EXPLORABLES

This explorable illustrates one of the most basic mechanisms for spontaneous pattern formation: *Local excitation* and *long range inhibition*. This mechanism or similar ones are responsible for patterns observed in many natural systems, such as neural tissue, animal fur and spatial heterogeneity in social systems.

** Press Play** and keep on reading….

# This is how it works:

The state of the two dimensional system is defined by a state variable $u(x,t)$ at each position $x$ at time $t$. The state variable can have values in the range $[-1,1]$. Think of this as a degree of polarization, the ends of the interval corresponding to extreme values and $0$ to a neutral state.

At each location $x$ the rate of change of state is governed by the equation

$$ \partial _t u(x,t) = - u(x,t) + H(x,t), $$

where the function $H(x,t)$ is the input that the system receives at location $x$. If $H(x,t)=0$ the state will go to the neutral state $u(x,t)=0$ due to the decay term. If $H(x,t)=H_0$ is a constant at $x$ then $u(x,t)\rightarrow H_0$.

## The mexican hat:

Interesting things emerge, when the input $H(x,t)$ at location $x$ is determined by the state or polarization in the vicinity. We define two regions around $x$, a small disk $U_1(x)$ centered at $x$ with radius $R_1$ and an annulus $U_2(x)$ bounded by the radii $R_2>R_1$. We now integrate the state of the field and compute

[ I(x,t) = \int _{U_1(x)} dy, u(y,t) -\int _{U _2(x)} dy, u(y,t) ]

This means we weigh the average state within $U_1$ positively (local excitation) and the average in $U_2$ negatively (long range inhibition). Because of the shape of the interaction, this is also known as a mexican hat interaction. Finally, we use the raw input $I(x,t)$ and plug it into a sigmoid function to get

[ H(x,t) = \frac{1-e ^{-\beta,I(x,t)}}{1+e ^{-\beta,I(x,t)}} ]

This function makes sure that $u(x,t)$ is restricted to the interval $[-1,1]$. The parameter $\beta$ controls the shape (* steepness*) of the sigmoid response function. Initially, the system is in the uniform state 0 plus a little bit of noise.

When you press the play button, you start the dynamical system and will find that slowly the initially uniform state will evolved into a striped pattern with a typical wavelength. The overall shape and wavelength depends on the choice of parameters,
you can explore how changes of the * radii* change the pattern.

The left arrow button resets initial conditions, the bottom right button resets the parameters.