BioMem-FWK4α: A Fractional Windkessel-Constant Phase Framework for Analog Simulation and Frequency-Domain Interpretation of Biological Ion Channels.

Background: Classical integer-order membrane and bioelectrical circuit models are useful for idealized ion-channel behavior, but they often fail to reproduce the distributed relaxation, power-law impedance, depressed Nyquist arcs, and memory-dependent phase dispersion observed in biological membranes and tissue interfaces. Objectives: This study aims to develop a fractional four-element Windkessel-constant phase framework, denoted BioMem-FWK4α, for analog simulation and frequency-domain interpretation of biological ion-channel and membrane-like impedance responses. Design: A theoretical, computational, and analog-circuit modeling study was conducted using normalized fractional circuit parameters. The study is intended as a mechanistic modeling framework rather than as a direct experimental validation study. Methods: The classical four-element Windkessel topology was reformulated by replacing ideal compliance with a constant-phase element of order α and by introducing fractional inertial behavior of order β and an active/passive resistance ratio γ. Caputo and Caputo-Fabrizio formulations were compared under explicit zero-initial frequency-domain assumptions. The model was analyzed through transfer functions, Bode plots, Nyquist loci, Nichols diagrams, RC-ladder approximation, sensitivity indicators, and biological interpretation tables. Results: The model predicts non-integer gain slopes approaching -20α dB per decade, phase plateaus near -90α degrees, and depressed Nyquist arcs whose depression increases as α decreases. The γ parameter controls the transition from passive leakage-dominated filtering to active or mixed active-passive regimes, while β modifies delayed ionic or kinetic effects. The RC-ladder approximation supports the physical realizability of the fractional element, although biological validation requires future fitting against patch-clamp and bioimpedance datasets. Conclusion: BioMem-FWK4α provides a compact and interpretable circuit framework for describing memory-rich membrane impedance and ion-channel-like dynamics. It does not replace Hodgkin-Huxley or Markov-state models for detailed nonlinear gating, but complements them by offering a reduced-order frequency-domain and analog-realizable representation of distributed membrane memory. Future work should validate the model using time-resolved patch-clamp, tissue bioimpedance, and voltage-dependent nonlinear parameter identification.