Fractional Soliton Dynamics in Coupled Myelinated Fibers: Comparative Modeling With Beta, Caputo, and Atangana–Baleanu Derivatives

Background: 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: 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.

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