INTRODUCTION

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PREVIOUS STUDIES HAVE SHOWN that the vasomotion induced by histamine in buffer-perfused rabbit ear resistance arteries is an endothelium-independent phenomenon that may be considered chaotic (11-13). This classification is based on observations of generic patterns of behavior that represent "universal" routes for the transition from regular to irregular dynamics and include a period-doubling cascade and intermittency (11). Estimates of a nonlinear statistical index, the correlation dimension, further indicate that vasomotion in rabbit ear arteries is of relatively low intrinsic complexity, being equivalent to the behavior of a nonlinear system with just four dynamic state variables (11). Analyses of the effects of specific pharmacological probes on experimental signals have identified two distinct components: 1) a "slow" cytosolic oscillation (period  minutes) generated by Ca²⁺ cycling between the cytosol and the sarcoplasmic reticulum (SR) and 2) a "fast" membrane oscillation (period  seconds) generated by voltage-dependent Ca²⁺ influx through L-type channels that is opposed by the coordinated activity of multiple membrane ion transport systems that promote hyperpolarization (8, 11, 12, 14).

Nonlinear interactions between these oscillators may permit the emergence of highly characteristic oscillatory behavior known as quasiperiodicity and mixed-mode dynamics, in addition to chaos (11, 13, 14). In the present study, we have used pharmacological interventions to gain insights into the intracellular coupling mechanisms ultimately responsible for these different patterns of response and to define the relationship between them. Although mixed-mode behavior may occasionally be observed in the presence of normal Ca²⁺ uptake by the SR, it appears consistently after pharmacological inhibition of the SR Ca²⁺-adenosinetriphosphatase (ATPase) pump with cyclopiazonic acid (CPA) or thapsigargin (11, 13, 14). In the present study, graded concentrations of CPA were used to induce mixed-mode behavior, and because the action of this agent is readily reversible on washout in rabbit ear arteries, specific protocols could be reversed or repeated (14). The role of Ca²⁺ influx in the maintenance of the patterns of response observed in the presence of CPA was investigated with the Ca²⁺ channel antagonist verapamil.

In their simplest form, mixed-mode complexes may be classified according to the notation Mⁿ, where M large amplitude excursions are associated with n small peaks. These represent the frequency-locked states of resonant oscillators that synchronize when the ratio of their "natural" frequencies is a rational fraction, i.e., p/q, where p and q are integers (26, 27). The dynamics of such behavior is therefore periodic and when represented in phase space can be visualized as a closed orbit on the surface of a three-dimensional torus. By contrast, when the dynamics is quasiperiodic, the frequency ratio of the coupled oscillators is irrational and the trajectories of their combined motion ultimately cover the surface of a torus completely because no orbit ever repeats twice. More complex chaotic orbits may arise when motion on the torus becomes unstable and its surface wrinkles and fragments, a scenario known as the quasiperiodic route to chaos (19, 27). The patterns of motion on the surface of a torus can be visualized in two dimensions by taking a planar section perpendicular to its axis (a Poincaré section) to construct a return map. When the behavior is periodic, the trajectories of the system intersect this plane at a finite number of points, whereas the intersections of quasiperiodic orbits lie on a closed curve that corresponds to the surface of the torus. Iterative one-dimensional maps constructed from Poincaré sections allow further simplification of the dynamics, the least complicated representation being the sine circle map, which models the progression of trajectories around a circle corresponding to the surface of a smooth torus in cross section.

Constructions of circle maps have provided useful representations of a wide range of experimental systems including oscillating Josephson junctions (4, 29), fluctuations in crystal electrical conductivity (17), and cardiac arrhythmias (9). In the present study, we show that phase space portraits of rabbit ear artery vasomotion and the iterative maps derived from them by Poincaré section similarly provide evidence for the existence of motion on the surface of a torus. When the coupling between two oscillators in a quasiperiodic system becomes sufficiently strong, their frequencies may become locked into a rational ratio. Plots of such frequency-locked states as a function of the ratio of the natural frequencies of the oscillators generate a staircase function. The derivation of such a staircase from Poincaré sections of experimental signals of rabbit ear artery vasomotion confirms that frequency-locked states and an associated quasiperiodic transition to chaos closely conform to the scenario described by discrete, iterative circle maps.

