INTRODUCTION

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

TISSUE PERFUSION is normally characterized by a marked degree of spatial and temporal heterogeneity. Average local perfusion in the heart, for example, may vary by at least sixfold under resting conditions, with the consequence that some microregions are permanently vulnerable to hypoxia (21, 62). Superimposed time-dependent fluctuations in perfusion that can encompass a wide range of frequencies and amplitudes within the same vascular bed are also apparent in myocardial elements as large as 1 cm³ and even at the segmental and lobar levels in the pulmonary circulation (18, 57). Such temporal variability results principally from spontaneous rhythmic fluctuations in vascular tone and diameter, a phenomenon known as vasomotion, which can be observed at all levels of the circulation. From a functional point of view, the asynchronous opening and closing of resistance arteries in different parts of the same vascular network ensures that all tissue elements receive intermittent perfusion, and at the microcirculatory level, vasomotion is thought to enhance fluid filtration and lymphatic drainage (see Refs. 23 and 34 for review). Theoretical simulations suggest that large-amplitude oscillations in vessel diameter can increase time-averaged hydraulic conductance (55), so that vasomotion may also represent an adaptive homeodynamic response in pathological states such as hypoxia and hemorrhagic shock, in which it becomes particularly pronounced (34).

Although the complex waveforms of vasomotion are often highly irregular, there is accumulating evidence that they are not simply random in origin but are generated by nonlinearity in the mechanisms regulating smooth muscle tone. Indeed, fluctuations in pressure and flow in isolated rabbit ear resistance arteries exhibit patterns that are generic in nonlinear systems, including period-doubling, quasiperiodic, and intermittent routes for the transition from regular to irregular dynamics, so that vasomotion may be classified as a deterministic or "chaotic" process (11, 14, 24-28). Nonlinearity may confer specific biological advantages as chaotic fluid flow accelerates diffusion-limited chemical reactions and mass-transfer processes and could therefore enhance the delivery of oxygen and nutrients and the removal of metabolites (23). Chaos may also allow large changes in state to occur with minimal energy expenditure through exploitation of the high sensitivity of nonlinear systems to perturbation, thus enhancing the flexibility and efficiency of vascular control (23, 26).

To gain insights into the nonlinear interactions that underlie vasomotion at the cellular level, in the present study we have constructed a mathematical model of the ion transport systems that participate in the regulation of vascular tone. The model simulates fluctuations in free cytosolic Ca²⁺ concentration ([Ca²⁺]_i) within a single myocyte, with changes in [Ca²⁺]_i translated into changes in force generation via the latch state hypothesis of Hai and Murphy (29). The mathematical formulation was assisted by the construction of a geometric phase space portrait of the dynamics of experimental signals from isolated rabbit ear resistance arteries by the technique of time-delayed coordinate embedding (11, 23). The complexity of this representation, which is known as an "attractor," can be quantified as a nonintegral fractal correlation dimension (here denoted as D_C) following nonlinear analysis with the algorithm of Grassberger and Procaccia (22). D_C provides an estimate of the minimum number of participating dynamic control variables when rounded to the nearest upper integer, a phase space of at least three dimensions, i.e., D_C > 2 being required to classify the behavior of a system as chaotic (22). In general, the correlation dimension of fluctuations in pressure and flow in rabbit arteries takes a value between 2 and 4 (14, 24-27), thus characterizing vasomotion as a low-dimensional chaotic process that can in theory be modeled with a "minimal" approach involving just four independent variables. This conclusion is confirmed by pharmacological analysis, which has revealed the participation of "slow" intracellular and "fast" membranous Ca²⁺ oscillators with periods on the order of 1-5 min and 5-30 s, respectively (11, 25). In the present model, two variables were therefore assigned to each subsystem to permit independent oscillatory activity. The selection of these variables was guided by the changes in D_C observed on administration of specific pharmacological probes (see below).

	EXPERIMENTAL BACKGROUND TO THE MODEL

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

Intracellular Oscillator

View larger version (33K): [in this window] [in a new window]   Fig. 1.   Schematic of key ion fluxes that contribute to 2 oscillatory subsystems of model. Internal stores [i.e., sarcoplasmic reticulum (SR)] possessing a ryanodine-sensitive Ca2+-induced Ca2+ release (CICR) mechanism are essential for genesis of intracellular Ca2+ oscillations, whereas stores possessing inositol 1,4,5-trisphosphate (InsP3) receptors are considered to sequester Ca2+ avidly and serve as a large "sink/source" that is permanently full. Agonists stimulate Ca2+ influx through voltage- and receptor-operated Ca2+ channels (VOCC and ROCC, respectively) and also cause InsP3-induced Ca2+ release (IICR) from stores into cytosol. Cyclic changes in membrane potential (Vm) and Ca2+ influx via VOCC underpin activity of membrane oscillator, with negative feedback on Vm provided by transport systems that promote hyperpolarization. These include Ca2+-activated K+ channels, Na+-K+-ATPase, and Cl channels. Ca2+-ATPase pumps () mediate uptake by stores and contribute to Ca2+ extrusion from cell. Na+/Ca2+ exchange can promote Ca2+ extrusion or influx, depending on instantaneous value of Vm. Coupling between 2 oscillatory subsystems is effected through cytosolic Ca2+ concentration.

Ca²⁺ release from stores. The sarcoplasmic reticulum (SR) of vascular smooth muscle is anatomically and functionally heterogeneous and possesses Ca²⁺- and InsP₃-sensitive Ca²⁺ stores (32, 38, 66). Compounds that block sequestration of Ca²⁺ by the SR Ca²⁺-ATPase, such as cyclopiazonic acid and thapsigargin, deplete both types of stores, whereas the alkaloid ryanodine selectively impairs the functionality of a specific Ca²⁺-release channel (7, 38). In rabbit ear arteries, ryanodine appears to target a subcomponent of the SR that is closely coupled to the contractile machinery of the smooth muscle cell as it suppresses the activity of an intracellular oscillator without affecting average perfusion pressure (25, 27). By contrast, cyclopiazonic acid and thapsigargin induce a depressor response and initially transform chaotic behavior to highly specific patterns of nonlinear response known as mixed-mode oscillations before high concentrations suppress vasomotion completely (11, 27). These observations suggest that Ca²⁺ is released from distinct store subtypes in rabbit ear arteries so that, in the formulation of the present model, CICR and IICR were assumed to originate from separate stores, as in the Goldbeter-Dupont-Berridge model of Ca²⁺ spiking (20).

