Continuation methods: application to a biophysical problem

The goal of this project is to study continuation methods and to learn how to implement them numerically. We will review the main points of the procedure and we will try to replicate these results numerically. Starting with some concepts learned during the degree like definitions or theorems, for example the "Implicit function Theorem", we will study basic concepts about curves to use them in solution techniques applying deformation, as well as conditions they have to satisfy to prevent the curve from running to infinity. Assuming that, we will undergo a study about predictor-corrector(PC) methods, which are the main objective of this project, understanding how they work and programming them in a software, in our case, MATLAB. Finally we will take advantage of the codes that we have programmed, to apply PC methods in a more particular case and draw some conclusions of it. The case that we have considered is a model of a paced cardiac cell with cytosolic and luminal spaces, each separated into different compartments,i.e dyadic, subsarcolemmal and cytosolic. We calculate the type of solution that we find as we change the total amount of calcium in the system and we discuss some conclusions of these solutions.