lecture 9 - department of informatics · 2015-12-14 · iteration, recursion unpredictability from...

[email protected]://informatics.indiana.edu/rocha/i-bic

biologicallyInspired

computing

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Informatics luis rocha 2015

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

http://informatics.indiana.edu/rocha/i-bic


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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 −=∨=

⇒>′⇒<′


),21()(xf

xfxrxf

⇒>⇒<

=−=′⇒=unstable1stable1

,|)21(|)(0r

rrxrxfx



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logistic map1 ≤ r ≤ 3

x0

1-1/r

rxx 110 −=∨=

⇒>′⇒<′


),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

⇒>′⇒<′


),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




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discrete dynamical systemsexamples

NK Boolean Network (N=13, K=3)DDLab (Andy wuensche): http://www.ddlab.com/



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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

lecture 9 - department of informatics · 2015-12-14 · iteration, recursion unpredictability from...

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