A DELAY EQUATION REPRESENTATION OF PULSE CIRCULATION ON A RING IN EXCITABLE MEDIA

This paper develops a theory for pulse circulation on a ring in a cont inuous excitable medium. Simulations of a partial differential equatio n (PDE) modeling propagation of electrical pulses on a one-dimensional ring of cardiac tissue are presented. The dynamics of the circulating pulse in this excitable medium are reduced to a single integral-delay equation. Stability conditions for steady circulation are obtained, a nd estimates are derived for the wavelength, growth rate, and asymptot ic amplitude of oscillating solutions near the transition from steady rotation to oscillatory pulse dynamics. The analytical results agree w ith simulations of the delay equation and the PDE model and uncover pr eviously uncharted solutions of the PDE equations.