# Determination of the edge of criticality in echo state networks through Fisher information maximization

arXiv: https://arxiv.org/abs/1603.03685 · HTML: https://mlsanity.com/paper/1603.03685/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2016.

## Built on by (most cited first, 12 of 12)

- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Exploiting Multiple Timescales in Hierarchical Echo State Networks](https://mlsanity.com/paper/2101.04223/) (2021)
- [Memory of recurrent networks: Do we compute it right?](https://mlsanity.com/paper/2305.01457/) (2023)
- [Quantifying Relevance in Learning and Inference](https://mlsanity.com/paper/2202.00339/) (2022)
- [Edge of stability echo state networks](https://mlsanity.com/paper/2308.02902/) (2023)
- [Fisher information flow in artificial neural networks](https://mlsanity.com/paper/2509.02407/) (2025)
- [ViR:the Vision Reservoir](https://mlsanity.com/paper/2112.13545/) (2021)
- [Minimal Deterministic Echo State Networks Outperform Random Reservoirs in Learning Chaotic Dynamics](https://mlsanity.com/paper/2507.06050/) (2025)
- [Toward a Physics of Deep Learning and Brains](https://mlsanity.com/paper/2509.22649/) (2025)
- [Phase Transition Adaptation](https://mlsanity.com/paper/2104.10132/) (2021)
- [Criticality Analysis: Bio-inspired Nonlinear Data Representation](https://mlsanity.com/paper/2305.14361/) (2023)
- [Federated Koopman-Reservoir Learning for Large-Scale Multivariate Time-Series Anomaly Detection](https://mlsanity.com/paper/2503.11255/) (2025)

## Abstract (arXiv)

It is a widely accepted fact that the computational capability of recurrent neural networks is maximized on the so-called "edge of criticality". Once the network operates in this configuration, it performs efficiently on a specific application both in terms of (i) low prediction error and (ii) high short-term memory capacity. Since the behavior of recurrent networks is strongly influenced by the particular input signal driving the dynamics, a universal, application-independent method for determining the edge of criticality is still missing. In this paper, we aim at addressing this issue by proposing a theoretically motivated, unsupervised method based on Fisher information for determining the edge of criticality in recurrent neural networks. It is proven that Fisher information is maximized for (finite-size) systems operating in such critical regions. However, Fisher information is notoriously difficult to compute and either requires the probability density function or the conditional dependence of the system states with respect to the model parameters. The paper takes advantage of a recently-developed non-parametric estimator of the Fisher information matrix and provides a method to determine the critical region of echo state networks, a particular class of recurrent networks. The considered control parameters, which indirectly affect the echo state network performance, are explored to identify those configurations lying on the edge of criticality and, as such, maximizing Fisher information and computational performance. Experimental results on benchmarks and real-world data demonstrate the effectiveness of the proposed method.