Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats

Abstract

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.

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

“…Using recently-developed TVTF and TVCF approaches (31, 32), we were able to critically examine the stationarity of the TGF and myogenic autoregulatory mechanisms, to assess the extent that nonstationarities contributes to autoregulatory effectiveness and dynamic complexity. Based on information from studies of nonlinear dynamics (4,6,7,21,24,28,30) and from time-frequency mapping (11,33), we predicted greater time-varying content in SHR than in SDR rats. The main findings of this study are that autoregulatory dynamics in both strains show significant variation over time, in the form of intermittent peaks and slow oscillations in transfer function gain.…”

“…The observation that both the TGF and myogenic autoregulatory systems exhibit substantial time variability is not surprising given that these two mechanisms interact via a nonlinear feedback process to minimize blood flow variations despite forced blood pressure fluctuations (7,14,21,24,27). A prominent feature of the TVTF is the very low frequency modulation of gain magnitude in the TGF frequency range in both SDR and SHR, and in the myogenic frequency range in SHR.…”

“…Similarly, both the myogenic mechanism (25) and TGF (31) are extensively modulated by ANG II, the generation of which is at least partly TGF dependent. Furthermore, there are nonlinear interactions between these two systems that produce spectral peaks (7,21,24) and changes in overall system behavior (19,20). In addition, the state of the TGF system appears to modulate the resonant frequency of the myogenic mechanism (26,27).…”

“…As a result, the time-resolution of time-invariant methods is very limited, and any nonstationarity that is present may distort the spectra and introduce artifactual spectral complexity (21). Nevertheless, time-invariant methods have played an important role in elucidating the dynamics of renal autoregulation.…”

“…In a simulation study, Ditlevsen et al (14) proposed that the nonlinear dynamics exhibited by SHR can be explained by relatively large and rapid random fluctuations in TGF system gain, a parameter chosen "to represent mechanisms not explicitly included in the model" (14). Layton et al (21) have suggested that a sufficient explanation for the irregular fluctuations in the TGF system in SHR may involve resetting and temporal variation in TGF gain and/or time delays in coupled nephrons that may result in abrupt switching between multiple stable oscillatory modes. It is interesting that both of these hypotheses involve parameter nonstationarity.…”