A numerical study of the cable equation in mathematical neuroscience

Abstract

The aim of this thesis is to compare di erent numerical methods for solving the cable equation for electrical signal propagation along dendrites with diameter varying in space. We solve the model with four di erent methods: a nite di erence scheme, nite element method, separation of variables combined with a nite di erence scheme and separation of variables combined with the nite element method. The di erent methods gives quite di erent solution even if the solutions main properties are the same. Separation of variables combined with the nite element method o ers a solution of much lower value than the other methods. This can be a result of an overestimate of the eigenvalue of the problem. The nite element method and the method of separation of variables combined with a nite di erence scheme gives almost the exact same solutions, a fact that was to expect during the derivations. The nite di erence scheme is the easiest method to use even if it is important for the schemes consistency how the derivatives was replaced by nite di erences. Finite di erence method is the method that give the less complicated programming in Matlab as well. The solutions for di erent diameter geometry are as expected from the mathematical analysis done in advance. The solutions stays symmetric about the mid point in space if the diameter and the initial condition have the same symmetry. The peak of the solutions for non-symmetric diameters move towards increasing space variable for a decreasing diameter and towards decreasing space variable for an increasing diameter.