	METHODS

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Experiments were performed on isolated ear preparations from male New Zealand White rabbits killed by injection of pentobarbital sodium (120 mg/kg iv) as previously described (11-14). First-generation vessels (1-1.5 cm long and ~150 µm in diameter) branching from the central artery were identified and perfused with oxygenated (95% O₂-5% CO₂) Holman's buffer (composition in mM: 120 NaCl, 5 KCl, 2.5 CaCl₂, 1.3 NaH₂PO₄, 25 NaHCO₃, 11 glucose, and 10 sucrose, pH 7.2-7.4) at 35°C in situ via a cannula secured in the central ear artery, which was itself ligated distally so as to divert all flow into the artery under study. The distal end of each vessel was cut to allow free outflow of perfusate, and side branches were ligated where possible. Average flow was maintained at 0.5 ml/min in all experiments, and superimposed fluctuations in flow and perfusion pressure were monitored continuously by a transonic flow probe and a pressure transducer connected via a side arm to the perfusion circuit. Most experiments were conducted in the presence of 50 µM N^G-nitro-L-arginine methyl ester (L-NAME) to inhibit nitric oxide synthesis and eliminate secondary effects dependent on nitric oxide production. Histamine hydrochloride, L-NAME, verapamil hydrochloride, and CPA were obtained from Sigma, and solutions were freshly prepared on the day of each experiment.

Theoretical considerations. The simplest circle map (the sine map) consists of the iteration

(1)

where _n is the angle (normalized to 2) that corresponds to the nth iterate around the circle, K is an index of the strength of the coupling between the two oscillators generating the dynamics and thus the nonlinearity present in the system, and  is the ratio of the frequencies of these oscillators in the absence of coupling (K = 0). This map may be considered as the prototype for more complex behavior generated by a general class of two-dimensional circle maps with which it possesses close dynamic similarity. In such maps, there are two iterated variables that couple a radial coordinate, r_n, with an angular coordinate, _n, such that _n+1 = f(_n, r_n) and r_n+1 = g(_n, r_n) where g and f are arbitrary functions but f is periodic such that f( + 1) = f() + 1 (19, 27). In the sine circle map, r_n is constant and equal to 1.

(2)

1

1

View larger version (35K): [in this window] [in a new window]   Fig. 1.   Properties of sine circle map (Eq. 1). K may be regarded as the strength of nonlinear coupling between 2 oscillators whose natural frequencies are in the ratio . A: an iteration that represents a 4-cycle is shown for K = 1 (left). For K > 1, map develops a local minimum and a local maximum and is no longer invertible, so that trajectories may be chaotic (right). The 2 branches of the map can be sketched within the square 0 < n < 1, 0 < n+1 < 1, where  is the angular coordinate, because of its periodic nature. B: schematic plot of K against . For K < 1 and  rational, there is frequency locking within regions known as Arnol'd tongues (shaded). These arise at every rational fraction p/q (p and q are integers) in the interval  = [0,1]; for clarity only a few examples are illustrated. As K increases, the tongues move together, but despite their finite width they do not intersect until K = 1, when overlap is complete. Above this critical value, chaos becomes possible but may coexist with order so that tongues above K > 1 may correspond to superstable, nonchaotic regimes associated with the previous frequency-locked regions. C: a plot of  against the set of rational fractions p/q generates a complete devil's staircase at K = 1, with each horizontal step corresponding to a frequency-locked state. The staircase is self-similar in the sense that small windows resemble the entire staircase when magnified, i.e., the overall structure of the staircase is fractal.

