Local Dynamics in Trained Recurrent Neural Networks

TL;DR

This paper develops a mean field theory for trained recurrent neural networks, revealing how their dynamics near attractors can be described by simple differential equations, which helps predict stability and response characteristics.

Contribution

The authors introduce a mean field theoretical framework for analyzing trained RNNs with multiple attractors, highlighting differences based on nonlinearities and predicting network stability and response features.

Findings

Dynamics near attractors are governed by low order linear ODEs.

Stability analysis predicts training success or failure.

Different nonlinearities lead to distinct attractor properties.

Abstract

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.