# Cellular Automata/Excitable media

## An introduction to excitable media

Excitable media are nonlinear dynamic systems known for exhibiting complex behavior that can be observed as pattern formation. They are usually defined by a reaction-diffusion differential equation.

$u_{t}(\mathbf {r} ,t)=D\nabla ^{2}u(\mathbf {r} ,t)+f(u(\mathbf {r} ,t))$ The diffusion part provides stability and propagation of information, the reactive part provides interesting local dynamics.

A common example of excitable media are prey-predator systems. Such systems are described by a system of differential equations, one function for each of the observed protagonists.

$u_{t}(\mathbf {r} ,t)=D_{u}\nabla ^{2}u(\mathbf {r} ,t)+f(u,v)$ $v_{t}(\mathbf {r} ,t)=D_{v}\nabla ^{2}v(\mathbf {r} ,t)+g(u,v)$ We will discuss two different approaches to modeling excitable media. Discretization of differential equations and modeling with cellular automata.

### Boundary conditions

There are different ways to define boundary conditions for the reaction-diffusion equation.

Dirichlet boundary conditions

The value of the function at the boundary is given explicitly $u(\mathbf {r_{0}} ,t)$ .

Cyclic boundaries

If the initial condition $u(x,0)$ is supposed to be periodic in space, cyclic boundary conditions can be used.

Zero-flux boundary conditions

If zero-flux is expected at the boundary than the component of the functions first derivative normal to the boundary is zero ${\vec {n}}\cdot \nabla {u}=0$ at the boundary. This can be achieved by reflecting function values from the inside over the boundary to the outside.

## Discretization of differential equations using the explicit FTCS method

The traditional method to simulate excitable media is discretization and numerical computation of the governing PDE. First the FTCS (forward-time centered-space method) discretization method is presented. Explicit methods are the simplest and the equations are similar to a cellular automaton, but are inadequate because of stability and convergence problems.

### Single PDE

We will first observe a single PDE describing a single function.

$u_{t}(\mathbf {r} ,t)=D\nabla ^{2}u(\mathbf {r} ,t)+f(u(\mathbf {r} ,t))$ #### One-dimensional problem

In the one dimensional case the space vector becomes a single variable $\mathbf {r} =x$ . The nabla operator becomes $\nabla ^{2}={\frac {\partial ^{2}}{\partial {x^{2}}}}$ .

${\frac {\partial u(x,t)}{\partial t}}=D{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}+f(u(x,t))$ The partial differential equation is discretized.

${\frac {u(x,t+\Delta {t})-u(x,t)}{\Delta {t}}}=$ $\quad D{\frac {u(x+\Delta {x},t)-2u(x,t)+u(x-\Delta {x},t)}{\Delta {x}^{2}}}$ $\quad +f(u(x,t))$ Each finite element at time $t+\Delta {t}$ is calculated from three neighboring elements at time $t$ (see figure at the right).

$u(x,t+\Delta {t})=u(x,t)+\,$ $\quad +d(u(x+\Delta x,t)-2u(x,t)+u(x-\Delta x,t))$ $\quad +\Delta {t}f(u(x,t))$ where the diffusion number is

$d=D{\frac {\Delta {t}}{\Delta {x}^{2}}}$ Stability

The FTCS method is stable if

$d\leq 1/2$ Boundary conditions

For periodic boundaries present values at the left boundary $x=0$ can be used to compute the future values at the right boundary $x=L_{x}$ and the other way round.

$u(0,t+\Delta {t})=u(0,t)+d(u(0+\Delta x,t)-2u(0,t)+u(L_{x},t))+\Delta {t}f(u(0,t))\,$ $u(L_{x},t+\Delta {t})=u(L_{x},t)+d(u(0,t)-2u(L_{x},t)+u(L_{x}-\Delta x,t))+\Delta {t}f(u(L_{x},t))\,$ If there is zero-flux $\nabla u(0,t)=0$ at the boundaries than values outside the boundary are reflections of values inside $u(0+\Delta x,t)=u(0-\Delta x,t)$ .

$u(0,t+\Delta {t})=u(0,t)+2d(u(0+\Delta x,t)-u(0,t))+\Delta {t}f(u(0,t))\,$ $u(L_{x},t+\Delta {t})=u(L_{x},t)+2d(u(L_{x},t)+u(L_{x}-\Delta x,t))+\Delta {t}f(u(L_{x},t))\,$ #### Two-dimensional problem

In the two-dimensional case, the space vector becomes a variable pair $\mathbf {r} =[x,y]$ . The nabla operator becomes $\nabla ^{2}={\frac {\partial ^{2}}{\partial {x^{2}}}}+{\frac {\partial ^{2}}{\partial {y^{2}}}}$ .

${\frac {\partial u(x,y,t)}{\partial t}}=D\left({\frac {\partial ^{2}u(x,y,t)}{\partial x^{2}}}+{\frac {\partial ^{2}u(x,y,t)}{\partial y^{2}}}\right)+f(u(x,y,t))$ The partial differential equation is discretized using the forward-time centered-space method.

${\frac {u(x,y,t+\Delta {t})-u(x,y,t)}{\Delta {t}}}=$ $\quad =D{\frac {u(x+\Delta {x},y,t)-2u(x,y,t)+u(x-\Delta {x},y,t)}{\Delta {x}^{2}}}+$ $\quad +D{\frac {u(x,y+\Delta {y},t)-2u(x,y,t)+u(x,y-\Delta {y},t)}{\Delta {y}^{2}}}+$ $\quad +f(u(x,y,t))$ Each finite element at time $t+\Delta {t}$ is calculated from five neighboring elements at time $t$ (see figure at the right).

$u(x,y,t+\Delta {t})=u(x,t)+\,$ $\quad +d_{x}(u(x+\Delta {x},y,t)-2u(x,y,t)+u(x-\Delta {x},y,t))+$ $\quad +d_{y}(u(x,y+\Delta {y},t)-2u(x,y,t)+u(x,y-\Delta {y},t))+$ $\quad +\Delta {t}f(u(x,y,t))$ where the diffusion numbers are

$d_{x}=D{\frac {\Delta {t}}{\Delta {x}^{2}}}\quad d_{y}=D{\frac {\Delta {t}}{\Delta {y}^{2}}}$ Stability

The FTCS method is stable if

$d_{x}\leq 1/2$ and $d_{y}\leq 1/2$ Boundary conditions

The same ideas as in the one dimensional case can be used for two dimensions.

### System of PDE

A system of PDE describes two functions that interact with each other (prey-predator).

$u_{t}(\mathbf {r} ,t)=D_{u}\nabla ^{2}u(\mathbf {r} ,t)+f(u,v)$ $v_{t}(\mathbf {r} ,t)=D_{v}\nabla ^{2}v(\mathbf {r} ,t)+g(u,v)$ The interaction is local, which means, the dispersion part can be computed separately for each equation, and than the reaction part is added to the result.