Lie symmetry analysis, soliton solutions, and bifurcation dynamics of ionic current model in intracellular environments

This study presents a unified analytical framework for investigating ionic currents along microtubules by integrating Lie symmetry analysis, exact soliton solutions, bifurcation dynamics, chaotic behavior, and conservation laws into a comprehensive mathematical model. The ionic current equation, which governs ion movement within intracellular environments, is systematically analyzed through Lie point symmetries, yielding an optimal system of generators that reduce the governing nonlinear partial differential equation to solvable ordinary differential equations via similarity transformations. A new extended direct algebraic method is employed to construct 37 distinct families of traveling wave solutions, encompassing dark, bright-dark, dark-bright, and combo-type solitons, each with specific biological relevance to microtubule electrophysiology. The qualitative dynamics of the system are rigorously examined through bifurcation analysis, revealing the dependence of equilibrium stability on the damping parameter \(\lambda \) , while the introduction of an external periodic perturbation uncovers quasi-periodic and chaotic regimes, quantitatively confirmed by the positive largest Lyapunov exponent ( \(\lambda _{\max }=0.097\) ). Sensitivity analysis demonstrates the system’s strong dependence on initial conditions, with numerical experiments clearly specifying the perturbation amplitudes used. Additionally, conservation laws are derived via the multiplier method, providing invariant quantities that govern the system’s dynamics. This integrated approach not only advances theoretical understanding of ionic transport mechanisms but also offers potential insights for investigating neurological disorders associated with abnormal ion flows and for designing bio-inspired nanodevices.