C-type period-doubling transition in nephron autoregulation

Abstract

The functional units of the kidney, called nephrons, utilize mechanisms that allow the individual nephron to regulate the incoming blood flow in response to fluctuations in the arterial pressure. This regulation tends to be unstable and to generate self-sustained oscillations, period-doubling bifurcations, mode-locking and other nonlinear dynamic phenomena in the tubular pressures and flows. Using a simplified nephron model, the paper examines how the regulatory mechanisms react to an external periodic variation in arterial pressure near a region of resonance with one of the internally generated mode-locked cycles. We show how the stable and unstable resonance cycles generated in this response undergo interconnected cascades of period-doubling bifurcations and how each period doubling leads to the formation of a new pair of saddle-node bifurcation curves along the edges of the resonance zone. We also show how period doubling of the resonance cycles is accompanied by a torus-doubling process in the quasiperiodic regime that exists outside of the resonance zone.

Synchronization of period-doubling oscillations in vascular coupled nephrons

“…Until now, however, few examples of this transition have been worked out for realistic systems. A recent study of the C-type transition for a single-nephron model subjected to a periodic variation of the arterial pressure 28 has confirmed the theoretical predictions for this kind of system and has also lead to a more detailed picture of the involved bifurcation structure. However, the structure we observe in the coupled nephron model is different in that the transition to synchronization occurs through the combination of a homoclinic and a torus bifurcation.…”

“…Along the upper left boundary of the resonance zone, the saddle-node bifurcation curves are seen to follow the demarcation lines between the brown quasiperiodic regime and the dark and light blue regions with period-1 and period-2 dynamics, respectively. This implies that (except for small corners around the points of appearance for the new saddlenode bifurcation curves [26][27][28] ) the synchronized periodic modes are born as pairs of stable nodes and singly-unstable saddle cycles. The same is found to be true for the following modes in the period-doubling cascade.…”

“…26,27 In studies of periodically forced period-doubling systems, one observes that the stable and unstable resonance cycles generated at the edge of the synchronization regime undergo interconnected cascades of period-doubling bifurcations and that each period doubling leads to the formation of a new pair of saddle-node bifurcation curves along the edges of the resonance zone. 27,28 Moreover, these saddle-node bifurcations accumulate to finally define the transition between phase synchronized chaos and non-synchronous chaos. Outside the resonance regions, one can observe the phenomenon of ergodic torus-doubling as discovered originally by Arnéodo et al 29 and by Kaneko,30 and as recently demonstrated in practice by Sekikawa et al…”

The transition to chaotic phase synchronization

“…By examining the bifurcation structure of a physiology-based model of the blood Àow regulation to the individual functional unit (nephron) of the kidney, we demonstrate how the same type of critical behavior may occur in the nephron's response to a periodic variation in the arterial blood pressure [25]. We ¿nally discuss a new type of transition from asynchronous to phase synchronized chaos that, besides the saddle-node bifurcations, also involves a dense set of torus bifurcation curves.…”

Advancing systems medicine and therapeutics through biosimulation

“…Both regulatory mechanisms tend to be unstable though and produce self-sustained oscillations of the nephron pressures and flows. Using a model that accounts for the main mechanisms behind these oscillations, the paper by Laugesen et al [11] examines how the regulatory mechanisms react to an external periodic variation in arterial pressure with frequencies close to one of the internally generated cycles. This leads the authors to develop relatively detailed diagrams for the bifurcation structure for the forced system.…”