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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.
We review attempts that have been made towards understanding the computational properties and mechanisms of input-driven dynamical systems like RNNs, and reservoir computing networks in particular. We provide details on methods that have been develop
We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor between th
Consistency is an extension to generalized synchronization which quantifies the degree of functional dependency of a driven nonlinear system to its input. We apply this concept to echo-state networks, which are an artificial-neural network version of
Continuous time recurrent neural networks (CTRNN) are systems of coupled ordinary differential equations that are simple enough to be insightful for describing learning and computation, from both biological and machine learning viewpoints. We describ
Echo State Networks (ESNs) are a special type of recurrent neural networks (RNNs), in which the input and recurrent connections are traditionally generated randomly, and only the output weights are trained. Despite the recent success of ESNs in vario