Laws of mutual spiral wave interaction in excitable media

Abstract

Interacting rotating spiral waves have been observed in complex systems, such as cardiac fibrillation, cognitive processing in the brain cortex and oscillating chemical reactions, during dynamical regimes that are still poorly understood. We present the equivalent of Newton's gravitational attraction law for spiral waves on planar reaction-diffusion systems. The spiral waves' phases and positions determine their regions of influence, separated by collision interfaces. At the collision interfaces, wave front deflections cause spiral drift that pushes the interfaces forward. As a result, the spiral wave drift velocity is proportional to the total force exerted on on it, which can be determined by a boundary integral over its region of influence. The proportionality factor between force and response is akin to the `mass' of the spiral. However, this spiral mass depends on the region of influence of the spiral and thus also varies over time. The forces between spiral wave pairs are not directed along the line connecting their centers, violating Newton's law of action and reaction. Our solution to the N-body interaction problem for spirals in extended excitable media encompasses both pairwise interactions and spiral wave drift in bounded domains, with application to cardiac fibrillation.

Figures (4)

• Figure 1: Examples of multiple interacting spirals in excitable media, in simulations Aliev:1996 (a-d) and in vitro experiments Harlaar:2021 (e-f). (a) Pair of oppositely rotating spirals, equivalent to a single spiral near a planar boundary. (b) Spiral pair with the same rotation sense. (c) Multiple spiral simulation. (d) Spiral wave simulation on a curved surface with the geometry of the atria of a human heart. (e) Culture of conditionally immortalized human atrial myocytes showing a spiral pair. (f) Same set-up, with multiple interacting spirals.
• Figure 2: Unraveling complex excitation pattern by segmenting the pattern of Fig. \ref{['fig:intro']}c into subdomains $\Omega_j$, each containing one spiral wave. The two waves meeting at the collision interfaces enclose the same angle $\beta$ at each side of the interface, which follows from the fact that their projected speeds along the interface must be equal.
• Figure 3: Velocity components for the case of 2-spiral interaction: symmetric cases (a-b) vs. asymmetric cases (c-d). (a) Oppositely rotating spirals with the same rotation phase. (b) Same-chirality spirals with opposite phases also create a straight collision interface. (c) Oppositely rotating spirals with different rotation phase. The spiral lagging ahead occupies a larger region of influence. (d) Same-chirality spirals with different rotation phases. In cases (c) and (d), the reaction of one spiral to the other is not reciprocal, violating Newton's third law.
• Figure 4: Representation of the multiple spiral state from Fig. \ref{['fig:intro']}c as a graph (blue). Every vertex represents a spiral center and a connecting edge is added if they interact directly, i.e. share a collision interface.