  In recent years, more and more works have appeared devoted to the analog
(hardware) implementation of artificial neural networks, in which neurons and
the connection between them are based not on computer calculations, but on
physical principles. Such networks offer improved energy efficiency and, in
some cases, scalability, but may be susceptible to internal noise. This paper
studies the influence of noise on the functioning of recurrent networks using
the example of trained echo state networks (ESNs). The most common reservoir
connection matrices were chosen as various topologies of ESNs: random uniform
and band matrices with different connectivity. White Gaussian noise was chosen
as the influence, and according to the way of its introducing it was additive
or multiplicative, as well as correlated or uncorrelated. In the paper, we show
that the propagation of noise in reservoir is mainly controlled by the
statistical properties of the output connection matrix, namely the mean and the
mean square. Depending on these values, more correlated or uncorrelated noise
accumulates in the network. We also show that there are conditions under which
even noise with an intensity of $10^{-20}$ is already enough to completely lose
the useful signal. In the article we show which types of noise are most
critical for networks with different activation functions (hyperbolic tangent,
sigmoid and linear) and if the network is self-closed.