Voltage-operated Ca²⁺ influx. VOCCs provide the dominant pathway for Ca²⁺ influx in vascular smooth muscle, and the L-type Ca²⁺ antagonist verapamil can suppress contraction in rabbit ear arteries (11, 25). The open state probability of VOCCs is a steep sigmoidal function of cell membrane potential (51), and their contribution to vasomotion was modeled as a noninactivating whole cell conductance with a Bolzmann-like activation term, the slope of which as a function of voltage was taken from Nelson et al. (51). In addition to regulating VOCC conductance by causing membrane depolarization, constrictor agonists may also promote the opening of such channels directly by shifting their activation curve to the left, such that open state probability is higher at a given voltage (51).

Receptor-operated Ca²⁺ influx. A receptor-gated cationic channel that is insensitive to membrane depolarization and to inhibitors of VOCCs, is not modulated by [Ca²⁺]_i or other second messengers, and possesses a threefold higher selectivity for Ca²⁺ than for Na⁺ has been identified in membrane patches from rabbit ear artery myocytes (6). Agonists such as ATP and norepinephrine thus promote an inward flux of monovalent and divalent cations, such that ~10% of the current stimulated under conditions of voltage clamp is carried by Ca²⁺ (3, 5, 6). Because the nature of the ROCC remains poorly defined (51), in the present model Ca²⁺ movements via this pathway were simplified as a constant term, the magnitude of which was directly related to agonist concentration. Under certain conditions, constrictor agonists may stimulate Ca²⁺ influx into vascular smooth muscle without causing depolarization (9), and it was therefore assumed that receptor-operated Ca²⁺ influx does not modulate membrane potential.

Ca²⁺ extrusion ATPase. In rabbit ear arteries, inhibition of the membrane Ca²⁺ extrusion pump with the vanadate ion evokes a large pressor response and a small decrease in D_C (14). In different simulations, linear or parabolic activation curves were used to approximate the dependence of the pump kinetics on [Ca²⁺]_i over the physiologically relevant range (15), and its voltage dependence was formulated as a linear function of membrane potential, the slope of which was such as to give a 40% greater rate of Ca²⁺ efflux at 0 mV than at 100 mV, in accord with observations that the rate of extracellular Na⁺ concentration ([Na⁺]_o)-independent Ca²⁺ efflux from vascular myocytes increases linearly with membrane potential (16). The pump exchanges Ca²⁺ for H⁺, but the stoichiometry of the exchange may vary with local pH (15, 16), and under different experimental conditions it has been reported to be electrogenic (16) or electroneutral (53). The possible influence of Ca²⁺ extrusion via this mechanism on membrane potential was not incorporated in the simulations presented.

Na⁺/Ca²⁺ exchange. Reductions in [Na⁺]_o inhibit Na⁺/Ca²⁺ exchange and progressively reduce D_C for vasomotion in rabbit ear arteries to <2 (14). The exchanger can act as an efflux (forward mode) or an influx (reverse mode) pathway for Ca²⁺ movements, with the instantaneous direction of the Ca²⁺ flux depending on the difference between membrane potential and the reversal potential of the exchanger, which may vary between 30 and 45 mV according to [Ca²⁺]_i (4, 54). Inasmuch as the stoichiometry of the exchange is 3 Na⁺:1 Ca²⁺, below its reversal potential this transport system will promote [Ca²⁺]_i extrusion associated with depolarization due to the net movement of charge; above the reversal potential these effects will change direction. Consequently, in mathematical simulations the signs for the contribution of this mechanism to the rate of change of [Ca²⁺]_i and membrane potential will always be opposed. Modulation of exchange activity by local changes in Na⁺ concentration was ignored in the formulation of the present model.

Membrane Oscillator

To generate oscillatory electrical membrane activity that contributes to rhythmic mechanical responses, mechanisms that promote hyperpolarization must operate to oppose voltage-dependent Ca²⁺ influx. Experimental evidence suggests that a spectrum of ion channels, pumps, and exchangers may act in concert to provide such negative feedback (14). Reductions in the activity of Ca²⁺-activated K⁺ (K_Ca) channels, Cl channels, the Na⁺-K⁺-ATPase, or Na⁺/Ca²⁺ exchange thus attenuate the membrane component of rabbit ear artery vasomotion, with each class of intervention separately causing D_C to fall to <2 (14). Although it is therefore clear that multiple channels, pumps, and exchangers contribute to the regulation of tone in these vessels, it is likely that the entrainment of weakly contributing variables results in the suppression of a number of degrees of freedom of the system with a corresponding reduction in its overall dynamic complexity (44). To construct a minimal four-dimensional model that conforms to experimental findings, the open state probability of the K_Ca channel was chosen as the fourth independent mathematical control variable. Membrane potential can fluctuate between 0 and 50 mV during vasomotion (48, 64), so that potassium ions, with reversal potential less than 80 mV (52), are the only species able to drive the membrane to sufficiently low potentials. Other membrane transport systems that influence D_C were included in the model as modulatory fluxes.

Ionic current through K_Ca channels. Multiple K_Ca channel isoforms may coexist in the same artery type (52). In a detailed study, Toro et al. (65) reconstructed the K_Ca channels present in pig coronary artery in lipid bilayers and distinguished different subtypes on the basis of their voltage and Ca²⁺ sensitivity. Frequency histograms of prevalence demonstrated two main populations of "maxi" K_Ca channels with conductances of 245 and 295 pS, each population consisting of two isoforms. The 245-pS subtypes had half-maximal activation potentials of 80 and 6 mV, whereas the 295-pS subtypes had half-maximal activation potentials of 28 and 66 mV (65). Subsidiary channels were also found at ~50 and ~400 pS and also exhibited differences in sensitivity to voltage and [Ca²⁺]_i. The voltage slope coefficients of these channels were generally similar, with the channel exhibiting greatest sensitivity to Ca²⁺ half-maximally activated at ~1.2 µM Ca²⁺ (65). In the present minimal model, representative values of these parameters were selected to define the activation characteristics of a single "composite" K_Ca subtype. The Hill coefficient for the Ca²⁺ dependence of the activation sigmoidal was taken as 2 to conform with experimental findings (65). The kinetics of channel inactivation were described by a first-order rate equation with an exponential decay constant. Membrane potential and [Ca²⁺]_i may also influence the rate of K_Ca channel closure (40, 52), but this additional complexity was not incorporated in the model. Although voltage-dependent K⁺ channels are functionally active and modulate perfusion pressure in rabbit ear arteries, they do not contribute to the dynamical complexity of vasomotion, perhaps reflecting their insensitivity to [Ca²⁺]_i, and were therefore not included in the model (14). ATP-sensitive K⁺ channels influence neither tone nor vasomotion in rabbit ear arteries under well-oxygenated conditions (25).

Ionic current through Na⁺-K⁺-ATPase. Under normal physiological circumstances the Na⁺-K⁺-ATPase is thought to contribute to vascular smooth muscle cell potential by hyperpolarizing the membrane by up to 10 mV (1). Inasmuch as changes in the local concentration of K⁺ and Na⁺ were not included as an integral dynamical feature of the model, the contribution of this pump was simplified as a constant hyperpolarizing flux.

Ionic current through Cl channels. A depolarizing Ca²⁺-activated outward Cl flux has been reported in rabbit ear arteries (3, 39), and reductions in extracellular Cl concentration and niflumic acid, an inhibitor of Ca²⁺-activated Cl channels, suppress the activity of the membrane oscillator involved in vasomotion in rabbit ear arteries (14). At membrane potentials more negative than the Nernst reversal potential (approximately 25 mV), efflux of Cl promotes depolarization of vascular smooth muscle, whereas at potentials more positive than this value, efflux of chloride ions results in hyperpolarization (39). Modulation of channel activity by Ca²⁺ concentration was not specifically incorporated in the present minimal model, however, and Cl fluxes were represented simply as a constant term multiplied by the electrochemical gradient for Cl.

	METHODS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

Experimental

Mathematical Formulation

Intracellular oscillator.

Ca²⁺ fluxes into the cytosol

Ca²⁺ uptake by stores

Membrane oscillator.

Relationship between ion fluxes and membrane potential

Open state probability of K_Ca channels

(1d)

All symbols are defined in Table 1, which summarizes the fixed physiological values of the parameters (e.g., slope functions and reversal potentials) used for the simulations.

Ca²⁺ Buffering and Volume Scaling

(1c')

Calcium ions that enter the cytosol are rapidly bound by intracellular proteins (60). Indeed, experiments in gonadotrophs and chromaffin cells suggest that ~1 in 100 calcium ions remains free in the cytosol (43). Changes in [Ca²⁺]_i may thus be related to the two electrogenic membrane Ca²⁺ currents included in the model, I_Ca and I_Na/Ca, by expressions of the form, d[Ca²⁺]_i/dt = fI, where f is the fraction of calcium ions that are not buffered,  = 1/ZV_cytF is a scaling factor that converts current to rate of change in [Ca²⁺]_i, V_cyt is the volume of the cytosol, F is Faraday's constant, and Z = 2 for I_Ca and 1 for I_Na/Ca (since the stoichiometry of this exchanger is 3 Na⁺:1 Ca²⁺). From Eq. 1, a and c', it follows that for each of these currents g = G/f = ZV_cytFG/f and that  = V_cytF/fC_m. The formulation of Eq. 1c thus introduces a scaling factor, , that involves the parameters of V_cyt, C_m, and the fraction of calcium ions that remains free (note that the divalent nature of Ca²⁺, i.e., Z = 2 is accounted for by the coefficient 2G_Ca in Eq. 1c; for other mechanisms included in the model, Z = 1). This "lumped" approach provides a convenient formulation for simulations, inasmuch as the contributing parameters are known only approximately. Indeed, to obtain realistic oscillations that matched experimental data, it was necessary to adjust  over the broad range of 0.2-8 (Table 2). In practice,  exerts an important influence on the relative periods of the two contributing oscillators, which can exhibit significant variation between different experimental preparations.

Buffering also occurs within the SR, and there is evidence from COS-7 cells that only ~1 in 400 calcium ions is free in this compartment, compared with 1 in 100 in the cytosol, because of the presence of specialized Ca²⁺-buffering proteins (36, 41). For the purposes of modeling, it therefore becomes necessary to define "effective volumes" available for Ca²⁺ movements (12) that are equivalent to 100 and 400 times the physical volumes of these compartments (V^e_cyt and V^e_SR), respectively, whereas the ratio of their actual volumes (V_SR/V_cyt) is <1:4, because the SR occupies <20% of the cytosol. It follows that V^e_SR ~ V^e_cyt, which permits the assumption that Ca²⁺ concentration changes in the cytosol and stores resulting from intracellular movements between these two compartments are directly related according to

(2a)

as is implicit in Eq. 1, a and b, if transmembrane fluxes are ignored. A more general relationship of the form

(2b)

could take into account structural differences between cells.

Parameter Estimation

5

12

12

1

In the model the contribution of Ca²⁺ influx through VOCCs to d[Ca²⁺]_i/dt is G_Ca(z  z_Ca1)/{1 + e^{[(zz_Ca2)/R_Ca]}}, which reduces to 0.02G_Ca µM/s for the fixed values z_Ca2 and R_Ca used in the simulations at an "average" membrane potential of 40 mV (Fig. 2). In practice, G_Ca was varied over the range 1-16 (Table 2), giving values of d[Ca²⁺]_i/dt = 0.02-0.32 µM/s, which encompass the value expected from the above order-of-magnitude calculation. To reflect the dominance of Ca²⁺ influx via VOCCs, this flux was generally made up to four times larger (depending on the instantaneous value of membrane potential) than the combined Ca²⁺ flux entering the cytosol via InsP₃-induced release from stores and receptor-operated channels.

View larger version (35K): [in this window] [in a new window]   Fig. 2.   A: phase plane plot for intracellular oscillator. Trajectories of free cytosolic Ca2+ concentration ([Ca2+]i) vs. Ca2+ concentration in SR ([Ca2+]SR; x vs. y) generate a limit cycle oscillation that is traversed in direction shown by arrows. Three principal regimens can be identified. Dashed and dotted curves, x and y nullclines, respectively; oscillations occur only when their intersection () is within "active ranges" of both (highlighted). B: oscillations in [Ca2+]i in time domain. In this simulation, membrane oscillator was "switched off" to reflect dynamics of [Ca2+]i oscillator in isolation. Consequently, signal generated is periodic in form. C and D: phase space plot for membrane oscillator and its associated temporal waveform, respectively. Dashed line, z nullcline; dotted line, w nullcline. , Equilibrium point; active range of each nullcline is highlighted. Limit cycle is traversed in direction shown by arrows. In this simulation, intracellular oscillator was switched off. KCa, Ca2+-activated K+ channel; Vm, membrane potential.

The magnitude of the Ca²⁺ flux via VOCCs employed in the above calculation can be used to estimate the rate of change of membrane potential during the upstroke of the vasomotion cycle. If it is assumed that the cell has a C_m of 1 µF/cm² (43) and a surface area of ~10⁵ cm², then total C_m  10⁵ µF. For a flux of n = 10⁶ ions/s, the current carried by Ca²⁺ is I_Ca = 2ne = 3.2 × 10¹³ C/s [where the electronic charge (e) = 1.6 × 10¹⁹ C]. It follows that dz/dt = I_Ca/C_m  32 mV/s. Given that opposing ion transport systems that promote membrane hyperpolarization are simultaneously active, this value is consistent with experimental measurements that range from 4 to 10 mV/s: ~4 mV/s (Fig. 1 in Ref. 64), ~5 mV/s (Fig. 2 in Ref. 17), and ~10 mV/s (Fig. 2 in Ref. 70).

A semiempirical approach was adopted to estimate the operating values of the remaining coefficients in Eq. 1c, inasmuch as, in practice, such variables as individual channel conductances and numbers of channel per cell are, in general, unknown. The model thus reproduces the experimental changes in membrane potential observed when the different ion transport systems are inhibited pharmacologically (see RESULTS). The coefficients chosen to reflect the magnitudes of net Ca²⁺ extrusion across the membrane by the Ca²⁺-ATPase and Na⁺/Ca²⁺ exchange and uptake by the SR also match relative estimates of these fluxes in vascular smooth muscle (see Inhibition of the membrane Ca²⁺-ATPase).

Integration of Ca²⁺ Dynamics by Latch Bridge Formation

View larger version (12K): [in this window] [in a new window]

(3a)

(3b)

(3c)

(3d)

(3e)

1

1

1

1

1

Numerical Methods

Analytic solution of Eq. 1, a-d, is impossible, inasmuch as the equations constitute a nonlinear fourth-order system. To gain insights into underlying mechanisms, each contributing oscillator was therefore first examined individually by reducing the four-dimensional system to two two-dimensional projections of the attractor. Phase space plots of the variable pairs x vs. y and z vs. w were thus obtained to evaluate the essential dynamical features of each subsystem and investigate systematically the consequences of changes in the coupling between them.

	RESULTS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

Stability Analysis

Equation 1, a and b. The equations describing the intracellular subsystem generate a limit cycle oscillation that exhibits three distinct regimens that are clearly identifiable in the phase plane plot of Fig. 2A and in the time domain representation of Fig. 2B. Along the slow section the dynamics are dominated by sequestration of Ca²⁺ entering the cell within the SR, with the result that [Ca²⁺]_i remains almost constant. This phase is followed by a "very fast" section, in which the SR empties abruptly into the cytosol. In the ensuing fast section, Ca²⁺ is extruded from the cell more rapidly than it enters from the extracellular space, so that [Ca²⁺]_i declines and [Ca²⁺]_SR remains low. When [Ca²⁺]_i becomes sufficiently low, uptake again dominates over extrusion, thus permitting sequestration of Ca²⁺ within the SR and allowing a new cycle to ensue.

(4a)

(4b)

(5)

(6a)

(6b)

(7a)

(7b)

Equation 1, c and d. A phase plane representation of the dynamics of the membrane oscillator given by Eq. 1, c and d, is shown in Fig. 2C. As in the case of the intracellular oscillator, the conditions necessary for oscillatory behavior can be obtained by considering the z and w nullclines, i.e., the loci of points where the rates of change of membrane potential and K_Ca channel open state probability are zero. The equilibrium point (z*,w*) is defined by (dz/dt)| = (dw/dt)| = 0.

(8a)

(8b)

(9)

Latch bridge dynamics. Figure 3 illustrates the integrating effect of the latch bridge mechanism on oscillations in [Ca²⁺]_i in which the parameters in Eq. 1 were selected to generate fast membrane oscillations with an amplitude ~20% of the slow intracellular oscillator. The top traces show how a high value of the latch bridge decay constant K₇ (0.2) selectively dampens fluctuations attributable to the membrane subsystem with little effect on the position of the extrema of the slow oscillations; i.e., the slow component of the force and signals remain almost in phase. Low values of K₇ (e.g., 0.01) cause further suppression of membrane oscillations, a phase shift to the right in the slow component and also a small rise in average tone. In physiological terms, high values of K₇ correspond to rapid decay of the latch bridge, so that the system is capable of responding to rapid changes in [Ca²⁺]_i. For low values of K₇ the latch decays slowly, allowing the system to function as an integrator of [Ca²⁺]_i.

View larger version (37K): [in this window] [in a new window]   Fig. 3.   Representative simulations showing force derived from Ca2+ oscillations by integrating system of Eq. 2 for different values of K7 (i.e., latch bridge decay constant).

Simulations of Pharmacological Interventions

Inhibition of CICR. Figure 4, A and D, illustrates the experimental effects of ryanodine on patterns of vasomotion that exhibit different relative contributions from the two participating oscillators. In Fig. 4A the responses are dominated by the intracellular subsystem, inasmuch as all activity was abolished by ryanodine. In Fig. 4D both subsystems are active and ryanodine selectively attenuated the contribution of the intracellular oscillator but not membrane activity. In neither artery was there a major change in mean perfusion pressure.

View larger version (33K): [in this window] [in a new window]   Fig. 4.   A-C: experimental trace and 2 simulations illustrating suppression of intracellular oscillator by ryanodine. D-F: experimental trace and 2 simulations showing effects of ryanodine when initial dynamics involved intracellular and membrane subsystems. Simulations in B and E were obtained by linearly decreasing coefficient C in Eq. 1, a and b, from 5,000 to 1,000 and from 6,250 to 1,250, respectively, over time period shown; in C and F, leak coefficient (L) was additionally increased from 0.025 to 0.055, again in a linear fashion. Traces in D-F illustrate a reverse period-doubling cascade (chaos  P2  P1). Changes in coefficients were initiated at points indicated by arrows.

(10)

View larger version (18K): [in this window] [in a new window]   Fig. 5.   Phase space interpretation of effects of ryanodine. On reducing C or reducing C and increasing L simultaneously (A and B, respectively), intersection of x and y nullclines () moves vertically upward until oscillatory behavior ceases relatively abruptly. L alters slopes of nullclines at high [Ca2+]i, but there is almost no change in shape and size of limit cycle.

