Double-wavelet approach to study frequency and amplitude modulation in renal autoregulation

Tubuloglomerular Feedback Synchronization in Nephrovascular Networks

To perform their functions, the kidneys maintain stable blood perfusion in the face of fluctuations in systemic BP. This is done through autoregulation of blood flow by the generic myogenic response and the kidney-specific tubuloglomerular feedback (TGF) mechanism. The central theme of this paper is that, to achieve autoregulation, nephrons do not work as single units to manage their individual blood flows, but rather communicate electrically over long distances to other nephrons via the vascular tree. Accordingly, we define the nephrovascular unit (NVU) to be a structure consisting of the nephron, glomerulus, afferent arteriole, and efferent arteriole. We discuss features that require and enable distributed autoregulation mediated by TGF across the kidney. These features include the highly variable topology of the renal vasculature which creates variability in circulation and the potential for mismatch between tubular oxygen demand and delivery; the self-sustained oscillations in each NVU arising from the autoregulatory mechanisms; and the presence of extensive gap junctions formed by connexins and their properties that enable long-distance transmission of TGF signals. The existence of TGF synchronization across the renal microvascular network enables an understanding of how NVUs optimize oxygenation-perfusion matching while preventing transmission of high systemic pressure to the glomeruli, which could lead to progressive glomerular and vascular injury.

The nephron-arterial network and its interactions

Tubuloglomerular feedback and the myogenic mechanism form an ensemble in renal afferent arterioles that regulate single-nephron blood flow and glomerular filtration. Each mechanism generates a self-sustained oscillation, the mechanisms interact, and the oscillations synchronize. The synchronization generates a bimodal electrical signal in the arteriolar wall that propagates retrograde to a vascular node, where it meets similar electrical signals from other nephrons. Each signal carries information about the time-dependent behavior of the regulatory ensemble. The converging signals support synchronization of the nephrons participating in the information exchange, and the synchronization can lead to formation of nephron clusters. We review the experimental evidence and the theoretical implications of these interactions and consider additional interactions that can limit the size of nephron clusters. The architecture of the arterial tree figures prominently in these interactions.

A gap junction inhibitor, carbenoxolone, induces spatiotemporal dispersion of renal cortical perfusion and impairs autoregulation

Renal autoregulation dynamics originating from the myogenic response (MR) and tubuloglomerular feedback (TGF) can synchronize over large regions of the kidney surface, likely through gap junction-mediated electrotonic conduction and reflecting distributed operation of autoregulation. We tested the hypotheses that inhibition of gap junctions reduces spatial synchronization of autoregulation dynamics, abrogates spatial and temporal smoothing of renal perfusion, and impairs renal autoregulation. In male Long-Evans rats, we infused the gap junction inhibitor carbenoxolone (CBX) or the related glycyrrhizic acid (GZA) that does not block gap junctions into the renal artery and monitored renal blood flow (RBF) and surface perfusion by laser speckle contrast imaging. Neither CBX nor GZA altered RBF or mean surface perfusion. CBX preferentially increased spatial and temporal variation in the distribution of surface perfusion, increased spatial variation in the operating frequencies of the MR and TGF, and reduced phase coherence of TGF and increased its dispersion. CBX, but not GZA, impaired dynamic and steady-state autoregulation. Separately, infusion of the Rho kinase inhibitor Y-27632 paralyzed smooth muscle, grossly impaired dynamic autoregulation, and monotonically increased spatial variation of surface perfusion. These data suggest CBX inhibited gap junction communication, which in turn reduced the ability of TGF to synchronize among groups of nephrons. The results indicate that impaired autoregulation resulted from degraded synchronization, rather than the reverse. We show that network behavior in the renal vasculature is necessary for effective RBF autoregulation.

Detecting Interactions between the Renal Autoregulation Mechanisms in Time and Space

OBJECTIVE

Our objective is to identify localized interactions between the renal autoregulation mechanisms over time.

METHODS

A time-varying phase-randomized wavelet bicoherence detector for quadratic phase coupling between tubuloglomerular feedback and the myogenic response is presented. Through simulations we show its ability to interrogate quadratic phase coupling. The method is applied to kidney blood flow and laser speckle imaging sequences of cortical perfusion from anesthetized rats before and after nonselective inhibition of nitric-oxide synthase.

