Stationary Action Principles for Neuronal Growth
Neuronal growth is guided by biochemical, mechanical, and geometrical cues while being influenced by stochastic fluctuations in cytoskeletal dynamics, adhesion, and growth cone sensing. Here, we develop two complementary variational descriptions of mechanically guided axonal growth. First, for neurons growing on periodic parallel micropatterns, we use Langevin and Fokker--Planck models to derive a stochastic path action incorporating deterministic substrate guidance and random fluctuations. The probability of a growth trajectory decreases exponentially with this action, and the most probable trajectories correspond to minimum action paths for fixed boundary conditions. Second, for axons growing on substrates with spatially varying stiffness, we show that stiffness dependent traction generates mechanical guidance and axonal turning at stiffness interfaces. In the quasi-static limit, this response leads to a Fermat-like spatial action with an effective axonal refractive index determined by the traction and substrate stiffness relationship. The resulting framework predicts axonal refraction across stiffness boundaries, including an approximate Snell-like relation and a critical angle condition. Although the stochastic and Fermat-like actions have distinct physical meanings and ranges of validity, both describe how substrate mechanics constrain axonal trajectories. These results connect measurable quantities, including drift, diffusion, stiffness, traction, and growth direction, to global properties of axonal paths and provide a framework for designing biomimetic substrates for controlled neuronal growth and regeneration.