Response of the Nephron Arterial Network and Its Interactions to Acute Hypertension: A Simulation

Authors: Donald J Marsh, Niels-Henrik Holstein-Rathlou

Abstract

We simulated the dynamics of a group of 10 nephrons supplied from an arterial network and subjected to acute increases in blood pressure. Arterial lengths and topology were based on measurements of a vascular cast. The model builds on a previous version exercised at a single blood pressure with 2 additional pressure diuresis and the effect of blood pressure on efferent arteriolar vascular resistance. The new version simulates autoregulation, and reproduces tubule pressure oscillations. Individual nephron dynamics depended on mean arterial pressure and the axial pressure gradient required to cause blood flow through the arteries. Rhythmic blood withdrawal into afferent arterioles caused blood flow fluctuations in downstream vessels. Blood pressure dependent changes in nephron dynamics affected synchronization metrics. The combination of vascular pressure gradients and oscillations created a range of arterial pressures at the origins of the 10 afferent arterioles. Because arterial blood pressure in conscious animals has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${1}/{f}$\end{document} dynamics, we applied an arterial pressure pattern with such dynamics to the model. Amplitude of tubule pressure oscillations were affected by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${1}/{f}$\end{document} blood pressure fluctuations, but the oscillation frequencies did not change. The pressure gradients required to deliver blood to all afferent arterioles impose a complexity that affects nephrons according to their locations in the network, but other interactions compensate to ensure the stability of the system. The sensitivity of nephron response to location on the network, and the constancy of the tubular oscillation frequency provide a spatial and time context.

Introduction

Terrestrial mammals maintain constant renal blood flow and glomerular filtration in response to blood pressure variation over a range of about 50 mmHg, a phenomenon known as autoregulation.^1^ Our goals in this study are to identify the tubular and vascular structures needed to provide autoregulation in a group of nephrons, and to evaluate how the interactions among nephrons and their arterioles change when blood pressure changes.

Tubuloglomerular feedback (TGF) and the myogenic mechanism regulate the diameter of each nephron’s afferent arteriole to control the vessel’s hydraulic conductance. Each of these mechanisms operates as an autonomous limit cycle oscillator; they interact with each other to form a bimodal process,^2–4^ entraining pressures and flows in the nephron and generating an electrical signal that propagates into the arterial network,^3^ where it interacts with similar signals from other nephrons.^2^^,^  ^5^^,^  ^6^

These signals contain information about the current operating state of a nephron and its vascular control mechanisms. Interactions between the 2 mechanisms in each nephron, and the interactions with signals from other nephrons, can lead to synchronization.^5^ The mechanisms adjust vascular conductance in arterioles, the periodic adjustments impose periodic withdrawals of blood from arteries supplying the arterioles causing the blood pressure and flow rates in the arteries to oscillate. The two electrical signals oscillate, the blood pressure at the origin of each afferent arteriole oscillates, and the result is a triad of interacting signals. These interactions operate over the arterial network and can lead to states of partial or global synchronization.^7–11^ In addition, both mechanisms depend on the mean arterial blood pressure, and a change in the mean pressure can affect the interactions.

The oscillation frequencies of these triads are sensitive to the dimensions of several structures that comprise the nephron arterial network. The macula densa is a collection of tubular epithelial cells at the end of the ascending limb of Henle’s loop that serves as the sensor for the TGF mechanism. The oscillations originating in the afferent arteriole initiate waves of tubular fluid whose wavelengths are functions of tubular length from glomerulus to macula densa; the longer the tubule, the lower the TGF frequency. Approximately 2/3 of all nephrons lie entirely within the renal cortex and outer medulla; the rest have descending and ascending limbs of Henle’s loop that extend variable distances into the inner medulla. All of the measurements of nephron dynamics that we cite in this work were made from the group of shorter nephrons. Nephron lengths in this group are not identical but variations of a few percent affect the oscillation frequencies and the degree of synchronization.^8^

The single renal artery branches irregularly into a series of smaller arteries that run parallel to the renal surface and end at the lateral edge of the kidney. Branches emerge from these smaller arteries and follow an orthogonal path toward the renal surface. Additional branchings occur from these orthogonal arteries; as few as 2 or as many as 7 branches may be present before an orthogonal artery ends at the renal surface.^12^ Afferent arterioles arise from the orthogonal arteries.

The renal circulation is a demand driven resource distribution network. The resource is oxygenated blood, nephrons provide the demand, and the arterial network is a group of tree networks. Tree networks, found frequently in natural and man-made resource distribution networks, have characteristics that are important for the stability of renal dynamics. The number of nodes is exactly one greater than the number of links, there are no redundant loops, and each node has a unique pathway to all other nodes in the network.^13^ The results from the vascular cast measured by computed tomography^12^ satisfy these requirements. A tree network in which the arteries serve as edges and nephrons with their afferent arteriole as leaves can represent the organization of the kidney. A characteristic of such graphs is that a unique path connects any pair of vertices, and that all vertices connect to each other. Other types of networks—small world, developed to characterize social interactions, and random, which describes the internet—do not restrict the number of connections at a given node. These networks usually have nodes of variable size, and this type of variability encourages dominant activity by the larger nodes. Multiple afferent arterioles would make it impossible for nephron development to match glomerular filtration and tubular flow rate with epithelial transport capacity. The tree network configuration of the renal network, because of its functional implication, is therefore fundamental to successful renal function.

