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.

如果对于给定的输入序列，递归神经网络（RNN）“忘记”了驱动（非自治）系统的任何内部状态，并且渐近地遵循唯一的，可能是复杂的轨迹，则它具有回波状态属性（ESP）。传统上将ESP的缺乏理解为RNN中缺乏可靠的行为。在这里，我们表明RNN可以在更通用的原则下可靠地执行计算，该原则仅考虑其在相空间中的局部行为。为此，我们制定了ESP的概括，并引入了回波索引来表征驱动RNN的同时稳定响应的数量。我们表明，回波索引可能随输入而变化，这突出了由于驱动动力学的输入特性而导致的RNN中计算错误的潜在来源。