Emergence of periodic soliton patterns in the time-fractional Hodgkin-Huxley model arising in modern neuroscience.

The time-fractional Hodgkin-Huxley Model (tfHHM) with Katugampola Fractional Derivatives (KFDs) is the main subject of the current study. This model was developed in modern neuroscience that imitate the way action potentials move along the giant axon in squids. Explicit solutions and in-depth details of soliton patterns are obtained by using the Riccati Modified Extended Simple Equation Method (RMESEM). The suggested approach converts the intended model into a nonlinear algebraic system by providing traveling solutions in closed form. Using the symbolic computation tool Maple, the resultant algebraic system is analytically solved to yield a new set of solutions in terms of trigonometric, exponential, rational, hyperbolic and rational-hyperbolic functions. A two-dimensional visual investigation of the dynamical characteristics of the derived solutions for fractional parameters shows that periodic solitons, such as periodic lump, hump, dark, and bright solitons, are formed by the obtained soliton solutions. These findings demonstrate the effectiveness of the proposed technique for solving complex nonlinear models and improve our knowledge of soliton behaviors in the context of tfHHM.