RESULTS

Quadratic phase coupling in kidney blood flow data was present in four out of nine animals during the control period for 13.0 ± 5.6% (mean ± SD) of time and in five out of nine animals during inhibition of nitric-oxide synthase for 15.8 ± 8.2% of time. Approximately 60% of time-series extracted from laser speckle imaging pixels of the renal cortex showed significant quadratic phase coupling. Pixels with significant coupling had a median coupling length of 10.8 ± 2.2% and 12.1 ± 3.1% of time with the 95th percentile of pixels being coupled for 25.5 ± 4.4% and 30.9 ± 6.4% of time during control and inhibition of nitric-oxide synthase, respectively.

CONCLUSION

These results indicate quadratic phase coupling exists in short time intervals between tubuloglomerular feedback and the myogenic response and is detected more often in local renal perfusion signals than whole kidney blood flow in anesthetized rats.

SIGNIFICANCE

Combining the detector and laser speckle imaging provides identification of coordination between renal autoregulation mechanisms that is localized in time and space.

Renal autoregulation in health and disease

Intrarenal autoregulatory mechanisms maintain renal blood flow (RBF) and glomerular filtration rate (GFR) independent of renal perfusion pressure (RPP) over a defined range (80-180 mmHg). Such autoregulation is mediated largely by the myogenic and the macula densa-tubuloglomerular feedback (MD-TGF) responses that regulate preglomerular vasomotor tone primarily of the afferent arteriole. Differences in response times allow separation of these mechanisms in the time and frequency domains. Mechanotransduction initiating the myogenic response requires a sensing mechanism activated by stretch of vascular smooth muscle cells (VSMCs) and coupled to intracellular signaling pathways eliciting plasma membrane depolarization and a rise in cytosolic free calcium concentration ([Ca(2+)]i). Proposed mechanosensors include epithelial sodium channels (ENaC), integrins, and/or transient receptor potential (TRP) channels. Increased [Ca(2+)]i occurs predominantly by Ca(2+) influx through L-type voltage-operated Ca(2+) channels (VOCC). Increased [Ca(2+)]i activates inositol trisphosphate receptors (IP3R) and ryanodine receptors (RyR) to mobilize Ca(2+) from sarcoplasmic reticular stores. Myogenic vasoconstriction is sustained by increased Ca(2+) sensitivity, mediated by protein kinase C and Rho/Rho-kinase that favors a positive balance between myosin light-chain kinase and phosphatase. Increased RPP activates MD-TGF by transducing a signal of epithelial MD salt reabsorption to adjust afferent arteriolar vasoconstriction. A combination of vascular and tubular mechanisms, novel to the kidney, provides for high autoregulatory efficiency that maintains RBF and GFR, stabilizes sodium excretion, and buffers transmission of RPP to sensitive glomerular capillaries, thereby protecting against hypertensive barotrauma. A unique aspect of the myogenic response in the renal vasculature is modulation of its strength and speed by the MD-TGF and by a connecting tubule glomerular feedback (CT-GF) mechanism. Reactive oxygen species and nitric oxide are modulators of myogenic and MD-TGF mechanisms. Attenuated renal autoregulation contributes to renal damage in many, but not all, models of renal, diabetic, and hypertensive diseases. This review provides a summary of our current knowledge regarding underlying mechanisms enabling renal autoregulation in health and disease and methods used for its study.

Synchronization of period-doubling oscillations in vascular coupled nephrons

The mechanisms by which the individual functional unit (nephron) of the kidney regulates the incoming blood flow give rise to a number of nonlinear dynamic phenomena, including period-doubling bifurcations and intra-nephron synchronization between two different oscillatory modes. Interaction between the nephrons produces complicated and time-dependent inter-nephron synchronization patterns. In order to understand the processes by which a pair of vascular coupled nephrons synchronize, the paper presents a detailed analysis of the bifurcations that occur at the threshold of synchronization. We show that, besides infinite cascades of saddle-node bifurcations, these transitions involve mutually connected cascades of torus and homoclinic bifurcations. To illustrate the broader range of occurrence of this bifurcation structure for coupled period-doubling systems, we show that a similar structure arises in a system of two coupled, non-identical Rössler oscillators.

