Noisy time series generation by feed-forward networks

We study the properties of a noisy time series generated by a continuous-valued feed-forward network in which the next input vector is determined from past output values. Numerical simulations of a perceptron-type network exhibit the expected broadening of the noise-free attractor, without changing the attractor dimension. We show that the broadening of the attractor due to the noise scales inversely with the size of the system, N, as 1/root N. We show both analytically and numerically that the diffusion constant for the phase along the attractor scales inversely with N. Hence, phase coherence holds up to a time that scales linearly with the size of the system. We find that the mean first passage time, t, to switch between attractors depends on N, and the reduced distance from bifurcation tau as t = a N/tau exp(brN(1/2)), where b is a constant which depends on the amplitude of the external noise. This result is obtained analytically for small tau and is confirmed by numerical simulations.