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biologically-inspired computinglecture 11
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course outlook
Assignments: 35% Students will complete 4/5 assignments based
on algorithms presented in class Lab meets in I1 (West) 109 on Lab
Wednesdays Lab 0 : January 14th (completed)
Introduction to Python (No Assignment) Lab 1 : January 28th
Measuring Information (Assignment 1) Graded
Lab 2 : February 11th
L-Systems (Assignment 2) Graded
Lab 3: March 11th
Cellular Automata and Boolean Networks (Assignment 3)
Sections I485/H400
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Readings until now
Class Book Nunes de Castro, Leandro [2006]. Fundamentals of Natural
Computing: Basic Concepts, Algorithms, and Applications. Chapman & Hall. Chapter 2, all sections Chapter 7, sections 7.3 – Cellular Automata Chapter 8, sections 8.1, 8.2, 8.3.10
Lecture notes Chapter 1: What is Life? Chapter 2: The logical Mechanisms of Life Chapter 3: Formalizing and Modeling the World Chapter 4: Self-Organization and Emergent
Complex Behavior posted online @ http://informatics.indiana.edu/rocha/i-
bic Optional
Flake’s [1998], The Computational Beauty of Life. MIT Press. Chapters 10, 11, 14 – Dynamics, Attractors and chaos
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highlightsLab 2: L-Systems
Tyler WojcikMatthew Remmel
Darlan Farias
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highlightsLab 2: L-Systems
Tyler WojcikMatthew Remmel
Darlan Farias
Michael Sorg
Kendall Ebley
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highlightsLab 2: L-Systems
Tyler WojcikMatthew Remmel
Darlan Farias
Michael Sorg
Kendall Ebley
Rafael Paiva
Edward Lautzenhiser
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highlightsLab 2: L-Systems
Tyler WojcikMatthew Remmel
Darlan Farias
Michael Sorg
Kendall Ebley
Rafael Paiva
Edward Lautzenhiser
Jonathan Stout
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Kauffman’s statistical analysis
Random networks Started with random initial conditions
Self-organization is not a result of special initial conditions
Statistical analysis K 2
Steady state, ordered, crystallization (5 K to ) K=N
Disordered, chaotic Mean length of cycles: 0.5 x 2N/2
Mean number of cycles: N/e High reachability, sensitive to perturbation
Number of other state cycles system can reach after perturbation
K=2 Mean length: n1/2
Mean number of cycles: n1/2
Low reachability Percolation of frozen clusters (isolated subsets) Not very sensitive to perturbation
Of NK-Boolean Networks
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edge of chaos on Boolean Networks
2 K 5 Good for evolvability? Some changes with large repercussions Best capability to perform information exchange
Information can be propagated more easily Problems with analysis
Network topology is random Not scale-free, as later explored by Aldana
Real genetic networks tend to have lower values of K (in ordered regime)
Genes as simply Boolean may be oversimplification Though a few states can approximate very well continuous
data
criticality
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dynamical behavior of ensembles of networkscriticality in Boolean networks
Aldana, M. [2003]. Physica D. 185: 45–66
Random topology
scale-free topology
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Criticality in Boolean Networks
Marques-Pita, Manicka, Teuscher & Rocha, [2015]. In Prep.
Current theory
kp 211
21
Aldana, M. [2003]. Physica D. 185: 45–66
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criticality in the presence of canalizationinput redundancy, effective connectivity
kFf f
r
nxk
2
max #
|
xkxkxk re )()(
ecrit k
p 32.0
kp 211
21
Marques-Pita, Manicka, Teuscher & Rocha, [2014]. In preparation.
Current Theory (CT): No redundancy is considered, only connectivity and bias of logical functions. Green: stable ensembles of BN. Red chaotic ensembles of BN. Blue line, criticality region for CT.
New Theory (NT): Redundancy is considered, as effective connectivity (a form of canalization and dynamical redundancy). Red chaotic ensembles of BN. Blue line, criticality region for NT.
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criticality in the presence of canalizationinput redundancy, effective connectivity
kFf f
r
nxk
2
max #
|
xkxkxk re )()(
ecrit k
p 32.0
kp 211
21
Marques-Pita, Manicka, Teuscher & Rocha, [2014]. In preparation.
Current Theory (CT): No redundancy is considered, only connectivity and bias of logical functions. Green: stable ensembles of BN. Red chaotic ensembles of BN. Blue line, criticality region for CT.