C-type period-doubling transition in nephron autoregulation

The functional units of the kidney, called nephrons, utilize mechanisms that allow the individual nephron to regulate the incoming blood flow in response to fluctuations in the arterial pressure. This regulation tends to be unstable and to generate self-sustained oscillations, period-doubling bifurcations, mode-locking and other nonlinear dynamic phenomena in the tubular pressures and flows. Using a simplified nephron model, the paper examines how the regulatory mechanisms react to an external periodic variation in arterial pressure near a region of resonance with one of the internally generated mode-locked cycles. We show how the stable and unstable resonance cycles generated in this response undergo interconnected cascades of period-doubling bifurcations and how each period doubling leads to the formation of a new pair of saddle-node bifurcation curves along the edges of the resonance zone. We also show how period doubling of the resonance cycles is accompanied by a torus-doubling process in the quasiperiodic regime that exists outside of the resonance zone.

Detection of low-frequency oscillations in renal blood flow

Detection of the low-frequency (LF; ∼0.01 Hz) component of renal blood flow, which is theorized to reflect the action of a third renal autoregulatory mechanism, has been difficult due to its slow dynamics. In this work, we used three different experimental approaches to detect the presence of the LF component of renal autoregulation using normotensive and spontaneously hypertensive rats (SHR), both anesthetized and unanesthetized. The first experimental approach utilized a blood pressure forcing in the form of a chirp, an oscillating perturbation with linearly increasing frequency, to elicit responses from the LF autoregulatory component in anesthetized normotensive rats. The second experimental approach involved collection and analysis of spontaneous blood flow fluctuation data from anesthetized normotensive rats and SHR to search for evidence of the LF component in the form of either amplitude or frequency modulation of the myogenic and tubuloglomerular feedback mechanisms. The third experiment used telemetric recordings of arterial pressure and renal blood flow from normotensive rats and SHR for the same purpose. Our transfer function analysis of chirp signal data yielded a resonant peak centered at 0.01 Hz that is greater than 0 dB, with the transfer function gain attenuated to lower than 0 dB at lower frequencies, which is a hallmark of autoregulation. Analysis of the data from the second experiments detected the presence of ∼0.01-Hz oscillations only with isoflurane, albeit at a weaker strength compared with telemetric recordings. With the third experimental approach, the strength of the LF component was significantly weaker in the SHR than in the normotensive rats. In summary, our detection via the amplitude modulation approach of interactions between the LF component and both tubuloglomerular feedback and the myogenic mechanism, with the LF component having an identical frequency to that of the resonant gain peak, provides evidence that 0.01-Hz oscillations may represent the third autoregulatory mechanism.

On the efficacy of the combined use of the cross-bicoherence with surrogate data technique to statistically quantify the presence of nonlinear interactions

The cross-bispectrum is an approach to detect the presence of quadratic phase coupling (QPC) between different components in bivariate signals. Quantification of QPC is by means of the cross-bicoherence index (CBI). The major limitations of the CBI are that it favors only the strongly coupled signals and its accuracy becomes compromised with noise and low coupling strength. To overcome this limitation, a statistical approach which combines CBI with a surrogate data method to determine the statistical significance of the QPC derived from bivariate signals is introduced. We demonstrate the accuracy of the proposed approach using simulation examples which are designed to test its robustness against noise contamination as well as varying levels of phase coupling and data lengths. Comparisons were made to the traditional CBI and the method based on the use of cross-bispectrum followed by a surrogate data technique. Our results show that the cross-bicoherence with surrogate data technique outperforms the two other methods compared in both sensitivity and specificity, and provides an unbiased and statistical approach to determining the presence of QPC in bivariate signals. These results are in contrast to our recent study where the auto-bispectrum combined with surrogate data approach had the best performance. Application of this approach to renal hemodynamic data was applied to renal stop flow pressure data obtained in the nephrons of the normotensive (N = 18) and hypertensive (N = 15) rats. We found significant nonlinear interactions between nephrons only when they are derived from the same cortical renal artery. The accuracy was 100% and verified by comparing the results to the known vascular connectivity between nephrons.

