  Echo state networks (ESN), a type of reservoir computing (RC) architecture,
are efficient and accurate artificial neural systems for time series processing
and learning. An ESN consists of a core of recurrent neural networks, called a
reservoir, with a small number of tunable parameters to generate a
high-dimensional representation of an input, and a readout layer which is
easily trained using regression to produce a desired output from the reservoir
states. Certain computational tasks involve real-time calculation of high-order
time correlations, which requires nonlinear transformation either in the
reservoir or the readout layer. Traditional ESN employs a reservoir with
sigmoid or tanh function neurons. In contrast, some types of biological neurons
obey response curves that can be described as a product unit rather than a sum
and threshold. Inspired by this class of neurons, we introduce a RC
architecture with a reservoir of product nodes for time series computation. We
find that the product RC shows many properties of standard ESN such as
short-term memory and nonlinear capacity. On standard benchmarks for chaotic
prediction tasks, the product RC maintains the performance of a standard
nonlinear ESN while being more amenable to mathematical analysis. Our study
provides evidence that such networks are powerful in highly nonlinear tasks
owing to high-order statistics generated by the recurrent product node
reservoir.
