Modeling cardiac pacemakers with relaxation oscillators

A modified van der Pol oscillator model was designed in order to reproduce the time series of the action potential generated by a natural pacemaker of the heart (i.e., the SA or the AV node). The main motivation was that the models published up to now were not altogether adequate for research on the heart. Based on either the classical van der Pol oscillator or other nonlinear oscillators, these models were interesting rather because of the physical phenomena that could be obtained (chaos and synchronization). However, they were unable to simulate many important physiological features of true physiological action potentials. We based our research on the experience of other groups which modeled neuronal oscillators. There complex nonlinear oscillators were used whose most important feature was a certain topology of the phase space. In our case, we modified the phase space of the classical van der Pol oscillator by adding two fixed points: a saddle and a node. In addition, a damping term asymmetric with respect to the voltage was introduced. Introduction of these new features into the van der Pol oscillator allowed to change the firing frequency of the pacemaker node without changing the length of the refractory period – an important physiological detail. We also show different ways of changing the pacemaker rhythm. A comparison of the properties of the signal obtained from our model with the features of the action potentials measured by other groups is made.