General mechanism for the 1/f noise

Abstract

We consider the response of a memoryless nonlinear device that converts an input signal ξ(t) into an output η(t) that only depends on the value of the input at the same time, t. For input Gaussian noise with power spectrum 1/f α , the nonlinearity modifies the spectral index of the output togive a spectrum that varies as 1/f α with α = α. We show that the value of α depends on the nonlinear transformation and can be tuned continuously. This provides a general mechanism for the ubiquitous '1/f ' noise found in nature.PACS numbers: 05.40. Ca, 05.10.Gg, 05.45.Tp In a very wide variety of natural systems, temporal fluctuations in observables are found to be long-ranged, with an approximately 1/f α divergence in the powerspectrum at small frequencies f . When the exponent α lies between 1 and 2, this is generally termed '1/f ' noise, and understanding the origin of such fluctuations as well as its ubiquity has been a long-standing problem in physics The difficulty of generating non-integer values of α for the noise exponent within linear response theory has contributed to keeping this problem somewhat enigmatic and unresolved. In the past nine decades that the problem has been extant, several different explanations have been proposed (see below), but none of these have been fully satisfactory. The search for a general mechanism of the 1/f noise was the main motivation behind the proposal of self-organized criticality (SOC) by Bak, Tang and Wiesenfeld in 1987 [13] that yields long-ranged correlations in time and thus can generate 1/f α dependence in the power spectrum of fluctuations. Several SOC models have been studied from this viewpoint [14][15][16][17][18][19], but since 1/f α noise may be found in models with only a small number of degrees of freedom, clearly self-organized criticality is not a necessary condition for obtaining 1/f noise.The 1/f α spectrum, with α ≥ 1 is not integrable near f = 0, and the variance of the signal would be infinite, if there were no cutoffs. In practice, one finds that if the signal is studied for a duration T , then if one doubles * Present address: Department of Physics, Indian Institute of Science, Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411 008, India the duration, the observed mean of the signal appears to drift, and the variance of the signal increases with T . Thus, if the lower cutoff on the frequencies is ∼ 1/T , the net power in the signal increases as T a , where a = α − 1. Our main observation in this paper is easily described. We consider a discrete-time Gaussian stochastic process ξ(t), which has the power-law spectral densitywith a lower-cutoff 1/T . The variance of the signal (which is also equal to the total power P ξ in the signal) increases as T α−1 when the cutoff T is increased. We generate the output signal η(t) by applying a instantaneous nonlinear transform to the input signal,where the function R is typically sigmoidal. A representative example of such a transform is given byThen the total power in the output process is P η = |η(t)| ...

Hopf Bifurcation in Mean Field Explains Critical Avalanches in Excitation-Inhibition Balanced Neuronal Networks: A Mechanism for Multiscale Variability

“…Figure 3C shows that the distance D is smallest when the network is poised near the Hopf bifurcation point predicted by the mean-field theory. Next, we confirmed the statistical significance of power-law distribution through truncated K-S test (see Materials and Methods) for dynamic states sufficiently close to the minimum of D. The existence of power-law distribution is only partial evidence of criticality, as other mechanisms could generate power-law distribution (Yadav et al, 2017). Finally, we further examined the scaling relation (Sethna et al, 2001) in the critical state (see Materials and Methods).…”

Linking space-time correlations for a class of self-organized critical systems

“…The SOC refers that a class of nonequilibrium systems respond nonlinearly in the form of critical avalanches when driven slowly. The response exhibits scaling in the form of power-law distribution for the avalanche sizes as well 1/f α type power spectral density [14][15][16][17][18][19][20][21][22]. Note that the emergence of such a critical state is spontaneous, caused by self-organization.…”

Scale invariance in a nonvibrating magnetic granular system

“…Often, this global behavior of complex systems is characterized by noise and multiple generating mechanisms have been proposed to model these fractal time series. Examples of such mechanisms include intermittency 39 , electronic circuits 40 , self-organized criticality 41 , lognormal distributions 42 , multiscaled randomness 43 , Fourier filters 38 , random matrix theory 36 , memoryless nonlinear transformations 44 , and anomalous diffusion in complex media 45 . Most of the proposed mechanisms are based on theoretical models, whereas the system which we studied here offers the possibility to experimentally generate time series that can smoothly transition over a wide range of scaling exponent values.…”