Cardiac arrhythmias are a common cause of sudden death worldwide. However, despite decades of thorough investigation the underlying biophysical mechanisms of cardiac arrhythmias are still... Show moreCardiac arrhythmias are a common cause of sudden death worldwide. However, despite decades of thorough investigation the underlying biophysical mechanisms of cardiac arrhythmias are still insufficiently understood due to incomplete theories and the lack of precise spatiotemporal control in experiments. In the last decade, the problem of insufficient spatiotemporal control has started to be tackled by means of a new technique, called optogenetics. This technique employs expression of light-activated proteins, which are activated or deactivated in time and space by switching on/off light (in the near-ultraviolet to near-infrared wavelength range) in specific patterns thus realizing fully biological spatiotemporal control. However, with a few notable exceptions, cardiac optogenetic studies have only confirmed previously known mechanisms and yielded no or little novel mechanistic insights. In this thesis, to fill this gap, we combined nonlinear dynamics theory, numerical simulations and optogenetic experiments with unique spatiotemporal control to theoretically predict and demonstrate novel arrhythmogenic phenomena in cardiac tissue. Thanks to the robustness of the optogenetics methods and generality of the applied theories and computations, this thesis uncovered novel mechanisms for the biophysics of cardiac tissue that are applicable to the functioning of excitable systems in general. Show less
The presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which... Show moreThe presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which vanishes on the spectrum of the linearization about the pattern. For certain 'slowly linear' prototype models it has been shown, via geometric arguments, that the Evans function factorizes in accordance with the scale separation. This leads to asymptotic control over the spectrum through simpler, lower-dimensional eigenvalue problems. Recently, the geometric factorization procedure has been generalized to homoclinic pulse solutions in slowly nonlinear reaction-di ffusion systems. In this thesis we study periodic pulse solutions in the slowly nonlinear regime. This seems a straightforward extension. However, the geometric factorization method fails and due to translational invariance there is a curve of spectrum attached to the origin, whereas for homoclinic pulses there is only a simple eigenvalue residing at 0. We develop an alternative, analytic factorization method that works for periodic structures in the slowly nonlinear setting. We derive explicit formulas for the factors of the Evans function, which yields asymptotic spectral control. Moreover, we obtain a leading-order expression for the critical spectral curve attached to origin. Together these approximation results lead to explicit stability criteria. Show less