Noisy time series generation by feed-forward networks

TL;DR

This paper investigates how noise affects the dynamics of a feed-forward network generating time series, revealing that noise-induced broadening of attractors diminishes with system size and influences phase coherence and switching times.

Contribution

It provides analytical and numerical insights into how noise impacts attractor properties and switching times in noisy time series generated by feed-forward networks.

Findings

Attractor broadening scales as 1/√N with system size N.

Phase diffusion constant scales inversely with N.

Mean first passage time depends exponentially on N and bifurcation distance.

Abstract

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.