Numerical integration in the reconstruction of cardiac action potentials using Hodgkin-Huxley-type models

A comparison between traditional numerical integration methods and a new hybrid integration method for the reconstruction of action potential activity is presented, using a mathematical model of the cardiac Purkinje fiber (MNT model). It is shown that the hybrid integration method reduces importantly the overall computation time required for solving the Hodgkin-Huxley differential equations describing membrane electrical events. To accomplish this, the particular form of the gating variable equations is exploited to reformulate the step-by-step computation. In this way, the time increment can be made much larger compared with traditional methods when the membrane potential changes slowly. A mathematical analysis of the hybrid integration method is presented also, together with a numerical verification of its performance both for the propagated and nonpropagated membrane action potential. It is shown that the local error, that is the error arising at each integration step, and the cumulative integration error are strictly controlled by the membrane potential offset. Using the MNT model, the nonpropagated cardiac Purkinje action potential can be reconstructed in real time with an accuracy of 1% for the potential and 5% for the time of occurrence of its main features. In reconstructing propagated events, the hybrid integration method allows computation time savings by a factor of 10 or more compared to accurate Runge-Kutta schemes.