A variety of measures determine the strength of a given connection. The distances separating the origins of nearest neighbor afferent arterioles follow an exponential distribution.^12^^,^  ^14^ Exponential distributions are typically seen in Poisson structures where the distances between objects are independent of each other. The arterial distances separating the origins of afferent arterioles function as electrical and blood flow resistances, and their random lengths will affect the propagation of the information streams throughout the network.

Micro computed tomography applied to a cast of the renal blood vessels^12^ revealed that 53% of all afferent arterioles arose from paired terminal arterial branches at the furthest extent of arterial trees. Physiological measurements have been made exclusively from this group of nephrons; 23% arose from unpaired sub-surface arteries, and the remaining afferent arterioles branched directly from subsurface arteries.

As blood flows from the renal artery to the periphery the hydrostatic pressure must decrease proceeding peripherally. In each separate tree structure the vascular pressure will be highest at the base, and will decline proceeding toward the surface. Afferent arterioles originate from all arterial branches in a tree, so the hydrostatic pressure at these points of origin will differ from each other. Glomerular filtration is a function of the arterial pressure at the origin of the afferent arteriole, and will vary accordingly.

We reported a simulation of the dynamics of a group of nephrons, their afferent arterioles, and the arteries that connect them.^8^ These simulations were conducted at a single arterial pressure, and predicted tubular dynamics, arterial network blood pressures, and nephron synchronization. Complete synchronization was found only between paired nephrons whose afferent arterioles originated from a common arterial site. Synchronization was partial when segments of arteries intervened. That work was designed to predict the effect of structural variation in the lengths of nephrons and arterial segments. We now turn to the problem of autoregulation induced by acute hypertension. Our goal is to determine whether the model can, by itself, autoregulate. It does not. Pressure diuresis^15–18^ and adjustments of efferent arteriolar resistance ^19^^,^  ^20^ are known responses to acute hypertension. We evaluate the role of these mechanisms in autoregulation. The evaluation examines the dynamics of the network’s individual components, the information streams they generate, and the interactions among those streams. The interactions involve triads of signals, each of which is a variable subject to modification by its participation in the network.

Methods

The simulations in this study are based on a model containing 10 single nephron models, each supplied by a single afferent arteriole originating from an arterial branch.^8^  Figure 1 shows the overall model structure. The equations of the single nephron-afferent arteriolar model and the network are available in the Supplementary Materials Section of this manuscript. Each of the 10 nephron models begins with a calculation of glomerular filtration rate for that nephron by solving an ordinary differential equation for plasma protein concentration as a function of distance along an idealized glomerular capillary. The glomerular filtration rate provides an initial flow condition for the tubular model, which calculates hydrostatic pressure, flow, and NaCl concentration as functions of time and distance along the nephron. The nephron is modeled as a compliant tubule of epithelial cells that transport water and solutes across their walls between the lumen and the interstitial fluid. The nephron model is divided into 3 proximal tubule, descending limb of Henle’s loop, and thick ascending limb of Henle’s loop. Each segment is provided with epithelial transport functions and compliance values determined from published experimental values. The terminal tubular flow rate and NaCl concentrations are the results of the epithelial transport processes while the terminal hydrostatic pressure is derived from a non-linear function of the terminal fluid flow. The system of equations describing tubular pressures, flows, and concentrations is solved iteratively at each time step.

Afferent arterioles were simulated using a set of 6 ordinary differential equations developed by Gonzalez-Fernandez and Ermentrout to predict blood flow dynamics in cerebral arteries.^21^ This model simulates voltage gated Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{2+}$\end{document} channels and voltage- and Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{2+}$\end{document}-sensitive K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{+}$\end{document} channels in smooth muscle cells. The interacting ionic currents are non-linear and produce autonomous oscillations of the membrane electrical potential difference and of intracellular Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{2+}$\end{document}. The Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{2+}$\end{document} oscillation induces oscillations in myosin light chain phosphorylation, cross bridge formation, and muscle stress. The model simulates hoop, elastic, and muscle stresses, and from them, the rate of change of the arteriole’s diameter.

The Gonzalez-Ermentrout model^21^ represents the action of the myogenic mechanism in arterioles throughout the body and is not specific to the kidney. TGF is a renal-specific mechanism whose signal originates in the macula densa and is transmitted through the mesangium to the afferent arteriole. We coupled the TGF signal to the membrane Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{2+}$\end{document} permeability of the arteriolar smooth muscle cell. The TGF signal arrives at the afferent arteriole at a site near the nephron’s glomerulus and its action propagates upstream. To simulate this non-uniform distribution we modeled the afferent arteriole as 2 separate segments in series coupled electrically, the segment closer to the glomerulus receiving the larger TGF signal.

