<p>The well-known mathematical model of nonlinear ionic currents in microtubules, which is a useful tool for examining the complex interrelationships among cytoskeletal dynamics, bioelectricity, and cellular function, is examined in this work. In addition to capturing the interaction of physical processes involved in the propagation of ionic currents along microtubules, its applications are diverse and include neuroscience, cancer research, nanotechnology, and systems biology. The invariant soliton solutions are essential for comprehending the dynamics of the problem under consideration. Thus, using the Lie symmetry analysis, we obtain the Lie invariance criteria. The proposed method provides a 2D Lie algebra where the conservation of mass and energy is corresponding to translation symmetries in space and time, respectively. For the ordinary differential equation generated from the microtubule partial differential model of ionic currents, we construct Lie-subalgebras and get invariant closed form solutions. In order to solve the microtubule equation driving ionic currents precisely, the analytical new auxiliary equation technique is employed, as the inverse scattering transform approach cannot handle the Cauchy problem. Numerous solitons, including kink, anti-kink, dark, brilliant, periodic, smooth topological periodic, unique, rational, and exponential, emerged as a result. By selecting the appropriate parametric parameters, the solution is visually shown in contour, two, and three dimensions. To discuss the sensitivity of the model, a dynamical system is constructed using the Galilean transformation. The graphical visualization of sensitivity is also displayed. For the model under consideration, the first order conservation laws are found. Because they provide profound insights into the behavior of physical systems, these concepts are crucial for comprehending and addressing complicated problems.</p> Graphical Abstract <p></p>