He produces some neat patterns formation with a simple agent-based model of self-propelled particles that (1) attempt to move generally forward, (2) release chemicals when they move (but not when they don't), and (3) turn toward the highest chemical gradient of their forward-3 directions. The chemicals diffuse and decay.
Lit review:
Nice description of Turing’s reaction-diffusion model:
“The question of how spatial patterns may arise from a
homogeneous initial state was initially addressed by Turing, who considered the
interactions of two hypothetical chemicals [51]. The method relied on the
autocatalytic production of an activator chemical, which enhanced the
production of a second chemical, which in turn inhibited the formation of the
activator chemical. Critically, the inhibitor diffused more quickly than the
activator, and a pattern of local activation (reaction) and lateral inhibition
(diffusion) was produced. Variations in the parameters of the differential
equations describing the interactions of the chemicals (which Turing called
morphogens) produced instabilities in the concentration profiles of the
reactants. By interpreting the concentrations of the two chemicals as, for
example, different colors, characteristic patterns of spots or stripes are
formed.” (p. 128)
- The RD approach has been the dominant theoretical model of pattern formation.
- However, the search for morphogen chemicals have yet to reveal definitive results.
- Also, the RD systems appear to be sensitive to perturbations, which cannot be the case, in embryonic development, e.g.
Other mechanisms can generate complex patterning.
- Mechanical models deform the substrate (e.g., cellular traction forces).
- Cellular models can use purely chemotactic stimuli so that cells migrate toward concentration gradients and initiate patterning. These approaches may be combined (mechanochemical).
All the above mentioned mechanisms use LALI.
Bonabeau suggests that ABM may be more suitable for modeling
systems where “low-level interactions produce emergent global behaviors.”
Top of p. 129 has a number of examples of biological pattern
formation. A good place to look for references.
“Some of these pattern formation mechanisms have certain
features in common: The patterning is an emergent phenomenon, qualitatively
different from, and not specified by, the individual behaviors. The patterns
are often composed of the “actors” themselves [10] from the “bottom up,” and
the patterns formed may be used for transport of substances within the organism
structure [7, 23].” (p. 129)
The plasmodium slime mole Physarum makes these “veins” which
transport food between the extremities of the slime mold. Its pretty cool. Very
different from my PD system (which looks the most like a “pretzel” slime mold).
“From a pattern formation perspective, Physarum can be
interpreted as a complex mechanism of planar pattern formation based upon the
two requirements of efficiency in foraging behavior (searching of a maximal
area) and efficiency in nutrient transport (minimal transport distance and
fault tolerance). The mechanisms used to fulfill these requirements are growth,
movement, and area reduction. During the growth-and-foraging stage the
plasmodium exhibits a default, broadly reticulated outward growth
pattern—albeit one that is influenced by substrate and gradient quality [46].
Once nutrients have been located, the topology of the pattern is influenced by
the nutrient distribution—the connectivity patterns (the protoplasmic tube
network) evolve to achieve a compromise between minimal transport costs and
fault tolerance. Since the plasmodium obviously cannot have any global
knowledge about the initial or optimal topology, the network must evolve by
physical forces acting on the protoplasmic transport.” (p. 130)
The Model
- There is some pre-set trail of chemical resources on a grid (though I think they usually start with none).
- Motor stage (all agents do this before they each move to sensory stage):
- Attempt to move forward in current direction
- If successful
- Deposit trail in new location.
- Else
- Choose random new orientation
- Sensory stage:
- Sample trail map values in forward cell (F), forward right (FR) and forward left (FL).
- If F is the biggest, stay facing same direction
- Else If FL and FR are both bigger than F, turn one direction randomly
- Else turn toward direction of greater signal
- The kernels of the chemical trait diffuse and decay.
Because agents only deposit chemoattractant after a
successful movement and because the agents have a forward-biased sensory
apparatus, static clustering of agents is avoided and a dynamics network is
formed.
Varying the parameters of the model changes some of the
types of patterns that emerge. I think the labrynthine pattern of Fig. 14 looks
the most like the PD patterns I’ve gotten.
If the agents are repelled
rather than attracted by the chemical signals, the results are similar to
Turing patterns. Here, the agent orients away from the source of the strongest
gradient.
There is a lot of analysis of the different types of
patterns you can get with this model. Pretty neat.
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