Cellular Automata Models of Cardiac Conduction

Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model" has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a "physiological state diagram," with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.

When the canine epicardium is stimulated, the spread of epicardial excitation is 2.4 times faster parallel to the long axes of the cardiac fibers than perpendicular to them. Likewise, gross tissue resistivity is lower parallel to fibers by a factor of 3.2, and the voltage across the depolarization wave is approximately three times as great in the longitudinal direction. Equations are presented which relate these variables. Theoretical considerations confirm the experimental finding that the potentials around a wave of depolarization cannot be accounted for by the conventional hypothesis that the wavefront is a uniform double-layer current source.