Spiking at the edge
Excitable media, ranging from bioelectric tissues and chemical oscillators to forest fires and competing populations, are nonlinear, spatially extended systems capable of spiking. Most investigations of excitable media consider situations where the amplifying and suppressing forces necessary for spiking coexist at every point in space. In this case, spiking requires a fine-tuned ratio between local amplification and suppression strengths. But, in Nature and engineered systems, these forces can be segregated in space, forming structures like interfaces and boundaries. Here, we show how boundaries can generate and protect spiking if the reacting components can spread out: even arbitrarily weak diffusion can cause spiking at the edge between two non-excitable media. This edge spiking is a robust phenomenon that can occur even if the ratio between amplification and suppression does not allow spiking when the two sides are homogeneously mixed. We analytically derive a spiking phase diagram that depends on two parameters: (i) the ratio between the system size and the characteristic diffusive length-scale, and (ii) the ratio between the amplification and suppression strengths. Our analysis explains recent experimental observations of action potentials at the interface between two non-excitable bioelectric tissues. Beyond electrophysiology, we highlight how edge spiking emerges in predator-prey dynamics and in oscillating chemical reactions. Our findings provide a theoretical blueprint for a class of interfacial excitations in reaction-diffusion systems, with potential implications for spatially controlled chemical reactions, nonlinear waveguides and neuromorphic computation, as well as spiking instabilities, such as cardiac arrhythmias, that naturally occur in heterogeneous biological media.