Detection of couplings in ensembles of stochastic oscillators

The problem of detection and quantitative characterization of directional couplings in an ensemble of noisy oscillators from a time series is addressed. We suggest estimators for the strengths of couplings which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint. Moreover, we present an analytic formula for a statistical significance level allowing to reveal an architecture of couplings reliably from a relatively short time series. The technique applies to weakly coupled nonsynchronized oscillators. It is introduced for oscillators with close basic frequencies but can be readily generalized to the case of arbitrary frequencies. Efficiency of the technique is demonstrated in numerical experiments.

Multistable dynamics mediated by tubuloglomerular feedback in a model of coupled nephrons

To help elucidate the causes of irregular tubular flow oscillations found in the nephrons of spontaneously hypertensive rats (SHR), we have conducted a bifurcation analysis of a mathematical model of two nephrons that are coupled through their tubuloglomerular feedback (TGF) systems. This analysis was motivated by a previous modeling study which predicts that NaCl backleak from a nephron's thick ascending limb permits multiple stable oscillatory states that are mediated by TGF (Layton et al. in Am. J. Physiol. Renal Physiol. 291:F79-F97, 2006); that prediction served as the basis for a comprehensive, multifaceted hypothesis for the emergence of irregular flow oscillations in SHR. However, in that study, we used a characteristic equation obtained via linearization from a single-nephron model, in conjunction with numerical solutions of the full, nonlinear model equations for two and three coupled nephrons. In the present study, we have derived a characteristic equation for a model of any finite number of mutually coupled nephrons having NaCl backleak. Analysis of that characteristic equation for the case of two coupled nephrons has revealed a number of parameter regions having the potential for differing stable dynamic states. Numerical solutions of the full equations for two model nephrons exhibit a variety of behaviors in these regions. Some behaviors exhibit a degree of complexity that is consistent with our hypothesis for the emergence of irregular oscillations in SHR.

Thermal oscillations in rat kidneys: an infrared imaging study

A high-resolution infrared (IR) camera was used to assess rhythmicity in localized renal blood flow, including the extent of regions containing nephrons with spontaneous oscillations in their individual blood flow. The IR imaging was able to follow changes in rat renal perfusion during baseline conditions, during occlusion of the main renal artery and during the administration of either saline or papaverine. Concurrent recordings were made of tubular pressure in superficial nephrons. Spontaneous vascular oscillations centred around 0.02-0.05 Hz and approximately 0.01 Hz could be detected reproducibly by IR imaging. Their spectral characteristics and their response to papaverine were in line with tubular pressure measurements. The intensity of and synchrony between thermal signals from different local areas of the kidney may allow, after surgical exposure, non-invasive imaging of functional clusters involved in renal cortical blood flow. Through visualization of the spatial extent of thermal oscillations, IR imaging holds promise in assessing kidney autoregulatory mechanisms.

Analysis of nonstationarity in renal autoregulation mechanisms using time-varying transfer and coherence functions

The extent to which renal blood flow dynamics vary in time and whether such variation contributes substantively to dynamic complexity have emerged as important questions. Data from Sprague-Dawley rats (SDR) and spontaneously hypertensive rats (SHR) were analyzed by time-varying transfer functions (TVTF) and time-varying coherence functions (TVCF). Both TVTF and TVCF allow quantification of nonstationarity in the frequency ranges associated with the autoregulatory mechanisms. TVTF analysis shows that autoregulatory gain in SDR and SHR varies in time and that SHR exhibit significantly more nonstationarity than SDR. TVTF gain in the frequency range associated with the myogenic mechanism was significantly higher in SDR than in SHR, but no statistical difference was found with tubuloglomerular (TGF) gain. Furthermore, TVCF analysis revealed that the coherence in both strains is significantly nonstationary and that low-frequency coherence was negatively correlated with autoregulatory gain. TVCF in the frequency range from 0.1 to 0.3 Hz was significantly higher in SDR (7 out of 7, >0.5) than in SHR (5 out of 6, <0.5), and consistent for all time points. For TGF frequency range (0.03-0.05 Hz), coherence exhibited substantial nonstationarity in both strains. Five of six SHR had mean coherence (<0.5), while four of seven SDR exhibited coherence (<0.5). Together, these results demonstrate substantial nonstationarity in autoregulatory dynamics in both SHR and SDR. Furthermore, they indicate that the nonstationarity accounts for most of the dynamic complexity in SDR, but that it accounts for only a part of the dynamic complexity in SHR.

