Kinematic Analysis of Wave Pattern Formation in Excitable Media

A homogeneous excitable medium supports the steady propagation of impulse waves of excitation in response to a spatially localized stimulus. In a one-dimensional medium, for which the nerve axon is a prototype, a periodic stimulus produces a train of impulse waves which propagates away from the stimulus site. Successive impulses in the train interact with each other during propagation and consequently form a spatio-temporal wave pattern. When an equally spaced impulse train, whose temporal period equals the period of the stimulus, develops as a consequence of the interaction, we say the medium is entrained to the input stimulus. In some cases however, the impulses exhibit bunching during propagation to form an unequally spaced impulse train. Here we investigate these phenomena by a kinematic analysis based upon an approximation of the wave interaction. We also illustrate that, in certain parameter ranges, the one-dimensional medium can support a center wave pattern in the absence of a maintained stimulus. This wave pattern is a one-dimensional analog of the target wave pattern which is observed in some two-dimensional excitable media such as the Belousov-Zhabotinskii reagent in a shallow dish.