Hyperscroll dynamics: Vortices in four-dimensional networks

We investigate a network of excitable nodes diffusively coupled to their neighbors along four orthogonal directions. This regular network effectively forms a four-dimensional reaction-diffusion system and has rotating wave solutions. We analyze some of the general features of these hyperscroll waves, which rotate around surfaces such as planes, spheres, or tori. The surfaces evolve according to local curvatures and a system-specific surface tension. They have associated local phases and phase gradients tend to decrease over time. We also discuss the robustness of these network states against the removal of random node connections and report an example of hyperscroll turbulence.