SOME MATHEMATICAL METHODS

1. SOME MATHEMATICAL METHODS • FUNCTION PROPERTIES AND CURVE FITTING • DIFFERENTIAL AND INTEGRAL CALCULUS • CALCULUS OF VARIATIONS • DIFFERENTIAL AND DIFFERENCE EQUATIONS • COMPUTATIONAL MODELING • STOCHASTIC METHODS

2. COMPUTATIONAL MODELING • NEURAL NETWORKS • CELLULAR AUTOMATA • DYNAMIC PROGRAMING • DYNAMIC STATE VARIABLE MODELS • GENETIC ALGORITHMS • SIMULATED ANNEALING • MONTE CARLO METHODS • STATISTICAL MECHANICS OF LEARNING

3. MARBLE IN A DOUBLE WELLDynamic model of choice

4. Duffing Equation

7. DIFFERENTIAL EQUATIONS AND DYNAMICS THE COURSE OF TRUE LOVE Suppose William is in love with Zelda, but Zelda is a fickle lover. The more William loves her, the more she dislikes him---but when he loses interest in her, her feelings for him warm up. On the other hand, William reacts to her: When she loves him, his love for her grows and when she loses interest, he also loses interest. Let w(t) = William’s feelings for Zelda at time t. Let z(t) = Zelda’s feelings for William at time t. • dw/dt = az w(0) = α • dz/dt = -bw z(0) = β

9. Interacting Contingencies • Predator-Prey • Competition • Mutualism/Symbiosis All involve non-linear processes. Can we model these in the operant laboratory through arranging interactive contingencies?

10. Example: Competition Imagine a two-compartment pigeon chamber with a partition. Each pigeon (P1 & P2) pecks for food under a schedule which delivers food according to a feed-back function where the rate of reinforcement is some positive function of the rate of response. (e.g.,VI t, VR n or some combination). The rate of responding of, say, P1 controls the rate of reinforcement for P2 and vice-versa by changing the schedule parameter for the other bird. For example, as P1’s rate increases, this leads to a decrease in P2’s rate of reinforcement, leading to decreased rate for P2, leading to a decreased rate for P1, etc.

11. Competition Model Pigeon 1 (VI t1) Pigeon 2 (VI t2) R1 increases R2 decreases  t2 increases t1 increases (so r2 decreases, (so r1 decreases, so R2 decreases, so R1 decreases, and vice versa). etc.)

12. Some Assumptions and Parameters 1. Rate of responding is a positive function of rate of reinforcement (r). 2.In the absence of a competitor, reinforcement rate (r) will increase logistically in proportion to a1 or a2 to asymptote at b1 or b2,respectively. 3. The parameters c12 and c21 specify the intensity of competition in the reduction of reinforcement rate contributed by an increase in response rate by each pigeon on the other.

14. Competitive System Predicted behavior depends on parameter values, most notably the level of competition (c12 & c21). Assuming all other parameters are equal for the two birds, if both c12 and c21 < 0, then a stable state with each bird settling for less than maximum possible reinforcement rate. Otherwise, one or the other will predominate.

18. ASSUME A SPHERICAL PIGEON Falling body in a viscous medium model of behavioral momentum. 1’ 1 m1 m2 v0 v0 m2 2 2 m1 vT2 vT1

20. Solution

21. m1 = 2 x m2 m1 v m2 t

22. SECOND-ORDER,SECOND-DEGREE DIFFERENTIAL EQUATION DOES THIS HAVE A CLOSED-FORMED SOLUTION?

23. HERRNSTEIN’S HYPERBOLA

25. HERRNSTEIN’S HYPERBOLA

27. Laplace’s Equation

28. O-rules, functional relations B = f (r) r: feedback B: output E-rules, feedback functions r = g (B) Figure 1. The behavior-environment feedback system Operant Conditioning

29. Ratio-Like Feedback Functions

30. Baum’s (1992) Proposed VI t Feedback Function

31. OTHER POSSIBILITIES

32. HOW FAST SHOULD A PIGEON EAT? • ASSUMPTION:REDUCTION OF FITNESS RELATED TO LEVEL OF DEPRIVATION AND THE RATE OF CHANGE OF DEPRIVATION.

33. Euler Condition (i.e. pigeon should feed at a gradually decreasing rate reducing deprivation exponentially.)