1/f Noise & Self-Organized Criticality

The ubiquity of 1/f^a noise in nature is one of the oldest problems in contemporary physics still lacking a generally accepted explanation, despite much effort. The phenomenon is characterized by a 1/f^a decay with a nontrivial a found in the power spectrum of a given time signal at low frequencies. In many cases, the time signal describes a transport process. Then 1/f^a noise is an indication of anomalous behavior compared to conventional transport, as for instance, equilibrium diffusion. Flicker noise has been observed in a huge number of diverse systems, many of which are far from equilibrium and show a bursty, avalanche dynamics. Examples include earthquakes, combustion fronts, chemical reactions, flux motion in superconductors, and Barkhausen noise, to name only a few.

Davidsen & Paczuski (2002) demonstrated explicitly that the concept of self-organized criticality can provide a dynamical mechanism which gives correlations between bursts leading to 1/f^a noise. The model studied exhibits a power-law distribution of avalanches, as well as 1/f^a fluctuations in the pattern of dissipation over the slow temporal domain of the external drive. Despite the fact that the time scales of the driving and of the avalanche events are completely separated, the critical behavior of the power spectrum is solely determined by the critical properties of the avalanche size distribution. They are linked by a scaling relation. These observations constitute a proof of principle that self-organized criticality mechanism can give low-frequency 1/f^a noise due to correlations between power-law distributed avalanches, without imposing temporal correlations in the external drive. 

Davidsen & Schuster (2002) presented a simple stochastic mechanism which generates pulse trains exhibiting a power-law distribution of the pulse intervals and a 1/f^a power spectrum over several decades at low frequencies with a close to 1. The essential ingredient of our model is a fluctuating thresh which performs a Brownian motion. Whenever an increasing potential V(t) hits the thresh, V(t) is reset to the origin and a pulse is emitted. We show that if V(t) increases linearly in time, the pulse intervals can be approximated by a random walk with multiplicative noise. Our model agrees with recent experiments in neurobiology and explains the high interpulse interval variability and the occurrence of 1/f^a noise observed in cortical neurons and earthquake data.