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In this paper, we present a numerical method for solving reaction-diffusion equations on one dimensional branched structures. Through the use of a simple domain decomposition scheme, the many branches are decoupled so that the equations can be solved as a system of smaller problems that are tri-diagonal. This technique allows for locally adaptive time stepping, in which the time step used in each branch is determined by local activity. Though the method is presented in the specific context of electrical activity in neural systems, it is sufficiently general that it can be applied to other classes of reaction-diffusion problems and higher dimensions. Information in neurons, which can be effectively modeled as one-dimensional branched structures, is carried in the form of electrical impulses called action potentials. The model equations, based on the Hodgkin-Huxley cable equations, are a set of reaction equations coupled to a single diffusion process. Locally adaptive time stepping schemes are well suited to neural simulations due to the spatial localization of activity. The algorithm significantly reduces the computational cost compared to existing methods, especially for large scale simulations.

A wave of apoptosis (programmed cell death) occurs normally in pancreatic β-cells of newborn mice. We previously showed that macrophages from non-obese diabetic (NOD) mice become activated more slowly and engulf apoptotic cells at a lower rate than macrophages from control (Balb/c) mice. It has been hypothesized that this low clearance could result in secondary necrosis, escalating inflammation and self-antigen presentation that later triggers autoimmune, Type 1 diabetes (T1D). We here investigate whether this hypothesis could offer a reasonable and parsimonious explanation for onset of T1D in NOD mice. We quantify variants of theCopenhagen model(Freiesleben De Blasioet al. 1999Diabetes48, 1677), based on parameters from NOD and Balb/c experimental data. We show that the original Copenhagen model fails to explain observed phenomena within a reasonable range of parameter values, predicting an unrealistic all-or-none disease occurrence for both strains. However, if we take into account that, in general, activated macrophages produce harmful cytokinesonlywhen engulfing necrotic (but not apoptotic) cells, then the revised model becomes qualitatively and quantitatively reasonable. Further, we show that known differences between NOD and Balb/c mouse macrophage kinetics are large enough to account for the fact that an apoptotic wave can trigger escalating inflammatory response in NOD, but not Balb/c mice. In Balb/c mice, macrophages clear the apoptotic wave so efficiently, that chronic inflammation is prevented.