Part 2D: Pattern Formation
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Differentiation
& Pattern Formation
•  A central problem in
development: How do cells
differentiate to fulfill
different purposes?
•  How do complex systems
generate spatial & temporal
structure?
•  CAs are natural models of
intercellular communication
B.
Pattern Formation
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Zebra
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figs. from Camazine & al.: Self-Org. Biol. Sys.
photos ©2000, S. Cazamine
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Vermiculated Rabbit Fish
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Activation & Inhibition
in Pattern Formation
figs. from Camazine & al.: Self-Org. Biol. Sys.
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Interaction Parameters
•  Color patterns typically have a characteristic length scale
•  Independent of cell size and animal size
•  Achieved by:
–  short-range activation ⇒ local uniformity
–  long-range inhibition ⇒ separation
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•  R1 and R2 are the interaction ranges
•  J1 and J2 are the interaction strengths
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Part 2D: Pattern Formation
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CA Activation/Inhibition Model
•
•
•
•
Let states si ∈ {–1, +1}
and h be a bias parameter
and rij be the distance between cells i and j
Then the state update rule is:
Demonstration of NetLogo
Program for Activation/Inhibition
Pattern Formation:
Fur
$
'
si ( t + 1) = sign&h + J1 ∑ s j ( t ) + J 2 ∑ s j ( t ))
&%
)(
rij <R1
R1 ≤rij <R 2
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RunAICA.nlogo
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€
Effect of Bias
Example
(h = –6, –3, –1; 1, 3, 6)
(R1=1, R2=6, J1=1, J2=–0.1, h=0)
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figs. from Bar-Yam
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Effect of Interaction Ranges
R2 = 6
R1 = 1
h = 0
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•  How can a system using strictly local
interactions discriminate between states at
long and short range?
•  E.g. cells in developing organism
•  Can use two different morphogens diffusing
at two different rates
R2 = 6
R1 = 1.5
h = –3
figs. from Bar-Yam
figs. from Bar-Yam
Differential Interaction Ranges
R2 = 8
R1 = 1
h = 0
R2 = 6
R1 = 1.5
h = 0
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–  activator diffuses slowly (short range)
–  inhibitor diffuses rapidly (long range)
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Part 2D: Pattern Formation
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Digression on Diffusion
Reaction-Diffusion System
•  Simple 2-D diffusion equation:
A ( x, y) = D∇ 2 A ( x, y)
•  Recall the 2-D Laplacian:
∂ 2 A( x, y ) ∂ 2 A( x, y )
∇ 2 A( x, y ) =
+
∂x 2
∂y 2
•  The Laplacian (like 2nd derivative) is:
€
diffusion
!
∂ ! A $ # DA
#
&=
∂ t " I % #" 0
0 $! ∇ 2 A
&#
DI &%#" ∇ 2 I
$ ! fA ( A, I )
&+#
& # f A, I
% " I( )
reaction
$
&
&
%
# A&
c˙ = D∇ 2c + f (c), where c = % (
$I'
–  positive in a local minimum
–  negative in a local maximum
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∂A
= DA ∇ 2 A + fA ( A, I )
∂t
∂I
= DI ∇ 2 I + fI ( A, I )
∂t
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€
Framework for Complexity
General Reaction-Diffusion System
# n
&
∂ci
= ∑ kαν iα % ∏ ckmkα ( − ∇ ⋅ ji
∂t
$ k=1
'
α

where ji = µi ci − div Di ci (flux)
•  change = source terms + transport terms
•  source terms = local coupling
= interactions local to a small region
•  transport terms = spatial coupling
= interactions with contiguous regions
= advection + diffusion
where kα = rate constant for reaction α
and ν iα = stoichiometric coefficient
and mkα = a non-negative integer

and µi = drift vector
–  advection: non-dissipative, time-reversible
–  diffusion: dissipative, irreversible and Di = diffusivity matrix
where div Dc = ∑ e j ∑ D jk ∂c
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j
k
∂xk
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Continuous-time
Activator-Inhibitor System
NetLogo Simulation of
Reaction-Diffusion System
•  Activator A and inhibitor I may diffuse at
different rates in x and y directions
•  Cell becomes more active if activator + bias
exceeds inhibitor
•  Otherwise, less active
∂A
∂2A
∂2A
= dAx 2 + dAy 2 + kA ( A + B − I ) (1− A)
∂t
∂x
∂y
∂I
∂ 2I
∂ 2I
= dIx 2 + dIy 2 + kI ( A + B − I ) (1− I )
∂t
∂x
∂y
