  A recurrent neural network (RNN) possesses the echo state property (ESP) if,
for a given input sequence, it ``forgets'' any internal states of the driven
(nonautonomous) system and asymptotically follows a unique, possibly complex
trajectory. The lack of ESP is conventionally understood as a lack of reliable
behaviour in RNNs. Here, we show that RNNs can reliably perform computations
under a more general principle that accounts only for their local behaviour in
phase space. To this end, we formulate a generalisation of the ESP and
introduce an echo index to characterise the number of simultaneously stable
responses of a driven RNN. We show that it is possible for the echo index to
change with inputs, highlighting a potential source of computational errors in
RNNs due to characteristics of the inputs driving the dynamics.
