  In recent years, more and more researchers in the field of neural networks
are interested in creating hardware implementations where neurons and the
connection between them are realized physically. The physical implementation of
ANN fundamentally changes the features of noise influence. In the case hardware
ANNs, there are many internal sources of noise with different properties. The
purpose of this paper is to study the peculiarities of internal noise
propagation in recurrent ANN on the example of echo state network (ESN), to
reveal ways to suppress such noises and to justify the stability of networks to
some types of noises.
  In this paper we analyse ESN in presence of uncorrelated additive and
multiplicative white Gaussian noise. Here we consider the case when artificial
neurons have linear activation function with different slope coefficients.
Starting from studying only one noisy neuron we complicate the problem by
considering how the input signal and the memory property affect the
accumulation of noise in ESN. In addition, we consider the influence of the
main types of coupling matrices on the accumulation of noise. So, as such
matrices, we take a uniform matrix and a diagonal-like matrices with different
coefficients called "blurring" coefficient.
  We have found that the general view of variance and signal-to-noise ratio of
ESN output signal is similar to only one neuron. The noise is less accumulated
in ESN with diagonal reservoir connection matrix with large "blurring"
coefficient. Especially it concerns uncorrelated multiplicative noise.