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1.  Diffuse activator in X and Y directions
2.  Diffuse inhibitor in X and Y directions
3.  Each patch performs:
stimulation = bias + activator – inhibitor + noise
if stimulation > 0 then
set activator and inhibitor to 100
else
set activator and inhibitor to 0
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Part 2D: Pattern Formation
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Demonstration of NetLogo
Program for Activator/Inhibitor
Pattern Formation
Demonstration of NetLogo
Program for Activator/Inhibitor
Pattern Formation
with Continuous State Change
Run Pattern.nlogo
Run Activator-Inhibitor.nlogo
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A Key Element of
Self-Organization
Turing Patterns
•  Alan Turing studied the mathematics of
reaction-diffusion systems
•  Activation vs. Inhibition
•  Turing, A. (1952). The chemical basis of
morphogenesis. Philosophical Transactions
of the Royal Society B 237: 37–72.
•  Amplification vs. Stabilization
•  Cooperation vs. Competition
•  Positive Feedback vs. Negative Feedback
–  Positive feedback creates
•  The resulting patterns are known as Turing
patterns
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•  Growth vs. Limit
–  Negative feedback shapes
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Reaction-Diffusion Computing
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Image Processing in BZ Medium
•  Has been used for image processing
–  diffusion ⇒ noise filtering
–  reaction ⇒ contrast enhancement
•  Depending on parameters, RD computing
can:
–  restore broken contours
–  detect edges
–  improve contrast
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•  (A) boundary detection, (B) contour enhancement, (C) shape enhancement, (D) feature enhancement
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Image < Adamatzky, Comp. in Nonlinear Media & Autom. Coll.
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Part 2D: Pattern Formation
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Voronoi Diagrams
Some Uses of Voronoi Diagrams
•  Given a set of generating
points:
•  Construct a polygon
around each generating
point of set, so all points
in a polygon are closer to
its generating point than to
any other generating
points.
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Image < Adamatzky & al., Reaction-Diffusion Computers
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Computation of Voronoi Diagram
by Reaction-Diffusion Processor
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Image < Adamatzky & al., Reaction-Diffusion Computers
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Path Planning via BZ medium:
No Obstacles
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Image < Adamatzky & al., Reaction-Diffusion Computers
•  Collision-free path planning
•  Determination of service areas for power
substations
•  Nearest-neighbor pattern classification
•  Determination of largest empty figure
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Mixed Cell Voronoi Diagram
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Image < Adamatzky & al., Reaction-Diffusion Computers
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Path Planning via BZ medium:
Circular Obstacles
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Image < Adamatzky & al., Reaction-Diffusion Computers
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Part 2D: Pattern Formation
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Mobile Robot with Onboard
Chemical Reactor
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Image < Adamatzky & al., Reaction-Diffusion Computers
Actual Path: Pd Processor
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Actual Path: Pd Processor
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Image < Adamatzky & al., Reaction-Diffusion Computers
Image < Adamatzky & al., Reaction-Diffusion Computers
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Actual Path: BZ Processor
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Image < Adamatzky & al., Reaction-Diffusion Computers
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Bibliography for
Reaction-Diffusion Computing
1.  Adamatzky, Adam. Computing in Nonlinear
Media and Automata Collectives. Bristol: Inst.
of Physics Publ., 2001.
2.  Adamatzky, Adam, De Lacy Costello, Ben, &
Asai, Tetsuya. Reaction Diffusion Computers.
Amsterdam: Elsevier, 2005.
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