Blockade of VOCCs. In some experimental preparations the Ca²⁺ antagonist verapamil abolishes all oscillatory activity, whereas in others the amplitude of the slow component of vasomotion is enhanced and membrane activity is attenuated (Fig. 6). These distinct classes of response are associated with a fall in mean perfusion pressure and could be successfully simulated by reducing the coefficient G_Ca, which determines the magnitude of Ca²⁺ influx via VOCCs in Eq. 1, a and c (Fig. 6, B and D). The variable outcomes in such simulations are explained by differences in the balance between Ca²⁺ influx via VOCCs and other pathways, i.e., via ROCCs, Ca²⁺ release from the InsP₃-sensitive store, or reverse-mode Na⁺/Ca²⁺ exchange (respectively represented by the coefficients A₀, A₁, and G_Na/Ca). When the net Ca²⁺ flux is small, blockade of VOCCs may prevent adequate replenishment of stores, so that vasomotion ceases. Conversely, if the net Ca²⁺ flux into the cytosol via VOCC-independent pathways is large, intracellular oscillations may persist even in the presence of verapamil. Indeed, the critical difference between the simulations shown in Fig. 6, B and D, is that the contribution of the Na⁺/Ca²⁺ exchanger is nearly fourfold greater and the contribution of A₀ + A₁ ~50% greater in the case where oscillatory behavior is maintained. Phase space analysis provides the dynamical explanation for this apparently paradoxical experimental behavior (Fig. 7). As a consequence of the hyperbolic shape of the active ranges of the x and y nullclines, verapamil moves the equilibrium point of Eq. 1, a, and b, to higher [Ca²⁺]_SR and lower [Ca²⁺]_i with two quite distinct effects: in Fig. 7A the equilibrium point is forced beyond the active oscillatory area, thus being transformed from an unstable focus to a stable focus or node that attracts trajectories, with the result that oscillations cease. Conversely, the equilibrium point may remain within the active region, as an unstable focus, even with G_Ca = 0 (Fig. 7B). Movement of the equilibrium point upward and to the left is accompanied by an increase in the amplitude of oscillatory behavior as mean [Ca²⁺]_i and force development decrease, consistent with the experimental trace shown in Fig. 6D. The increase in the size of the limit cycle is also consistent with the slight increment in the period of the slow component of vasomotion observed in Fig. 6, C and D.

View larger version (33K): [in this window] [in a new window]   Fig. 6.   A and C: experimental traces illustrating effects of verapamil, which may abolish vasomotion or induce large-amplitude oscillations. B and D: model-generated force oscillations simulating this paradoxical action of verapamil. These were obtained by linearly decreasing coefficient GCa in Eq. 1a over range 11.2 to 0 in B and over range 9.9 to 0 in D. Changes in GCa were initiated at points indicated by arrows.

View larger version (16K): [in this window] [in a new window]   Fig. 7.   Phase space interpretation of effects of verapamil. In A, equilibrium point is transformed from an unstable focus to a stable focus or node () with loss of oscillatory behavior on reducing GCa; in B, equilibrium point remains an unstable focus, even for GCa = 0, and amplitude of oscillations is increased.

Activation by constrictor agonists. Figure 8A shows a typical experimental response in which irregular rhythmic activity results from the administration of histamine in an otherwise quiescent artery. As shown in Fig. 8B for a different preparation, further increments in histamine concentration may nevertheless result in overstimulation and the suppression of oscillatory behavior at high levels of perfusion pressure. In the former case the pharmacological induction of vasomotion by histamine was simulated simply by increasing the combined contribution of A₀ + A₁ (Fig. 9A) and is readily interpreted by a phase plane plot of Eq. 1, a and b (Fig. 9B). For low values of A₀ + A₁, there is subcritical Ca²⁺ flux into the cytosol, which cannot sustain an adequate rate of Ca²⁺ sequestration in the ryanodine-sensitive component of the SR and thus oscillatory behavior (cf. initial part of Fig. 8A). This situation is denoted by  in Fig. 9B, which represents a stable node. Increases in A₀ + A₁ move the equilibrium point through a short segment, where it is a stable focus, into an active regimen, where it represents an unstable focus that permits oscillatory behavior ( and  in Fig. 9B).

View larger version (17K): [in this window] [in a new window]   Fig. 8.   Experimental pressure traces after administration of 2.5 µM histamine (Hist) and 2.5 and 5 µM histamine.

View larger version (15K): [in this window] [in a new window]   Fig. 9.   A: simulation of experimental trace in Fig. 8A obtained by increasing sum of coefficients A0 and A1 linearly over range 0-1.05 from point indicated by arrow. B: phase space interpretation. Increasing concentrations of histamine translate intersection of x and y nullclines from a stable fixed point () through a regimen of oscillatory activity ( and ) until high levels of activation cause oscillatory behavior to cease (). Note progressive increase in [Ca2+]i.

View larger version (43K): [in this window] [in a new window]   Fig. 10.   Simulations of experimental trace in Fig. 8B illustrating how high concentrations of histamine may suppress vasomotion. A was obtained by increasing A0 + A1 linearly over range 1.05-1.95; B was obtained by additionally increasing GCa over range 9.57-12.57. Changes in A0 + A1 and GCa were initiated at arrows. C and D: phase space projections onto z-w plane. Increasing A0 + A1 alone progressively decreases size of limit cycle of membrane oscillator, which spirals into region of plane contained within box as it approaches a stable focus. In this simulation, there is little effect on average membrane potential, whereas increases in GCa in parallel with A0 + A1 cause substantial depolarization.

Inhibition of the membrane Ca²⁺-ATPase. Two cases that illustrate the effect of the vanadate ion, which blocks the membrane Ca²⁺ extrusion ATPase, are presented to differentiate between Ca²⁺ extrusion by the membrane Ca²⁺-ATPase and by the Na⁺/Ca²⁺ exchanger. Figure 11, A and B, shows experimental and simulated traces exhibiting a prominent slow component in which vanadate caused a marked rise in perfusion pressure but only minor effects on the superficial form of the signals until oscillatory activity was abolished. By contrast, in the experimental and simulated signals of Fig. 11, D and E, fast and slow oscillations coexist and vasomotion persisted when Ca²⁺-ATPase inhibition with vanadate caused a rise in pressure, although the oscillations became simpler in form. In Fig. 11B, extrusion was dominated by the Ca²⁺-ATPase, whereas Na⁺/Ca²⁺ exchange was the principal extrusion mechanism in Fig. 11E (Table 2). The phase plane analysis of Fig. 11, C and F, confirms the presence of the typical triangular x-y projection of the attractor attributable to CICR whether extrusion is dominated by either mechanism. The simulation of Fig. 11E demonstrates how the cytosolic and membrane subsystems can become entrained if Na⁺/Ca²⁺ exchange becomes the sole mechanism for Ca²⁺ extrusion, resulting in relatively simple patterns of dynamic behavior.

View larger version (31K): [in this window] [in a new window]   Fig. 11.   A and B: experimental trace and simulation illustrating effects of vanadate in signals dominated by intracellular oscillator. C: phase plane plot of model-generated data of B illustrating a gradual rise in average [Ca2+]i on linearly decreasing coefficient D in Eq. 1a from 6.25 to 0.5 from point indicated. Arrow in C shows direction of temporal evolution of attractor, which initiates in upper left quadrant and grows in size before shifting, contracting, and reaching a stable equilibrium point. D and E: experimental trace and simulation illustrating effects of vanadate in a time series exhibiting a mixture of fast and slow oscillations. F: phase plane plot of model-generated data of E obtained by linearly decreasing coefficient D from 6.25 to 0 from point indicated. As in C, attractor evolves from left to right (initial dynamics, dotted lines; final dynamics, continuous lines), but oscillations are maintained by forward-mode Na+/Ca2+ exchange, even when D = 0. Differences between simulations in B and E principally reflect differences in value of GNa/Ca.

