Analyzing the neural wave structures in the field of neuroscience

Soliton theory research has a substantial impact on the application of nonlinear sciences in different fields. This has led to a significant increase in the focus of researchers on the study of solitary waves in recent years. This study explores the diverse dynamic behaviors exhibited by soliton solutions within the framework of the soliton neuron model. In neuroscience, this model is regarded as an important tool for comprehending the initiation and propagation of action potentials along axons through the application of a thermodynamic theory of nerve pulse transmission. The model proposed herein suggests that signals that propagate through the cell membrane can be represented as solitons, or solitary sound pulses. In order to analyze these soliton solutions, the nonlinear differential equation is transformed into the corresponding ordinary differential equation using a wave transformation. The wave profiles of the soliton neuron model are derived by using the Kumar-Malik method, multivariate generalized exponential rational integral function method, and Riccati modified extended simple equation method. These methods are implemented to extract a diverse array of soliton solutions, such as mixed, dark, bright-dark, singular, bright, complex, and combined solitons. This examination is focused on specific nonlinear phenomena of the proposed model. Numerous graphs are incorporated to clarify the behavior of solutions across a diverse range of parameter values. By validating the effectiveness of current methodologies and elucidating the nonlinear dynamic characteristics of a system, this research makes a substantial contribution to the domains of nonlinear science and higher-dimensional nonlinear wave fields. The insights presented in this paper can be implemented to address analytical challenges in various nonlinear systems in biological, technological, and physical systems to facilitate the comparison of computational and experimental data.