Fractional Cable Equation Models for Anomalous Electrodiffusion in Nerve Cells: Finite Domain Solutions

Abstract

In recent work we introduced fractional Nernst–Planck equations and related fractional cable equations to model electrodiffusion of ions in nerve cells with anomalous subdiffusion along and across the nerve cells. This work was motivated by many computational and experimental studies showing that anomalous diffusion is ubiquitous in biological systems with binding, crowding, or trapping. For example, recent experiments have shown that anomalous subdiffusion occurs along the axial direction in spiny dendrites due to trapping by the spines. We modeled the subdiffusion in two ways leading to two fractional cable equations and presented fundamental solutions on infinite and semi-infinite domains. Here we present solutions on finite domains for mixed Robin boundary conditions. The finite domain solutions model passive electrotonic properties of spiny dendritic branch segments with ends that are voltage clamped, sealed, or killed. The behavior of the finite domain solutions is similar for both fractional cable equations. With uniform subdiffusion along and across the nerve cells, the solution approaches the standard nonzero steady state, but the approach is slowed by the anomalous subdiffusion. If the subdiffusion is more anomalous along the axial direction, then (boundary conditions permitting) the solution converges to a zero steady state, whereas if the subdiffusion is less anomalous along the axial direction, then the solution approaches a spatially linear steady state. These solutions could be compared with realistic electrophysiological experiments on actual dendrites.