Fractional Soliton Dynamics in Coupled Myelinated Fibers: Comparative Modeling With Beta, Caputo, and Atangana–Baleanu Derivatives
Wulfran Fendzi Mbasso, Ambe Harrison, Monia Ferchichi, Muhammad Suhail Shaikh, Zokir Mamadiyarov, Saad F. Al-Gahtani, Z. M. S. ElbarbaryBackground
fractional-order modeling provides a powerful framework for representing memory-dependent conduction in excitable biological media. However, existing soliton-based models of myelinated nerve fibers are often theoretical, operator-specific, and insufficiently benchmarked in terms of numerical reproducibility, physiological plausibility, and computational cost.
Objectives
This study aims to compare the Liouville-Caputo, Atangana-Baleanu, and Beta fractional operators for modeling soliton-like action-potential propagation in ephaptically coupled myelinated nerve fibers, with emphasis on waveform stability, energy retention, biological consistency, computational efficiency, and adaptive parameter learning.
Design
A comparative computational modeling study was conducted using a coupled fractional nonlinear partial differential equation framework, physiological parameter mapping, numerical sensitivity analysis, and physics-informed neural network-based parameter estimation.
Methods
A coupled fractional Korteweg-de Vries-type system was solved under identical initial and boundary conditions for the three fractional operators. The time-fractional order α was varied over [0.6, 1.0], while the space-fractional order β was varied over [1.5, 2.0]. Simulations used a uniform spatial grid, fixed time step, localized sech 2 initial pulse, and Neumann boundary conditions. The operators were compared using soliton-like velocity, amplitude, pulse width, normalized energy retention, residual error, RMSE, MAE, and CPU runtime. A physics-informed neural network was further used to estimate model parameters while enforcing the fractional PDE residual.
Results
The Beta derivative produced the most localized and stable soliton-like pulses, with stronger amplitude preservation, lower energy loss, and shorter runtime than the Liouville-Caputo and Atangana-Baleanu formulations under the tested settings. Increasing ephaptic coupling strength reduced pulse amplitude, whereas increasing α improved propagation velocity and increasing β enhanced waveform localization. Quantitative residual and error analyses confirmed that the Beta-based formulation maintained low numerical error while preserving biologically plausible conduction behavior.
Conclusion
The results support the Beta derivative as a biologically plausible and computationally efficient approximation for soliton-like nerve-pulse propagation in coupled myelinated fibers. The Liouville-Caputo and Atangana-Baleanu operators remain valuable for long-memory and fading-memory regimes, respectively. Future work should integrate literature-constrained biological consistency assessment, stochastic ion-channel dynamics, and heterogeneous multidimensional nerve-bundle geometries.