We solved the 10 nephron model assuming that blood flow and electrical currents were conserved at each vascular node. We further assumed that the combination of TGF and the myogenic mechanism acting on the afferent arteriole were responsible for all adjustments to arteriolar diameter in each nephron. Lengths of arterial segments were based on measurements made from a vascular cast of a rat kidney.^12^ Optical methods for measuring vascular radii lack sufficient resolution for use in the model. The goal of each simulation in this study was to achieve a stable state of the system, and then to apply a step increase in the root pressure at Node 11. The periodic actions of the afferent arterioles impose a set of stresses on the arterial segments from which they originate, and achieving a stable state means that the arterial segments undergo a series of relaxations until no further changes in their radii occur. To establish a function whose minimum represents the stable state we constructed a vector whose elements were the blood pressures at each of the arterial nodes, and calculated its Euclidian norm. At the end of each iteration we applied a correction to each radius by applying Murray’s law,^22^ and continued to iterate until the norm of the vector reached an asymptotic limit. Murray’s law calculates the relationship among the diameters of a source vessel and its branches that minimizes the energy cost of driving fluid flow through the branch, and is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $r^\varepsilon =r^\varepsilon _1 + r^\varepsilon _2$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $r$\end{document} is the radius of the source vessel and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $r_1$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $r_2$\end{document} are the radii of the first and second branches. Originally Murray predicted a value of 3 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varepsilon$\end{document}.^22^ Later work has shown that the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varepsilon$\end{document} depends on such factors as branch angles and unequal branch diameters and also to vary among vascular beds.^23^ These factors are not known in sufficient detail for the renal arterial network to assign a value a priori for the model. We therefore conducted a series of simulations varying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varepsilon$\end{document} from 2.5 to 3, to determine the sensitivity of arterial radii and autoregulatory efficacy. After reaching the asymptote, the arterial diameters were held constant during an acute increase in arterial pressure at the root of the network, Node 11.

In this study, we assumed a mean arterial blood pressure of 100 mmHg, and estimated a pressure drop to 85 mmHg from the renal artery to the root of the network. After achieving the asymptote we used a hyperbolic tangent function to raise the Node 11 pressure acutely to the desired level. Experimental induction of acute hypertension caused a rise of mean arterial blood pressure to 130 mmHg, which, given the pressure drop to from the renal artery, represented a root pressure of 115 mmHg.^15^^,^  ^24^

In the initial presentation of the model,^8^ we used a random number generator to vary the lengths of nephrons and arterial segments in 15 simulations, each simulation using a unique seed value. This strategy was designed to evaluate the effects of geometric variation on synchronization among the information streams. For the current work, we selected a single one of the 15 simulations and applied 7 different levels of arterial pressure at Node 11. We used 4 different variations of the model to determine which one best predicted the original model, the original model with a function simulating pressure diuresis; the original model with a function simulating the efferent arteriolar response to blood pressure variation; and the original model with both the pressure diuresis and efferent arteriolar responses.

The original model derives the behavior of NaCl in the proximal tubule from the assumptions that NaCl is the only solute in the tubular fluid and that active NaCl reabsorption drives fluid reabsorption, leaving the tubular fluid concentration unchanged along the tubule’s length. The local rate of fluid reabsorption, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $J_v$\end{document}, was approximated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $J_v = \kappa \,\, exp(-\theta z)$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\theta$\end{document} are coefficients, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $z$\end{document} is distance along the proximal tubule. Pressure diuresis is a decrease in proximal tubule reabsorption of NaCl and water in response to an increase in blood pressure.^15–18^ The original model used the same equation for each of the 10 nephron simulations, with a parameter set selected to simulate tubule pressure dynamics measured in a basal state.^5^^,^  ^24–27^ The arterial pressure at Node 11 was set at 85 mmHg to achieve this state. To simulate the pressure diuresis effect we developed a function of the arterial pressure at the origin of each nephron’s afferent

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varphi$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\theta$\end{document} are coefficients, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Delta P(i)$\end{document} is the difference between the arterial pressure at the origin of the ith nephron’s afferent arteriole and 85 mmHg. Chou and Marsh measured the change in proximal tubule fluid reabsorption during acute hypertension.^15^ Their experimental result was used to provide the value for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varphi$\end{document}, 0.018 mmHg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{-1}$\end{document}.

The vascular resistance of the efferent arteriole, a known determinant of the glomerular filtration rate, is represented in our original model with a single parameter, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $R_E$\end{document}. The efferent arteriole’s vascular resistance is known to vary inversely with arterial pressure.^19^^,^  ^20^ We modeled this pressure dependence

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varrho$\end{document} is a coefficient and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $R_e (i)$\end{document} is the resistance of the ith efferent arteriole. The value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varrho$\end{document}, 0.010 mmHg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{-1}$\end{document}, was established by sensitivity analysis, using a predicted autoregulatory index of 0.0 as the selection criterion.

Simulation Protocols

All simulations were begun with the blood pressure at Node 11, the root of the arterial tree, set to 85 mmHg. Initial simulations ran for 1000 s during which the arterial radii were adjusted until an asymptotic value of the arterial nodal pressure vector was reached, as described above. Acute hypertension was then induced using a hyperbolic tangent function that reached its half maximum value at 22 simulated seconds. These simulations were allowed to run for 4000 simulated seconds, with a time interval between data points of 0.25 s, corresponding to a total of 16000 data points in each run. Presentation of the results and data analysis were begun at 1000 simulated seconds, allowing the first 1000 s to serve as a period of relaxation from the initial increase in arterial pressure. We exercised the model at 7 different values of Node 11 85, 90, 95, 100, 105, 110, and 115 mmHg. No adjustments of arterial diameters were made once the acute change in blood pressure started.

