Turing suggested that the mechanisms of morphogenesis can be explained by ``reaction-diffusion'' equations (see [11]). These are systems of partial differential equations of the form

where the vector-valued at each time **t** describes the
changing spatial concentrations of a set of chemical substances called
``morphogens''. The resulting * Turing patterns* for various
reaction specifications have been widely simulated and have
been shown to produce regular patterns reminiscent of zebra stripes
and leopard spots. Whether or not reaction-diffusion mechanisms are
the actual phenomena in such biological systems,
simulation of such systems does tend to generate regular patterns that
appear as standing-wave solutions to the equations.

Interestingly, the resulting Turing patterns are often independent
of the initial state of the system. Starting from a random
configuration and evolving the equation eventually results in regular
patterns in the concentrations of the morphogens.
Figure 2 shows one such set of patterns,
obtained from a random initial distribution in a system
that evolves according to a system of reaction-diffusion equations called the
* Gray-Scott* model. We have built a simulation environment for
exploring these kinds of reaction models. Figure 2
shows a sample result.

**Figure 2:**
Simulation of the Gray-Scott model starting from a random initial
configuration (left) and evolving to regular spots after 20,000 time
steps (right). There are 2500 particles randomly packed in a square
of size 1, starting with values of the and
morphogens randomly assigned between 0 and 1. Each particle's neighbors are those
within a distance of 0.05 from it. The constants for the Gray-Scott
model here are , , , , and
integration is performed using the forward-Euler method with . The different colors of the particles indicate different values
of .

This suggests the intriguing possibility of organizing amorphous computing systems by starting with the particles in a random state and solving a discrete analog of a reaction-diffusion system. The resulting regularly spaced ``blobs'' of particles could then serve as organizing centers for subsequent stages of computing.

Thu Jun 27 16:56:19 EDT 1996