Solution of a multinephron,multisolute model of the mammalian kidney by Newton and continuation methods

A method of numerically solving the differential equations specifying solute and water flow in a multinephron, multisolute model of the mammalian kidney by a combination of Newton and continuation techniques is described. This method is used to generate a connected component of the steady state solution manifold of the model. A three dimensional section of this manifold is shown to be convoluted, with upper and lower sheets of stable solutions connected by an unstable middle sheet. Two dimensional sections of this surface are followed from a trivial constant profile of concentrations in the nonconcentrating kidney to the profiles of the maximally concentrating kidney. Study of these sections shows that for a given choice of model parameters there may exist no solution, there may be a unique solution, or there may be multiple solutions. A study of the time dependent solutions shows that the dynamic transition from the lower to the upper state and return may be via a hysteresis loop.