Biomimicry and Fuzzy Modeling: A Match Made in Heaven

1. Biomimicry and Fuzzy Modeling: A Match Made in Heaven Michael Margaliot School of Electrical EngineeringTel Aviv University, Israel SCIS&ISIS’08, Nagoya University, Japan, Sep. 2008.

2. Outline • Biomimicry • Fuzzy modeling: from words to equations • Fuzzy modeling of animal behavior: two examples • Advantages of fuzzy modeling: • A synergy between words, a fuzzy rule-base, and the mathematical model • Interpretability • Verifying the verbal description

3. Biomimicry Definition: Biomimicry is the development of artificial products or machines that mimic (or are inspired by) biological phenomena.

4. Motivation for Biomimcry Living 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.

5. Motivation for Biomimicry Scientists 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.

6. 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

7. 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.

8. mathematical model fuzzy rule-base simulation/analysis Problem 1 & Fuzzy Modeling Extensive research suggests that fuzzy modeling is the most suitable tool for addressing Problem 1. Fuzzy modeling process: verbal description

9. Fuzzy Modeling of Animal Behavior Input: Verbal description of the behavior. • Identify the state variables • Restate the verbal data as If-Then rules • Define the fuzzy terms • Inference the fuzzy rule base to obtain a well-defined mathematical model

10. Fuzzy Modeling of Animal Behavior • Territorial behavior of fish (Tron & Margaliot, 2004). • Flocking behavior (Lebar Bajec, Zimic, & Mraz, 2004). • Orientation to light in a planarian (Tron & Margaliot, 2005). • Foraging behavior of ants (Rozin & Margaliot, 2007).

11. Fuzzy Modeling of Animal Behavior 5. Population dynamics in flies (Rashkovsky & Margaliot, 2007). 6. The Lambda switch (Laschov & Margaliot, 2008).

12. Two Detailed Examples • Territorial behavior in the stickleback (Lorenz) • Orientation to light in the Dendrocoleum lacteum (flat worm)(Ullyott, Fraenkel & Gunn)

13. "a real stickleback fight can be seen only when two males are kept together in a large tank where they are both building theirnests. 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 thevictor 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)

14. Fuzzy Modelling State variables: • • •• c1 x1 x2 c2 Fuzzy rule-base: Then If If Then Then and If and Then If

15. Fuzzy Modelling Inferencing yields the mathematical model:

16. Simulations territory2 “The pursuit is repeated a few times inalternating directions, swinging back and forth like a pendulum which at last reaches a state of equilibrium at a certain point.” [Lorenz] territory1

17. Simulations (3D) • oscillatory behaviour • convergence to equilibrium (proof via linearization and eigenvalue analysis)

18. dim light bright light After a couple of hours: Orientation to Light in the Dendrocoleum lacteum

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

20. r.c.d. and Light Intensity adaptation

21. Klino-Kinesis • “An increase in stimulating intensity produces an increase in r.c.d. • This initial increase in r.c.d. falls off under constant stimulation owing to adaptation. • 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.)

22. A B D C The “Average Animal”* bright light dim light light Increases  r.c.d increases  AB short adaptation  r.c.d. decreases  CD long (* Fraenkel & Gunn. The Orientation of Animals, 1961)

23. Fuzzy Modeling L(t) – light intensity A(t) – level of adaptation to light R(t) – r.c.d. B – basal r.c.d. Fuzzy rule-base: 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

24. Fuzzy Modeling

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

26. Simulation 2 Trajectory (x(t),y(t)). Light intensity is L(x,y)=x

27. Advantages of Fuzzy Modeling • The knowledge is represented in three forms: • The initial verbal description • The fuzzy rule-base • The mathematical model • This provides a synergistic overview of the • system.

28. Advantage 1: Interpretability A fuzzy model is interpretable; each parameter has a perceivable meaning. Example 1: Consider the parameter in the stickleback model. Recall: As decreases, the Gaussian becomes more centered, so Fish becomes “less aggressive.”

29. Advantage 1: Interpretability This links the parameter with the verbal description. The equilibrium points of the mathematical model are: If the equilibrium position is no longer symmetric; eventually fish 1 will have a larger territory than fish 2.

30. Advantage 1: Interpretability k1 =1, k2 =0.5 first fish is “more aggressive”

31. Advantage 1: Interpretability • “.. the relative fighting potential of the individual is shown by the size of the territory he keeps clear of rivals.” (Lorenz)

32. 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.

33. Advantage 2: Verification Example: The planarian model includes the rule: 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.

34. Advantage 2: Verification For the mathematical model yields: so If then Recall that the right-hand turns take place at times such that: Hence, a periodic trajectory without gradually moving to the shadier parts.

35. Fuzzy Modeling and Animal Behavior • Animal (and human) actions are “fuzzy”: “… no sharp distinction is possible between intention movements and more complete responses; they form a continuum.” (Heinroth, 1910) Compare with: “… a class of objects with a continuum of grades of membership.” (Zadeh, 1965)

36. Fuzzy Modeling and Animal Behavior 2. Verbal (and therefore vague) information: “A high degree of contact causes low activity.” (Fraenkel & Gunn, 1961) “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)

37. Summary • Fuzzy modeling seems very suitable for transforming words to equations. • Numerous potential applications in the “soft sciences”: psychology, economy, animal behavior and more.

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

39. 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.