Lecture 7 - Pattern Formation II

Pattern Formation II

This lecture continued the discussion on pattern formation we started in the previous lecture. We discussed again the Majority Model explored in the last lecture and generalized the idea to a spatially continuous model.

We introduced the concept of a field $u(\mathbf{x},t)$ as a function of the spatial coordinate $\mathbf{x}=(x,y)$ and time $t$ and the dynamics of such a field

$$ \partial_t u(\mathbf{x},t) ) = F[u(\mathbf{x},t)] $$

We began with a simple model (specification of the rhs. of the above equation):

$$ \partial_t u(\mathbf{x},t) ) = h(\mathbf{x},t) - u(\mathbf{x},t) $$

where $$ h(\mathbf{x},t) = \sigma(m(\mathbf{x},t)) $$

and $$ m(\mathbf{x},t) = \int_{\Omega(\mathbf{x})}d\mathbf{z} u(\mathbf{x},t) $$

is the mean field in the neighborhood $\Omega(\mathbf{x})$ of $\mathbf{x}$ and the function $\sigma$ is a sigmoid function.

We used this as a foundation for a model that can generate stable patterns based on the mechanism of local excitation and long-range inhibition that is in detail explained in the explorable:

“Keith Haring’s Mexican Hat”

Reaction Diffusion Systems

We also discussed briefly the ingredients needed for understanding reaction diffusion systems:

• diffusion
• activator-inhibitor dynamics

and started talking about the foundations of diffusion processes: random walks.

A script on the topic will be made available soon.