(3)

(4)

(5)

Derivation of circle maps from experimental data. Standard techniques from dynamic systems theory were used to analyze the interaction of the two contributing oscillators after preprocessing of experimental data by singular value decomposition. This transformation is a noise-reduction technique that "disentangles" phase space trajectories without altering winding numbers because it is a one-to-one linear mapping that preserves the topology of phase space (1). Measurements of flow or pressure at time t, z(t), were embedded in an m-dimensional space by means of a time delay, , whose optimal value was found by inspection to be of the order of ~4 s. This procedure generated a state matrix {Z(t), Z(t + )...., Z[t + (m  1)]} where Z(t) = [z(t), z(t + 1),..., z(t + n)] for signal lengths varying between n = 500-2,000 points. Because the nearly periodic trajectories used to derive winding numbers lie on the surface of a torus, which on small scales of measurement has a topological dimension of 2 (8), an embedding dimension of m = 4 was considered sufficiently large.

Fractal dimension of the devil's staircase. An index of the global structure of the staircase can be obtained by calculating the dimension of the intervals between the steps as follows. For a given pair of intervals p₁/q₁ and p₂/q₂, the length of the interval between them is denoted S_T. If the daughter state (p₁ + p₂)/(q₁ + q₂) is found and the gaps between this state and its parents are denoted as S₁ and S₂, respectively, the fractal dimension of the staircase D may be obtained from

(6)

if the construction is continued until a sufficiently large number of gaps of sizes S_i are found (18, 27). This formulation of fractal dimension may be regarded as a generalization of the Hausdorff box counting algorithm (19). Numerical estimates of the fractal dimension of the staircases generated by the sine circle map and more general two-dimensional circle maps provide evidence for universal scaling behavior with D  0.87 at the onset of criticality (4-6, 15). Estimates of D for the sine circle map based on only two gaps in the staircase have been found to be accurate to within a few percent (15).

Statistics. The principal frequencies of the mixed-mode oscillations observed at different concentrations of CPA and verapamil were determined by fast Fourier transform and, in the case of verapamil, were compared by the Student's t-test, P < 0.05 being considered as significant. It was not always possible to identify definitively the frequency corresponding to the fast membrane oscillator because of overlapping harmonics generated by the slow component.

	RESULTS

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Mixed-mode behavior was observed in two arteries after administration of histamine or the combination of histamine plus L-NAME (see, e.g., Figs. 2 and 5A). In the example of Fig. 2, administration of 50 µM L-NAME transformed periodic mixed-mode complexes induced by histamine into a quasiperiodic signal in association with a rise in perfusion pressure. Phase space portraits constructed from these time series confirmed that the trajectories of the dynamics could be visualized as motion on the surface of a torus, with a Poincaré section approximating a closed loop in the quasiperiodic case and thus providing evidence for the existence of two interacting oscillators with an incommensurate frequency ratio (Fig. 2). In a further 34 arteries 1 or 2.5 µM histamine induced obviously chaotic oscillations in the presence of 50 µM L-NAME. CPA consistently (i.e., in >95% of all preparations) transformed such responses into behavior possessing the typical characteristics of mixed-mode dynamics before ultimately abolishing rhythmic activity at high concentrations and causing a concentration-dependent fall in mean perfusion pressure (Figs. 3-6). Chaotic behavior tended to persist at low CPA concentrations (see, e.g., Figs. 3A and 6B) but was sometimes transformed (~70% of all preparations) into quasiperiodicity before the appearance of mixed-mode complexes (see, e.g., Fig. 4A). Chaotic patterns could nevertheless occasionally emerge from simpler patterns of quasiperiodic or mixed-mode dynamics on administration of CPA (see, e.g., Fig. 4B) or increases in the concentration of CPA present in the perfusate (see, e.g., Fig. 6B).

View larger version (27K): [in this window] [in a new window]   Fig. 2.   A: pressure trace in which a 23 mixed-mode response induced by 2.5 µM histamine (Hist) was transformed into a quasiperiodic signal (Q) by 50 µM NG-nitro-L-arginine methyl ester (L-NAME). B and C: 2-dimensional projections of 3-dimensional phase portraits of these signals. D and E: Poincaré sections showing intersection of positively directed trajectories with planes indicated by lines in B and C. In the case of the mixed-mode signal, the intersections locate to discrete regions of the section (D); in the quasiperiodic case (E) they approximate to a closed loop that corresponds to the surface of the torus shown in C.

View larger version (26K): [in this window] [in a new window]   Fig. 3.   A: representative traces from an artery perfused with 1 µM Hist and 50 µM L-NAME in which chaotic fluctuations in flow (top) and pressure (bottom) were transformed into mixed-mode behavior on administration of cyclopiazonic acid (CPA) with an associated fall in perfusion pressure. Chaos and mixed-mode states are identified as  and as 12 and 10, respectively, and are evident in both signals. CPA progressively slowed oscillatory behavior (and in this example increased its amplitude) before vasomotion was finally abolished. Frequencies indicated were calculated from period between large-amplitude components of mixed-mode complexes (note 8-min time gap between peaks in traces for 9 µM CPA). [CPA], CPA concentration. B: plots of frequencies of principal slow () and fast () components of typical mixed-mode signals as a function of log [CPA] (dotted lines). Frequency of 10-type mixed-mode complexes is plotted separately because slow and fast oscillators then behave synchronously (, solid line). In each case, a linear least-squares approximation was applied to the data points.

View larger version (32K): [in this window] [in a new window]   Fig. 4.   Trace showing complex effects of CPA in arteries perfused with 2.5 µM Hist and 50 µM L-NAME. A: administration of 2 µM CPA initially transformed chaotic behavior () into quasiperiodic dynamics (Q) before emergence of stable 12 mixed-mode complexes. B: preparation in which graded increases in CPA transformed a quasiperiodic signal into chaos before inducing mixed-mode behavior. Vasomotion ceased at higher [CPA] in both preparations, with perfusion pressure declining towards preconstriction levels (not shown).

View larger version (34K): [in this window] [in a new window]   Fig. 5.   Traces illustrating a spectrum of mixed-mode patterns in arteries perfused with 1 µM Hist and 50 µM L-NAME. A: aperiodic dynamics. B: high-order concatenations of frequency-locked states in presence of CPA. C: frequency-locked behavior showing unstable drift between 12 and 11 patterns in presence of 2 µM CPA. These complexes were stable both at higher and lower [CPA].

View larger version (36K): [in this window] [in a new window]   Fig. 6.   Experiments illustrating hysteresis and sensitivity to initial conditions. A: preparation perfused with 2.5 µM Hist and 50 µM L-NAME in which mixed-mode behavior was abolished by 7 µM CPA but reappeared when its concentration was decreased. Identical patterns were not observed at similar [CPA] when protocol was reversed. Final response with 1 and 3 µM CPA could be classified either as 01 or 10. Bracketed 13 segment was unstable. B: experiment repeated after washout of CPA for 30 min (W) in continuous presence of 1 µM Hist and 50 µM L-NAME. There were substantial differences in patterns of dynamics observed at equivalent [CPA].

In most instances, mixed-mode complexes were periodic or nearly periodic and exhibited stable Mⁿ patterns (Figs. 2-6), whose frequency decreased progressively with CPA concentration before vasomotion was abolished (see, e.g., Fig. 3A). The number of small oscillations, n, present in each complex was inversely related to the concentration of CPA administered, so that the simplest patterns, i.e., 1¹ or 1⁰, became apparent just before the disappearance of all oscillatory activity (Figs. 3 and 6). Fast Fourier transforms were used to estimate the "effective" (rather than natural) frequencies of the contributing slow and fast subsystems during typical periodic mixed-mode behavior in which both components could clearly be distinguished (n = 44; Fig. 3B). Both oscillators exhibited an approximately linear decline in frequency as a function of the log of CPA concentration, and a linear least-squares fit to the experimental data showed that this was approximately sixfold more pronounced in the case of the fast subsystem, thus reflecting convergence of the two oscillators toward a single frequency (i.e., an exactly synchronized 1⁰ state) before the abolition of rhythmic activity. Plots of the frequency of such 1⁰ complexes, which emerged over a range of CPA concentrations, also exhibited a linear falloff that was similar to that observed for the slow component of more complex mixed-mode patterns (n = 19).

Concatenations of periodic mixed-mode states conforming to the sequences of rational numbers related by Farey arithmetic could also be identified as the concentration of CPA was varied (Figs. 5B and 6B). Theoretically, there may be daughter concatenations of the form 1^p1^q between states of the form 1^p and 1^q, but these were not always identified experimentally. Moreover, in some instances the dynamics was clearly mixed mode in form but the overall behavior was aperiodic because the number of small oscillations present was variable (see, e.g., Fig. 5A). Transitions between states satisfying Farey arithmetic were sometimes unstable, with the system drifting reversibly between different frequency-locked modes after a few cycles (e.g., 1² and 1¹; Fig. 5C). In some arteries it was not possible to classify the observed complexes according to the standard method of notation, and they are most appropriately described as chaotic while nevertheless retaining features reminiscent of mixed-mode dynamics (see, e.g., Fig. 6B).

In five experiments, the concentration of CPA perfusing the arteries was first increased in a graded fashion to the point at which vasomotion was abolished and then decreased to restore oscillatory activity (see, e.g., Fig. 6A). The precise patterns of mixed-mode dynamics observed were generally found to differ during these reciprocal experimental protocols at equivalent concentrations of CPA. This is likely to reflect hysteresis rather than transient behavior, because at least 20 min of response were always recorded at each CPA concentration to ensure that a steady state had been attained. In four arteries, CPA was administered in a graded fashion until vasomotion was abolished and was then washed out with Holman's buffer containing the same concentration of histamine and L-NAME for 30-40 min. When the experimental protocol with CPA was repeated, identical patterns of mixed-mode dynamics were usually not seen at equivalent concentrations of CPA in such consecutive experiments (see, e.g., Fig. 6B). The unpredictable nature of these observations can be interpreted as being characteristic of a chaotic system.

Two-dimensional projections of the attractor of a nearly periodic 1² mixed-mode state induced by administration of CPA are shown in Fig. 7, B and C, and a Poincaré section through the associated attractor is shown in Fig. 7D. Careful inspection of the experimental trace shows that the peaks of the signal vary slightly from complex to complex. The spatial distribution of the points on the corresponding Poincaré section confirms that the dynamics is not strictly periodic, because the data points cluster into regions of the plane in such a way as to suggest the existence of an underlying torus whose surface is not smooth, but fragmented. This behavior is to be expected generically in two-dimensional circle maps close to the transition to chaos (4). In the case of fluctuations in flow that were clearly chaotic (Fig. 8), phase space portraits show that the trajectories of the dynamics were much less well localized to specific regions of phase space than those of the mixed-mode dynamics depicted in Fig. 7, and the corresponding Poincaré section also has a more complex structure. One-dimensional circle maps derived from the Poincaré sections of these signals are shown in Figs. 7E and 8E and in the irregular case exhibit a well-defined minimum and maximum. This noninvertibility confirms that the underlying behavior is chaotic.

View larger version (27K): [in this window] [in a new window]   Fig. 7.   Derivation of a circle map from a nearly periodic flow signal. A: trace exhibiting characteristics of a 12 mixed-mode oscillation (drug concentrations: CPA, 1 µM; Hist, 1 µM; L-NAME, 50 µM). B and C: 2 projections of a 3-dimensional phase portrait of dynamics rotated through ~90°. D: Poincaré section showing intersection of positively directed trajectories with plane normal to the paper indicated by line in B and C. This suggests the cross section of a torus, although the spatial localization of the points indicates that the surface of the torus has fragmented, as would be typical near the onset of chaos in a 2-dimensional circle map. E: circle map constructed from Poincaré section, showing clustering in 3 regions of plane. F: angular rotation as a function of iteration number. Slope of this line gives winding number.

