These equations have an explicit dependence on the reservoir weight matrix W, the input
weight matrix V, the autocorrelation of input signal ui ⊗ uj, and correlation of input and output
ui ⊗ y
t, at various time scales. In the specific case of training the network to reproduce its
inputs from τ time steps ago, these can be analytically evaluated. Let ut be random variables
drawn from identical and independent uniform distributions on the interval [−1, 1]. Then we
have: