Fractional Calculus and Memory in Biophysical Time Series

Abstract

The simplest and probably the most familiar model of statistical processes in the physical sciences is the random walk. This model has been applied to all manner of biophysical phenomena, ranging from DNA sequences to the firing of neurons. Herein we extend the random walk model to include long-time memory in the dynamics and find that this gives rise to a fractional-difference stochastic process. The continuum limit of this latter random walk is a fractional Langevin equation, that is, a fractional differential equation driven by random fluctuations. We show that the index of the inverse power-law spectrum in many biophysical processes can be related to the order of the fractional derivative in the fractional Langevin equation. This fractional stochastic model suggests that a scale-free process guides the dynamics of many complex biophysical phenomena.