Nonlinear analysis of renal autoregulation in rats using principal dynamic modes

Synchronization in renal microcirculation unveiled with high-resolution blood flow imaging

Internephron interaction is fundamental for kidney function. Earlier studies have shown that nephrons signal to each other, synchronise over short distances, and potentially form large synchronised clusters. Such clusters would play an important role in renal autoregulation, but due to the technological limitations, their presence is yet to be confirmed. In the present study, we introduce an approach for high-resolution laser speckle imaging of renal blood flow and apply it to estimate frequency and phase differences in rat kidney microcirculation under different conditions. The analysis unveiled spatial and temporal evolution of synchronised blood flow clusters of various sizes, including the formation of large (>90 vessels) long-lived clusters (>10 periods) locked at the frequency of the tubular glomerular feedback mechanism. Administration of vasoactive agents caused significant changes in the synchronisation patterns and, thus, in nephrons' co-operative dynamics. Specifically, infusion of vasoconstrictor angiotensin II promoted stronger synchronisation, while acetylcholine caused complete desynchronisation. The results confirm the presence of the local synchronisation in the renal microcirculatory blood flow and that it changes depending on the condition of the vascular network and the blood pressure, which will have further implications for the role of such synchronisation in pathologies development.

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.

Methodology of Recurrent Laguerre-Volterra Network for Modeling Nonlinear Dynamic Systems

In this paper, we have introduced a general modeling approach for dynamic nonlinear systems that utilizes a variant of the simulated annealing algorithm for training the Laguerre-Volterra network (LVN) to overcome the local minima and convergence problems and employs a pruning technique to achieve sparse LVN representations with l₁ regularization. We tested this new approach with computer simulated systems and extended it to autoregressive sparse LVN (ASLVN) model structures that are suitable for input-output modeling of nonlinear systems that exhibit transitions in dynamic states, such as the Hodgkin-Huxley (H-H) equations of neuronal firing. Application of the proposed ASLVN to the H-H equations yields a more parsimonious input-output model with improved predictive capability that is amenable to more insightful physiological/biological interpretation.

Understanding spike-triggered covariance using Wiener theory for receptive field identification

Receptive field identification is a vital problem in sensory neurophysiology and vision. Much research has been done in identifying the receptive fields of nonlinear neurons whose firing rate is determined by the nonlinear interactions of a small number of linear filters. Despite more advanced methods that have been proposed, spike-triggered covariance (STC) continues to be the most widely used method in such situations due to its simplicity and intuitiveness. Although the connection between STC and Wiener/Volterra kernels has often been mentioned in the literature, this relationship has never been explicitly derived. Here we derive this relationship and show that the STC matrix is actually a modified version of the second-order Wiener kernel, which incorporates the input autocorrelation and mixes first- and second-order dynamics. It is then shown how, with little modification of the STC method, the Wiener kernels may be obtained and, from them, the principal dynamic modes, a set of compact and efficient linear filters that essentially combine the spike-triggered average and STC matrix and generalize to systems with both continuous and point-process outputs. Finally, using Wiener theory, we show how these obtained filters may be corrected when they were estimated using correlated inputs. Our correction technique is shown to be superior to those commonly used in the literature for both correlated Gaussian images and natural images.

Linear and Nonlinear Modeling of Cerebral Flow Autoregulation Using Principal Dynamic Modes

Cerebral Flow Autoregulation (CFA) is the dynamic process by which cerebral blood flow is maintained within physiologically acceptable bounds during fluctuations of cerebral perfusion pressure. The distinction is made with “static” flow autoregulation under steady-state conditions of perfusion pressure, described by the celebrated “autoregulatory curve” with a homeostatic plateau. This paper studies the dynamic CFA during changes in perfusion pressure, which attains critical clinical importance in patients with stroke, traumatic brain injury and neurodegenerative disease with a cerebrovascular component. Mathematical and computational models have been used to advance our quantitative understanding of dynamic CFA and to elucidate the underlying physiological mechanisms by analyzing the relation between beat-to-beat data of mean arterial blood pressure (viewed as input) and mean cerebral blood flow velocity(viewed as output) of a putative CFA system. Although previous studies have shown that the dynamic CFA process is nonlinear, most modeling studies to date have been linear. It has also been shown that blood CO2 tension affects the CFA process. This paper presents a nonlinear modeling methodology that includes the dynamic effects of CO2 tension (or its surrogate, end-tidal CO2) as a second input and quantifies CFA from short data-records of healthy human subjects by use of the modeling concept of Principal Dynamic Modes (PDMs). The PDMs improve the robustness of the obtained nonlinear models and facilitate their physiological interpretation. The results demonstrate the importance of including the CO2 input in the dynamic CFA study and the utility of nonlinear models under hypercapnic or hypocapnic conditions.

