Conservation laws in coupled multiplicative random arrays lead to $1/f$ noise

Abstract

We consider the dynamic evolution of a coupled array of N multiplicative random variables. The magnitude of each is constrained by a lower bound w_0 and their sum is conserved. Analytical calculation shows that the simplest case, N=2 and w_0=0, exhibits a Lorentzian spectrum which gradually becomes fractal as w_0 increases. Simulation results for larger $N$ reveal fractal spectra for moderate to high values of w_0 and power-law amplitude fluctuations at all values. The results are applied to estimating the fractal exponents for cochlear-nerve-fiber action-potential sequences with remarkable success, using only two parameters.