We analyze the system of reaction-diffusion equations of Fitzhugh-Nagumo (FHN) type on open bounded convex domains $\mathcal{D} \subset \mathbb{R}^{d} (d \leq 3)$ with Robin boundary conditions. Classical Faedo-Galerkin approach \cite{lions1969quelques} and compactness arguments are utilised to show the existence, uniqueness, and continuous dependence on initial data of weak and strong solutions. Moreover, we introduce a complete case study for the application of this approach to Fitzhugh-Nagumo system.