Simple model for ${1/f}^{\alpha }$ noise

We present a simple stochastic mechanism which generates pulse trains exhibiting a power-law distribution of the pulse intervals and a ${1/f}^{\alpha }$ power spectrum over several decades at low frequencies with α close to 1. The essential ingredient of our model is a fluctuating threshold which performs a Brownian motion. Whenever an increasing potential $V\left(t\right)\mathrm{}$ hits the threshold, $V\left(t\right)\mathrm{}$ is reset to the origin and a pulse is emitted. We show that if $V\left(t\right)\mathrm{}$ 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 ${1/f}^{\alpha }$ noise observed in cortical neurons and earthquake data.