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<p>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 <span>Maple</span>, 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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年7月18日

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  • Mati ur Rahman,
  • Sonia Akram,
  • Laila A. AL-Essa,
  • Khalid Aldawsari

摘要

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.