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Modeling NatureMarch 2009

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Modeling Nature

LECTURE 6: Self-Organisation

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Overview

• Percolation and Ising spin models

• Definition of self-organization

• Local and global interaction

• Natural patterns

• Flocking and Boids

• Synchronization of clapping and fireflies

• Self-organization in social dynamics

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Percolation and Ising Model revisited

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Percolation

• Given the probability p of an occupied site

• What is the size M(p) of the largest connected cluster?

• Clearly, M(p) grows with p

p = 0.30 p = 0.45

p = 0.55 p = 0.70

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next neighbor interaction

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Forest fires on a square lattice

Number of rows L

8scale Large scaleSmall scale

p < pc

p pc

p > pc

zooming out

At pC the forest is a fractal

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The Ising model

N

S N

S

coupling

+1

-1 +1

+1

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2D Ising model

Spontaneous magnetization

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Phase transition

m

T

+1

-1

Tc

Zero temperature

(deterministic)

High temperature

(stochastic)

• At a critical value of T, Tc, the behaviour of the systems changes (it becomes critical)

• The critical behaviour is characteristic for the system (and other systems)

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Ising and Percolation models

Macroscopic Effects are entirely determined by LOCAL INTERACTIONS

– Local microscopic interactions (with nearest neighbour)

– Global macroscopic structure/patterns

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Self-Organization and

Global or Local interactions

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Self-organizationDefinition

• Self-organization is a process where the organization of a system spontaneously increases, i.e., without this increase being controlled by the environment or an encompassing or otherwise external system.

Principia Cybernetica Webhttp://pespmc1.vub.ac.be/SELFORG.html

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Self-Organization

1. GLOBAL INTERACTIONS

A director/manager/conductor coordinates and controls the numerous microcopic processes such that global order arises

Perhaps in a hierarchical network

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Hierarchical organization of cooperations

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Five-Year Plans for the National Economy of the Soviet Union

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Organization of consciousness: Homunculus

Diagram from Descartes' Treatise of Man (1664), showing the formation of inverted retinal images in the eyes, and the transmission of these images, via the nerves so as to form a single, re-inverted image (an idea) on the surface of the pineal gland.

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Organization of consciousness:

Homunculus

Problem with Descartes' Homunculus: how is the consciousness of the homunculus itself organized?

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Self-Organization

2. LOCAL INTERACTIONS

There is no director/manager/conductor that coordinates or controls the global order, it ‘spontaneously’ arises from local interactions, e.g. next-neighbor-interactions.

THERE IS NO CENTRAL ORGANIZATION !

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Patterns from local interactions in inanimate Nature: snow flakes.

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‘Metal leaves’ produced during the electrochemical deposition

of ZnSO4

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Paenibacillus vortex

Pattern of colonial bacteria cooperative self-organization

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Colonial cooperative bacteria: patterns of self-organization

Paenibacillus dendritiformis

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Self-Organization in

Insect Societies

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Biology: Ant Self-Organization

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Biology: Flocking

http://www.red3d.com/cwr/boids/

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Boids

Software Robots that emulate flocking behaviour with simple

local rules

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Self-organization in the brain

The brain consists of +/- 1011 neurons connected in a large network where each neuron is connected to only a handfull other neurons

There is no central organization, only local interactions.

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Self-organization in the brain

• Orientation sensitivity

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Self-organization in the brain

• Direction tuning

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Lion King sequence

• Simulation stampede simulated using “boid” techniques

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Self-Organization: synchronization

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Synchronization of clapping

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Self-organization of clapping

Global noise intensity

Local noise intensity

Correlation parameter

Average noise intensity

Clapping period

Period doubling

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Period doubling

• Transition from asynchronous to rhythmic clapping: skip every second beat

• Yields a clapping mode with a double period

Audience of 73 persons Audience of single person

asynchronousrhythmic asynchronousrhythmic

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Firefly Synchronization

• Fireflies will synchronize their flashes based on their observations of their neighbor’s rhythm.

• Entire riversides thus flash simultaneous

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Menstrual cycle Synchronization Women who live together tend to have

synchronized menstrual periods. The phenomenon was first presented

to the scientific community almost 40 years ago, and has since remained a matter of much debate.

There is still much controversy over what is the cause.

• Martha McClintock, Menstrual synchrony and suppression. Nature. 1971; 291: pp. 244-245.

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Self-Organization in social dynamics

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Mexican Wave

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Mexican-Wave model

Critical mass is needed for onset of MW

Mexican-Wave demo http://angel.elte.hu/wave

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Sociology: Escape Dynamics

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Sociology: Escape Dynamics

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Sociology: Escape Dynamics

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Lane formation and other demo’s

http://www.helbing.org/

Mexican-Wave demo

http://angel.elte.hu/wave

Steering behaviour for Autonomous Characters

www.red3d.com/cwr/steer

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Issues in self-organisation

• Self-organizing processes underlie patterns in nature, society, and culture

• What is the relation between individual behavior and collective behavior?– Not always obvious, e.g., termite behaviour

• Nature evolves group behaviors that improve fitness

• Society evolves group behaviors that improve social acceptance

• …

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Relation to the Tasks

Task 6a. Emergent Structure – How does global and macroscopic structure arise from interactions that are strictly local?

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Relation to the Tasks

Task 6b. Complex models of self-organisation

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END of COURSE