In some of the simulations we used a blood pressure input with a 1/f noise pattern. The 1/f time series were created using the routine dsp.ColoredNoise in Matlab.^28^ The intent in these simulation was to use a blood pressure forcing with the same statistical properties as those found in unrestrained conscious rats with implanted units containing arterial catheters, pressure transducers, and radiotransmitters.^29^ To be more precise, the general formula for the pattern is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${1}/{f^{\alpha }}$\end{document}. The measured value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha$\end{document} was 1.45, and this value was used to generate the blood pressure time series used in the simulations.

Data Analysis

Power spectral densities were computed using Welch’s overlapped segment averaging estimator as implemented in the routine pwelch in Matlab.^28^ The first 1000 points of each time series were discarded (pressure transient). The remaining points were divided into 8 segments, with a 50 % overlap, and each segment was windowed using a Hamming window.

The continuous wavelet transforms were done using the routine cwt in Matlab,^28^ with the default analytic Morse wavelet.

Group synchrony was assessed using the method of Richardson et al.^30^ A value of 0 indicates no synchrony between the series, whereas a value of 1 indicates perfect synchrony among all time series.

Results

Our first goal was to identify the components of a model of nephrons and the arterial network needed to achieve autoregulation of renal blood flow. Each version of the model was also required to simulate proximal tubule pressure oscillations in surface nephrons comparable in frequency and spectral power to those found experimentally.^31^ As explained in the section “Methods,” values of arterial radii are calculated by the model, and are adjusted in a series of iterations by applying Murray’s law. Because of some uncertainty about the appropriate value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varepsilon$\end{document}, the exponent in Murray’s formula, we performed a parameter sensitivity study, using 3 different values of the parameter. At the same time we added other known features of the renal response to acute hypertension.

Experimental results used for model validation came from a variety of tubular pressure measurements at basal blood pressure,^5^^,^  ^25–27^ Doppler flow measurements from single efferent arterioles made at control and acutely elevated blood pressure,^24^ laser speckle contrast spectrometry at basal blood pressure,^32–34^ and tubular microperfusion at control and acutely elevated blood pressure,^15^ all in anesthetized male rats.

The results of these simulations are in Table 1. Increasing the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varepsilon$\end{document} reduced arterial radii, increased blood flow through the system at the resting blood pressure, but had no systematic effect on the autoregulatory index or the blood flow during acute hypertension. The results in Table 1 did not establish a preferred value for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\varepsilon$\end{document}, and because recent experimental and theoretical studies indicate a value less than Murray’s initial prediction and lying in the interval [2.50,3.0],^23^ we elected to use a value of 2.75 throughout the remainder of the study.

Figure 2 shows the predicted blood flow over a range of blood pressures at Node 11. The blood pressure was increased from the basal level of 85 mmHg. The results were calculated for all 10 nephrons of the model at each of 7 target blood pressure levels. We tested 4 versions of the model. The first was the original network model.^8^ With increasing blood pressure at Node 11, the amplitude of tubular pressure oscillations exceeded values greater than have been observed experimentally,^24^ and the model failed to converge at node 11 pressures greater than 110 mmHg. The excessive amplitude of the tubular pressure oscillations and the failure to achieve convergence at a blood pressure level where autoregulation is known to occur means that the original model, by itself, fails.

Model 2 began with the original model and added a function that simulates pressure diuresis, equation 1. An increase in arterial blood pressure inhibited proximal tubule reabsorption of NaCl and water.^15–18^ The result was an increase of proximal tubular outflow leading to an increase in the amplitude of the TGF signal emanating from the macula densa, and vasoconstriction of the afferent arteriole. While the model produced an expected response, that response was not autoregulation, so that Model 2 failed to simulate known experimental results, and is therefore not an adequate representation of network dynamics.

Model 3 also began with the original model and added a function that simulated the increased hydraulic conductance of the efferent arteriole resulting from acute hypertension.^19^^,^  ^20^ This increased conductance increases blood flow but does not simulate autoregulation. Like Model 1 it failed to achieve convergence at a pressure of 115 mmHg. Model 3 thus also failed.

Model 4 combined the original model with both the pressure diuresis and the efferent arteriolar conductance functions, and achieved an autoregulatory index of −0.035. We consider this model a suitable tool with which to evaluate the signal interactions in the nephron arterial network, and we used it exclusively in the following sections of the paper.

Figure 3 presents simulated time series results for nephron 10 at 3 different levels of Node 11 blood pressure. The variables are proximal tubular pressure, arterial blood pressure at the origin of the nephron’s afferent arteriole, and TGF signal strength. The signal strength is calculated from a function of the TGF action that multiplies the membrane \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Ca^{2+}$\end{document} permeability, and serves to control the feedback gain.

The top row of Figure 3 shows the predicted proximal tubule hydrostatic pressure. The amplitude and frequency of the tubular pressure oscillations with the Node 11 pressure set at 85 mmHg are consistent with measured in vivo values.^5^^,^  ^25^^,^  ^27^ The blood pressure variation at Node 20 is a fraction of 1 mmHg; all three variables track each other with the Node 11 pressure at this level.

