Computational efficiency of fractional diffusion using adaptive time step memory and the potential application to neural glial networks

Numerical solutions to fractional differential equations can be extremely computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In finite difference methods this has been approximated using the "short memory effect" where it is assumed that previous events prior to some certain time point are insignificant and thus not calculated. Here, an "adaptive time" method is presented for smooth functions that is computationally efficient and results in smaller errors during numerical simulations. Sampled points along the system's history at progressively longer intervals are assumed to reflect the values of neighboring time points. By including progressively fewer points as a function of time, a temporally "weighted" history is computed that includes contributions from the entire past of the system. This results in increased accuracy, and with fewer points actually calculated, also ensures computational efficiency. The end goal is to eventually incorporate this time-saving fractional method into a neural glial network that better-describes the mechanism of extracellular ATP diffusion that results in calcium signaling between astrocyte