Oscillation nephron

Neighboring nephrons also communicate with one another. Experiments performed by Holstein-Rathlou show how nephrons that share a common interlobular artery tend to adjust their TGF mediated pressure oscillations so as to produce a state of in-phase synchronization [7]. Holstein-Rathlou also demonstrated how microperfusion of one nephron (with artificial tubular fluid) affects the amplitude of the pressure oscillation in a neighboring nephron. This provides a method to determine the strength of the nephron-nephron interaction. 

Fig. 12.2 (a) Experimental recording of the proximal tubular pressure in a single nephron of a normotensive rat. The power spectrum (b) clearly shows the TGF-mediated oscillations at fsiow(t) 0.034 Hz and the myogenic oscillations at ffasi(t) 0.16 Hz. The spectrum also displays harmonics and subharmonics of the TGF-oscillations. A plot... 

For T = 16 s, the single nephron model undergoes a supercritical Hopf bifurcation at a = 11 (outside the figure), fn this bifurcation, the equilibrium point loses its stability, and stable periodic oscillations emerge as the steady-state solution. For a = 19.5, at the point denoted PDla 2 in Fig. 12.5, this solution undergoes a period-... 

For normotensive rats, the typical operation point around a = 10—12 and T = 16 s falls near the Hopf bifurcation point. This agrees with the experimental finding that about 70% of the nephrons perform self-sustained oscillations while the remaining show stable equilibrium behavior [22]. We can also imagine how the system is shifted back and forth across the Hopf bifurcation by variations in the arterial pressure. This explains the characteristic temporal behavior of the nephrons with periods of self-sustained oscillations interrupted by periods of stable equilibrium dynamics. 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

Fig. 12.6 (a) Temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model fora = 12 and T = 16 s. (b) Corresponding phase plot. With the assumed parameters the model displays self-sustained oscillations in good agreement with the behavior observed for normotensive rats. The unstable equilibrium point falls in the middle of the limit cycle, and the motion along the cycle proceeds in the clockwise direction. 

Fig. 12.9 Two-dimensional bifurcation diagram for the single-nephron model. The diagram illustrates the complicated bifurcation structure in the region of 1 1, 1 2, and 1 3 resonances between the arteriolar dynamics and the TGF-mediated oscillations. In the physiologically interesting regime around T = 16 s, another set of complicated period-doubling and saddle-node bifurcations occur. Here, we are operating close to the 1 4...

As demonstrated by the power spectra in Figs. 12.2a and 12.3b, regulation of the blood flow to the individual nephron involves several oscillatory modes. The two dominating time scales are associated with the period Tsiow 30—40 s of the slow TGF-mediated oscillations and the somewhat shorter time scale Tjast 5—10 s defined by the myogenic oscillations of the afferent arteriolar diameter. The two modes interact because they both involve activation of smooth muscle cells in the arteriolar wall. Our model describes these mechanisms and the coupling between the two modes, and it also reproduces the observed multi-mode dynamics. We can, therefore, use the model to examine some of the phenomena that can be expected to arise from the interaction between the two modes. 

Fig. 12.11 Two-mode oscillatory behavior in the single nephron model. Black colored regions correspond to a chaotic solution. The figure shows different regions in which 1 4, 1 5 and 1 6 synchronization occurs in the interaction between the fast myogenic oscillations...

Let us examine the situation for large values of a where the individual nephron exhibits chaotic dynamics. Figure 12.15a shows a phase plot for one of the nephrons in our two-nephron model for a = 32, T = 16 s, e = 0.0, and y = 0.2. Here we have introduced a slight mismatch AT = 0.2 s in the delay times between the two nephrons and, as illustrated by the tubular pressure variations of Fig. 12.15b, the nephrons follow different trajectories. However, the average period is precisely the same. This is a typical example of phase synchronization of two chaotic oscillators. 

Fig. 12.15 (a) Phase plot for one of the nephrons and (b) temporal variation of the tubular pressures for both nephrons in a pair of coupled chaotically oscillating units, a = 32, T = 16 s, and e = y = 0.2. The figure illustrates the phenomenon of chaotic phase synchronization. By virtue of their mutual coupling the two chaotic oscillators adjust their (average) periods to be identical. The amplitudes, however, vary incoherently and in a chaotic manner [27],... 

