I was studying the FitzHugh-Nagumo model with diffusion and I quite do not understand the meaning of it.
If we consider the system without diffusion,
\begin{equation}\label{FHN}\begin{cases} \dot{u}=\mu\left( u-\frac{u^3}{3}+v+z \right), \\ \dot{v}=-\frac{1}{\mu}\left( u-a+bv \right), \end{cases}\end{equation}
where $u=u(t)$ and $v=v(t)$, then it models the control of the electrical potential across a cell membrane. It models the action potential in neurons, for example. But this happens because in the system there is a stimulus current, $z(t)$. The system with diffusion
\begin{equation*}\begin{cases} \frac{\partial u}{\partial t} = \epsilon D_u \nabla ^2u+\frac{1}{\epsilon}(u-\frac{u^3}{3}+v), \\ \frac{\partial v}{\partial t} = \epsilon^2D_v \nabla ^2v-(u-a+bv), \end{cases}\end{equation*}
where $u=u(x,y,t)$ and $v=v(x,y,t)$, models exactly the same but just adding diffusion and removing the stimulus current.
- I've simulated numericaly the system with diffusion using finite differences and I wanted to know if one can get a stability condition for it.
Thank you so much!