Increasing Node 11 pressure to 100 mmHg led to an increase in the amplitude of tubular pressure oscillations, Node 20 pressure, and TGF signal strength. Each of the time series at 100 mmHg appeared to be regular, with a constant amplitude, a description different from that with Node 11 pressure at 85 mmHg.

The simulation with Node 11 pressure raised to 115 mm Hg showed greater amplitudes in tubular and arterial nodal pressures. Additional fluctuations in tubular and arterial pressure appeared at the highest Node 11 pressure, reflecting the emergence of a new dynamic state.

Thus, the simulation results in Figure 3 show that the system can operate over a range of dynamic states, depending on the mean arterial blood pressure.

Time series of blood pressures at each of Nodes 12-20, are presented in Figure 4, with simulations run at each of the 3 different root pressures used in Figure 3. The root pressure at Node 11 was held constant during each of the 3 simulations, and the time varying pressures at the other nodes were the result of periodic withdrawal of blood into afferent arterioles. The pressure gradient from Node 12 to the nodes closest to the renal surface was approximately the same for each of the 3 simulations, reflecting the fact that the simulations predicted autoregulation. At each of the 3 root pressures, the lowest pressures were predicted to occur at the origins of the afferent arterioles serving Nephrons 6-10, the set of surface nephrons; afferent arterioles serving sub-surface Nephrons, 1-5, experienced higher nodal pressures. Except for the paired afferent arterioles originating from common arterial Nodes—2 and 3, 6 and 7, and 9 and 10, the nodal pressures differed from each other.

The nodal blood pressure time series showed periodic behavior in each of the 3 series of Figure 4, but the amplitudes of the fluctuations varied with the root pressure and with distance from Node 11. Of all the afferent arterioles of the system, Nephron 1 is exposed to the highest blood pressure, both in the basal state and during acute hypertension, because it is closest to the network’s root. The amplitude of the Node 13 pressure oscillations is smaller than the Node 20 oscillations. The blood flows in arteries supplying the surface nephrons are smaller than in the upstream arteries nearer the root, because of blood withdrawal into the sub-surface nephrons. These withdrawals are periodic and increase the amplitude of pressure and flow oscillations as blood flow travels toward the terminal end of the network.

The most regular of the 3 sets was produced with a root pressure of 100 mmHg, a result also shown in Figure 3. A root pressure of 85 mm Hg, our nominal resting state, generated TGF signal strengths inadequate to achieve full synchronization. The highest root pressure, 115 mmHg, produced fluctuations in nodal pressures, but these were more irregular than the other two. Although the nodal pressures near the renal surface appear to be noisy, this system is deterministic.

For Figure 5, we calculated spectral power of nephron blood flow fluctuations in 2 different frequency bands. The total power encompasses the frequency band 15.0-24.4 mHz, and includes both the TGF oscillation and the lower frequency fluctuations shown in Figure 3. The TGF frequency is 23.4 mHz at Node 11 pressures up to 100 mmHg. Low frequency power in the frequency band 15.0-21.5 mHz begins to increase at Node 11 pressures above 100 mm Hg and provides an increasing fraction of total power at higher pressures. The spectral power of tubular pressure with Node 11 pressure at 115 mmHg, 20.5 mHz, is provided entirely by the lower frequency oscillation.

Figure 6 shows the dependence of the synchronization metric for the tubular pressures, as described in section “Methods,” on Node 11 blood pressure. Synchronization is incomplete with the Node 11 Pressure at 85 mmHg. The paired nephrons (2 and 3, 6 and 7, 9 and 10) formed fully synchronized pairs, while nephrons 1, 4, 5, and 8, were not fully synchronized with other nephrons. Each member of the paired nephron sets are supplied by afferent arterioles arising from a common arterial site while the unpaired nephrons are separated from adjacent neighbors by segments of arteries.^8^ The interposition of these arterial segments inserts an electrical resistance that dissipates the propagated electrical signals and leads to partial synchronization. Because increasing blood pressure increases TGF signal strength, the electrical signal increases with pressure, and achieves levels sufficient to achieve near full synchronization at Node 11 blood pressures in the range of 90-100 mmHg. Synchronization falls off at Node 11 pressures greater than 100 mmHg. Figure 3, column 3 shows irregular oscillations in tubule pressure, Node 20 blood pressure, and Nephron 10 TGF signal strength that account for the decline in synchronization. The decrease in synchronization at 105 and 110 mmHg is consistent with the transition to a lower frequency transition in the TGF oscillation, as shown in Figure 5.

The TGF oscillation in the 10 nephrons of the model is the major determinant of its dynamics, but the results presented thus far point to more than synchronization as a result of the interactions taking place. To evaluate the impact of these interactions we studied the phase plane plots of the tubule pressure attractor in the sub-surface Nephron 1 and the surface Nephron 10 to changes in mean arterial pressure at Node 11. Phase plane plots are created by plotting a time series, in this case the tubular pressure, against a time-shifted version of the same time series. The phase plane plots of a periodic signal will retrace itself, while unsynchronized periodicities will emerge as a series of different trajectories. We used a time shift of 10 s to produce the results in Figure 7.

