  The heartbeat is mediated between cardiac cells by waves of electrical
depolarisation. During cardiac arrhythmias, electrical activity was found to be
organised in scroll waves which rotate around a dynamical filament curve. In
this thesis, a curved-space approach is used to mathematically capture
anisotropy of wave propagation. We derive for the first time the covariant laws
of motion for traveling wave fronts and scroll wave filaments in anisotropic
excitable media such as cardiac tissue. We show that locally varying anisotropy
yields non-zero Riemann tensor components, which may alter the stability of
scroll wave filaments. The instability of scroll wave filaments has been linked
to transition from ventricular tachycardia to fibrillation.
