Modelling axon growing using CTRW

The main goal of this study is to propose a mathematical model describing paths of the axon growth cones and differences in the behavior of normal and mutant axons. We introduce a probabilistic model for axon growing, such that each family of axons is described as an ensemble of trajectories of a continuous time random walk (CTRW) model under different parameters in the case of normal and mutant axons. We describe different regimes in the model and conclude how the behavior of axons depends on the parameters of the model. Biological observations of the axonal growth process say us that the guiding development of axons to their targets is operated by chemical signals from the cellular environment. To simulate this control mechanism we propose the CTRW model, where a random waiting time reflects a reaction time of the growth cones on the neighboring chemical environment. Continuous-time random walks (CTRW) are a natural generalization of the usual random walk. The mathematical analysis can be dated back to the pioneering work of Montroll and Weiss in the sixties [1]. At present they are extensively used in applications to physics, chemistry, and other sciences, see e.g. [2], [3]. We consider three families of (static) trajectories associated with axon growth cones. One of the family is for normal axons, the other two are for two different axon mutations. We don’t have dynamical data, and any trajectory presents positions of an axon tip during the observation time. The observation time is the same for all axons, but we don’t have information about the stopping time for development of each axon. On figures 1-3 we present three sets of trajectories of axon growth cones. In fact, axons on real data images have different starting positions, but we shifted all of them to have the same starting point.