# Local Dynamics in Trained Recurrent Neural Networks

arXiv: https://arxiv.org/abs/1511.05222 · HTML: https://mlsanity.com/paper/1511.05222/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Nov 2015.

## Built on by (most cited first, 5 of 5)

- [Learning Reservoir Dynamics with Temporal Self-Modulation](https://mlsanity.com/paper/2301.09235/) (2023)
- [Astrocytes as a mechanism for meta-plasticity and contextually-guided network function](https://mlsanity.com/paper/2311.03508/) (2023)
- [Dynamical stability for dense patterns in attractor neural networks](https://mlsanity.com/paper/2507.10383/) (2025)
- [Beyond the Edge of Chaos: Stability-Expressivity Transfer in Reservoir Forecasting](https://mlsanity.com/paper/2607.17909/) (2026)
- [Learning-Induced Dynamical Transition in Recurrent Neural Networks](https://mlsanity.com/paper/2609.19288/) (2026)

## Abstract (arXiv)

Learning a task induces connectivity changes in neural circuits, thereby changing their dynamics. To elucidate task related neural dynamics we study trained Recurrent Neural Networks. We develop a Mean Field Theory for Reservoir Computing networks trained to have multiple fixed point attractors. Our main result is that the dynamics of the network's output in the vicinity of attractors is governed by a low order linear Ordinary Differential Equation. Stability of the resulting ODE can be assessed, predicting training success or failure. As a consequence, networks of Rectified Linear (RLU) and of sigmoidal nonlinearities are shown to have diametrically different properties when it comes to learning attractors. Furthermore, a characteristic time constant, which remains finite at the edge of chaos, offers an explanation of the network's output robustness in the presence of variability of the internal neural dynamics. Finally, the proposed theory predicts state dependent frequency selectivity in network response.