When a cardiac cell is stimulated at rapid rates, the action potential duration (APD) can exhibit period-doubling oscillations referred to as "alternans", where the APD alternates in a long-short-long-short (LSLS) temporal pattern [1,2]. A number of studies have demonstrated that alternans is causally related to the onset of turbulent electrical activity known as ventricular fibrillation, which has been shown to lead to sudden cardiac death [3]. Theoretical studies have described alternans using a nonlinear mapping between the time spent at the resting potential at beat n, referred to as the diastolic interval DI(n), and the APD at beat n+1, APD(n+1). This functional relationship, referred to as the restitution relation, is given by APD(n+1)=f(DI(n)), which has been shown to accurately predict the onset and evolution of alternans in cardiac cells. Experimental measurements of alternans on the surface of animal hearts have revealed complex spatiotemporal patterns of APD alternans [4]. These spatial patterns have been shown to lead directly to wave instabilities via the formation of a nonuniform substrate of APD.

Given the importance of cardiac alternans, an important question to address is whether spatiotemporal alternans in cardiac tissue can be controlled. Theoretical and experimental studies [5,6] have demonstrated that alternans can be abolished at the single cell level via a control scheme in which the pacing interval is modulated according to feedback from the last two APD measurements. However, applying this scheme to single cell in a cardiac tissue yields control over only a small piece of tissue [7]. This result is based on an amplitude equation analysis of small amplitude alternans, which reveals that unstable modes of a wave-equation cannot be controlled using a point stimulus.

In this paper, we extend the existing analysis of Echebarria and Karma [7] by exploring the amplitude equation with a point stimulus control as a boundary controlled PDE problem. We consider the modal representation of parabolic PDE describing dissipative properties of the alternans evolution along a cable [8]. In particular, we show that by measuring the APD along several sites on a cable, it is possible to construct a novel control scheme which can suppress alternans over a much larger spatial scale. Results of preliminary simulation studies demonstrate successful stabilization of alternans given by the amplitude equation within the state boundary feedback control framework. Finally, we discuss the feasability of applying this control scheme within an experimental setting.

[1] G. R. Mines, J. Physiology, vol.46, p. 349, 1913.

[2] J. B. Nolasco and R. W. Dahlen, J. Appl. Physiology, vol. 25, p. 191, 1968.

[3] A. Karma, ``Electrical alternans and spiral wave breakup in cardiac tissue,'' Chaos, vol. 4(3), p. 461-472, 1994.

[4] J. M. Pastore, S. D. Girouard, K. R. Laurita, F. G. Akar, and D. S. Rosenbaum, ``Mechanism linking T-wave alternans to the genesis of cardiac fibrillation,'' Circ., vol. 99, p. 1385--1394, 1999.

[5] D. J. Christini, M. L. Riccio, C. A. Culianu, J. J. Fox, A. Karma, and R. F. Gilmour, ``Control of electric alternans in canine cardiac purkinje fibers,'' Phys. Rev. Lett., vol. 96, p. 104101, 2006.

[6] G. M. Hall and D. J. Gauthier, ``Experimental control of cardiac muscle alternans,'' Phys. Rev. Lett., vol. 88, p. 198102, 2002.

[7] B. Echebarria and A. Karma, ``Spatiotemporal control of cardiac alternans,'' Chaos, vol. 12, p. 923-930, 2002.

[8] Christofides, P. D., ``Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Applications to Transport Reaction Processes,'' Birkhauser, Boston, 2001.