  Supralinear and sublinear pre-synaptic and dendritic integration is
considered to be responsible for nonlinear computation power of biological
neurons, emphasizing the role of nonlinear integration as opposed to nonlinear
output thresholding. How, why, and to what degree the transfer function
nonlinearity helps biologically inspired neural network models is not fully
understood. Here, we study these questions in the context of echo state
networks (ESN). ESN is a simple neural network architecture in which a fixed
recurrent network is driven with an input signal, and the output is generated
by a readout layer from the measurements of the network states. ESN
architecture enjoys efficient training and good performance on certain
signal-processing tasks, such as system identification and time series
prediction. ESN performance has been analyzed with respect to the connectivity
pattern in the network structure and the input bias. However, the effects of
the transfer function in the network have not been studied systematically.
Here, we use an approach tanh on the Taylor expansion of a frequently used
transfer function, the hyperbolic tangent function, to systematically study the
effect of increasing nonlinearity of the transfer function on the memory,
nonlinear capacity, and signal processing performance of ESN. Interestingly, we
find that a quadratic approximation is enough to capture the computational
power of ESN with tanh function. The results of this study apply to both
software and hardware implementation of ESN.