Linear and nonlinear modeling of cerebral flow autoregulation using principal dynamic modes

Cerebral Flow Autoregulation (CFA) is the dynamic process by which cerebral blood flow is maintained within physiologically acceptable bounds during fluctuations of cerebral perfusion pressure. The distinction is made with “static” flow autoregulation under steady-state conditions of perfusion pressure, described by the celebrated “autoregulatory curve” with a homeostatic plateau. This paper studies the dynamic CFA during changes in perfusion pressure, which attains critical clinical importance in patients with stroke, traumatic brain injury and neurodegenerative disease with a cerebrovascular component. Mathematical and computational models have been used to advance our quantitative understanding of dynamic CFA and to elucidate the underlying physiological mechanisms by analyzing the relation between beat-to-beat data of mean arterial blood pressure (viewed as input) and mean cerebral blood flow velocity(viewed as output) of a putative CFA system. Although previous studies have shown that the dynamic CFA process is nonlinear, most modeling studies to date have been linear. It has also been shown that blood CO2 tension affects the CFA process. This paper presents a nonlinear modeling methodology that includes the dynamic effects of CO2 tension (or its surrogate, end-tidal CO2) as a second input and quantifies CFA from short data-records of healthy human subjects by use of the modeling concept of Principal Dynamic Modes (PDMs). The PDMs improve the robustness of the obtained nonlinear models and facilitate their physiological interpretation. The results demonstrate the importance of including the CO2 input in the dynamic CFA study and the utility of nonlinear models under hypercapnic or hypocapnic conditions.

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.

Limbic dysregulation is associated with lowered heart rate variability and increased trait anxiety in healthy adults

OBJECTIVES

We tested whether dynamic interaction between limbic regions supports a control systems model of excitatory and inhibitory components of a negative feedback loop, and whether dysregulation of those dynamics might correlate with trait differences in anxiety and their cardiac characteristics among healthy adults.

EXPERIMENTAL DESIGN

Sixty-five subjects received fMRI scans while passively viewing angry, fearful, happy, and neutral facial stimuli. Subjects also completed a trait anxiety inventory, and were monitored using ambulatory wake ECG. The ECG data were analyzed for heart rate variability, a measure of autonomic regulation. The fMRI data were analyzed with respect to six limbic regions (bilateral amygdala, bilateral hippocampus, Brodmann Areas 9, 45) using limbic time-series cross-correlations, maximum BOLD amplitude, and BOLD amplitude at each point in the time-series.

PRINCIPAL OBSERVATIONS

Diminished coupling between limbic time-series in response to the neutral, fearful, and happy faces was associated with greater trait anxiety, greater sympathetic activation, and lowered heart rate variability. Individuals with greater levels of trait anxiety showed delayed activation of Brodmann Area 45 in response to the fearful and happy faces, and lowered Brodmann Area 45 activation with prolonged left amygdala activation in response to the neutral faces.

CONCLUSIONS

The dynamics support limbic regulation as a control system, in which dysregulation, as assessed by diminished coupling between limbic time-series, is associated with increased trait anxiety and excitatory autonomic outputs. Trait-anxious individuals showed delayed inhibitory activation in response to overt-affect stimuli and diminished inhibitory activation with delayed extinction of excitatory activation in response to ambiguous-affect stimuli.