K_Ca channel inhibition. Figure 12A shows an original pressure trace from an experiment in which TEA, which blocks a spectrum of K_Ca channels, promoted constriction and suppressed membrane oscillations and decreased the amplitude of slow activity attributable to the intracellular oscillator. Model-generated oscillations in force development simulating the effects of TEA are shown in Fig. 12B, with a phase space interpretation in Fig. 12C. Reductions in the coefficient S in Eq. 1d depress the w nullcline and cause the equilibrium point to move to the right, consistent with membrane depolarization, enhanced Ca²⁺ influx, and a rise in mean [Ca²⁺]_i and force development, as observed experimentally. This increase in [Ca²⁺]_i interactively reduces the maximum in the z nullcline, translates the w nullcline to the left, and decreases the active range over which oscillations are possible. The equilibrium point consequently moves to a region of progressively smaller-amplitude voltage oscillations, until they are ultimately abolished at a stable focus or node under conditions of relative depolarization and high levels of constriction. However, until oscillations stop, the period of the limit cycle remains almost constant. Inasmuch as K_Ca channel blockers do not significantly alter the frequency of the membrane subsystem experimentally (14) (Fig. 12A), this justifies modeling the effects of TEA by reducing the activation-to-inactivation ratio (S) rather than the scaling factor (). Also the model predicts that large values of S (i.e., high K_Ca channel activity) may suppress oscillatory activity by converting the equilibrium point to a stable focus or node (Fig. 12C). Experimentally, the change in average membrane potential that follows administration of high concentrations of K_Ca channel blockers such as TEA may vary between ~10 and ~25 mV, which is consistent with the range over which the equilibrium point shifted (~20 mV) after reductions in S in Fig. 12C (19, 70).

View larger version (30K): [in this window] [in a new window]   Fig. 12.   A: experimental pressure trace showing attenuation of membrane activity and increases in pressure after administration of tetraethylammonium (TEA). B: simulated time series obtained by linearly decreasing coefficient S in Eq. 1d over range 1 to 0.8 from point indicated. C: phase space interpretation showing depolarization and an associated decrease in size of limit cycle as position of intersection point of z and w nullclines changes. Trajectories may also converge to a stable node or focus for high values of S ().

Role of Cl channels. Figure 13A shows an experimental pressure trace before and after administration of 30 µM niflumic acid, which inhibits Cl channel activity. This demonstrates selective loss of the membrane oscillator, with the slow intracellular component relatively unaffected. At the concentration of niflumic acid administered, there was a small fall in perfusion pressure. Figure 13B shows model-generated tension oscillations that closely simulate these experimental findings on reducing the term G_Cl in Eq. 1c. A phase space plot of Eq. 1, c and d, shows that oscillations are abolished when the equilibrium point of Eq. 1, c and d, becomes a stable focus. In rat cerebral arteries, administration of Cl channel blockers has been shown to promote hyperpolarization of 10-15 mV in vessels pressurized to induce depolarization and myogenic tone but to have no effect on membrane potential if intravascular pressure is low (50). In the simulation of Fig. 13 the value of G_Cl was selected such that the contribution of Cl fluxes to net membrane potential was on the order of ~5 mV.

View larger version (30K): [in this window] [in a new window]   Fig. 13.   A: experimental pressure trace illustrating effects of niflumic acid, which decreases open state probability of Cl channels. B: simulated time series was obtained by linearly decreasing GCl in Eq. 1c over range 35.0 to 0 from point indicated. C: phase space interpretation showing that intersection of z and w nullclines moves to a region where oscillations stop at a stable focus.

Na⁺-K⁺-ATPase inhibition. Figure 14A shows an experimental pressure trace after administration of ouabain, which inhibits the membrane Na⁺-K⁺-ATPase and preferentially attenuates the membrane oscillator while it also reduces the amplitude of the intracellular oscillator. The action of ouabain was simulated by reducing the term F_Na/K in Eq. 1c and successfully reproduced the general form of the experimental responses (Fig. 14B). Phase plane analysis shows that ouabain causes the z nullcline to translate vertically on the ordinate, such that the equilibrium point moves to the right in association with membrane depolarization (Fig. 14C). This in turn enhances Ca²⁺ influx and elevates [Ca²⁺]_i in association with a change in the nullcline configuration, such that the active range over which oscillations are possible is reduced; i.e., there is a translation of the w nullcline to the left. As a consequence of the combined nullcline movement, membrane oscillations become progressively smaller in amplitude until they are ultimately abolished at a stable focus or node. Experimentally, blockade of the Na⁺-K⁺-ATPase with high concentrations of ouabain may result in arterial depolarization of up to ~10 mV (1), which is comparable to the change in membrane potential observed in the model simulation on reducing the value of the coefficient F_Na/K selected to zero.

View larger version (32K): [in this window] [in a new window]   Fig. 14.   A: experimental pressure trace illustrating effects of Na+-K+-ATPase inhibition with ouabain. B: simulation obtained by linearly decreasing coefficient FNa/K in Eq. 1c over range 0.32 to 0 from point indicated. C: phase space interpretation. On decreasing FNa/K, intersection of z and w nullclines moves to a region of decreased oscillatory amplitude with an associated rise in pressure, ultimately leading to loss of membrane activity at a stable focus.

Attenuated Na⁺/Ca²⁺ exchange. Figures 15A and 16A show experimental pressure traces during graded reductions in [Na⁺]_o and illustrate how low [Na⁺]_o, which depresses Na⁺/Ca²⁺ exchange, may abolish vasomotion or selectively attenuate membrane activity while it amplifies the contribution of the intracellular oscillator. Such apparently paradoxical behavior can be simulated by reducing G_Na/Ca in Eq. 1, a and c, with the existence of two distinct patterns of response explained by the ability of Na⁺/Ca²⁺ exchange to mediate net efflux or net influx of Ca²⁺ across the cell membrane, depending on whether average membrane potential is below or above the reversal potential of the exchanger. At high levels of depolarization, VOCCs and reverse-mode Na⁺/Ca²⁺ exchange mediate Ca²⁺ influx into the cytosol, thus elevating x* and pressure; when the cell hyperpolarizes, the direction of Na⁺/Ca²⁺ exchange allows forward-mode efflux of Ca²⁺.

