Curvature-dependent onset of oscillations in excitable tissue

In cardiac tissue, the sinoatrial node (SAN) is responsible for initiating the periodic electrical pulses underlying heart beats. However, other heterogeneities (e.g., ischemic regions) can act as rogue pacemakers and produce oscillations in neighboring tissue that compete with the natural pacemaking of the SAN and cause potentially life-threatening arrhythmia. Thus, it is important to understand the physiological conditions that enable local regions of tissue to form pathological rhythms. It is well known that a small heterogeneity (a source) should not be able to easily activate a large area of excitable tissue (a sink). On a local level, this source-sink balance implies that positive curvature of a pacemaking region reduces the source-sink ratio and the ability to drive the neighboring tissue. However, while numerous studies provide evidence that supports the source-sink balance relationship, other studies have shown that for some depolarized heterogeneities, oscillations preferentially emerge from corners and other areas of high curvature. Here, we use an idealized two-domain reaction-diffusion system and a corresponding two-cell model to bridge the gap between these seemingly opposing viewpoints. In doing so, we identify the conditions for which curvature of a pacemaking region promotes or obstructs the production of oscillations in the neighboring tissue. Through our findings, we argue that the seemingly opposing views are, in fact, not contradictory, and the standard notion of source-sink balance is upheld, as long as we adapt a modified description of source and sink.