Robert D. Kirkpatrick

A realization approach to stochastic model reduction

The problem of discrete-time stochastic model reduction (approximation) is considered. Using the canonical correlation analysis approach of Akaike (1975), a new order-reduction algorithm is developed. Furthermore, it is shown that the inverse of the reduced-order realization is asymptotically stable. Next, an explicit relationship between canonical variables and the linear least-squares estimate of the state vector is established. Using this, a more direct approach for order reduction is presented, and also a new design for reduced-order Kalman filters is developed. Finally, the uniqueness and symmetry properties for the new realization—the balanced stochastic realization—along with a simulation result, are presented.

State-specific asymmetries in EEG slow wave activity induced by local application of TNFα

Sleep is posited to be a fundamental property of groups of highly interconnected neurons and regulated in part by activity-dependent sleep regulatory substances such as tumor necrosis factor alpha (TNFalpha). We show that the unilateral local application of TNFalpha onto the somatosensory cortex of rats induced state- and frequency-dependent EEG asymmetries. In contrast, the unilateral injection of a TNFalpha inhibitor, a TNFalpha soluble receptor, attenuated sleep deprivation-enhanced EEG slow wave power ipsilaterally during non-rapid eye movement sleep (NREMS) but not during REMS or waking. Results are consistent with the notion that sleep begins with state changes occurring within small groups of highly interconnected neurons and is driven in part by the local production of sleep regulating substances.

Nonlinear Myofilament Regulatory Processes Affect Frequency-Dependent Muscle Fiber Stiffness

To investigate the role of nonlinear myofilament regulatory processes in sarcomeric mechanodynamics, a model of myofilament kinetic processes, including thin filament on-off kinetics and crossbridge cycling kinetics with interactions within and between kinetic processes, was built to predict sarcomeric stiffness dynamics. Linear decomposition of this highly nonlinear model resulted in the identification of distinct contributions by kinetics of recruitment and by kinetics of distortion to the complex stiffness of the sarcomere. Further, it was established that nonlinear kinetic processes, such as those associated with cooperative neighbor interactions or length-dependent crossbridge attachment, contributed unique features to the stiffness spectrum through their effect on recruitment. Myofilament model-derived sarcomeric stiffness reproduces experimentally measured sarcomeric stiffness with remarkable fidelity. Consequently, characteristic features of the experimentally determined stiffness spectrum become interpretable in terms of the underlying contractile mechanisms that are responsible for specific dynamic behaviors.

Myofilament kinetics in isometric twitch dynamics.

AbstractTo better understand the relationship between kinetic processes of contraction and the dynamic features of an isometric twitch, studies were conducted using a mathematical model that included: (1) kinetics of cross bridge (XB) cycling; (2) kinetics of thin filament regulatory processes; (3) serial and feedback interactions between these two kinetic processes; and (4) time course of calcium activation. Isometric twitch wave forms were predicted, morphometric features of the predicted twitch wave form were evaluated, and sensitivities of wave form morphometric features to model kinetic parameters were assessed. Initially, the impulse response of the XB cycle alone was analyzed with the findings that dynamic constants of the twitch transient were much faster than turnover number of steady-state XB cycling, and, although speed and duration of the twitch wave form were sensitive to XB cycle kinetic constants, parameters of wave shape were not. When thin filament regulatory unit (RU) kinetics were added to XB cycle kinetics, the system impulse response was slowed with only little effect on wave shape. When cooperative neighbor interactions between RU and XB were added, twitch wave shape (as well as amplitude, speed and duration) proved to be sensitive to variation in cooperativity. Importantly, persistence and shape of the falling phase could be strongly modified. When kinetic coefficients of XB attachment were made to depend on sarcomere length, changes in wave shape occurred that did not occur when only sliding filament mechanisms were operative. Indeed, the force–length relationship proved to be highly sensitive to length-dependent XB attachment in combination with cooperative interactions. These model findings are the basis of hypotheses for the role of specific kinetic events of contraction in generating twitch wave form features.

Short-time-scale left ventricular systolic dynamics. Evidence for a common mechanism in both left ventricular chamber and heart muscle mechanics.

