Dynamics and Stability of Spatially Extended Patterns

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This research project aims to further our understanding of cardiac arrhythmias by analysing the properties of spiral waves, which are anomalous electrochemical waves in cardiac tissue that may lead to fibrillation. A particular focus will be on studying how robust spiral waves behave under spatial perturbations in specific model systems. In addition, the principal investigator will study the role of different mechanisms that can lead to complex spatio-temporal dynamics in models of cardiac tissue. The project will include research activities that will train both graduate and undergraduate students.The project focuses on the spectral and linear stability of spiral waves in reaction-diffusion systems. Building on earlier work in one spatial dimension, the goal is to elucidate spatio-temporal responses to spatial perturbations of spiral waves in the core and far field. The main approach will involve deriving estimates of the resolvent kernel using spatial dynamics and the construction of the spatio-temporal Green's function about a spiral wave. The second part of the project will focus on an analytical approach to backfiring and pulse-splitting in the FitzHugh-Nagumo model using steady-state approximations and geometric singular perturbation theory.