Spiral waves are observed in a variety of natural systems, including cardiac arrhythmias and chemical oscillations, both of which are commonly modeled with reaction-diffusion systems. We are interested in the stability of spiral waves, and how spectral properties of spirals on bounded domains can provide insight into understanding the instabilities from a mathematical perspective.
Ventricular tachycardia, a dangerous fast-paced heart rate, is a result of a sustained spiral wave rotating on the surface of the heart. After the spiral destabilizes, unorganized electrical activity can lead to ventricular fibrillation and sudden cardiac arrest (SCA) – the leading natural cause of death in the US. Clinically, the onset of SCA has been linked to the alternans instability, a beat-to-beat alteration in the action potential duration of the spiral bands. We seek to understand the role of spectra in the formation of the alternans instability.
Related Poster: Spectral Properties of Spiral Waves in the Karma Model, SIAM Conference on Applications of Dynamical Systems, May 2017, Snowbird, UT.
Reaction-diffusion systems often have one or more diffusion-less species (such as ion-channel models). Numerical results indicate that removing diffusion from a slowly diffusing component dramatically changes the spectra of a 2D spiral; a property that is not observed in 1D. The figure shows the change in the spectra of a spiral in the Barkley model from the slow species slightly diffusing (δ = 0.2) to not diffusing (δ = 0). We are working to analytically and numerically understand why these differences occur and what the implications are.
Spiral waves in the Rossler Model exhibit stationary line defects. It has been hypothesized that these defects emerged from instabilities of the boundary sink, but until now, this theory has not been directly tested. A comparison of the spectra of spiral waves and the boundary sinks suggests that the line defects are a result of unstable point eigenvalues from the boundary.
In collaboration with the Cohen Lab at Harvard University, we developed a PDE model for isradipine Optopatch Spiking Human Embrionic Kidney Cells (iOS-HEK cells), a synthetic excitable tissue. The iOS-HEK cells have simple and well understood ionic channels, but exhibit complex electrical activity similar to cardiac cells. Additionally, the model and experiment highlight how the geometry of the tissue impacts the behavior of propagating electrical waves.
Preprint available on bioRxiv:
Dr. Sandstede's reseach group participates in weekly meetings and each semester discusses a topic in dynamical systems. The topics covered are below, as well as my contributions.
|Spring 2016||Nonlinear Waves: Spatial Dynamics and Fredholm Approaches|
|Fall 2016||Vegetation Patterns||Localised pattern formation in a model for dryland vegetation||Slides|
|Spring 2017||Data Science||Data Science in Cooking||Slides|
|Fall 2017||Dynamics and Statistics||Introduction to Parameter Estimation|
|Spring 2018||Probability and Statistics||Information Theory|