Autonomic nervous nonlinear interactions lead to frequency modulation between low- and high-frequency bands of the heart rate variability spectrum

Cardiac sympathetic and parasympathetic neural activities have been found to interact with each other to efficiently regulate the heart rate and maintain homeostasis. Quantitative and noninvasive methods used to detect the presence of interactions have been lacking, however. This may be because interactions among autonomic nervous systems are nonlinear and nonstationary. The goal of this work was to identify nonlinear interactions between the sympathetic and parasympathetic nervous systems in the form of frequency and amplitude modulations in human heart rate data. To this end, wavelet analysis was performed, followed by frequency analysis of the resultant wavelet decomposed signals in several frequency brackets defined as very low frequency ( f < 0.04 Hz), low frequency (LF; 0.04–0.15 Hz), and high frequency (HF; 0.15–0.4 Hz). Our analysis suggests that the HF band is significantly modulated by the LF band in the heart rate data obtained in both supine and upright body positions. The strength of modulations is stronger in the upright than supine position, which is consistent with elevated sympathetic nervous activities in the upright position. Furthermore, significantly stronger frequency modulation than in the control condition was also observed with the cold pressor test. The results with the cold pressor test, as well as the body position experiments, further demonstrate that the frequency modulation between LF and HF is most likely due to sympathetic and parasympathetic nervous interactions during sympathetic activations. The modulation phenomenon suggests that the parasympathetic nervous system is frequency modulated by the sympathetic nervous system. In this study, there was no evidence of amplitude modulation among these frequencies.

Synchronization among mechanisms of renal autoregulation is reduced in hypertensive rats

We searched for synchronization among autoregulation mechanisms using wavelet transforms applied to tubular pressure recordings in nephron pairs from the surface of rat kidneys. Nephrons have two oscillatory modes in the regulation of their pressures and flows: a faster (100–200 mHz) myogenic mode, and a slower (20–40 mHz) oscillation in tubuloglomerular feedback (TGF). These mechanisms interact; the TGF mode modulates both the amplitude and the frequency of the myogenic mode. Nephrons also communicate with each other using vascular signals triggered by membrane events in arteriolar smooth muscle cells. In addition, the TGF oscillation changes in hypertension to an irregular fluctuation with characteristics of deterministic chaos. The analysis shows that, within single nephrons of normotensive rats, the myogenic mode and TGF are synchronized at discrete frequency ratios, with 5:1 most common. There is no distinct synchronization ratio in spontaneously hypertensive rats (SHR). In normotensive rats, full synchronization of both TGF and myogenic modes is the most probable state for pairs of nephrons originating in a common cortical radial artery. For SHR, full synchronization is less probable; most common in SHR is a state of partial synchronization with entrainment between neighboring nephrons for only one of the modes. Modulation of the myogenic mode by the TGF mode is much stronger in hypertensive than in normotensive rats. Synchronization among nephrons forms the basis for an integrated reaction to blood pressure fluctuations. Reduced synchronization in SHR suggests that the effectiveness of the coordinated response is impaired in hypertension.

Vascular coupling induces synchronization, quasiperiodicity, and chaos in a nephron tree

The paper presents a study of synchronization phenomena in a system of 22 nephrons supplied with blood from a common cortical radial artery. The nephrons are assumed to interact via hemodynamic and vascularly propagated coupling, both mediated by vascular connections. Using anatomic and physiological criteria, the nephrons are divided into groups: cortical nephrons and medullary nephrons with short, intermediate and long Henle loops. Within each of these groups the delay parameters of the internal feedback regulation are given a random component to represent the internephron variability. For parameters that generate simple limit cycle dynamics in the pressure and flow regulation of single nephrons, the ensemble of coupled nephrons showed steady state, quasiperiodic or chaotic dynamics, depending on the interaction strengths and the arterial blood pressure. When the solutions were either quasiperiodic or chaotic, cortical nephrons synchronized to a single frequency, but the longer medullary nephrons formed two clusters with different frequencies. Under no physiologically realistic combination of parameters did all nephrons assume a common frequency. Our results suggest a greater variability in the nephron dynamics than is apparent from measurements performed on cortical nephrons only. This variability may explain the development of chaotic dynamics in tubular pressure records from hypertensive rats.