The phase plane plots for both nephrons increased in amplitude with increasing Node 11 Pressure. The data in Figure 3 show this effect for Nephron 10. With Node 11 at 85 mmHg, Nephron 1 displays an image consistent with that of a limit cycle oscillator, but Nephron 10 does not, as can also be seen in Figure 3. The arterial node supplying Nephron 10 is farther downstream from the root than Nephron 1, and its lower arterial pressure generates a weaker TGF signal. When Node 11 pressure is raised to 95 mmHg, both nephrons generate increased TGF signals, and both are capable of operating as limit cycle oscillators. Both images also show the smaller and more frequent oscillations of the myogenic mechanism, synchronized with the TGF oscillation in a 1 ratio as predicted^17^^,^  ^22^, and confirmed experimentally^35^. The phase plane plots of both nephrons show multiple trajectories with Node 11 at 105 mmHg. This result reflects the transition in dynamics that also appears in Figures 3, 5, and 6. At a still higher Node 11 pressure, Nephron 1 shows a reduced trajectory dispersion and synchronization of TGF and myogenic oscillations in a 1 ratio. The image of Nephron 10's phase space behavior is similar to that of Nephron 1, but less fully developed because of the differences in arterial pressure at their respective afferent arteriolar origins.

The results show that increasing the input pressure increases the strengths of a number of signals, and alters the interactions on the network. The only experimental data available for a comparison with this prediction is from a single study using a focused laser beam aimed at an efferent arteriole emerging on the renal surface.^24^ The simulation prediction is from a deterministic system, while the experimental data have system and measurement noise, preventing a direct comparison. The experimental results show an increase in spectral power with increased pressure, a result consistent with the simulation result, cf. Figure 6.

The results presented to this point apply a sudden increase of arterial pressure at the root of the network. The data used to validate the model were collected from anesthetized rats. These animals are normally active at night and at rest during the times the experiments were conducted. In another study we measured blood pressure in conscious rats with an implanted unit containing an arterial catheter, a pressure transducer, and a radiotransmitter.^29^ In addition to the well known diurnal rhythm in blood pressure we detected epochs of 2-3 h duration with 1/f noise, the characteristic behavior of which is that small events occur more frequently, larger events less frequently and correlation between events is not required. We simulated a blood pressure time series with a 1/f pattern and an initial value of 85 mmHg and applied it to the model at Node 11. The result is shown in Figure 8. The top panel contains the time series of Nephron 10’s tubule pressure, the middle panel Nephron 10’s blood flow, and the bottom panel the blood pressure at Node 20, the origin site of Nephron 10’s afferent arteriole. The tubule flow and nephron blood flow continue to oscillate, but the oscillations reflect the irregularities imposed by the blood pressure fluctuations applied at Node 11. The results in Figure 8 are thus a prediction of tubule dynamics in a conscious animal.

Figure 9 presents a wavelet transform of the Nephron 10 tubule pressure when forced with the 1/f pattern at 85 (upper panels) or 115 mmHg (lower panels). Wavelet transforms show spectral power as a function of time and frequency. The power of the TGF oscillation, shown in light blue (85 mm Hg) or yellow and red (115 mm Hg), varies with the Node 11 pressure, but the frequency remains unchanged for the duration of the simulations.

Discussion

Long term survival of mammals requires stable renal function in the face of the challenges of daily existence. Many of these challenges cause arterial blood pressure to change and renal autoregulation is an outcome of the responses the kidney generates. Mathematical models of the renal circulation must therefore be able to predict autoregulation. Experiments demonstrating autoregulation of renal blood flow are usually conducted over intervals of minutes to hours, and successful simulations must reflect physiological responses over similar time scales. We developed a dynamic model of the nephron arterial network that achieves autoregulation over a range of 30 mmHg blood pressure increases lasting for periods of 1 h. The model contains 10 nephrons each of which simulates afferent arterioles equipped with a myogenic mechanism and TGF, proximal tubular reabsorption sensitive to local arterial pressure, a phenomenon known as pressure diuresis, and a dependence of efferent arteriolar diameter on local arterial pressure. Each of these components has been demonstrated experimentally, and all of them are required in each nephron to enable the simulation to produce autoregulation.

The results of the model illustrate the effects of interactions occurring on the network. The structure contains examples of each of 3 different patterns of afferent arteriolar distribution revealed by analysis of a vascular cast of a rat kidney.^12^ The axial distances separating adjacent afferent arterioles vary randomly, and the exact structure we used was therefore chosen arbitrarily. The whole kidney likely contains nephron clusters with nephron arrangements and nephron numbers different from those we chose. The model was based on experimental results from surface cortical nephrons on the ventral surface of the left kidney of young adult male rats, and not on the longer juxtamedullary nephrons that enter the inner medulla. There are no reported measures of tubular dynamics in the inner medulla, nor can regional variations among cortical nephron populations be excluded. The network section we simulated represents a subsurface artery that branches into terminal arteries supplying nephrons on or near the renal surface. The renal cast revealed a tree network whose branches reached the surface in as many as 7 or as few as 2 branches. The results we present are therefore only illustrative.

The model is minimal in the sense that it identifies the fewest components required to achieve both autoregulation and measured tubular dynamics, and except for vascular smooth muscle cells, the dynamics of these components are entered as approximations of experimental results. Renal blood vessels respond to a large number of signals that affect vascular performance, including neurotransmitters, agents produced by endothelial cells, and hormones.^36^ Nephrons and blood vessels form a physical structure within which material flows occur, and the actions of these additional processes take place within a context established by the dynamics of that structure.

