  Recurrent networks with transfer functions that fulfill the Lipschitz
continuity with K=1 may be echo state networks if certain limitations on the
recurrent connectivity are applied. It has been shown that it is sufficient if
the largest singular value of the recurrent connectivity is smaller than 1. The
main achievement of this paper is a proof under which conditions the network is
an echo state network even if the largest singular value is one. It turns out
that in this critical case the exact shape of the transfer function plays a
decisive role in determining whether the network still fulfills the echo state
condition. In addition, several examples with one neuron networks are outlined
to illustrate effects of critical connectivity. Moreover, within the manuscript
a mathematical definition for a critical echo state network is suggested.
