Transitions in echo index and dependence on input repetitions

TL;DR

This paper explores how the echo index, indicating the number of stable responses in input-driven dynamical systems, varies with input parameters and amplitude, revealing complex dependencies in neural network models.

Contribution

It provides a theoretical analysis of the echo index's dependence on input repetitions and amplitude in nonautonomous systems with multiple attractors, extending understanding of input-driven dynamics.

Findings

For small forcing amplitude, the echo index equals the number of attractors.

For large forcing amplitude, the echo index reduces to one.

Intermediate forcing regimes show complex dependencies on input properties.

Abstract

The echo index counts the number of simultaneously stable asymptotic responses of a nonautonomous (i.e. input-driven) dynamical system. It generalizes the well-known echo state property for recurrent neural networks - this corresponds to the echo index being equal to one. In this paper, we investigate how the echo index depends on parameters that govern typical responses to a finite-state ergodic external input that forces the dynamics. We consider the echo index for a nonautonomous system that switches between a finite set of maps, where we assume that each map possesses a finite set of hyperbolic equilibrium attractors. We find the minimum and maximum repetitions of each map are crucial for the resulting echo index. Casting our theoretical findings in the RNN computing framework, we obtain that for small amplitude forcing the echo index corresponds to the number of attractors for the input-free system, while for large amplitude forcing, the echo index reduces to one. The intermediate regime is the most interesting; in this region the echo index depends not just on the amplitude of forcing but also on more subtle properties of the input.