<p>This work uses novel analytical computational methods to examine the complex. By a suitable wave transformation, the governing partial differential equation has been reduced to an ordinary differential equation which is in turn solved by three advanced methods namely the modified F-expansion method, the generalized projective Riccati equation method and the new generalized exponential rational function method. These methods produce an abundant supply of precise soliton structures, such as bright, dark, kink, bright-dark and combined solitons and thus indicate the outstanding multiplicity of wave forms that the neuronal system can sustain. The solutions obtained, in addition to displaying the computational power and versatility of such techniques in the context of complex nonlinear systems, give more detailed information on the mechanics of neural signal propagation. In addition, the physical behavior of the solutions is fully explored by choosing the various parametrical values of the solutions in relation to the various physiological conditions which are verified using detailed graphical forms. The general analysis contributes to the better comprehension of complicated wave process in neurophysical conditions and to the importance of the contemporary analytical methods in the development of nonlinear science and its biological implementations.</p>