View larger version (39K): [in this window] [in a new window]   Fig. 15.   A and B: experimental trace and simulation illustrating loss of all vasomotion in presence of low extracellular Na+ concentration ([Na+]o). Model signal was obtained by linearly decreasing GNa/Ca in Eq. 1, a and c, from an initial value of 73.8 to 51.0, with changes initiated at point indicated. C and D: nullcline analysis and evolution of full 4-dimensional attractor projected onto x-y plane demonstrating how vasomotion may cease at a stable focus corresponding to high levels of [Ca2+]i. E: on z-w plane, initially unstable intersection of z and w nullclines moves to a stable focus and then a stable node, so that oscillatory membrane activity is also suppressed. F: z-w phase plane projection of 4-dimensional attractor underlying dynamics.

View larger version (53K): [in this window] [in a new window]   Fig. 16.   A and B: experimental trace and simulation illustrating appearance of sustained large-amplitude oscillations in presence of low [Na+]o. Model signal was obtained by linearly decreasing GNa/Ca in Eq. 1, a and c, from an initial value of 26.4 to 5.4, with changes initiated at point indicated. Fast activity attributable to membrane oscillator persists only during low pressure/force phases of vasomotion cycle. C and E: phase space analysis. D and F: simultaneous projections of 4-dimensional attractor on x-y and z-w planes.

Universal Routes to Chaos

Period doubling and integral periodicity. In the Feigenbaum period-doubling "route to chaos," progressive increases in a single control parameter cause a steady state to bifurcate first into two alternating solutions, thereby generating period 2 dynamics; subsequent bifurcations produce period 4, 8, and 16 oscillations, and so on until the period-doubling cascade ultimately leads to a region of fully developed chaos. Figure 4D illustrates a period-doubling bifurcation in an artery in which ryanodine converted chaos into periodic membrane behavior, and Fig. 4, E and F, shows that this scenario can be reproduced in simulations. Figure 17, A and B, illustrates experimental and model-generated examples of period 8 behavior, which is again consistent with period doubling. In nonlinear systems, narrow windows of exactly integral periodic behavior, which may be odd in nature, are found within the chaotic regimen, with period 3 (Fig. 17, C and D) and period 5 dynamics (Fig. 17, E and F) typically being the patterns most commonly observed experimentally in isolated rabbit ear arteries and in simulations with the present model.

View larger version (42K): [in this window] [in a new window]   Fig. 17.   Representative examples of exactly integral periodic oscillations observed experimentally (A, C, and E) and in model simulations (B, D, and F). A and B: period 8; C and D: period 3; E and F: period 5. , 1 period.

Quasiperiodicity and mixed-mode behavior. The intracellular and the membrane subsystems can generate quasiperiodic patterns that typically consist of two distinct oscillations possessing an incommensurate frequency ratio (11). Experimental and simulated examples are illustrated in Fig. 18, A and B. Related patterns known as mixed-mode responses, which in their simplest form represent the frequency-locked states of coupled oscillators that resonate when the ratio of their frequencies is a rational fraction, may also be observed (11). These may be classified according to the notation k^m, where k large excursions are followed by m smaller ones. Figure 18, C and D, provides experimental and simulated examples consisting of a mixture of oscillations of 1^m type.

View larger version (50K): [in this window] [in a new window]   Fig. 18.   Routes to chaos observed experimentally (A, C, and E) and in model simulations (B, D, and F). A and B: quasiperiodicity; C and D: mixed-mode dynamics; E and F: intermittency. P3, period 3-type behavior; I, intermittent chaos.

Intermittency. Another recognized route to chaos, which involves the gradual conversion of a periodic signal to sustained chaos under the variation of a single system parameter, is known as intermittency, in which periodic behavior is interrupted by bursts of irregular behavior of increasing duration (24, 28). Figure 18, E and F, illustrates experimental signals in the vicinity of a period 3 window in which the periodic signal is interrupted by chaos. This pattern is typical of so-called type I intermittency, which is associated with the destabilization of an equilibrium point on the first-return map constructed from the successive maxima present in the signal (31). In phase space the trajectories of nearly periodic oscillations gradually move away from this equilibrium point and enter a chaotic region but are eventually reinjected into the neighborhood of the equilibrium point, thereby generating episodes of nearly periodic behavior of varying duration.

	DISCUSSION

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

Correlation dimension analysis suggests that a model with at least four independent control variables is required to reproduce the dynamics of vasomotion in isolated rabbit ear arteries. The present system of equations achieves this degree of complexity by assigning two independent control variables to each of two autonomous, but nevertheless coupled, nonlinear subsystems. Variations in the parameters that determine the magnitudes of the ion fluxes included in the model permit realistic simulations of a range of pharmacological interventions, including apparently paradoxical responses, and allow the emergence of hallmark patterns of nonlinear behavior. Such versatility could not be obtained in a linear system in which the contributions of different control mechanisms were simply summated.

Intracellular Oscillator

The model provides a systematic explanation for differences between interventions that modulate CICR and other transport systems, inasmuch as the coefficients of the terms contributing to the intracellular oscillator can be classified into two principal subgroups: 1) those common to Eq. 1, a and b (i.e., B, C, and L), and 2) those that appear in Eq. 1a but not in Eq. 1b (A₀, A₁, G_Ca, G_Na/Ca, and D). Changes in the first set of coefficients preserve the relative position of the x and y nullclines, so that alterations in CICR (e.g., coefficients C and L) do not modulate the amplitude and frequency of the CICR limit cycle or average [Ca²⁺]_i, which reflects ambient force development, until oscillations stop relatively abruptly. This closely matches the pharmacological effects of ryanodine observed experimentally. By contrast, interventions that cause changes in the second set of coefficients affect the position of the x but not the y nullcline. The hyperbolic shapes of the nullclines over the range that permits oscillatory activity then maintain an inverse relationship between average [Ca²⁺]_i and the duration of the slow SR refilling phase, which is the principal determinant of the amplitude and frequency of the intracellular limit cycle (Fig. 2A). This analysis suggests that interventions that increase tone will be associated with decreases in the amplitude and period of intracellular oscillations, whereas dilator stimuli that act by reducing [Ca²⁺]_i will cause the opposite effect. These predictions were supported by experiments with the Ca²⁺-channel antagonist verapamil and low [Na⁺]_o, which directly influence transmembrane Ca²⁺ fluxes, and agents such as TEA and ouabain, which promote voltage-dependent Ca²⁺ influx secondary to membrane depolarization. The analysis also predicts that a small fall in net Ca²⁺ influx may allow oscillations to persist with increased amplitude, whereas after more severe reductions the system may no longer be able to sustain the CICR mechanism. This explains the existence of apparently paradoxical experimental responses to verapamil and low [Na⁺]_o, which can promote the emergence of slow large-amplitude oscillations or abolish vasomotion completely. Similar arguments explain how histamine stimulates vasomotion at low concentrations, whereas high concentrations may subsequently result in the loss of rhythmic behavior. The model predicts that these experimental observations can be reproduced simply by increasing the combined flux of Ca²⁺ represented by the terms A₀ and A₁, even though this approach to the action of histamine did not simulate the net depolarization of vascular smooth muscle that generally accompanies administration of constrictor agonists. To obtain this effect of agonists on membrane potential, it was necessary to increase the coefficient that determines voltage-dependent Ca²⁺ influx, G_Ca, in parallel with A₀ and A₁, although this strategy did not greatly affect the form of the oscillations obtained.

