Exponential dispersion relation and its effects on unstable propagating pulses in unidirectionally coupled symmetric bistable elements

► We show the dispersion relation in a traveling pulse in unidirectionally coupled bistable elements. ► We derive a kinematical description of the propagation of transient pulses. ► The duration of transient pulses increases exponentially with the number of elements. ► Noise sustains the propagation of pulses. ► Properties of pulse transmission qualitatively differ from those in other bistable systems.