Action potential-like modes as modulated waves in an extended soliton model for biomembranes and nerves

By extending the Heimburg–Jackson soliton model for neural signals that considers the effects of higher-order nonlinearities, the dynamics of modulated waves characterizing electromechanical density pulses is described in the form of soliton-like pulse signals representing nerve impulses, well-known as action potential pulses (Appulses). The investigation is performed both analytically and numerically, where a comprehensive picture of higher-order nonlinearities effects on the generation and evolution of nerve impulses is provided. Within the framework of a multiple-scale-expansion analysis and the reductive perturbation method, while considering third- and fourth-order nonlinearities, the electromechanical area-density pulse propagation is investigated, leading to the generation of a localized Appulse. Accordingly, the analytical theory uses a perturbative technique, and a damped cubic–quintic nonlinear Schrödinger equation is derived, which admits a single-pulse-type solitary solution that possesses different phase characteristics of the typical neuronal Appulse structure, representative of nerve impulse profiles. A modulational instability (MI) analysis demonstrates the increase of the modulation gain in the system with increasing fourth-order nonlinearity, indicating that the higher-order nonlinearities influence the MI in the proposed extended soliton model. Furthermore, a numerical analysis is performed, and consistent agreement with the analytical prediction is achieved, confirming a localized typical longitudinal single pulse-like solitary wave solution for the extended soliton model. Importantly, the appearance of a typical longitudinal single-solitary pulse-type structure can evolve uniformly with increasing fourth-order nonlinearity, leading to the splitting of the single-pulse-soliton signal and resulting in the appearance of a double asymmetric localized pulse-like mode or bisoliton-pulse structure, characteristic of a coupled Appulse.