As an M.Sc. student, I worked on a model describing the early stages of
type 1 diabetes. Type 1 diabetes occurs when the immune system attacks and destroys the insulin producing beta-cells in the pancreas. Our
experimental collaborator had observed that macrophages (a part of the immune system that clears away dead cells from tissues) from mice susceptible to type 1 diabetes were less efficient than macrophages from healthy mice. It was also observed that not every mouse susceptible to the disease developed it. Based on these observations, we developed a model to answer two questions:
1. Can the difference in macrophage efficiency account for the differences observed between healthy and susceptible mice?
2. Can a naturally occurring wave of beta-cell death associated with normal development in all mice be a triggering event that leads to type 1 diabetes in susceptible mice?
Within our modeling framework, the healthy state corresponds to a steady-state solution representing no inflammation, while the diseased state corresponds to a steady-state solution representing chronic inflammation. It is then assumed that during the chronic inflammation state the immune system will become primed to attack and kill the pancreatic beta-cells. Since not all susceptible mice develop type 1 diabetes, the healthy state should exist and be stable for both strains of mice being modeled. In addition, the steady state corresponding to chronic inflammation should exist and be stable for the susceptible mice. Through the use of dynamical systems approaches (especially phase-plane analysis) we determined that the initial model could not satisfy these requirements for biologically reasonable parameter values. The model was then expanded to include additional cell populations as well as the toxic effect of harmful cytokines released by the macrophages. This expanded model, shown below, demonstrated that differences in macrophage efficiency could explain the difference between healthy and susceptible mice.
To answer the second question, the wave of beta-cell death was incorporated into the model. The model demonstrated that for healthy mice the temporary inflammation quickly died down, and the model returned to the non-inflamed steady state. In the case of the susceptible mice, the wave of beta-cell death was sufficient to push the system to chronic inflammation, suggesting that the naturally occurring cell death could indeed be a triggering stimulus for the development of type 1 diabetes.
As a Ph.D. candidate, my research focused on the development of an efficient computational algorithm suitable for simulations of electrical impulses in nerve cells. Current research in computational neuroscience involves simulations of electrical impulses that travel through large computational domains, such as the example shown to the left from the Blue Brain Project. In many cases there is a spatial localization of activity, with a small region of the cell (or network of cells) changing rapidly while the majority of the system evolves very little. By taking advantage of this spatial localization of activity, I was able to develop an algorithm that can be more efficient than the standard approach. In a traditional simulation algorithm, the entire cell is treated as a single large system that is solved simultaneously. Thus in order to obtain an accurate solution, the entire system must be updated using a time step size that is sufficiently small to capture the fastest evolution, even though most of the system could be accurately updated using a much larger time step.
To take advantage of the spatial localization of activity, I developed an algorithm for locally adaptive time stepping (LATS). Within this scheme, the system is split into subdomains, and each subdomain is updated with an adaptive time step most appropriate for the local level of activity, as shown in the figure below. The challenge of localized adaptive time stepping is in maintaining accurate flow of information and stability of the solution. Through the application of domain decomposition techniques, I was able to computationally connect the subdomains through boundary conditions obtained through a conservation of flux. To address the stability concerns, I replaced the time stepping scheme that had been used for neuroscience simulations since the 1960’s with a method that provides better stability and proved better suited to the LATS algorithm.
Evaluating the LATS algorithm is not as simple as stating an X% reduction in computational time. The underlying numerical scheme is comparable to the standard approach, but the major benefit of the LATS method is that the computational cost scales with the level of activity in the system, rather than the physical size of the domain. Thus in situations where there is sparse activity in a large computational domain, the LATS method provides a significant reduction in computational cost by focusing computational resources where they are most needed. The LATS method was developed within the context of computational neuroscience, but is applicable to any system with sparse activity.
In the video below, an electrical impulse is initiated in a cell, and propagates through two cells. The colors represent the membrane voltage, and the sections of the cell become transparent as the step size increases.