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We examine the dynamics of a spatially extended excitable neuron model between phase state and stable/unstable equilibrium point depending on the parameter regimes. The solitary wave profiles in the excitable medium are characterized by an improved Hindmarsh-Rose (HR) spiking-bursting neuron model with an injected decaying current function. Linear stability and the nature of deterministic system dynamics are analyzed. Further investigation for the existence of wave using the reaction-diffusion HR system and the criteria for diffusion-driven instabilities are performed. An approximation method is introduced to analyze traveling wave profiles for the oscillatory neuron model that allows the explicit analytical treatment of both the speed equations and shape of the traveling wave solution. The solitary wave profiles exhibited by the system are explored. The analytical expression for the solution scheme is …