The sprouting, stabilization and growth of neurites is a dynamic process by which developing neurons establish connections with the other elements of the nervous system; this process is under the control of extracellular cues, among which neurotrophic factors play a crucial role. Due to the complexity of the spatiotemporal evolution of the neurite network, particularly in the early stages of growth, it is not easy to obtain information about the relevant parameters from qualitative observations. We have developed a quantitative description of the dynamics of production and stabilization of neuritic processes in a well-characterized experimental model of peripheral neurons in culture, and we have combined it with a simulation approach to extract the differences between the behaviour in the absence and in the presence of the neurotrophic factor basic Fibroblast Growth Factor (bFGF). We show that the factor rapidly stabilizes the neuronal morphology to a bipolar phenotype, by shifting the sprouting process from a disordered phase to a more ordered and organized one.