Based on the premise that short-time-scale, small-amplitude pressure/volume/outflow behavior of the left ventricular chamber was dominated by dynamic processes originating in cardiac myofilaments, a prototype model was built to predict pressure responses to volume perturbations. In the model, chamber pressure was taken to be the product of the number of generators in a pressure-bearing state and their average volumetric distortion, as in the muscle theory of A.F. Huxley, in which force was equal to the number of attached crossbridges and their average lineal distortion. Further, as in the muscle theory, pressure generators were assumed to cycle between two states, the pressure-bearing state and the non-pressure-bearing state. Experiments were performed in the isolated ferret heart, where variable volume decrements (0.01-0.12 ml) were removed at two commanded flow rates (flow clamps, -7 and -14 ml/sec). Pressure responses to volume removals were analyzed. Although the prototype model accounted for most features of the pressure responses, subtle but systematic discrepancies were observed. The presence or absence of flow and the magnitude of flow affected estimates of model parameters. However, estimates of parameters did not differ when the model was fitted to flow clamps with similar magnitudes of flows but different volume changes. Thus, prototype model inadequacies were attributed to misrepresentations of flow-related effects but not of volume-related effects. Based on these discrepancies, an improved model was built that added to the simple two-state cycling scheme, a pathway to a third state. This path was followed only in response to volume change. The improved model eliminated the deficiencies of the prototype model and was adequate in accounting for all observations. Since the template for the improved model was taken from the cycling crossbridge theory of muscle contraction, it was concluded that, in spite of the complexities of geometry, architecture, and regional heterogeneity of function and structure, crossbridge mechanisms dominated the short-time-scale dynamics of left ventricular chamber behavior.

Overall cardiac functional effect of positive inotropic drugs with differing effects on relaxation.

Recent interest in so-called calcium-sensitizing positive inotropic drugs has highlighted the potential problem of a positive effect on force development being offset, at least partially, by the negative effect that many of these drugs have on relaxation. The purpose of this study was to examine the interplay of contraction and relaxation in determining the overall cardiac effect of different positive inotropic drugs. Using a buffer-perfused isolated rabbit heart preparation, we studied four drugs (calcium, dobutamine, EMD 57033, and CGP 48506) that were given at doses sufficient to increase similarly left ventricular pressure-generating capability by approximately 20%. We show that, even though they produce equivalent changes in pressure-generating capability, these four agents produce dissimilar changes in relaxation capability, with dobutamine speeding relaxation, EMD 57033 slowing relaxation, and calcium and CGP 48506 having little effect of relaxation. Similar relative effects were observed for drug-induced changes in the timing of pressure-generation events. These effects combine to produce different drug-induced changes in overall cardiac pump function judged by the relation between cardiac output and heart rate. Dobutamine shifted the maximal cardiac output to a higher heart rate. In contrast, both calcium sensitizers shifted the maximum in cardiac output to a lower heart rate, whereas calcium had no effect. Thus even though positive inotropic drugs may have similar effects on left ventricular pressure generation, the overall benefit of such drugs on ventricular pump function will depend on how the drug also affects ventricular relaxation and ejection capabilities.

Relaxation effect of CGP-48506, EMD-57033, and dobutamine in ejecting and isovolumically beating rabbit hearts

Because it is not known whether ejection influences the negative effect of the Ca(2+)-sensitizing drugs on ventricular relaxation, we extended our previous analysis of stress-dependent relaxation in isovolumic beats to encompass ejecting beats and evaluated the relationships between both the time of onset of relaxation and the rate of relaxation and wall stress in a broader analysis framework. Furthermore, because the sites of action of the Ca(2+)-sensitizing drugs CGP-48506 and EMD-57033 may be different, and thus CGP-48506 may have fewer adverse effects on resting muscle length or force, we compared these two drugs to test the hypothesis that CGP-48506 would have less effect than EMD-57033 on relaxation in the isolated buffer-perfused rabbit heart. This analysis of stress-dependent relaxation in both ejecting and isovolumic beats readily differentiates between the negative lusitropic effect of 2 x 10(-6) M EMD-57033, the negligible lusitropic effect of 6 x 10(-6) M CGP-48506, and the positive lusitropic effect of 1.25 x 10(-6) M dobutamine. Furthermore, comparison of the effect of the two Ca(2+)-sensitizing drugs in ejecting versus isovolumic contractions shows that CGP-48506 affects relaxation differently in ejecting contractions than it does in isovolumic contractions, whereas EMD-57033 affects relaxation similarly in both ejecting and isovolumic contractions.Because it is not known whether ejection influences the negative effect of the Ca2+-sensitizing drugs on ventricular relaxation, we extended our previous analysis of stress-dependent relaxation in isovolumic beats to encompass ejecting beats and evaluated the relationships between both the time of onset of relaxation and the rate of relaxation and wall stress in a broader analysis framework. Furthermore, because the sites of action of the Ca2+-sensitizing drugs CGP-48506 and EMD-57033 may be different, and thus CGP-48506 may have fewer adverse effects on resting muscle length or force, we compared these two drugs to test the hypothesis that CGP-48506 would have less effect than EMD-57033 on relaxation in the isolated buffer-perfused rabbit heart. This analysis of stress-dependent relaxation in both ejecting and isovolumic beats readily differentiates between the negative lusitropic effect of 2 × 10-6 M EMD-57033, the negligible lusitropic effect of 6 × 10-6 M CGP-48506, and the positive lusitropic effect of 1.25 × 10-6 M dobutamine. Furthermore, comparison of the effect of the two Ca2+-sensitizing drugs in ejecting versus isovolumic contractions shows that CGP-48506 affects relaxation differently in ejecting contractions than it does in isovolumic contractions, whereas EMD-57033 affects relaxation similarly in both ejecting and isovolumic contractions.

