Transitions in echo index and dependence on input repetitions
Peter Ashwin, Andrea Ceni · Physica D Nonlinear Phenomena · 2024
The echo index counts the number of simultaneously stable asymptotic responses of a nonautonomous (i.e. input-driven) dynamical system. It generalises 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. • We discuss a generalisation of the “echo state property” to the “echo index” enumerating stable responses of an input-driven discrete time dynamical system. • For inputs that take only a finite set of values, lower bounds on the echo index are given in terms of attractors of the individual maps and input repetitions. • We explore changes in the echo index for an example of a simple recurrent neural network on changing repetition properties for the input sequence.