  Geometrical cues play an essential role in neuronal growth. Here, we quantify
axonal growth on surfaces with controlled geometries and report a general
stochastic approach that quantitatively describes the motion of growth cones.
We show that axons display a strong directional alignment on micro-patterned
surfaces when the periodicity of the patterns matches the dimension of the
growth cone. The growth cone dynamics on surfaces with uniform geometry is
described by a linear Langevin equation with both deterministic and stochastic
contributions. In contrast, axonal growth on surfaces with periodic patterns is
characterized by a system of two generalized Langevin equations with both
linear and quadratic velocity dependence and stochastic noise. We combine
experimental data with theoretical analysis to measure the key parameters of
the growth cone motion: angular distributions, correlation functions, diffusion
coefficients, characteristics speeds and damping coefficients. We demonstrate
that axonal dynamics displays a cross-over from an Ornstein-Uhlenbeck process
to a non-linear stochastic regime when the geometrical periodicity of the
pattern approaches the linear dimension of the growth cone. Growth alignment is
determined by surface geometry, which is fully quantified by the deterministic
part of the Langevin equation. These results provide new insight into the role
of curvature sensing proteins and their interactions with geometrical cues.
