Computational methods and diffusion theory in triangulation sensing to model neuronal navigation

Computational methods are now recognized as powerful and complementary approaches in various applied sciences such as biology. These computing methods are used to explore the gap between scales such as the one between molecular and cellular. Here we present recent progress in the development of computational approaches involving diffusion modeling, asymptotic analysis of the model partial differential equations, hybrid methods and simulations in the generic context of cell sensing and guidance via external gradients. Specifically, we highlight the reconstruction of the location of a point source in two and three dimensions from the steady-state diffusion fluxes arriving to narrow windows located on the cell. We discuss cases in which these windows are located on the boundary of a two-dimensional plane or three-dimensional half-space, on a disk in free space or inside a two-dimensional corridor, or a ball in three dimensions. The basis of this computational approach is explicit solutions of the Neumann-Green's function for the mentioned geometry. This analysis can be used to design hybrid simulations where Brownian paths are generated only in small regions in which the local spatial organization is relevant. Particle trajectories outside of this region are only implicitly treated by generating exit points at the boundary of this domain of interest. This greatly accelerates the simulation time by avoiding the explicit computation of Brownian paths in an infinite domain and serves to generate statistics, without following all trajectories at the same time, a process that can become numerically expensive quickly. Moreover, these computational approaches are used to reconstruct a point source and estimating the uncertainty in the source reconstruction due to an additive noise perturbation present in the fluxes. We also discuss the influence of various window configurations (cluster vs uniform distributions) on recovering the source position. Finally, the applications in developmental biology are formulated into computational principles that could underly neuronal navigation in the brain.