Renal blood flow and dynamic autoregulation in conscious mice

Autoregulation of renal blood flow (RBF) occurs via myogenic and tubuloglomerular feedback (TGF) mechanisms that are engaged by pressure changes within preglomerular arteries and by tubular flow and content, respectively. Our understanding of autoregulatory function in the kidney largely stems from experiments in anesthetized animals where renal perfusion pressure is precisely controlled. However, normally occurring variations in blood pressure are sufficient to engage both myogenic and TGF mechanisms, making the assessment of autoregulatory function in conscious animals of significant value. To our knowledge, no studies have evaluated the dynamics of RBF in conscious mice. Therefore, we used spectral analysis of blood pressure and RBF and identified dynamic operational characteristics of the myogenic and TGF mechanisms in conscious, freely moving mice instrumented with ultrasound flow probes and arterial catheters. The myogenic response generates a distinct resonance peak in transfer gain at 0.31 +/- 0.01 Hz. Myogenic-dependent attenuation of RBF oscillations, indicative of active autoregulation, is apparent as a trough in gain below 0.3 Hz (-6.5 +/- 1.3 dB) and a strong positive phase peak (93 +/- 9 deg), which are abolished by amlodipine infusion. Operation of TGF produces a local maximum in gain at 0.05 +/- 0.01 Hz and a positive phase peak (62.3 +/- 12.3 deg), both of which are eliminated by infusion of furosemide. Administration of amlodipine eliminated both myogenic and TGF signature peaks, whereas furosemide shifted the myogenic phase peak to a slower operational frequency. These data indicate that myogenic and TGF dynamics may be used to investigate the effectiveness of renal autoregulatory mechanisms in conscious mice.

Representation of time-varying nonlinear systems with time-varying principal dynamic modes

System identification of nonlinear time-varying (TV) systems has been a daunting task, as the number of parameters required for accurate identification is often larger than the number of data points available, and scales with the number of data points. Further, a 3-D graphical representation of TV second-order nonlinear dynamics without resorting to taking slices along one of the four axes has been a significant challenge to date. In this paper, we present a TV principal dynamic mode (TVPDM) method which overcomes these deficiencies. The TVPDM, by design, reduces one dimension, and by projecting PDM coefficients onto a set of basis functions, both nonstationary and nonlinear dynamics can be characterized. Another significant advantage of the TVPDM is its ability to discriminate the signal from noise dynamics, and provided that signal dynamics are orthogonal to each other, it has the capability to separate them. The efficacy of the proposed method is demonstrated with computer simulation examples comprised of various forms of nonstationarity and nonlinearity. The application of the TVPDM to the human heart rate and arterial blood pressure data during different postures is also presented and the results reveal significant nonstationarity even for short-term data recordings. The newly developed method has the potential to be a very useful tool for characterizing nonlinear TV systems, which has been a significant, challenging problem to date.

Interactions contributing to kidney blood flow autoregulation

PURPOSE OF REVIEW

Autoregulation of renal blood flow has traditionally been considered to stabilize glomerular filtration, and thus tubular load, in the face of blood pressure fluctuations. This view arose because of the contribution of tubuloglomerular feedback, which senses distal tubular fluid composition, to regulation and autoregulation of renal blood flow. Studies have indicated a more important role for the myogenic mechanism. It has been proposed that the 'purpose' of autoregulation is to defend glomerular structure. Both these views may be incomplete because neither takes into consideration the complex interactions between tubuloglomerular feedback and the myogenic mechanism and among nephrons whose afferent arterioles derived from a common interlobular artery.

RECENT FINDINGS

Recent findings indicate that it is now indisputable that effective autoregulation is necessary for defense of glomerular structure. Extensive modulation of the myogenic mechanism by tubuloglomerular feedback has been shown using a variety of experimental designs that have illuminated one pathway (neuronal nitric oxide synthase at the macula densa) by which this occurs.

SUMMARY

These findings indicate that the myogenic mechanism can no longer be considered as a purely vascular mechanism in the kidney and instead receives information via tubuloglomerular feedback about the status of renal function.

