Three axes are explored. 1) Derivation of mathematical models of electrophysiological phenomena applied to cardiology. Asymptotic analysis methods allow to derive simplified models from three-dimensional complex atrial ones, while controlling approximation errors. We construct a bilayer surface model that allows to simulate three-dimensional phenomena for a bi-dimensional computational load, and to investigate 3D atrial patterns involved in atrial arrhythmia such as electrical dissociation or transmural heterogeneities. We prove the convergence of the bilayer model, and an optimization strategy to improve the model outside the asymptotic regime is formalised. Homogeneisation methods are also used to construct a homogenized continuous model of the electrical activity of the myocytes that includes the non linear behavior of gap junctions. 2) Triggers of atrial arrhythmia. Proofs of concept of arrhythmogenic mechanisms are given by using numerical models of the pulmonary veins. The first mechanism is based on a unidirectional conduction block triggered by a discontinuity of the fibre distribution. The second one comes from a different propagation pattern during the depolarization and the repolarization when two layer of fibres are superimposed. 3)Atrial arrhythmia perpetuation. A bilayer model of the atria is constructed from a semi automatic method that we developed. We investigate the influence of transmural heterogeneities of the distribution of fibrosis on the perpetuation of atrial arrhythmia. Several ablation protocols are assessed. Finally, a method of personalization of the model is given.
