Transitions in echo index and dependence on input repetitions
Peter Ashwin Andrea Ceni
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.