Membrane Oscillator

Although a relatively wide spectrum of pharmacological interventions is able to abolish membrane activity in rabbit arteries, in general the frequency of the membrane oscillator remains at ~0.06 Hz until its contribution is no longer detectable in experimental signals (14). This finding can also be explained by phase plane analysis of the model. In contrast to the hyperbolic shapes of the x and y nullclines of the intracellular oscillator, which generate an asymmetric limit cycle, the size of which determines frequency and amplitude, the active ranges of the z and w nullclines are effectively straight-line segments with an almost circular limit cycle, the period of which changes little during simulations even when its size varies substantially. Dynamically, this is analogous to the behavior of a simple harmonic oscillator, such as a pendulum, the limit cycle of which is also circular with a period independent of oscillation amplitude.

Routes to Chaos and the Role of Coupling

Vasomotion in rabbit ear arteries also displays the so-called quasiperiodic route to chaos, in which the strength of the coupling between two oscillators and the ratio of their frequencies influence the nature of the dynamics of the coupled system (11). If the frequency ratio is commensurate, there is a frequency-locked resonance and the trajectories of the system trace out a closed orbit on the surface of a three-dimensional torus that is exactly periodic. By contrast, if the frequency ratio of the oscillators is irrational, the trajectories of their combined motion ultimately cover the surface of a torus completely, because no orbit ever repeats twice, and the dynamics are termed quasiperiodic. Chaotic orbits arise if the motion on the torus becomes unstable. Weak coupling between the two oscillators favors the emergence of quasiperiodicity or frequency-locked states, which in the case of vasomotion can be identified as mixed-mode responses (11). Increasing levels of coupling permit the appearance of chaos but may ultimately cause the oscillators to become entrained, resulting in relatively simple patterns of behavior.

All these possibilities were evident during simulations that manipulated the coefficients of the transmembrane ion fluxes that are common to Eq. 1, a and c, and therefore regulate the coupling between the two oscillators. Thus a transition from chaos to quasiperiodicity became evident on reducing terms that appear in both equations (e.g., G_Ca in Fig. 6D or G_Na/Ca in Fig. 16B), with the waveforms corresponding to the fast and the slow components then becoming almost linearly superimposed. On the other hand, chaos may also be transformed to periodic behavior if coupling increases, e.g., if there is only a single mechanism (Na⁺/Ca²⁺ exchange) mediating extrusion of Ca²⁺ from the cell after inhibition of Ca²⁺-ATPase by vanadate (Fig. 11E). In this situation the two oscillators become entrained, because Na⁺/Ca²⁺ exchange is common to Eq. 1, a and c. Analogously, simulations in which efflux via the Ca²⁺-ATPase was assumed to be electrogenic, and thus the full term containing the coefficient D in Eq. 1a included in Eq. 1c, resulted in patterns of vasomotion that were often simpler in form than the electroneutral case, inasmuch as increased coupling between the two subsystems then reduced the geometric complexity of the underlying attractors (not shown).

Compartmentalized Function: One-Pool Models and the Role of Diffusion

The different microenvironments of the peripheral and central SR result in significant gradients in Ca²⁺ concentration between the subplasmalemmal space and the bulk of the cytosol, so that localized elevations in near-membrane Ca²⁺ concentration modulate ion transport systems in a spatially regulated fashion (38). "Sparks" of high Ca²⁺ concentration generated by spontaneous focal release from the peripheral SR may thus selectively activate K_Ca channels and promote hyperpolarization and relaxation (38). There is also a spatially regulated interaction between the SR and the cell membrane that allows calcium ions to be released vectorially from the peripheral SR into the subplasmalemmal space and extruded from the cell via the Na⁺/Ca²⁺ exchanger, which is located within closely adjacent caveolar membrane domains (38). Inasmuch as the exchanger is thought to be colocalized with the Na⁺-K⁺-ATPase (38), high levels of Ca²⁺ concentration in the subplasmalemmal space may stimulate exchange of Ca²⁺ for Na⁺, elevate intracellular Na⁺ concentration, and thereby increase the activity of the Na⁺-K⁺-ATPase, which is not saturated at normal levels of intracellular Na⁺ concentration (38). The scenario could contribute to oscillatory release of Na⁺ and K⁺ from isolated arteries during rhythmic activity (58) and promote hyperpolarization that exerts a negative feedback on Ca²⁺ influx.

Explicit representation of such sources of spatial heterogeneity and compartmentalized function would require the introduction of not only further ionic species (such as Na⁺ and K⁺) as independent dynamic variables but also second-order partial differential equations to model diffusion of Ca²⁺ and/or second messengers within the intracellular domain. Indeed, waves of elevated [Ca²⁺]_i are known to propagate through the cytoplasm of vascular smooth muscle cells after activation by agonists (33, 61). Although the present minimal model readily reproduced the characteristics of vasomotion in isolated rabbit ear arteries, it is clear that additional factors would confer greater dynamical flexibility. Preliminary simulations, incorporating a further constant term in Eq. 1c to simulate the depolarizing inward Na⁺ current that is thought to contribute to the activation of vascular smooth muscle by agonists (51), demonstrate a significant expansion in the range of oscillatory behavior possible and greater flexibility in the selection of the coefficients representing other membrane ion transport systems, i.e., G_Ca, G_Na/Ca, G_Cl, and F_Na/K. The assumption that changes in [Ca²⁺]_i and [Ca²⁺]_SR are directly related is also a simplification, and preliminary simulations with a more general relationship in which the ratio of the "effective" volumes of the cytosol and SR, V^e_SR/V^e_cyt, was different from 1 (Eq. 2) generated patterns of vasomotion with increased complexity as a result of the asymmetry introduced between Eq. 1a and Eq. 1b.

Mechanical Models

Conclusions