A Mathematical Model for Neuron Reorientation and Axonal Growth on a Cyclically Stretched Substrate

Experiments have shown that mechanical cues play a central role in determining the direction and rate of axonal growth. In particular, neurons seeded on planar substrates undergoing periodic stretching have been shown to reorient and reach a stable equilibrium orientation corresponding to angles within the interval $\left({60}^{\circ },,,{90}^{\circ }\right)$ with respect to the main stretching direction. In this work, we present a new model that considers both the reorientation and growth of neurons in response to cyclic stretching. Specifically, a linear viscoelastic model for the growth cone reorientation with the addition of a stochastic term is merged with a moving-boundary model for tubulin-driven neurite growth to simulate the axonal pathfinding process. Various combinations of stretching frequencies and strain amplitudes have been tested by numerical simulation of the proposed model. The simulations show that neurons tend to reorient toward an equilibrium angle that falls in the experimentally observed range. Moreover, the model captures the relation between the stretching condition and the speed of reorientation. Indeed, numerical results show that neurons tend to reorient faster as the frequency and amplitude of oscillation increase.