Chaotic activity in a mathematical model of the vagally driven sinoatrial node.

Abstract

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,

Alternans and higher-order rhythms in an ionic model of a sheet of ischemic ventricular muscle

“…While we have as yet no firm evidence of chaotic behavior in our simulations, there are several other modeling studies on spatially-extended electrophysiological systems showing apparently chaotic behavior, [56][57][58]61,90,92,108,[129][130][131] sometimes as the result of a period-doubling cascade. It is also quite possible that the spiral-or scroll-wave breakup seen in modeling work [56][57][58][59][60][61]113,132 might be an example of ''extensive chaos,'' 133 perhaps corresponding to ventricular fibrillation.…”

“…The existence of period-2 and period-4 rhythms in ischemia, together with the experimental 42,45-47,88,109 and modeling 97,108,128,129 evidence for the period-doubling route to chaotic dynamics in cardiac tissue, leads very naturally to consideration of the possibility that the close temporal association of ventricular fibrillation with alternans might be an indication that ventricular fibrillation is a chaotic phenomenon somehow arising out of a cascade of period-doubling bifurcations. 9,32,37,38,52,53,111 Our results in the larger sheet ͑4 cmϫ4 cm, with a 2 cmϫ2 cm ischemic area͒ and the corresponding strand might be taken as providing some support for this hypothesis, in that we see the sequence ͕period-1→period-2→period-4→period-8→¯͖.…”

Propagation of bursting oscillations

“…In this regard, our system proposes a natural extension to a two-dimensional system (with both diffusions on the potential and on the recovery variable). This relates to physiological experiments made in cardiodynamics (Chialvo & Jalife 1987;Michaels et al 1989) in which a central nucleus of pacemaker is vagally driven. Our interpretation in terms of propagating bursting oscillations seems novel.…”

Challenging Recently Published Parameter Sets for Entropy Measures in Risk Prediction for End-Stage Renal Disease Patients

“…Furthermore, this complex optimization tends to exhibit chaotic properties [34,35]. Because entropy is a measure of unpredictability and thus quantifies these chaotic properties, it can be concluded that a reduced HRV entropy reflects an impairment of the heart rate regulation.…”