Traveling pulses and its wave solution scheme in a diffusively coupled 2D Hindmarsh-Rose excitable systems

In this article, an analytical approach is demonstrated to show the emerging traveling pulses for the local evolution of a set of diffusively coupled dynamical equations representing neuronal impulses. The derived dynamics governing the traveling pulses solution is described in a space-time reference frame with a two-dimensional excitable Hindmarsh-Rose (H-R) type oscillator. We deduce the conditions that allow us to describe explicitly the nature of propagating traveling pulses. We have constructed the detailed analytical results using semi-discrete approximation method with numerical simulations illuminating possible traveling pulses that include dispersion relations and group velocity equations. We show that the diffusive network can be expressed by the complex Ginzburg-Landau equation. The extended excitable medium with a homogeneous diffusive connection exhibits envelope solitons and multipulses. We observe how the series expansion parameter and coupling play key roles for the appearance of different traveling pulses. The transition phases and amplitude modulations are reported. The obtained results in the form of single solitary pulses and multipulses, reveal the possibility of collective behavior for information processing in excitable system.