Analysis of TGF-mediated dynamics in a system of many coupled nephrons
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Previously, Pitman et al. (2002) derived an integral model of the tubuloglomerular feedback system. This integral model (also known as reduced model) describes the fluid flow up the thick ascending limb of a single, short-looped model nephron, dependent on characteristic model parameters representing a gain and a time delay present in the system. More recently, Pitman et al. (2004) examined the effect of coupling on two nephrons, each independently described by the partial differential equation (PDE) model (also known as minimal model) of Layton et al. (1991). Analysis of these model systems indicates that, depending on the parameters, a Hopf bifurcation may lead to solutions exhibiting sustained limit cycle oscillations. In the present study, the integral model to which we will refer either as the single-nephron model or the 1-Nephron model is extended to a system of many coupled nephrons. Specifically, a nearest neighbor coupling of a collection of N nephrons branching from the same cortical radial artery is proposed. In the special case when all model parameters are identical, explicit bifurcation analysis of the equations can be performed. This analysis shows that the region of parameter space where limit cycle oscillations occurs is larger when coupling of nephrons is present. In addition, if the time delays of the individual nephrons do not differ by too much, then numerical simulations suggest that frequency entrainment and phase synchronization occur in the flow oscillations of the nephron systems.