View larger version (23K): [in this window] [in a new window]   Fig. 8.   Derivation of a circle map from a chaotic flow signal. A: time series (drug concentrations: CPA, 2.5 µM; Hist, 1 µM; L-NAME, 50 µM). B and C: phase space portraits show that trajectories of irregular experimental signal are less well localized than in Fig. 7, although global structure of underlying attractor is similar. D and E: Poincaré section is complex, and circle map derived from this section exhibits a minimum and a maximum. This implies noninvertibility and confirms that the underlying dynamics is chaotic. F: it was still possible to derive a winding number from Poincaré section, although chaotic nature of flow signal precludes derivation of a firing number.

Plots of winding number against firing number, which in the present study is defined as the fraction M/(M + n) for a nearly periodic mixed-mode oscillation of the form Mn and (M1 + M2)/(M1 + M2 + n1 + n2) for a concatenation of the form Mn11Mn21, reveal a devil's staircase-type structure, although the staircase was incomplete (Fig. 9). In theory, the staircase could be constructed indefinitely for circle maps at criticality if there was sufficiently high experimental resolution, because between any two periodic states there will be an infinite number of other periodic states and concatenations. Then the firing numbers would form an infinite self-similar staircase devil's staircase as illustrated in Fig. 1 for the sine circle map. However, evidence for the existence of mode-locked states that correspond to narrow Arnol'd tongues and thus narrow steps on the staircase was found only in a few preparations [e.g., (11)3 10 (Fig. 5B) and (12)2(11)2 (Fig. 6B)]. An estimate of the fractal dimension D of the staircase was obtained from the intervals between the 13 and 11 states and the mediant 12 state as described in METHODS, the value obtained being 0.63.

View larger version (10K): [in this window] [in a new window]   Fig. 9.   Winding number plotted against firing number for nearly periodic mixed mode complexes, revealing presence of a devil's staircase. Fractal dimension of staircase was estimated as 0.63 from intervals S1, S2, and ST relating the 13, 12, and 11 steps (see METHODS).

In six preparations, the contribution of Ca²⁺ influx to the genesis of mixed-mode dynamics was investigated with verapamil. High concentrations abolished rhythmic activity completely and reduced perfusion pressure, but low concentrations tended to transform mixed-mode complexes into quasiperiodic or nearly periodic sinusoidal responses, although chaos occasionally became evident as a transient phenomenon (Figs. 10 and 11). In three of six preparations, low concentrations of verapamil (0.03 µM) induced a "paradoxical" constrictor response that could subsequently be reversed by administration of higher concentrations (see, e.g., Fig. 10). Although verapamil influenced the form and amplitude of the overall dynamic behavior, in marked contrast to CPA, it did not influence the frequency of either contributing oscillator until vasomotion abolished. The peak in the power spectrum corresponding to the slow oscillator was 0.014 ± 0.001 Hz in the presence of CPA alone and 0.014 ± 0.001 Hz in the additional presence of 0.3 µM verapamil. In the case of the fast oscillator, these frequencies were 0.034 ± 0.004 Hz in the presence of CPA alone and 0.032 ± 0.006 Hz in the additional presence of 0.3 µM verapamil. In neither case were these values significantly different from each other.

View larger version (20K): [in this window] [in a new window]   Fig. 10.   Effects of verapamil in a preparation in which CPA initially transformed a quasiperiodic signal (Q) into 11-type complexes. A low concentration of verapamil (0.03 µM; A) stimulated appearance of chaos () and "paradoxically" elevated perfusion pressure, whereas higher concentrations (B) induced dilatation and ultimately abolished chaotic vasomotion without emergence of simpler mixed-mode patterns of response.

