A Numerical Simulation of Anomalous Electro-Diffusion of Ions in Spiny Dendrites Using a Local Petrov-Galerkin Method

The cable equation is one the most fundamental mathematical models in the neuroscience, which describes the electro-diffusion of ions in denderits. New findings indicate that the standard cable equation is inadequate for describing the process of electro-diffusion of ions. So, recently, the cable model has been modified based on the theory of fractional calculus. In this paper, the two dimensional time fractional nonlinear cable equation as an improved mathematical model in neuronal dynamics, is investigated numerically. An efficient and powerful computational technique based on the combination of time integration scheme and local weak form meshfree method has been formulated and implemented to solve the underlying problem. An implicit difference scheme with second order accuracy is used to discretize the model in the temporal direction. Then a meshless method based on the local Petrov-Galerkin technique is employed to fully discretize the model. The proposed numerical technique is performed to approximate the solutions of three examples. Presented results through the Tables and figures confirm the high efficiency and accuracy of the method.