A piecewise-linear adaptive exponential integrate-and-fire neuron model with emerging traveling waves using analytical scheme

The diverse firing characteristics of the excitable dynamical models are spontaneous organized responses often observed in networks. The adaptive exponential integrate-and-fire (AdEx) model has been widely used to study functional properties in neuronal networks and information processing. The dynamics of piecewise-linear approximation to the AdEx model is explored to enhance mathematical and computational efficiency and accuracy. Analytical calculations and numerical schemes are implemented to solve coupled nonlinear diffusive dynamical equations for traveling wave solutions. More specifically, we explore the diversity among the types of wave we encounter. We determined the approximate solutions using our proposed method and derived the solutions from variations in the diffusion parameters. The traveling front and pulses are explicitly specified and analyzed to validate our analytical scheme. This study contributes to understanding the role of self-diffusion and single cross-diffusion in piecewise linear AdEx models, providing insights into the mechanisms underlying action potential generation. We analyze the effects of different types of diffusion on the model’s behavior, including the generation of wave propagation, wave speed, and shape. The findings have implications for modeling neuronal activity, particularly in replicating complex behavior in neuronal computation.