A Delay Equation Representation of Pulse Circulation on a Ring in Excitable Media

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Abstract
This paper develops a theory for pulse circulation on a ring in a continuous excitable medium. Simulations of a partial differential equation (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, and estimates are derived for the wavelength, growth rate, and asymptotic amplitude of oscillating solutions near the transition from steady rotation to oscillatory pulse dynamics. The analytical results agree with simulations of the delay equation and the PDE model and uncover previously uncharted solutions of the PDE equations.