Stable bound states of pulses in an excitable medium

The interaction of one-dimensional pulses is studied in the excitable regime of a two variable reaction–diffusion model. The model is capable of exhibiting long range attraction of pulses and formation of stable bound pulse states. The important features of pulse interactions can be captured by a combination of various analytical and numerical methods. A kinematic ansatz treating pulses as particle-like interacting structures is described. Their interaction is determined using the dispersion relation for pulse trains, which gives the dependence of the speed c(d) of the wavetrain on its wavelength d. Anomalous dispersion for large d, i.e. a negative slope of c(d), corresponds to long range pulse attraction. Stable bound pairs are possible if the medium exhibits long range attraction and there is at least one maximum of the dispersion curve. We compare predictions of the kinematic theory with numerical simulations and stability analysis. If the slope of the dispersion curve changes sign, branches of non-equidistant pulse train solutions bifurcate and may lead to bound pulse states. The transition from normal long range dispersion, typical in excitable media, to the anomalous dispersion studied here can be understood through a multiscale perturbation theory for pulse interactions. We derive the relevant equations, which yield an analytic expression for non-monotonic dispersion curves with a finite number of extrema.