# Distributions of covariances as a window into the operational regime of neuronal networks

arXiv: https://arxiv.org/abs/1605.04153 · HTML: https://mlsanity.com/paper/1605.04153/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2016.

## Built on by (most cited first, 3 of 3)

- [Optimal input representation in neural systems at the edge of chaos](https://mlsanity.com/paper/2107.05709/) (2021)
- [Neuronal correlations shape the scaling behavior of memory capacity and nonlinear computational capability of reservoir recurrent neural networks](https://mlsanity.com/paper/2504.19657/) (2025)
- [Hierarchical temporal receptive windows and zero-shot timescale generalization in biologically constrained scale-invariant deep networks](https://mlsanity.com/paper/2601.02618/) (2026)

## Abstract (arXiv)

Massively parallel recordings of spiking activity in cortical networks show that covariances vary widely across pairs of neurons. Their low average is well understood, but an explanation for the wide distribution in relation to the static (quenched) disorder of the connectivity in recurrent random networks was so far elusive. We here derive a finite-size mean-field theory that reduces a disordered to a highly symmetric network with fluctuating auxiliary fields. The exposed analytical relation between the statistics of connections and the statistics of pairwise covariances shows that both, average and dispersion of the latter, diverge at a critical coupling. At this point, a network of nonlinear units transits from regular to chaotic dynamics. Applying these results to recordings from the mammalian brain suggests its operation close to this edge of criticality.