Memory-Loss is Fundamental for Stability and Distinguishes the Echo State Property Threshold in Reservoir Computing & Beyond

Reservoir computing, a highly successful neuromorphic computing scheme used to filter, predict, classify temporal inputs, has entered an era of microchips for several other engineering and biological applications. A basis for reservoir computing is memory-loss or the echo state property. It is an open problem on how design parameters of the reservoir can be optimized to maximize reservoir freedom to map an input robustly and yet have its close-by-variants represented in the reservoir differently. We present a framework to analyze stability due to input and parameter perturbations and make a surprising fundamental conclusion, that the echo state property is \emph{equivalent} to robustness to input in any nonlinear recurrent neural network that may or may not be in the gambit of reservoir computing. Further, backed by theoretical conclusions, we define and find the difficult-to-describe \emph{input specific} edge-of-criticality or the echo state property threshold, which defines the boundary between parameter related stability and instability.