View larger version (30K): [in this window] [in a new window]   Fig. 11.   A: "quiescent" preparation in which CPA induced 11 mixed-mode complexes that were subsequently transformed into a quasiperiodic signal by 1 µM verapamil. B: preparation in which verapamil abolished 11-type mixed-mode complexes.

	DISCUSSION

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The present study has provided insights into the dynamic patterns generated by nonlinear interactions between the cytosolic and membrane oscillators that contribute to vasomotion in rabbit ear arteries. These include chaos, quasiperiodicity, and sequences of mixed-mode complexes that conform to Farey arithmetic and can be organized into a devil's staircase-type structure. The findings thus provide evidence for a quasiperiodic route to chaos via overlapping frequency-locked resonances that can be represented by iterative maps on the circle.

Physiological perspectives. In all experiments, histamine was used to induce rhythmic activity and the resulting fluctuations in pressure and flow were usually chaotic, although simpler quasiperiodic and mixed-mode patterns were occasionally apparent. The contribution of Ca²⁺ sequestration by internal stores to these distinct dynamic patterns was investigated by manipulating the activity of the vascular smooth muscle SR Ca²⁺-ATPase with CPA. Intermediate concentrations of this inhibitor almost always transformed chaotic responses into mixed-mode patterns whose frequency and complexity were inversely related to the concentration of CPA administered, and in ~70% of preparations quasiperiodic dynamics could also be identified. Experiments with verapamil, which modulates L-type Ca²⁺ channel activity, confirmed that oscillatory behavior was sustained by Ca²⁺ influx from the extracellular space, because concentrations 1 µM generally suppressed vasomotion completely (12). Low concentrations of verapamil often converted mixed-mode responses to simpler nearly sinusoidal or quasiperiodic behavior, although chaos also sometimes became evident as a transient phenomenon. These observations confirm that the interaction of the participating oscillators is nonlinear and suggest that coupling between them is effected via Ca²⁺ movements in the subplasmalemmal space that are influenced by the buffering capacity of the SR and influx of Ca²⁺ via voltage-operated channels.

Quasiperiodic transition to chaos via frequency-locked resonances. The complex patterns of vasomotion observed experimentally were reduced to one-dimensional iterative circle maps after construction of phase space portraits by time-delayed embedding, singular value decomposition, and Poincaré section. When the experimental signals were irregular, these circle maps were noninvertible, confirming that the dynamics could lie beyond criticality within a chaotic regime. The form of the Poincaré sections suggests, however, that rabbit ear artery vasomotion is of intrinsically higher complexity than the behavior of the sine circle map, because they were typical of higher-order circle maps that nevertheless exhibit identical universal scaling properties at criticality (4-6, 27). In the case of nearly periodic mixed-mode signals, Poincaré sections thus often revealed intersection points that clustered in specific regions of the sectioning plane in such a way as to suggest motion on the surface of a fragmented torus. In the context of chemical reactions, similar experimental observations have been documented by Argoul et al. (2), Maselko and Swinney (18), and Richetti et al. (25) for the coexisting chaos, quasiperiodicity, and mixed-mode dynamics that can be observed in the Belousov-Zhabotinskii (B-Z) reaction under small variations in reaction conditions. Below criticality, quasiperiodicity and frequency-locked states are generic in circle maps, and transitions between them often became evident during the experimental protocols of the present study. The more frequent occurrence of mixed-mode dynamics may simply reflect the fact that frequency-locked regions are statistically more probable near criticality in circle maps, whereas the probability of quasiperiodicity increases as the coupling strength between the oscillators decreases to zero. Indeed, above criticality, chaos and frequency-locked regions are densely interwoven but quasiperiodic behavior is no longer permitted as the Arnol'd tongues overlap completely (see METHODS).