  Echo state networks (ESNs) are a powerful form of reservoir computing that
only require training of linear output weights whilst the internal reservoir is
formed of fixed randomly connected neurons. With a correctly scaled
connectivity matrix, the neurons' activity exhibits the echo-state property and
responds to the input dynamics with certain timescales. Tuning the timescales
of the network can be necessary for treating certain tasks, and some
environments require multiple timescales for an efficient representation. Here
we explore the timescales in hierarchical ESNs, where the reservoir is
partitioned into two smaller linked reservoirs with distinct properties. Over
three different tasks (NARMA10, a reconstruction task in a volatile
environment, and psMNIST), we show that by selecting the hyper-parameters of
each partition such that they focus on different timescales, we achieve a
significant performance improvement over a single ESN. Through a linear
analysis, and under the assumption that the timescales of the first partition
are much shorter than the second's (typically corresponding to optimal
operating conditions), we interpret the feedforward coupling of the partitions
in terms of an effective representation of the input signal, provided by the
first partition to the second, whereby the instantaneous input signal is
expanded into a weighted combination of its time derivatives. Furthermore, we
propose a data-driven approach to optimise the hyper-parameters through a
gradient descent optimisation method that is an online approximation of
backpropagation through time. We demonstrate the application of the online
learning rule across all the tasks considered.
