize to classes of diverse physical systems. Following the RG idea, scaling laws have been reported in populations of spiking neurons at microscopic scales. Whether similar scaling principles govern large-scale neural activity in the human brain and how they relate to underlying neural physiology remain unresolved. Here, we analyze large-scale electrophysiological recordings (MEG) of human resting-state brain activity and apply a RG-inspired coarse-graining approach to track collective neural dynamics across spatial scales. We find that multiple observables exhibit robust scale-invariant behavior under coarse-graining: activity variance and correlations grow according to power laws, covariance eigenspectra follow a characteristic scaling relation, and neuronal avalanche statistics remain invariant. Using an analytically tractable neural network model, we show that the observed scaling signaturesarise when the system operates slightly below criticality, and that the scaling exponents depend on the excitation–inhibition balance. These findings demonstrate that RG-inspired scaling analysis can uncover signatures of critical dynamics in non-invasive human electrophysiology and suggest a principled route toward estimating excitation–inhibition balance from large-scale brain recordings.

This repository contains Python scripts implementing the coarse-graining procedure introduced by Meshulam et al. in "Coarse Graining, Fixed Points, and Scaling in a Large Population of Neurons". These scripts were used for the coarse-graining analysis described in our manuscript/preprint.

Files

• cg_funcs.py performs iterative coarse-graining steps and computes cluster-level covariance eigenspectra.
• analyse_quantities.py include functions for computing silence probability, cluster variance, and autocorrelation properties of coarse-grained variables across renormalization steps.

Requirements

Python 3 with standard scientific libraries:

• numpy
• scipy
• matplotlib

Citation

If you use this code, please use:

@article{https://arxiv.org/pdf/2602.17820,
  title={Scaling and tuning to criticality in resting-state human magnetoencephalography},
  author={Topal, Irem and Poggialini, Anna and Maschio, Marco Dal and De Martino, Daniele and Shriki, Oren and Lombardi, Fabrizio},
  journal={arXiv preprint arXiv:2602.17820},
  year={2026}
}

License

This project is licensed under the MIT License.