Maximum-Entropy Model of Colored Noise in Superdiffusive Axonal Growth

Abstract

We develop a coarse-grained stochastic theory for axonal growth on micropatterned substrates using the Shannon--Jaynes maximum entropy principle. Starting from a Langevin description of growth cone motion, we infer the effective distribution of traction force relaxation rates from experimentally motivated constraints rather than postulating the colored noise directly. The resulting relaxation rate distribution generates a stationary colored acceleration process with power-law temporal correlations and yields analytical predictions for the axonal mean squared displacement and velocity autocorrelation. The long-time behavior is controlled by the slow-relaxation part of the inferred distribution, corresponding physically to broadly distributed clutch or adhesion engagement times. For biologically relevant parameters, the model predicts a negative correlation exponent $α=-1/2$. This prediction is in close quantitative agreement with measurements on cortical neurons cultured on micropatterned poly-D-lysine-coated PDMS substrates, which are well described by $α\simeq -0.6$ and exhibit superdiffusive mean squared displacement scaling with exponent $1.4$. The same framework accounts for the crossover from early diffusive behavior to long-time anomalous growth and for the corresponding power law decay of the velocity autocorrelation. These results show how entropy-constrained active fluctuations can connect microscopic force generation processes to emergent growth laws in neuronal systems and, more broadly, in active matter.