Stability in a mathematical model of neurite elongation

We have developed a continuum partial differential equation model of tubulin-driven neurite elongation and solved the steady problem. For non-zero values of the decay coefficient, the authors identified three different regimes of steady neurite growth, small, moderate and large, dependent on the strength of the tubulin flux into the neurite at the soma. Solution of the fully time-dependent moving boundary problem is, however, hampered by its analytical intractibility. A linear instability analysis, novel to moving boundary problems in this context, is possible and reduces to finding the zeros of an eigen-condition function. One of the system parameters is small and this permits solutions to the eigen-condition equation in terms of asymptotic series in each growth regime. Linear instability is demonstrated to be absent from the neurite growth model and a Newton-Raphson root-finding algorithm is then shown to corroborate the asymptotic results for some selected examples. By numerically integrating the fully non-linear time-dependent system, we show how the steady solutions are non-linearly stable in each of the three growth regimes with decay and oscillatory behaviour being as predicted by the linear eigenvalue analysis.