Dynamics of path aggregation in the presence of turnover

Path aggregation has been mathematically studied mainly in two classes of models. One of them is known as the active-walker models [6] in which each walker in course of its passage through the system changes the surrounding environment locally, which in turn influences the later walkers. An example of such a process is the ant trail formation [2,3]. While walking, an ant leaves a chemical trail of pheromones which other ants can sense and follow. The mechanism of human and animal trail formations is mediated by the deformation of vegetation that generates an interaction between earlier and later walkers [6]. A mathematical formalism to study the formation of such trails has been developed in Ref. [5,6]. The other class of models showing path aggregation deals with non-interacting random walkers moving through a fluctuating environment. In Ref. [7], condensation of trails of particles moving in an environment with Gaussian spatial and temporal correlation is demonstrated analytically. Another example of this model class is the Scheidegger river model [1] (and related models [8]) which describes the formation of a stream net-work by aggregation of streams flowing downhill on a slope with local random elevations.

In this Letter we analyze the dynamics of path aggregation using a simple model that belongs to the class of active walker systems discussed above. The model is similar to the one used to study path localization in Ref. [9]. In our model, however, we take into account the aging of the paths, an important aspect of the active walker models. For instance, in ant trail systems, the pheromone trails age due to evaporation. In the mammalian trail formation the deformations of the vegetation due to the movement of a mammal decays continuously with time [6]. In our model, the individual paths do not age gradually, but rather maintain their full identity until they are abruptly removed from the system. This particular rule for path aging is chosen to allow application of our model to the process of axon fasciculation, which we discuss next.

During the development of an organism, neurons located at peripheral tissues (e.g. the retina or the olfactory epithelium) establish connections to the brain via growing axons. The growth cone structure at the tip of the axon interacts with other axons or external chemical signals and can bias the direction of growth when spatially distributed chemical signals are present [10]. In the absence of directional signals, the growth cone maintains an approximately constant average growth direction, while exploring stochastically the environment in the transverse direction [11]. The interaction of growth cones with the shafts of other axons commonly leads to fasciculation of axon shafts [12]. During development a significant portion of fully grown neurons die and get replaced by newborn neurons with newly growing axons. For certain types of neurons (such as the sensory neurons of the mammalian olfactory system) the turnover persists throughout the lifespan of an animal. In mice, the average lifetime of an olfactory sensory neuron is 1-2 months [13], which is less than one tenth of the mouse lifespan. The mature connectivity pattern is fully established only after several turnover periods [14].

The model we propose in this Letter captures the basic ingredients of the process of axon fasciculation, i.e. attractive interaction of growth cones with axon shafts, as well as neuronal turnover. The main contribution of this Letter is a detailed discussion of the slow time scales that emerge from the dynamics of our model. Using Monte-Carlo simulations we characterize the time scale for the approach to steady state and the correlation time within the steady state, and show that they can exceed the average axonal life time by orders of magnitude. To understand these results we formulate an analytically tractable effective single fascicle dynamics. This allows us to relate the observed slow time scales to the dynamics of the basins of the fascicles. From the effective fascicle dynamics we derive three time scales which we compare to the time scales extracted from the Monte-Carlo simulation of the full system.

For clarity, we stress that the dynamics of our model differs substantially from one-dimensional coalescence (A + A → A) [15] or aggregation (mA + nA → (m + n)A) [16]. In our model, there is no direct inter-walker interaction; rather, each random walker interacts locally with the trails of other walkers. While the stationary properties of the system (such as the fascicle size distribution in the steady state) may be approximately understood using an analogy to one-dimensional diffusi

…(Full text truncated)…