Three oscillator model of the heartbeat generator

The sinoatrial (SA) node is a group of self-oscillatory cells in the heart which beat rhythmically and initiate electric potentials, producing a wave of contraction that travels through the heart resulting in the circulation of blood. The SA node is an inhomogeneous collection of cells which have varying intrinsic frequencies. Experimental measurements of these frequencies have shown that the peripheral cells of the SA node have a higher natural frequency than do the interior cells. This is surprising to us since in 1:1 phase-locked motion of two oscillators of different frequency, the oscillator with the higher frequency leads the other oscillator by a phase angle. If the wave originates in the center of the SA node as one expects, then the interior cells would be leading in a 1:1 phase-locked motion and should therefore have a higher frequency than the peripheral cells. Our objective in this work is to explain this discrepancy between intuition and the measured results, and to determine possible advantages of having cells of lower frequency in the interior. Using a model of the SA node consisting of three coupled phase-only oscillators, we show that increased robustness of synchronized behavior (represented by a larger region of parameter space) comes as a result of the experimentally observed distribution of frequencies in the SA node. Associated with the loss of synchronized behavior is a complicated series of bifurcations called the “devil’s staircase”. We use our system to derive a 1D discontinuous map which exhibits the devil’s staircase, and we analyze its dynamics.