There have been many models of nephron dynamics, most of which have dealt with a single nephron.^2^^,^  ^29^^,^  ^37^^,^  ^38^ Theoretical studies on groups of coupled oscillators often reveal that the structure of the network affects its dynamics.^39^ Single nephron models do not include networks and therefore cannot simulate the dynamics of a network.

Nephron function regulates the excretion of water and a variety of solutes, among which are sodium ions and several inorganic anions, of which Cl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{-}$\end{document} is the most abundant. The excretion of these inorganic ions is the principal means by which the body regulates the volume and composition of extracellular fluids. The epithelial transport mechanisms responding to extra-renal signals are located in tubular segments downstream from the macula densa. The macula densa is the sensor for TGF, a non-linear negative feedback mechanism that acts on afferent arteriolar radius to regulate nephron blood flow. The macula densa senses the concentrations of Na\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^+$\end{document}, K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^+$\end{document}, and Cl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^-$\end{document} in the tubular fluid flowing by, and generates a signal it sends to the afferent arteriole.^1^^,^  ^35, 40, 41^ The signal acts on plasma membrane ionic permeabilities in the arteriole, and generates a periodic electrical signal that propagates through the arterial system, interacting with similar signals from neighboring nephrons. The adjustments to afferent arteriole diameter regulate blood flow, but also affect blood pressure throughout the vascular network. The epithelial transport mechanisms responsible for electrolyte transport responsive to whole body needs operate over narrow dynamic ranges. The TGF mechanism, with the macula densa as its receptor, constrains the delivery of these ions to a functionally useful range.

In the absence of periodic forcing, non-linear dynamical systems of order 2 or higher can generate autonomous oscillations.^42^ The response of the TGF mechanism to ionic concentration levels in tubular fluid at the macula densa is non-linear. These oscillations result from the non-linear interactions between the system variables, and represent a stable state of the system. Following a perturbation, the system will return to the trajectory of the oscillation.

The emergence of an autonomous oscillation depends on the magnitude of the parameter values of the system, including, in this case, the input arterial pressure. It is possible to identify one or more parameters that, when changed, can cause the oscillation to emerge and are designated as bifurcation parameters. We have shown that for single nephron units of our model the coupling of the TGF signal to the Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{2+}$\end{document} permeability of arteriolar smooth muscle cells can serve as a bifurcation parameter.^6^

The myogenic mechanism also operates as a limit cycle oscillator. Interactions between cell membrane ionic permeabilities underlie its oscillation.^21^ TGF and the myogenic mechanism share a single set of smooth muscle and elastic mechanisms that provide the contraction and relaxation responsive to changes in mechanical forces and intracellular Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^{2+}$\end{document} concentrations. These shared mechanisms provide the opportunity for interactions. Simulations predict that TGF modulates the amplitude and frequency of myogenic oscillations;^2^^,^  ^3^ the predictions have been confirmed in experiments.^43^ The role of the myogenic mechanism in autoregulation will therefore strengthen at lower blood pressures, and decrease as blood pressure rises.^3^

The membrane electrical potential differences generated in each nephron’s afferent arteriole are bimodal, containing identifiable oscillations of both TGF and the myogenic mechanism. These bimodal changes in arteriolar membrane potentials constitute an information stream and the interactions among information streams from different nephrons underlie the synchronization of the oscillations that have been observed.^5^^,^  ^24–27^^,^  ^32^^,^  ^33^ And, as noted above, the periodic variation in afferent arteriolar diameter affects the local blood pressure in the arterial network.

The distribution of blood from the renal artery to terminal arteries supplying surface nephrons requires an axial hydrostatic pressure gradient, a consequence of which is that the pressure at the origin of each afferent arteriole along the route differs from that of all other afferent arterioles arising from the same arteries. Figure 4 shows the model predictions for these pressures, at 3 different pressures applied to Node 11. The pressure difference from Node 12 to Nodes 18 and 20 is approximately the same for each of the 3 Node 11 pressures. This similarity occurs because the network achieves autoregulation at all 3 input pressures.

Figure 4 also reveals that the magnitude of pressure oscillations at each of the nodes varies with the pressure applied to Node 11. As shown in Figure 3, the TGF signal strength varies with the Node 11 pressure, so that the magnitude of the TGF-nodal pressure interaction increases with increasing root blood pressure.

As can be seen in Figure 3, the tubular responses to changes in Node 11 pressure exhibit 3 different patterns. In the resting state, at 85 mmHg, predicted pressure oscillations in a surface nephron are consistent with in vivo measurements. Nephron 10 shares a site of afferent arteriolar origination with Nephron 9. The oscillations in the two nephrons are fully synchronized with each other, but only partially synchronized with the other 8 nephrons.^8^ Increasing mean arterial pressure at Node 11 increases the amplitude of the TGF signal in all nephrons, leading to increased synchronization throughout the network, as shown in Figure 6, and also among the vascular nodes, as shown in Figure 4. As Node 11 pressure is increased to 105 mmHg and higher values, oscillations with irregular amplitudes begin to emerge in surface nephron tubular pressures, in the blood pressures of the vascular nodes supplying the surface nephrons, and in the TGF signal produced by the surface nephrons. Figure 7 shows that this process begins at lower pressures in Nephron 1, whose afferent arteriole is exposed to the highest nodal pressure of all the nephrons. Figure 4 shows that the nodal pressure supplying Nephron 1, Node 13, has smaller amplitude oscillations than the higher number nodes. Thus, the major factor driving this break in synchrony is the higher mean pressure. Additional increases in the Node 11 pressure propagate more completely through the system, and lead to further loss of synchrony.

