ar X iv : 0 80 7 . 15 14 v 1 [ nl in . P S ] 9 J ul 2 00 8 epl draft Generation of finite wave trains in excitable media

Spatiotemporal control of excitable media is of paramount importance in the development of new applications, ranging from biology to physics. To this end we identify and describe a qualitative property of excitable media that enables us to generate a sequence of traveling pulses of any desired length, using a one-time initial stimulus. The wave trains are produced by a transient pacemaker generated by a one-time suitably tailored spatially localized finite amplitude stimulus, and belong to a family of fast pulse trains. A second family, of slow pulse trains, is also present. The latter are created through a clumping instability of a traveling wave state (in an excitable regime) and are inaccessible to single localized stimuli of the type we use. The results indicate that the presence of a large multiplicity of stable, accessible, multi-pulse states is a general property of simple models of excitable media. Excitable media are characterized by a large (finite amplitude) response to a supra-threshold perturbation of the rest state, followed by decay back to the same rest state [1]. Such behavior is frequently found in biological [2], chemical [3], and physical [4] systems. As suggested independently by FitzHugh and Nagumo [5], temporal excitable behavior can be understood qualitatively via a prototype two-component ordinary differential equation of Bonhoeffer-van der Pol type with a cubic nonlinearity in the activator field [1]. In spatially extended excitable media with activator diffusion, such a perturbation results in the formation of a solitary traveling wave, hereafter referred to as a traveling pulse [1]. Since the medium returns to the rest state after the passage of the pulse, repeated stimulation is required to generate a sequence of pulses [6, 7]. Here we identify a novel property of excitable systems of this type that allows us to generate economically sequences of traveling pulses through a onetime initial stimulus, whose shape controls the number of pulses within the wavetrain. In a comoving frame, the one-dimensional (1D) profile of a traveling pulse corresponds, in many systems, to a Shil’nikov homoclinic orbit [8–12]. Such orbits are known to be accompanied by (an infinite number of) additional multi-pulse homoclinic orbits [7, 9, 13], and these in turn correspond to spatially localized groups of pulses (hereafter finite pulse trains) traveling together with common speed. This speed differs in general from the speed of a single pulse. All of these states form close to the parameter values at which the primary homoclinic orbit is present [9, 11] and have similar speeds. In contrast, there are also spatially extended states (hereafter infinite pulse trains) consisting of copies of the single pulse state, in which each pulse is locked to the oscillatory tail of the preceding pulse [10, 14]. Many different states consisting of equally or unequally spaced pulses are possible, each traveling at its own speed [6, 7, 10, 13, 15]. Although solutions of either type are readily constructed using ideas from spatial dynamics, their stability properties are in general unknown [7, 10, 12, 13]. Moreover, even when the solutions are stable, no robust procedures are known for generating a pulse train of a desired type or length without continued input. In this article, we demonstrate that in spatially extended homogeneous excitable media, distinct families of multi-pulse states can be simultaneously stable, and explain how the different states may be generated. Of these the fast pulse trains can be generated with great selectivity by suitably tuned one-time perturbations, via the formation of a transient pacemaker. We show how to mold the initial perturbation to achieve the desired result. The basic idea is simple: a one-time perturbation evolves first in