Numerical techniques for dynamic resistive networks

Abstract

We consider approaches to the numerical difficulties posed by modelling resistive networks with dynamically changing resistances according to a set of coupled ordinary differential equations (odes). The prototype problem is autoregulation on a cerebral microvascular network. In this network the amount of perfusion, or tissue blood supply, is determined by the resistance of the vascular network. This resistance can be dynamically altered to regulate the amount of blood flow and hence maintain a balance of chemicals and nutrients in the tissue. The mechanisms responsible for this are primarily local, and are due to various chemical pathways to the vascular smooth muscle cells lining the arterioles and arteries in the vasculature. These cells contract and dilate, which alters the radii and hence resistance of the vessels. Contents C172Because of the global coupling induced by the resistive network, the resulting stiff system of odes has a dense Jacobian which precludes the direct application of traditional implicit numerical solution methods. We consider a means of remedying this problem by taking a block diagonal Jacobian approximation which allows use of an implicit method but retains the desirable property of explicit solvers of linear solution time scaling with problem size. Subject class: 65L99

Neurovascular coupling: a parallel implementation

“…The parallel implementation follows the work of Brown ( 2013 ), however we give a brief explanation here for brevity. Because of the global coupling induced by the resistive network, the resulting stiff system of ODEs has a dense Jacobian which precludes the direct application of traditional implicit methods for numerically solving the differential equations.…”

“…We choose the approximation to have block diagonal structure, the factorization and solution can be decomposed into as many independent tasks as there are blocks, and hence the decomposition yields a natural parallelization of the problem. Details of this procedure can be found in Brown ( 2013 ).…”