# FitzHugh–Nagumo system with diffusion

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,

$$\label{FHN}\begin{cases} \dot{u}=\mu\left( u-\frac{u^3}{3}+v+z \right), \\ \dot{v}=-\frac{1}{\mu}\left( u-a+bv \right), \end{cases}$$

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.

1. How can it model the same biological behaviour if the stimulus current disappears?
2. Therefore, which is the exact meaning of this sistem (the second one)?

I've implemented numerically this system, and ploting $u$ in terms of $x$ and $y$ in a specific time, I get something similar to the following images:

https://chemoton.files.wordpress.com/2010/07/fitzhugh-nagumo-type-reaction-diffusion-system-i.png

Where the first image corresponds to $t=0$, the second one $t=10$ and the last one $t=100$.

1. What does this pattern formation mean? I do not know how action potential (which is modelled in this system) is related with these images...

2. Does this happen because the activated cells activate the ones in the neighbourhood (since diffusion acts)? Or this is wrong because the system only takes into account one single neuron?

Thank you so much!

• The FHN equation describes a dynamical system which is only dependent on time using a system of non-linear ordinary differential equations. When you allow diffusion (of the current) then you'll get a partial differential equation with spatial as well as temporal variable. You will get travelling waves along the spatial co-ordinates. Also have a look at the cable equation. – WYSIWYG May 22 '16 at 8:53
• Thanks for your comment. The system with diffusion also models the behaviour of a single neuron, like the system without diffusion? Or it goes a step further and models a group of them? – S. Proa May 22 '16 at 19:45
• It is also of a single neuron. Perhaps you can add details to clarify the premise of your question. – WYSIWYG May 22 '16 at 20:10
• Mmmm... But in the output of the numerical implementation of the system with diffusion (the three images in the link) we observe a kind of pattern formation, which seems to represent that activated cells activate the cells near them. Therefore, many cells should be taken into account, in this model, or not? If it only models the behavior of a single neuron, what represents the "pattern formation" of the images? – S. Proa May 22 '16 at 20:51
• Are you talking about the wikipedia article? I am not sure what exactly you are referring to. Can you add the right reference and also include the diagram that you were talking about. AFAIK, the diffusion is within a cell. – WYSIWYG May 23 '16 at 4:35