Frequency modulation between low- and high-frequency components of the heart rate variability spectrum

Interactions among physiological mechanisms are abundant in biomedical signals, and they may exist to maintain efficient homeostasis. For example, sympathetic and parasympathetic neural activities interact to either elevate or depress the heart rate to maintain homeostasis. There has been considerable effort devoted to developing algorithms that can detect interactions between various physiological mechanisms. However, methods used to detect the presence of interactions between the sympathetic and parasympathetic nervous systems, to take one example, have had limited success. This may be because interactions in physiological systems are non-linear and non-stationary. The goal of this work was to identify non-linear interactions between the sympathetic and parasympathetic nervous systems in the form of frequency and amplitude modulations in human heart-rate data (n=6). To this end, wavelet analysis was performed, followed by frequency analysis of the resultant wavelet decomposed signals in several frequency brackets we define as: very low frequency (f<0.04 Hz), low frequency (0.04-0.15 Hz) and high frequency (0.15-0.4 Hz). Our analysis suggests that the high-frequency bracket is modulated by the low-frequency bracket in the heart rate data obtained in both upright and sitting positions. However, there was no evidence of amplitude modulation among these frequencies.

Unraveling cell processes: interference imaging interwoven with data analysis

The paper presents results on the application of interference microscopy and wavelet-analysis for cell visualization and studies of cell dynamics. We demonstrate that interference imaging of erythrocytes can reveal reorganization of the cytoskeleton and inhomogenity in the distribution of hemoglobin, and that interference imaging of neurons can show intracellular compartmentalization and submembrane structures. We investigate temporal and spatial variations of the refractive index for different cell types: isolated neurons, mast cells and erythrocytes. We show that the refractive dynamical properties differ from cell type to cell type and depend on the cellular compartment. Our results suggest that low frequency variations (0.1-0.6 Hz) result from plasma membrane processes and that higher frequency variations (20-26 Hz) are related to the movement of vesicles. Using double-wavelet analysis, we study the modulation of the 1 Hz rhythm in neurons and reveal its changes under depolarization and hyperpolarization of the plasma membrane. We conclude that interference microscopy combined with wavelet analysis is a useful technique for non-invasive cell studies, cell visualization, and investigation of plasma membrane properties.

Frequency modulation between low- and high-frequency components of the heart rate variability spectrum may indicate sympathetic-parasympathetic nonlinear interactions

Interactions among physiological mechanisms are abundant in biomedical signals, and they may exist to maintain efficient homeostasis. For example, sympathetic and parasympathetic neural activities interact to either elevate or depress the heart rate, to maintain homeostasis. There has been considerable effort devoted to developing algorithms that can detect interactions between various physiological mechanisms. However, methods used to detect the presence of interactions between the sympathetic and parasympathetic nervous systems, to take one example, have had limited success. This may be because interactions in physiological systems are nonlinear and nonstationary. The goal of this work was to identify nonlinear interactions between the sympathetic and parasympathetic nervous systems in the form of frequency and amplitude modulations in human heart rate data. To this end, wavelet analysis was performed, followed by frequency analysis of the resultant wavelet decomposed signals in several frequency brackets we define as: very low frequency (f<0.04 Hz), low frequency (0.04-0.15 Hz) and high frequency (0.15-0.4 Hz). Our analysis suggests that the HF bracket is significantly modulated by the LF bracket in the heart rate data obtained in both supine and upright body positions. Furthermore, the strength of modulations is stronger in the upright than supine position, which is consistent with elevated sympathetic nervous activities in the upright position. However, there was no evidence of amplitude modulation among these frequencies.

Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats

Single-nephron proximal tubule pressure in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the corresponding complex power spectra in SHR. A bifurcation analysis of the TGF model equations, for nonzero thick ascending limb (TAL) NaCl permeability, was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. These techniques revealed four parameter regions, consistent with TGF gain and delays in SHR, where multiple stable model solutions are possible: 1) a region having one stable, time-independent steady-state solution; 2) a region having one stable oscillatory solution only, of frequency f1; 3) a region having one stable oscillatory solution only, of frequency f2, which is approximately equal to 2f1; and 4) a region having two possible stable oscillatory solutions, of frequencies f1 and f2. In addition, we conducted simulations in which TAL volume was assumed to vary as a function of time and simulations in which two or three nephrons were assumed to have coupled TGF systems. Four potential sources of spectral complexity in SHR were identified: 1) bifurcations that permit switching between different stable oscillatory modes, leading to multiple spectral peaks and their respective harmonic peaks; 2) sustained lability in delay parameters, leading to broadening of peaks and of their harmonics; 3) episodic, but abrupt, lability in delay parameters, leading to multiple peaks and their harmonics; and 4) coupling of small numbers of nephrons, leading to multiple peaks and their harmonics. We conclude that the TGF system in SHR may exhibit multistability and that the complex power spectra of the irregular TGF fluctuations in this strain may be explained by switching between multiple dynamic modes, temporal variation in TGF parameters, and nephron coupling.

Interference microscopy under double-wavelet analysis: a new approach to studying cell dynamics

This Letter combines a novel experimental approach to the study of intracellular processes with a newly developed technique for multimode time-series analysis. Experiments are performed on isolated pond snail (Lymnaea stagnalis) neurons. Local variations in the cellular refractive index as detected by laser interference microscopy are related to the processes in the cell. A wavelet analysis shows the presence of several identifiable modes in the membrane and intracellular dynamics, and a double-wavelet analysis reveals nonlinear interactions between the regulatory processes in the form of mutual frequency and amplitude modulations.

Frequency encoding in renal blood flow regulation

With a model of renal blood flow regulation, we examined consequences of tubuloglomerular feedback (TGF) coupling to the myogenic mechanism via voltage-gated Ca channels. The model reproduces the characteristic oscillations of the two mechanisms and predicts frequency and amplitude modulation of the myogenic oscillation by TGF. Analysis by wavelet transforms of single-nephron blood flow confirms that both amplitude and frequency of the myogenic oscillation are modulated by TGF. We developed a double-wavelet transform technique to estimate modulation frequency. Median value of the ratio of modulation frequency to TGF frequency in measurements from 10 rats was 0.95 for amplitude modulation and 0.97 for frequency modulation, a result consistent with TGF as the modulating signal. The simulation predicted that the modulation was regular, while the experimental data showed much greater variability from one TGF cycle to the next. We used a blood pressure signal recorded by telemetry from a conscious rat as the input to the model. Blood pressure fluctuations induced variability in the modulation records similar to those found in the nephron blood flow results. Frequency and amplitude modulation can provide robust communication between TGF and the myogenic mechanism.

Double-wavelet approach to studying the modulation properties of nonstationary multimode dynamics

On the basis of double-wavelet analysis, the paper proposes a method to study interactions in the form of frequency and amplitude modulation in nonstationary multimode data series. Special emphasis is given to the problem of quantifying the strength of modulation for a fast signal by a coexisting slower dynamics and to its physiological interpretation. Application of the approach is demonstrated for a number of model systems, including a model that generates chaotic dynamics. The approach is then applied to proximal tubular pressure data from rat nephrons in order to estimate the degree to which the myogenic dynamics of the afferent arteriole is modulated by the slower tubulo-glomerular dynamics. Our analysis reveals a significantly stronger interaction between the two mechanisms in spontaneously hypertensive rats than in normotensive rats.

Mechanism-Based Modeling of Complex Biomedical Systems

Mechanism-based modeling is an approach in which the physiological, pathological and pharmacological processes of relevance to a given problem are represented as directly as possible. This approach allows us (i) to test whether assumed hypotheses are consistent with observed behaviour, (ii) to examine the sensitivity of a system to parameter variation, (iii) to learn about processes not directly amenable to experimentation, and (iv) to predict system behavior under conditions not previously experienced. The paper illustrates different aspects of the application of mechanism-based modeling through three different examples of relevance to the treatment of diabetes and hypertension: subcutaneous absorption of insulin, pulsatile insulin secretion in normal young persons, and synchronization of the pressure and flow regulation in neighbouring nephrons. The underlying ideas are that each regulatory mechanism represents the target for intervention and that the development of new and more effective drugs must be based on a deeper understanding of the biological processes.