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Biomimicry and Fuzzy Modeling: A Match Made in Heaven

Michael MargaliotSchool of Electrical EngineeringTel Aviv University, Israel

SCIS&ISIS’08, Nagoya University, Japan, Sep. 2008.

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OutlineBiomimicryFuzzy modeling: from words to

equationsFuzzy modeling of animal behavior: two

examplesAdvantages of fuzzy modeling:

A synergy between words, a fuzzy rule-base, and the mathematical model

Interpretability Verifying the verbal description

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Biomimicry

Definition: Biomimicry is the

development of artificial products or

machines that mimic (or are inspired

by) biological phenomena.

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Motivation for BiomimcryLiving systems developed efficient

solutions to various problems they

encounter in their natural habitat.

For example, foraging animals learned

how to address the challenge of

efficiently navigating and searching in

an unknown terrain.

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Motivation for BiomimicryScientists are interested in many

problems that living systems address.

For example: navigation in an unknown

terrain is a major challenge in the design

of autonomous robots. A natural idea is to

follow the solutions already developed by

living systems.

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Examples of Biomimicry Biological Agent

foraging animals

insects

evolution

trees

immune system

social insects

Artificial Design

autonomous robots

walking robots

genetic algorithm

artificial structures

computer security

clustering algorithms

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Biomimcry & Fuzzy Modeling Biomimcry requires “reverse engineering.”

In many cases, biologists have already provided a verbal description and explanation of the relevant biological behavior. This reduces biomimicry to the following problem.

Problem 1 Transform a given verbal description into a mathematical model or algorithm.

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Problem 1 & Fuzzy ModelingExtensive research suggests that fuzzy

modeling is the most suitable tool for

addressing Problem 1.

verbal description

mathematical model

fuzzy rule-base

simulation/analysis

Fuzzy modeling process:

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Fuzzy Modeling of Animal Behavior

Input: Verbal description of the behavior.

1. Identify the state variables

2. Restate the verbal data as If-Then rules

3. Define the fuzzy terms

4. Inference the fuzzy rule base to obtain a well-defined mathematical model

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Fuzzy Modeling of Animal Behavior

1. Territorial behavior of fish (Tron & Margaliot, 2004).

2. Flocking behavior (Lebar Bajec, Zimic, & Mraz, 2004).

3. Orientation to light in a planarian (Tron & Margaliot, 2005).

4. Foraging behavior of ants (Rozin & Margaliot, 2007).

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Fuzzy Modeling of Animal Behavior

5. Population dynamics in flies (Rashkovsky & Margaliot, 2007).

6. The Lambda switch (Laschov & Margaliot, 2008).

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Two Detailed Examples 1. Territorial behavior in the stickleback

(Lorenz)

2. Orientation to light in the Dendrocoleum lacteum (flat worm) (Ullyott, Fraenkel & Gunn)

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"a real stickleback fight can be seen only when two males are kept together in a large tank where they are both building their nests. The fighting inclinations of a stickleback, at any given moment, are in direct proportion to his proximity to his nest… The vanquished fish invariably flees homeward and the victor chases the other furiously, far into its domain. The farther the victor goes from home, the more his courage ebbs, while that of the

vanquished rises in proportion.

Arrived in the precincts of his nest, the fugitive gains new strength, turns right about and dashes with gathering fury at his pursuer.” (King Solomon’s Ring, p. 44)

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Fuzzy Modelling

• • • •

c1 x1 x2 c2

1 1w 1 1w

1 2 1x x x 1 1 1x c x

If If

If If

1 1( , )near x c Then 1 1( , )far x c Then

Then Then

1 1( , )near x c and and 1 1( , )far x c

1( )high w1( )low w

State variables:

Fuzzy rule-base:

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( ) (1 tanh( )) / 2

( ) 1 ( )

i

i

i i

whigh wa

low w high w

2 2( , ) exp( / )

( , ) 1 ( , )i i

i i

near x y x y k

far x y near x y

2 22 exp( ( ) / )( )( )

i i

i i i i

i i i j i

i i

w p x c k px c x high w x c

Inferencing yields the mathematical model:

Fuzzy Modelling

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Simulations

“The pursuit is repeated a few times in alternating directions, swinging back and forth like a pendulum which at last reaches a state of equilibrium at a certain point.” [Lorenz]

territory 1

territory 2

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Simulations (3D)

oscillatory behaviour convergence to equilibrium (proof via

linearization and eigenvalue analysis)

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Orientation to Light in the Dendrocoleum lacteum

dim light bright light

After a couple of hours:

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Rate of Change of Direction (r.c.d)

. . defelections (in a given time unit)r c d

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r.c.d. and Light Intensity

adaptation

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Klino-Kinesis(1)“An increase in stimulating intensity produces

an increase in r.c.d.