Synchronization is a prominent feature of nephron dynamics, providing stability that would not be available to components acting independently. Absent interactions, for example, the blood flow distributed to surface nephrons would be more heavily dependent on the independent behavior of sub-surface nephrons. At lower blood pressures, surface nephrons could then fail to receive sufficient blood flow to sustain glomerular filtration. Absent sufficient blood flow, medullary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $pO_2$\end{document} could become too low to maintain epithelial cell viability. Interactions leading to synchronization allow these surface nephrons to participate in a dialogue that helps to ensure adequate perfusion of all nephrons. This scenario can be seen in Figure 9, where despite a nodal pressure at one point of 72 mmHg that affects the amplitude of the TGF signal, mean nephron blood flow remains unchanged.

The results of this study point to increased stability as an outcome of the various interactions taking place within the network. Autoregulation of total blood flow in the group of 10 nephrons is an example of this stability. Analysis of the single nephron model supports the conclusion that both TGF and the myogenic mechanism function as limit cycle oscillators. As a general class of non-linear systems, and the single nephron model as a particular example, limit cycle oscillators follow a single trajectory. As can be seen in Figure 7, single trajectories do not occur consistently in the nephron group represented in the model. Limit cycle oscillators, when perturbed, return to their original trajectory after the perturbation is removed, and thus are not sensitive to initial conditions. Although individual nephrons in our model can not function as classical limit cycle oscillators, the system in its entirety is insensitive to initial conditions, as can be seen in Figures 2 and 9. The axial hydrostatic pressure gradient in the arteries is an asymmetry of the system responsible for preventing individual nephrons from functioning as limit cycle oscillators. Interactions on the network enable a robust stability to persist despite the effects of this symmetry breaking.

We have previously compared the behavior of one- and two-nephron models in two different ways. First, we created bifurcation diagrams and found that the phase space domains of both limit cycle and chaotic dynamics were greater in the two-nephron mode.^6^ Second, we examined the effect of variation of a single parameter, the probability of open K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $^+$\end{document} channels in smooth muscle cells on the frequency and amplitude of TGF and myogenic oscillations.^40^ The range of values for this parameter that allowed unchanged dynamics was 13.4 times as great in the two-nephron model. This result illustrates another aspect of the stability that arises in the synchronized renal network.

Finally, as shown in Figures 4, 5, 8, and 9, variations in blood pressure applied to the root of the tree affects the strengths of interactions and the amplitudes of the oscillations, but the frequency of the TGF oscillation remains unchanged. The constancy of the frequency provides a stable basis for interactions with other components not included in this model. For example, Schurek and Johns recorded \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $pO_{2}$\end{document} from several locations on the renal surface of anesthetized rats and found fluctuations at the frequencies of the TGF oscillation.^44^ Their observation suggests that the periodic variation in the mass flow of NaCl through the thick ascending limbs caused a periodic demand on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $O_{2}$\end{document} consumption, an effect that would impact other energy dependent processes and intracellular pH both in the thick ascending limb and downstream in the nephron. In addition, at the frequencies we examined in this study, the transfer function of the whole kidney circulation, using blood pressure as the input, gain magnitude was negative, a finding consistent with TGF as a negative feedback system.^33^ At the frequency of the diurnal rhythm, however, the gain magnitude at the TGF frequency becomes positive, indicating a modulating effect on the nephron-arterial network by systems responding to the geophysical rhythm.^45^ The constancy of the network’s frequency could provide a stable basis for modulations of this sort.

In summary, we have developed a model of renal blood flow regulation in a cluster of 10 nephrons supplied from an arterial network. The network facilitates communication among nephrons by electric information streams and also provides the arterial hydrostatic pressure gradients required to drive blood flow. The models of the 10 nephrons are the same as those used previously to study single nephron dynamics;^2^ the structure of the arterial network was derived from measurements of a renal vascular cast.^6^ Pressure diuresis and a dependence of efferent arteriolar vascular resistance were added to achieve autoregulation by the cluster. Periodic withdrawal of blood into afferent arterioles generated oscillations of pressure at the arterial nodes, providing an additional information stream. Each of the nephron models developed tubular pressure fluctuations like those found in vivo in mice, rats and man.^26^^,^  ^46^^,^  ^47^ Acute hypertension affected the amplitude of tubular pressure oscillations, the TGF signal to the afferent arteriole, and oscillations of nephron blood flow. The magnitude of the changes varied with the applied blood pressure. The increase of the TGF signal to the afferent arteriole saturated after moderate elevations of blood pressure. Aperiodic low frequency tubular pressure changes emerged at higher blood pressures. Synchronization among nephrons increased with arterial blood pressure until the lower frequency oscillations emerged. Variable blood pressures with the 1/f distribution found in vivo provoked effective autoregulatory responses while maintaining the TGF frequency. The overall effect of the cluster was to provide greater stability than could be achieved by single nephrons.

Supplementary Material