  Electrical waves in the heart form rotating spiral or scroll waves during
life-threatening arrhythmias such as atrial or ventricular fibrillation. The
wave dynamics are typically modeled using coupled partial differential
equations, which describe reaction-diffusion dynamics in excitable media. More
recently, data-driven generative modeling has emerged as an alternative to
generate spatio-temporal patterns in physical and biological systems. Here, we
explore denoising diffusion probabilistic models for the generative modeling of
electrical wave patterns in cardiac tissue. We trained diffusion models with
simulated electrical wave patterns to be able to generate such wave patterns in
unconditional and conditional generation tasks. For instance, we explored the
diffusion-based i) parameter-specific generation, ii) evolution and iii)
inpainting of spiral wave dynamics, including reconstructing three-dimensional
scroll wave dynamics from superficial two-dimensional measurements. Further, we
generated arbitrarily shaped bi-ventricular geometries and simultaneously
initiated scroll wave patterns inside these geometries using diffusion. We
characterized and compared the diffusion-generated solutions to solutions
obtained with corresponding biophysical models and found that diffusion models
learn to replicate spiral and scroll waves dynamics so well that they could be
used for data-driven modeling of excitation waves in cardiac tissue. For
instance, an ensemble of diffusion-generated spiral wave dynamics exhibits
similar self-termination statistics as the corresponding ensemble simulated
with a biophysical model. However, we also found that diffusion models {produce
artifacts if training data is lacking, e.g. during self-termination,} and
`hallucinate' wave patterns when insufficiently constrained.
