Simplification of the quasiperiodic route to chaos in agonist-induced vasomotion by iterative circle maps

We have shown that the patterns of vasomotion induced by histamine in isolated rabbit ear resistance arteries can be described in terms of iterative circle maps that model the dynamics of coupled nonlinear oscillators. Cyclopiazonic acid (CPA), an inhibitor of the sarcoplasmic reticulum Ca(2+)-adenosinetriphosphatase pump, consistently transformed chaotic behavior into characteristic periodic oscillations known as mixed-mode responses, which consist of mixtures of large- and small-amplitude excursions and represent frequency-locked states. Quasiperiodicity, which reflects the interaction of oscillators with incommensurate frequencies, was also observed, although in a smaller number of experiments. The patterns of mixed-mode complexes found at different CPA concentrations allowed the derivation of firing numbers, i.e., number of large oscillations/sum of number of small and large oscillations, and the sequences in which they emerged conformed to Farey arithmetic. Two-dimensional return maps derived by Poincaré section of phase space representations of the dynamics were used to compute the mean number of rotations per iteration on the circle, i.e., the winding number. Plots of winding number against firing number revealed a devil's staircase-type structure. Experiments with verapamil, a voltage-operated L-type Ca(2+)-channel antagonist, confirmed that influx of extracellular Ca2+ was essential to sustain chaos, quasiperiodicity, and mixed-mode responses. Nonlinear coupling between cytosolic and membrane events in rabbit ear arteries thus results in a self-organized dynamics that collapses to that predicted by the theory of simple circle maps.