(2) This initial increase in r.c.d. falls off under

constant stimulation owing to adaptation.

(3) There is a basal r.c.d., which is an expression of

the fact that turning movements occur even in

absolute darkness or at complete adaptation.”

(P. Ullyott, J. Experimental Biology, 1936.)

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The “Average Animal*”

light Increases r.c.d increases AB short

adaptation r.c.d. decreases CD long

(* Fraenkel & Gunn. The Orientation of Animals, 1961)

dim light

bright light

A B

CD

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Fuzzy Modeling L(t) – light intensity

A(t) – level of adaptation to light

R(t) – r.c.d. B – basal r.c.d.

1( )A t c

1( )A t c

2( )R t c

3( )R t c

If (L(t)-A(t)) is positive then

If (L(t)-A(t)) is negative then

If (R(t)-B) is large then

If (L(t)-A(t)) is high then

Fuzzy rule-base:

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Fuzzy Modeling1 1

2 2 3 3

tanh( ( ))

( ) ( )cos( ), sin( )

k k

A c k L A

R c S R B c S L Ax v y v

/ 21

i

i it t

( )i

t

t R d q

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Simulation 1

R(t) as a function of time. Light is switched on at t=1.

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Simulation 2

Trajectory (x(t),y(t)). Light intensity is L(x,y)=x

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Advantages of Fuzzy Modeling

The knowledge is represented in three forms:

1. The initial verbal description

2. The fuzzy rule-base

3. The mathematical model

This provides a synergistic overview of the

system.

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Advantage 1: Interpretability

A fuzzy model is interpretable; each

parameter has a perceivable meaning.

Example 1: Consider the parameter in the

stickleback model. Recall: 2 2( , ) exp( / ).i in x y x y k

ik

As decreases, the Gaussian becomes

more centered, so Fish becomes “less

aggressive.”

ik

i

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Advantage 1: Interpretability

This links the parameter with the verbal

description.

The equilibrium points of the mathematical

model are: 1 2

1 2ln 2, ln 2.x c k x c k

1 2 ,k kIf the equilibrium position is no

longer symmetric; eventually fish 1 will have

a larger territory than fish 2.

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k1 =1, k2 =0.5

Advantage 1: Interpretability

first fish is “more aggressive”

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“.. the relative fighting potential of the individual is shown by the size of the territory he keeps clear of rivals.” (Lorenz)

Advantage 1: Interpretability

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Advantage 2: Verification

The mathematical model can be examined

using both simulations and rigorous

analysis.

This can be used, to some extent,

to verify the original verbal description.

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Advantage 2: Verification

Example: The planarian model includes the

rule:3( ) .R t c

3 0.c

If is high, then

Consider the case The r.c.d. will not

increase, and we may expect that the

model’s behavior will change substantially.

( ) ( )L t A t

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Advantage 2: VerificationFor the mathematical model yields:

22 ( ).kR c S R B

IfRecall that the right-hand turns take place at

times such that:

3 0,c

(0) ,R B then 0,R so ( ) .R t B

1 ( ) .i

i

t

t R d q

Hence, a periodic trajectory without

gradually moving to the shadier parts.

kt

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Fuzzy Modeling and Animal Behavior1. Animal (and human) actions are “fuzzy”:

“… a class of objects with a continuum of

grades of membership.” (Zadeh, 1965)

“… no sharp distinction is possible between

intention movements and more complete

responses; they form a continuum.”

(Heinroth, 1910)

Compare with:

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Fuzzy Modeling and Animal Behavior2. Verbal (and therefore vague) information:

“Nor shall I here discuss the various definitions which have been given of the term species. No one definition has as yet satisfied all naturalists; yet every naturalist knows vaguely what he means when he speaks of a species.” (Darwin, 1859)

“A high degree of contact causes low activity.” (Fraenkel & Gunn, 1961)

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Summary

Fuzzy modeling seems very suitable for transforming words to equations.

Numerous potential applications in the “soft sciences”: psychology, economy, animal behavior and more.

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Summary Fuzzy modeling seems particularly suitable for modeling animal behavior and for biomimcry:Start with a verbal description of abiological system (e.g., foraging ants); use fuzzy modeling to derive an analytical model which can then be implemented byartificial systems (e.g., autonomous robots).

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The Humpback Flippers*“Flippers with tubercles produced as much as 32% lower drag than the sleek flipper.”

*Miklosovic, Murray, Howlea & Fish, Physics of Fluids, May 2004.