Dethiophalloidin increases Ca2+ responsiveness of skinned cardiac muscle.

Phalloidin, an F-actin stabilizing peptide, is known to enhance Ca2+ responsiveness in skinned cardiac muscle. Here we studied the effects of dethiophalloidin (DTPH), a phalloidin derivative which, binding much more weakly to F-actin, has no phalloidin-specific ability to stabilize F-actin, on skinned bovine left ventricle muscle. When added to activated skinned muscle, DTPH (15–80μm), similarly to phalloidin, caused a rapid (within several minutes) enhancement of active force; the relative force enhancement by DTPH became greater as Ca2+ concentration was decreased. Unlike phalloidin, DTPH effects were reversible. Using a value of the force enhancement at 15μm DTPH (76% of maximum), an apparent equilibrium constant for DTPH binding to myofilaments was estimated at about 5μm. Force–pCa plots showed that DTPH (80μm) brought about a 10% increase in the maximal Ca2+-activated force and a 0.34 pCa units increase in the Ca2+sensitivity. Both changes are stronger than those caused by phalloidin in similar conditions (6% and 0.2pCa units, respectively). As with phalloidin, DTPH did not change the value of the Hill coefficient in the fit to the force–pCa curve. DTPH and phalloidin interacted as follows: (1) pre-treatment with phalloidin entirely prevented the response to DTPH, indicating the absence of any non-specific DTPH action; and (2) when added after DTPH, phalloidin decreased the force enhancement due to DTPH, reflecting a stronger effect of DTPH to increase force. In conclusion, the stabilization of F-actin structure is not a major factor in the mechanism by which phalloidin modifies contraction.

Left ventricular pressure response to small-amplitude, sinusoidal volume changes in isolated rabbit heart

The objective was to determine the dynamics of contractile processes from pressure responses to small-amplitude, sinusoidal volume changes in the left ventricle of the beating heart. Hearts were isolated from 14 anesthetized rabbits and paced at 1 beats/s. Volume was perturbed sinusoidally at four frequencies (f) (25, 50, 76.9, and 100 Hz) and five amplitudes (0.50, 0.75, 1.00, 1.25, and 1.50% of baseline volume). A prominent component of the pressure response occurred at the f of perturbation [infrequency response, delta Pf(t)]. A model, based on cross-bridge mechanisms and containing both pre- and postpower stroke states, was constructed to interpret delta Pf(t). Model predictions were that delta Pf(t) consisted of two parts: a part with an amplitude rising and falling in proportion to the pressure around that which delta Pf(t) occurred [Pr(t)], and a part with an amplitude rising and falling in proportion to the derivative of Pr(t) with time. Statistical analysis revealed that both parts were significant. Additional model predictions concerning response amplitude and phase were also confirmed statistically. The model was further validated by fitting simultaneously to all delta Pf(t) over the full range of f and delta V in a given heart. Residual errors from fitting were small (R2 = 0.978) and were not systematically distributed. Elaborations of the model to include noncontractile series elastance and distortion-dependent cross-bridge detachment did not improve the ability to represent the data. We concluded that the model could be used to identify cross-bridge rate constants in the whole heart from responses to 25- to 100-Hz sinusoidal volume perturbations.The objective was to determine the dynamics of contractile processes from pressure responses to small-amplitude, sinusoidal volume changes in the left ventricle of the beating heart. Hearts were isolated from 14 anesthetized rabbits and paced at 1 beats/s. Volume was perturbed sinusoidally at four frequencies ( f ) (25, 50, 76.9, and 100 Hz) and five amplitudes (0.50, 0.75, 1.00, 1.25, and 1.50% of baseline volume). A prominent component of the pressure response occurred at the f of perturbation [in-frequency response,[Formula: see text] ( t)]. A model, based on cross-bridge mechanisms and containing both pre- and postpower stroke states, was constructed to interpret[Formula: see text] ( t). Model predictions were that[Formula: see text] ( t) consisted of two parts: a part with an amplitude rising and falling in proportion to the pressure around that which[Formula: see text] ( t) occurred [Pr( t)], and a part with an amplitude rising and falling in proportion to the derivative of Pr( t) with time. Statistical analysis revealed that both parts were significant. Additional model predictions concerning response amplitude and phase were also confirmed statistically. The model was further validated by fitting simultaneously to all[Formula: see text] ( t) over the full range of f and ΔV in a given heart. Residual errors from fitting were small ( R 2 = 0.978) and were not systematically distributed. Elaborations of the model to include noncontractile series elastance and distortion-dependent cross-bridge detachment did not improve the ability to represent the data. We concluded that the model could be used to identify cross-bridge rate constants in the whole heart from responses to 25- to 100-Hz sinusoidal volume perturbations.