Quantifying cardiac sympathetic and parasympathetic nervous activities using principal dynamic modes analysis of heart rate variability

The ratio between low-frequency (LF) and high-frequency (HF) spectral power of heart rate has been used as an approximate index for determining the autonomic nervous system (ANS) balance. An accurate assessment of the ANS balance can only be achieved if clear separation of the dynamics of the sympathetic and parasympathetic nervous activities can be obtained, which is a daunting task because they are nonlinear and have overlapping dynamics. In this study, a promising nonlinear method, termed the principal dynamic mode (PDM) method, is used to separate dynamic components of the sympathetic and parasympathetic nervous activities on the basis of ECG signal, and the results are compared with the power spectral approach to assessing the ANS balance. The PDM analysis based on the 28 subjects consistently resulted in a clear separation of the two nervous systems, which have similar frequency characteristics for parasympathetic and sympathetic activities as those reported in the literature. With the application of atropine, in 13 of 15 supine subjects there was an increase in the sympathetic-to-parasympathetic ratio (SPR) due to a greater decrease of parasympathetic than sympathetic activity ( P = 0.003), and all 13 subjects in the upright position had a decrease in SPR due to a greater decrease of sympathetic than parasympathetic activity ( P < 0.001) with the application of propranolol. The LF-to-HF ratio calculated by the power spectral density is less accurate than the PDM because it is not able to separate the dynamics of the parasympathetic and sympathetic nervous systems. The culprit is equivalent decreases in both the sympathetic and parasympathetic activities irrespective of the pharmacological blockades. These findings suggest that the PDM shows promise as a noninvasive and quantitative marker of ANS imbalance, which has been shown to be a factor in many cardiac and stress-related diseases.

Use of meixner functions in estimation of Volterra kernels of nonlinear systems with delay

Volterra series representation of nonlinear systems is a mathematical analysis tool that has been successfully applied in many areas of biological sciences, especially in the area of modeling of hemodynamic response. In this study, we explored the possibility of using discrete time Meixner basis functions (MBFs) in estimating Volterra kernels of nonlinear systems. The problem of estimation of Volterra kernels can be formulated as a multiple regression problem and solved using least squares estimation. By expanding system kernels with some suitable basis functions, it is possible to reduce the number of parameters to be estimated and obtain better kernel estimates. Thus far, Laguerre basis functions have been widely used in this framework. However, research in signal processing indicates that when the kernels have a slow initial onset or delay, Meixner functions, which can be made to have a slow start, are more suitable in terms of providing a more accurate approximation to the kernels. We, therefore, compared the performance of Meixner functions, in kernel estimation, to that of Laguerre functions in some test cases that we constructed and in a real experimental case where we studied photoreceptor responses of photoreceptor cells of adult fruitflies (Drosophila melanogaster). Our results indicate that when there is a slow initial onset or delay, MBF expansion provides better kernel estimates.

Nonlinear Analysis of the Separate Contributions of Autonomic Nervous Systems to Heart Rate Variability Using Principal Dynamic Modes

This paper introduces a modified principal dynamic modes (PDM) method, which is able to separate the dynamics of sympathetic and parasympathetic nervous activities. The PDM is based on the principle that among all possible choices of expansion bases, there are some that require the minimum number of basis functions to achieve a given mean-square approximation of the system output. Such a minimum set of basis functions is termed PDMs of the nonlinear system. We found that the first two dominant PDMs have similar frequency characteristics for parasympathetic and sympathetic activities, as reported in the literature. These results are consistent for all nine of our healthy human subjects using our modified PDM approach. Validation of the purported separation of parasympathetic and sympathetic activities was performed by the application of the autonomic nervous system blocking drugs atropine and propranolol. With separate applications of the respective drugs, we found a significant decrease in the amplitude of the waveforms that correspond to each nervous activity. Furthermore, we observed near complete elimination of these dynamics when both drugs were given to the subjects. Comparison of our method to the conventional low-frequency/high-frequency ratio shows that our proposed approach provides more accurate assessment of the autonomic nervous balance. Our nonlinear PDM approach allows a clear separation of the two autonomic nervous activities, the lack of which has been the main reason why heart rate variability analysis has not had wide clinical acceptance.