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## Morphological processes and reaction-diffusion systems

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.

Next: Getting started: Hardware Up: Getting started: Programming Previous: Example: Using approximate

Gerald Jay Sussman
Thu Jun 27 16:56:19 EDT 1996