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biologically-inspired computinglecture 9
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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) Due February 25th
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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The logistic map
Demographic model introduced by Pierre François Verhulst in 1838
Continuous state-determined system Memory of the previous state only
Observations X=0: population extinct X=1: Overpopulation, leads to extinction
)1(1 ttt xrxx −=+
quadratic equation
Reproduction rate
Population size
]4,0[]1,0[
∈∈
rx
positive feedback negative
feedback
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xt
xt+1
logistic mapplot
)1(1 ttt xrxx −=+
]4,0[]1,0[
∈∈
rx
1/2
r/4
1-1/r
)1()( xrxxf −=xxf =)(
rxx 110 −=∨=
Fixed-point attractors
xxf =)(
⇒>′⇒<′
−=′unstable1)(stable1)(
),21()(xf
xfxrxf
( ) 01)1()1( =+−⇔=−⇔ xrxxxrx
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logistic mapr ≤ 1
x0
rxx 110 −=∨=
⇒>′⇒<′
−=′unstable1)(stable1)(
),21()(xf
xfxrxf
⇒>⇒<
=−=′⇒=unstable1stable1
,|)21(|)(0r
rrxrxfx
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logistic map1 ≤ r ≤ 3
x0
1-1/r
rxx 110 −=∨=
⇒>′⇒<′
−=′unstable1)(stable1)(
),21()(xf
xfxrxf
rxfx =′⇒= )(0 |2|)(11 rxfr
x −=′⇒−=
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logistic map3 ≤ r ≤ 4 (r ≤ 3.44)
x0
rxx 110 −=∨=
xxff =))((
Limit cycle
⇒>′⇒<′
−=′unstable1)(stable1)(
),21()(xf
xfxrxf
|2|)(11 rxfr
x −=′⇒−=
rxfx =′⇒= )(0
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logistic map3 ≤ r ≤ 4 (3.44 ≤ r ≤ 3.54)
x0
rxx 110 −=∨=
Limit cycle
xxffff =))))((((
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logistic mapr = 4
x0
rxx 110 −=∨=
ChaoticDeterministic
Sensitiveergodic
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logistic mapmovie
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logistic mapbifurcation map
r11−
Period doubling
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logistic mapbifurcation map
r11−
Period doubling
ChaoticDeterministic (not random)Sensitiveergodic
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logistic mapbifurcation map: cycle of 3
r11−
Period doubling
xxfff =)))(((
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Natural design principles
self-similar structures Trees, plants, clouds, mountains
morphogenesis Mechanism
Iteration, recursion Unpredictability
From limited knowledge or inherent in nature? Mechanism
Chaos (sensitivity to initial conditions, ergodicity) Collective behavior, emergence, and self-organization
Complex behavior from collectives of many simple units or agents cellular automata, ant colonies, development, morphogenesis, brains,
immune systems, economic markets Mechanism
Parallelism, multiplicity, redundancy, attractor beghavior, emergent computation
Adaptation Evolution, learning, social evolution Mechanism
Reproduction, transmission, variation, selection, Turing’s tape Network causality (complexity)
Behavior derived from many inseparable sources Environment, embodiment, epigenetics, culture
Mechanism Modularity, connectivity, stigmergy
exploring similarities across nature
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discrete dynamical systemsexamples
xx-1 x+1
Cellular Automata
xt
NK Boolean Network (N=13, K=3)
RBNLAB (Carlos Gershenson):http://student.vub.ac.be/~cgershen/rbn/RBN.html
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NK-networksStuart Kauffman
NK Boolean Network (N=13, K=3)
#nodes (Boolean variables)
# of inputs per node
2K → possible input combinations for an automaton node
→ possible Boolean functions of k inputsK22x1 x20011
0101
0001
x1 x20011
0101
K=2
x1 ∧ x2
Self-organization: solely dependent on its own rules
p: bias, or proportion of “1’s” (or “0’s”) in output
p = 0.25
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simple Boolean networkSmall NK-network
n1
n2n3
and
or orp q p ˅ q0011
0101
0111
p q p ˄ q0011
0101
0001
t t+1
n1 n2 n3 n1 n2 n3
0 0 0 0 0 0 0 01 0 0 1 0 1 0 22 0 1 0 0 0 1 13 0 1 1 1 1 1 74 1 0 0 0 1 1 35 1 0 1 0 1 1 36 1 1 0 0 1 1 37 1 1 1 1 1 1 7
State space
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simple Boolean networkExample: attractors
n1
n2n3
and
or or
t t+1
n1 n2 n3 n1 n2 n3
0 0 0 0 0 0 0 0
1 0 0 1 0 1 0 2
2 0 1 0 0 0 1 1
3 0 1 1 1 1 1 7
4 1 0 0 0 1 1 3
5 1 0 1 0 1 1 3
6 1 1 0 0 1 1 3
7 1 1 1 1 1 1 7
0: 000
1: 001 2: 010 3: 011 7: 111
6: 110
5: 101
4: 100
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simple Boolean networkstate-space (State-transition Graph)
1
32
4 5
6 7
0: 000
1: 001 2: 010
3: 011 7: 111
6: 110
5: 101
4: 100
In discrete dynamical systems there must always exist at least one cycle, because there is only a finite number of states, eventually the system must repeat a previous state.
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SimpleNetSmall Boolean network
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State-transition graph (basins of attraction)dynamical landscape of SimpleNet
There are 28=256 possible states but only a small set of attractors
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SimpleNet: dynamical landscape (basins of attraction)Small Boolean network
There are 28=256 possible states but only a small set of attractors
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discrete dynamical systemsexamples
NK Boolean Network (N=13, K=3)DDLab (Andy wuensche): http://www.ddlab.com/
There are 213=8192 possible states but only a small set of attractors
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attractor behavior
The 213=8192 states in state space are organized into 15 basins attractor periods ranging between 1 and 7. The number of states in each basin is: 68, 984, 784, 1300,
264, 76,316, 120, 64, 120, 256, 2724,604, 84, 428.
self-organization
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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