A model of electrically coupled sinus node cells was used to investigate pacemaker coordination and conduction. Individual cells were simulated using differential equations describing transmembrane ionic currents. Intrinsic cycle lengths (periods) were adjusted by applying constant depolarizing or hyperpolarizing bias current, and cells were coupled through ohmic resistances to form two-dimensional arrays. Activation maps of 81-225 coupled cells showed an apparent wavefront conducting from a leading pacemaker region to the rest of the matrix even though the pattern actually resulted from mutual entrainment of all spontaneously beating cells. Apparent conduction time increased with increasing intercellular resistance. Appropriate selection of pacemaker cycle lengths and intercellular resistances permitted the accurate simulation of the activation sequence seen experimentally for the rabbit sinus node. Furthermore, a simulated acetylcholine pulse applied to a randomly selected 20% of the cells in this model produced a pacemaker shift that lasted several beats. These results support the hypothesis that sinus node synchronization occurs through a "democratic" process resulting from the phase-dependent interactions of thousands of pacemakers.

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Dynamic interactions and mutual entrainment of coupled sinoatrial pacemaker cells with different intrinsic frequencies were investigated using a computerized mathematical model. Transmembrane potentials were simulated using equations of individual membrane currents based on voltage clamp data for the sinoatrial node. The intrinsic frequency of a given cell was altered by applying bias hyperpolarizing current, or by changing the amount of slow inward current. Cells were coupled through simple ohmic resistances to form linear arrays of two or more cells. Simulations closely reproduced previous experimental work showing that the mutual interactions between pacemakers are mediated electrotonically and show phase dependence. Results from the present simulations provide an explanation for the ionic basis of these phase-dependent interactions. In addition, it is demonstrated that the mutual entrainment of coupled pacemakers can lead to their coordinated behavior (synchronization). Two pacemaker cells can synchronize at simple harmonic (i.e., 1:1, 2:1, etc.) or more complex ratios (3:2, 5:3, etc.), depending on the differences in intrinsic frequencies and the degree of electrical coupling between cells. Simulations using larger numbers of linearly connected cells yielded various patterns of pacemaker activity including 2:1 sinoatrial block and complex dysrhythmic activity. The overall results may be used to predict higher order interactions of thousands of cells comprising the sinus node. Under such a scheme, synchronization occurs not by the conducted influence of a dominant pacemaker cell, but by the mutual "democratic" interaction of individual pacemaker cells.

SUMMARY. A mathematical model of dynamic vagus-sinus interactions was devised based on Hodgkin and Huxley-type equations of time-and voltage-dependent membrane currents. Brief vagal pulses were modeled with a concentration-dependent, acetylcholine-activated, potassium current. Single acetylcholine ('vagal") pulses scanning the sinus cycle induced changes in pacemaker rhythm that depended on pulse magnitude, duration, and time of occurrence during the cycle. Phase-response curves summarizing these effects are strikingly similar to experimental results. Notably, appropriately timed acetylcholine pulses could produce an acceleratory response. With repetitive acetylcholine input, the model produced various patterns of synchronization of the sinus pacemaker. There was stable entrainment at harmonic (i.e., 1:1, 2:1, etc.) relations, as well as more complex arrhythmic patterns that depended on the relationship between the acetylcholine cycle length and the sinus pacemaker period. In some cases, shortening of the acetylcholine input cycle length led to "paradoxical" acceleration of the sinus pacemaker. Simulations suggest that many clinically observed sinus rhythm disturbances can be explained by dynamic vagus-sinus interactions, (tire Res 55: 89-101, 1984)

Phase-locking behavior and irregular dynamics were studied in a mathematical model of the sinus node driven with repetitive vagal input. The central region of the sinus node was simulated as a 15 x 15 array of resistively coupled pacemakers with each cell randomly assigned one of 10 intrinsic cycle lengths (range 290-390 msec). Coupling of the pacemakers resulted in their mutual entrainment to a common frequency and the emergence of a dominant pacemaker region. Repetitive acetylcholine (ACh; vagal) pulses were applied to a randomly selected 60% of the cells. Over a wide range of stimulus intensities and basic cycle lengths, such perturbations resulted in a large variety of stimulus/response patterns, including phase locking (1 ; 1,3 : 2, 2 : 1 , etc.) and irregular (i.e., chaotic) dynamics. At a low ACh concentration (1 fiM), the patterns followed the typical Farey sequence of phase-locked behavior. At a higher concentration (5 ftM), period doubling and aperiodic patterns were found. When a single pacemaker cell was perturbed with repetitive ACh pulses, qualitatively similar results were obtained. In both types of simulation, chaotic behavior was investigated using phase-plane ( Received December 7, 1988; accepted June 2,1989. randomly selected 20% of the cells. Such external perturbation resulted in a shift of the dominant pacemaker site and a decrease in apparent conduction velocity within the pacemaker array, which mimicked very accurately the response patterns of the mammalian heart rate to vagus nerve stimulation. 6 -7 Repetitive vagal input is capable of entraining the already mutually entrained pacemaker array in a harmonic fashion similar to that seen previously 4 for single cells. In fact, depending on the frequency of the ACh input, zones of stable phase-locking rhythms (1 : 1,2:1, etc.) may be separated by regions in which extremely irregular behavior can predominate. Several questions arise from the study of these phenomena that may have importance in the quantitative description of the heart rhythm and its alterations. First, is it possible to analyze and describe this very complex aperiodic behavior in a quantitative but nonstatistical form? Second, can the transition ("road") between ordered (i.e., 1 : 1,