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Talk Abstract
Bifurcations in Models of the Tubuloglomerular Feedback Mechanism

Niels-Henrik Holstein-Rathlou, M.D., Ph.D.
Department of Medical Physiology
University of Copenhagen
Copenhagen N, Denmark

The tubuloglomerular feedback (TGF) is an intrarenal mechanism that stabilizes renal blood flow, GFR, and the tubular flow rate. The anatomical basis for TGF is the return of the tubule (the ascending limb of the loop of Henle (ALH)) to its own glomerulus. The macula densa, which is the sensor mechanism for the TGF, is a plaque of specialized epithelial cells in the wall of the ALH. It is localized at the site where the tubule establishes contact with the glomerulus. Because of a flow dependency of NaCl reabsorption in the ALH, a change in tubular flow rate, elicited for example by a change in the arterial pressure, will lead to a change in the NaCl concentration of the tubular fluid. This is sensed by the macula densa, and through unknown mechanisms results in a change in the hemodynamic resistance of the afferent arteriole. The dynamic properties of the TGF system has been characterized in experimental studies in both normo-and hypertensive rats. In normotensive rats, TGF displays autonomous self-sustained regular oscillations, whereas in spontaneously hypertensive rats (SHR) highly irregular, "chaotic" fluctuations are present. Several attempts has been made to formulate mathematical models of the TGF system that is able to reproduce both the regular oscillations, and the irregular fluctuations. However, in most cases the models have been successful in describing the regular oscillations, but have failed to reproduce the irregular fluctuations. It has only been possible to achieve irregular (chaotic) fluctuations in TGF through nonlinear extensions of the equations that describe the afferent arteriole. Although physiologically justified, these extensions has lacked a firm experimental basis. To overcome the shortcomings of the previous models, we have in the present work extended a model of the TGF mechanism with a model of the response of the afferent arteriole. To examine the bifurcation structure of this highly complex model we have applied one-and two dimensional continuation techniques. The results show that a Hopf bifurcation causes the system to perform self-sustained regular oscillations if the feedback gain is sufficiently strong. If the feedback gain is increased further, a folded structure of overlapping sheets of period-doubling cascades appear, leading ultimately to the appearance of classical chaotic behavior.

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1998-1999 Mathematics in Biology

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