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A numerical study of the cable equation in mathematical neuroscience

Larsen, Hilde
Master thesis
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Larsen2012.pdf (2.293Mb)
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http://hdl.handle.net/11250/188845
Utgivelsesdato
2012-09-17
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  • Master's theses (RealTek) [1402]
Sammendrag
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.
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Norwegian University of Life Sciences, Ås

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