Localized numerical impulses solutions in diffuse neural networks modeled by the complex fractional Ginzburg-Landau equation

We investigate a network of diffusively Hindmarsh-Rose neurons with long-range synaptic coupling. By means of a specific perturbation technique, we show by using the Lienard form of the model that it can be governed by a complex fractional Ginzburg-Landau (CFGL) equation where analytical as well as numerical nonlinear wave solutions can be obtained. We propose the semi implicit Riesz fractional finite-difference scheme to solve efficiently the obtained CFGL equation. From numerical simulations, it is found that the fractional solutions for the nerve impulse are well-localized impulses whose shape and stability depend on the value of the long-range parameter.