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The formation of neuron networks is a complex phenomenon of fundamental importance for understanding the development of the nervous system, and for creating novel bioinspired materials for tissue engineering and neuronal repair. The basic process underlying the network formation is axonal growth, a process involving the extension of axons from the cell body towards target neurons. Axonal growth is guided by environmental stimuli that include intercellular interactions, biochemical cues, and the mechanical and geometrical features of the growth substrate. The dynamics of the growing axon and its biomechanical interactions with the growing substrate remains poorly understood. In this paper, we develop a model of axonal motility which incorporates mechanical interactions between the axon and the growth substrate. We combine experimental data with theoretical analysis to measure the parameters that describe axonal growth on surfaces with micropatterned periodic geometrical features: diffusion (cell motility) coefficients, speed and angular distributions, and axon bending rigidities. Experiments performed on neurons treated Taxol (inhibitor of microtubule dynamics) and Blebbistatin (disruptor of actin filaments) show that the dynamics of the cytoskeleton plays a critical role in the axon steering mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which high-curvature geometrical features impart high traction forces to the growth cone. These results have important implications for our fundamental understanding of axonal growth as well as for bioengineering novel substrates that promote neuronal growth and nerve repair.