New Theory (NT): Redundancy is considered, as effective connectivity (a form of canalization and dynamical redundancy). Red chaotic ensembles of BN. Blue line, criticality region for NT.
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Boolean networks, sound, art, and educationself-organization and the cybernetics of life
Chaos et al [2006]. “From Genes to Flower Patterns and Evolution: Dynamic Models of Gene Regulatory Networks”. Journal of Plant Growth Regulation. 25(4): 278-289.
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Cellular automatahomogenous lattice of state-determined systems
xx-1 x+1
Cellular Automata
xt
2-D
},...2,1,0{
,...,..., 1
sxxxxfx t
riiriti
1-D
x
1,,,, ,...,..., t
rjrijirjrit
ji yxyxxxxfx
Toroidal LatticeToroidal LatticeToroidal Lattice
Space-time diagram
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cellular automata
Parallel updating Artificial physics
Local interactions only No actions at a distance
Homogeneous Unpredictable global behavior
Emergence 2-levels: rules (micro-level) and
attractor behavior (macro-level) Irreversible
Self-organization Example rules
Rug (diffusion) 256 states Average of 8 neighbors in 2-d grid, if
state is 255 -> 0. Vote/majority
binary
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elementary CA rules Radius 1
Neighborhood =3 Binary
23 = 8 input neighborhoods 28 = 256 rules
http://mathworld.wolfram.com/CellularAutomaton.html
xx-1 x+1
Cellular Automata
xt
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state-determined transitionsCellular automata
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Living patterns easily replicated in CA
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What’s a CA?more formally
D-dimensional lattice L with a finite automaton in each lattice site (cell)
State-determined system finite number of states Σ: K=| Σ|
E.g. Σ = {0,1} finite input alphabet α transition function Δ: α→Σ
uniquely ascribes state s in Σ to input patterns α
Neighborhood templateN
NN K ,Number of possible
neighborhood states
NKKD Number of possible
transition functions
ExampleK=8N=5|α|=37,768D 1030,000
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Langton’s parameter
Statistical analysis Identify classes of transition functions with similar behavior
Similar dynamics (statistically) Via Higher level statistical observables
Like Kauffman The Lambda Parameter (similar to bias in BN)
Select a subset of D characterized by λ Arbitrary quiescent state: sq
Usually 0 A particular function Δ has n transitions to this state and (KN-n)
transitions to other states s of Σ (1-λ) is the probability of having a sq in every position of the rule table
Finding the structure of all possible transition functions
Langton, C.G. [1990]. “Computation at the edge of chaos: phase transitions and emergent computation”. Artificial Life II. Addison-Wesley.
N
N
KnK
λ = 0: all transitions lead to sq (n =KN)λ = 1: no transitions lead to sq (n =0)λ = 1-1/K: equally probable states ( n=1/K . KN)
Range: from most homogeneous to most heterogeneous
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Langton’s observations λ only correlates well with dynamic behavior for fairly large values
of K and N E.g. K≥4 and N≥5
Experiments K=4, N=5
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Langton’s results
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Langton’s results
Approximate time when density is within 1% of long-term behavior
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Langton’s results
Approximate time when density is within 1% of long-term behavior
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Langton’s results
Approximate time when density is within 1% of long-term behavior
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Edge of chaos
Transient growth in the vicinity of phase transitions Length of CA lattice only relevant around phase transition (λ=0.5)
Conclusion: more complicated behavior found in the phase transition between order and chaos Patterns that move across the lattice
A phase transition?
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Next lectures
Class Book Nunes de Castro, Leandro [2006]. Fundamentals of Natural
Computing: Basic Concepts, Algorithms, and Applications. Chapman & Hall. Chapter 2, all sections Chapter 7, sections 7.3 – Cellular Automata Chapter 8, sections 8.1, 8.2, 8.3.10
Lecture notes Chapter 1: What is Life? Chapter 2: The logical Mechanisms of Life Chapter 3: Formalizing and Modeling the World Chapter 4: Self-Organization and Emergent Complex
Behavior posted online @ http://informatics.indiana.edu/rocha/i-bic
Papers and other materials Optional
Flake’s [1998], The Computational Beauty of Life. MIT Press. Chapters 10, 11, 14 – Dynamics, Attractors and chaos
readings