Myocardial contractile depression from high-frequency vibration is not due to increased cross-bridge breakage

Experiments were conducted in 10 isolated rabbit hearts at 25°C to test the hypothesis that vibration-induced depression of myocardial contractile function was the result of increased cross-bridge breakage. Small-amplitude sinusoidal changes in left ventricular volume were administered at frequencies of 25, 50, and 76.9 Hz. The resulting pressure response consisted of a depressive response [ΔPd( t), a sustained decrease in pressure that was not at the perturbation frequency] and an in-frequency response [ΔP f ( t), that part at the perturbation frequency]. ΔPd( t) represented the effects of contractile depression. A cross-bridge model was applied to ΔP f ( t) to estimate cross-bridge cycling parameters. Responses were obtained during Ca2+ activation and during Sr2+ activation when the time course of pressure development was slowed by a factor of 3. ΔPd( t) was strongly affected by whether the responses were activated by Ca2+ or by Sr2+. In the Sr2+-activated state, ΔPd( t) declined while pressure was rising and relaxation rate decreased. During Ca2+ and Sr2+ activation, velocity of myofilament sliding was insignificant as a predictor of ΔPd( t) or, when it was significant, participated by reducing ΔPd( t) rather than contributing to its magnitude. Furthermore, there was no difference in cross-bridge cycling rate constants when the Ca2+-activated state was compared with the Sr2+-activated state. An increase in cross-bridge detachment rate constant with volume-induced change in cross-bridge distortion could not be detected. Finally, processes responsible for ΔPd( t) occurred at slower frequencies than those of cross-bridge detachment. Collectively, these results argue against a cross-bridge detachment basis for vibration-induced myocardial depression.Experiments were conducted in 10 isolated rabbit hearts at 25 degrees C to test the hypothesis that vibration-induced depression of myocardial contractile function was the result of increased cross-bridge breakage. Small-amplitude sinusoidal changes in left ventricular volume were administered at frequencies of 25, 50, and 76.9 Hz. The resulting pressure response consisted of a depressive response [delta Pd(t), a sustained decrease in pressure that was not at the perturbation frequency] and an infrequency response [delta Pf(t), that part at the perturbation frequency]. delta Pd(t) represented the effects of contractile depression. A cross-bridge model was applied to delta Pf(t) to estimate cross-bridge cycling parameters. Responses were obtained during Ca2+ activation and during Sr2+ activation when the time course of pressure development was slowed by a factor of 3. delta Pd(t) was strongly affected by whether the responses were activated by Ca2+ or by Sr2+. In the Sr(2+)-activated state, delta Pd(t) declined while pressure was rising and relaxation rate decreased. During Ca2+ and Sr2+ activation, velocity of myofilament sliding was insignificant as a predictor of delta Pd(t) or, when it was significant, participated by reducing delta Pd(t) rather than contributing to its magnitude. Furthermore, there was no difference in cross-bridge cycling rate constants when the Ca(2+)-activated state was compared with the Sr(2+)-activated state. An increase in cross-bridge detachment rate constant with volume-induced change in cross-bridge distortion could not be detected. Finally, processes responsible for delta Pd(t) occurred at slower frequencies than those of cross-bridge detachment. Collectively, these results argue against a cross-bridge detachment basis for vibration-induced myocardial depression.