Let us consider the case of a = 30 corresponding to a weakly developed chaotic attractor in the individual nephron. The coupling strength y = 0.06 and the delay time T2 in the second nephron is considered as a parameter. Three different chaotic states can be identified in Fig. 12.16. For the asynchronous behavior both of the rotation numbers ns and n f differ from 1 and change continuously with T2. In the synchronization region, the rotation numbers are precisely equal to 1. Here, two cases can be distinguished. To the left, the rotation numbers ns and n/ are both equal to unity and both the slow and the fast oscillations are synchronized. To the right (T2 > 14.2 s), while the slow mode of the chaotic oscillations remain locked, the fast mode drifts randomly. In this case the synchronization condition is fulfilled only for one of oscillatory modes, and we speak of partial synchronization. A detailed analysis of the experimental data series reveals precisely the same phenomena [31]. 

Figure 12.17 shows an example of the tubular pressure variations that one can observe for adjacent nephrons for a normotensive rat. For one of the nephrons, the pressure variations are drawn in black, and for the other nephron in gray. Both curves show fairly regular variations in the tubular pressures with a period of approximately 31s. The amplitude is about 1.5 mmHg and the mean pressure is close to 13 mmHg. Inspection of the figure clearly reveals that the oscillations are synchronized and remain nearly in phase for the entire observation period (corresponding to 25 periods of oscillation). 

Figures 12.18a and b show examples of the tubular pressure variations in pairs of neighboring nephrons for hypertensive rats. These oscillations are significantly more irregular than the oscillations displayed in Fig. 12.17 and, as previously discussed, it is likely that they can be ascribed to a chaotic dynamics. In spite of this irregularity, however, one can visually observe a certain degree of synchronization between the interacting nephrons.

Fig. 12.17 Tubular pressure variations for a pair of coupled nephrons in a normotensive rat. The pressure variations remain nearly in phase forthe entire observation time (or 25 periods of oscillation).

Fig. 12.18 Two examples (a and b) of the tubular pressure variations that one can observe in adjacent nephrons for hypertensive rats. In spite of the irregularity in the dynamics, one can see a certain degree of synchronization in the phases of the two oscillations. This synchronization is supported by formal investigations, e.g., by means of the wavelet technique.

In a recent study [34], we made use of wavelet and double-wavelet analysis to examine the relative occurrence of various states of synchronization in pairs of interacting nephrons. We showed that both full and partial synchronization occur for normotensive as well as for hypertensive rats, and that the partial synchronization can involve only the slow oscillations or only the fast oscillations. We also used... 

In order to examine the synchronization phenomena that can arise in larger ensembles of nephrons, we recently developed a model of a vascular-coupled nephron tree [35], focusing on the effect of the hemodynamic coupling. As explained above, the idea is here that, as one nephron reduces its arterioler diameter to lower the incoming blood flow, more blood is distributed to the other nephrons in accordance with the flow resistances in the network. An interesting aspect of this particular coupling is that the nephrons interact both via the blood flow that controls their tendency to oscillate and via the oscillations in this blood flow that control their tendency to synchronize. We refer to such a structure as a resource distribution chain, and we have shown that phenomena similar to those that we describe here... 

Fig. 12.19 Left sketch of a vascular-coupled nephron tree including the interlobular artery, the afferent arterioles and the glomeruli. Right oscillation amplitudes as function of the arterial pressure and the position of the branching point along the vascular tree.

Depending on the choice of control parameters, the amplitudes of the pressure oscillations in the nephron tree are found to be different at different positions in the tree. Due to model symmetry, two nephrons connected to the same node have the same oscillation amplitudes. Thus, we can refer to the number of the branching point to describe the amplitude properties. Branching points 2, 3, and 4 may correspond to deep nephrons and branching points 6 and 7 to superficial nephrons. Experimentally, only the pressure oscillations in nephrons near the surface of the kidney have been investigated. However, we suppose that deep (juxtamedullary) nephrons can exhibit oscillations in their pressures and flows as well. 

In collaboration with Alexander Gorbach, NIH, we have initiated a study of the spatial patterns in the nephron synchronization. This study involves the use of infrared cameras or other types of equipment that can measure variations in the blood supply by small (0.01°C) fluctuations in the temperature at the surface of the kidney. It is also of interest to study how the large amplitude oscillations in pressure, fluid flow, and salt concentration at the entrance of the distal tubule influence the delicate hormonal processes in that part of the kidney, to establish a more quantitative description of some of the mechanisms involved in the development of hypertension, and to examine the effects of various drugs. 

N.-H. Holstein-Rathlou, Synchronization of Proximal Intratubular Pressure Oscillations Evidence for Interaction between Nephrons, Pflugers Archiv 408,438-443 (1987). 

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