Fractional cable theory with Riemann-Liouville dynamics: memory-dependent action potential propagation and analytical validation through neural scaling laws

Action potential propagation in neurons exhibits complexspatiotemporal dynamics that classical integer-order cablemodels cannot fully capture. Extensive experimental evidence demonstrates that neural processes exhibit memory-dependentphenomena arising from molecular crowding(20-30% cytoplasmic volume fraction), power-law ionchannel kinetics, and anomalous diffusion in dendrites.We develop a comprehensive fractional cable theory thatextends the Hodgkin--Huxley framework by incorporating these experimentally observed phenomena through Riemann--Liouville (RL) fractional derivatives, which areadvantageous over the Caputo formulation for α>0 because they require no integrability constraint on the initial data and yield cleaner convolution-based Green'sfunctions. Dimensional correction parameters σ _x (length) and σ _t (time) are introduced following Gȯmez-Aguilar et al. to maintain physical consistencyfor all fractional orders. We derive analytical solutionsusing Fourier--Laplace transform techniques, obtain exact Green's functions in terms of Fox H -functions and Mittag-Leffler functions, and establish dispersion relationsrevealing frequency-dependent propagation velocities. The framework's validity is established analytically throughits reproduction of multiple empirically observed scalinglaws: the f ^-4 power spectral density of corticalsurface potentials, the distinct diameter--velocity relationships for myelinated and unmyelinated axons arisingfrom different physical mechanisms, and the spectrum of1/f ^v noise exponents observed across neural recordings. Theoretical predictions for neurological disorders arederived analytically from the validated scaling relationships:epilepsy manifests as a transition of the temporal fractional order toward α=2; multiple sclerosis produces systematically increased σ _x values; and Alzheimer'sdisease corresponds to decreased spatial fractional order β . Detailed numerical simulations validating these analytical predictions are reserved for follow-up studies.