A phase response curve based model: effect of vagal and sympathetic stimulation and interaction on a pacemaker cell

This study introduces a simple mathematical model for a pacemaker cell affected by an external parasympathetic and/or sympathetic input. The model presented is based on the two most important functional properties of the cardiac pacemaker cells. The first property is the intrinsic pacemaker cycle length, an "internal" parameter of the cell. The second basic property is the phase response curve (PRC), a function which reflects the various interactions of the pacemaker cell with the outside world (i.e. interaction with surrounding cells, external stimulus). The vagal stimulus is simulated as affecting the pacemaker cycle length via a PRC, while the sympathetic input is expressed in the model as a continuous reduction in the pacemaker cycle length. When combined vagal and sympathetic activation is allowed, our model shows that autonomic systems are also capable of interacting. First, we studied the entrainment phenomena resulting from a repetitively applied vagal stimulus. Various complex patterns of dynamic interaction between the pacemaker cell and the vagal input were simulated. The PRC parameters appear to be an important factor in the prediction of the entrainment phenomena. Specifically, they permit a quantitative description of the limits of a 1:1 synchronization zone. Next, we apply this model to qualitatively investigate the phenomenon of "accentuated antagonism" between parasympathetic and sympathetic autonomic branches. We examined the various options for this interaction in regulating the pacemaker periodicity. Although this model is a simplified reflection of the biological system, we conclude that it can mimic many aspects of the dynamic autonomic control and of the possible interactions between vagal and sympathetic stimulation of a pacemaker cell.