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 $1/ \sqrt{N}$. 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 $t = a {N \over \tau} \exp(b \tau N^{1/2})$, where $b$ is a constant which depends on the amplitude of the external noise. This result is obtained analytically for small $\tau$ and confirmed by numerical simulations.