Analytical construction of lump, rogue, and multi-wave structures in a nonlinear neuron membrane model via bilinear neural network approach.

In this article, we study a nonlinear neuron membrane model describing the propagation of action potentials along nerve fibers, incorporating nonlinear elastic effects and higher-order dispersion. By applying the Hirota bilinear transformation, the given equation is converted into an equivalent bilinear form, which provides a suitable analytical framework for systematic construction of exact solutions. To enrich the functional solution space, we introduce a bilinear neural network method (BNNM), where neural network architectures are used as structured symbolic generators rather than numerical approximators. Both single-hidden-layer and double-hidden-layer configurations are constructed to generate exact analytical solutions. Through symbolic coefficient matching assisted by MAPLE, multiple admissible parameter sets are obtained. The presented framework yields a diverse family of exact wave structures, involving lump solutions, breather-type oscillatory waves, soliton-lump interaction states, double-period lump superpositions, three-wave interaction patterns, and hybrid lump-rogue wave excitations. The derived solutions are expressed in compact Hirota form and signified via three-dimensional, density, and contour visualizations, revealing strong spatial localization, temporal modulation, nonlinear energy redistribution, and coherent phase-locked propagation. The results represent that the neural-bilinear approach offers a powerful and systematic mechanism for constructing rich nonlinear wave families in neuron-type models. These analytical structures contribute to understanding localized pulse transmission, multi-wave interaction dynamics, and transient amplification phenomena in excitable biological media.