Decision Dynamics and Decision States in the Leaky Competing Accumulator Model

1. Decision Dynamics and Decision States in the Leaky Competing Accumulator Model Jay McClellandStanford University

2. Is the rectangle longer toward the northwest or longer toward the northeast?

3. Longer toward the Northeast! 1.99” 2.00”

4. A Classical Model of Decision Making:The Drift Diffusion Model of Choice Between Two Alternative Decisions • At each time step a small sample of noisy information is obtained; each sample adds to a cumulative relative evidence variable y. • Mean of the noisy samples is +m for when one alternative is correct, –m when the other, with standard deviation s. • When a bound is reached, the corresponding choice is made. • Alternatively, in ‘time controlled’ or ‘interrogation’ tasks, respond when signal is given • Choose choice 1 if y is positive • Choose choice 2 if y is negative y

5. Easy Prob. Correct Hard A Problem with the DDM • Accuracy should gradually improve toward ceiling levels, but this is not what is observed in data. • Two possible fixes: • Trial-to-trial variance in the direction of drift (Ratcliff) • Evidence accumulation may reach a bound and stop, even if more time is available (Shadlen and colleagues)

6. Usher and McClelland (2001)Leaky Competing Accumulator Model y2 y1 I1 I2 • Proposes accumulators of noisy evidence, with leakage, and mutual inhibition: dy1/dt = I1-gy1–bf(y2)+x1 dy2/dt = I2-gy2–bf(y1)+x2 f(y) = [y]+ • In time controlled tasks, choose response 1 iff y1-y2 > 0 • Let y = (y1-y2). While y1 and y2 are positive, the model reduces to: dy/dt = I-ly+x [I=I1-I2; l = g-b; x=x1-x2]

7. Time course of stimulus sensitivity in the linear approximation:

8. Time-accuracy curves for different |k-b| or |l| |k-b| = 0 |k-b| = .2 |k-b| = .4

9. The Full Non-Linear LCAi Model y1 Although the value of the differencevariable is not well-captured by thelinear approximation, the sign of thedifference is approximated very closely. y2

10. Result of fitting the full model to individual participant data (Usher & McClelland, 2001) Prob. Correct

11. Distinguishing Leak Dominance From Inhibition Dominance

12. Kiani, Hanks and Shadlen 2008 Random motion stimuli of different coherences. Stimulus duration follows an exponential distribution. ‘go’ cue can occur at stimulus offset; response must occur within 500 msec to earn reward.

13. The earlier the pulse, the more it matters(Kiani et al, 2008)

14. These results rule out leak dominance Still viable X The bounded DDM and the full non-linear LCAi are also still viable.

15. Plan for the Rest of the Talk • Discuss several interesting features of decision states in the non-linear LCAi • Describe three experiments combining experiment and simulation that address these features.

16. Quasi-Continuous, Quasi-Discrete, Reversible Decision States in the Non-Linear LCAi Non-Linear Linear

17. v Distribution of winner’s activationswhen correctalternative wins Distribution of winner’s activationswhen incorrectalternative wins v

18. Predictions • We should be able to find signs of differences in decision states associated with correct and incorrect responses. • We should be able to see signs of bifurcation even if we ask for a continuous response. • We should be able see evidence of rebound of suppressed alternatives if the input changes.

19. Predictions • We should be able to find signs of differences in ‘strength’ of decision states associated with correct and incorrect responses. • We should be able to see signs of bifurcation when we ask for a continuous response. • We should be able to see evidence of recovery of suppressed alternatives if the input changes.

20. Integration of reward and stimulus informationGao, Tortell & McClelland PLoS One, 2011

21. Proportion of Choices toward Higher Reward

22. Fits based on full LCAi

23. Relationship between response speed and choice accuracy

24. An Account:High-Threshold LCAi Gao & McClelland, (in preparation)

25. v Distribution of winner’s activationswhen correctalternative wins Distribution of winner’s activationswhen incorrectalternative wins v

26. v Distributionof activationswhen correctalternative wins Distribution of activationswhen incorrectalternative wins v

27. Predictions • We should be able to find signs of differences in ‘strength’ of decision states associated with correct and incorrect responses. • We should be able to see signs of bifurcation even when we ask for a continuous response. • We should be able to see evidence of recovery of suppressed alternatives if the input changes.

28. Toward Continuous Measures of Decision States Lachter, Corrado, Johnston & McClelland (in progress)

30. Can participants give a continuous readout when they have as much time to respond as they would like? • To test: • Participant observes display as long as desired, moves joystick to desired position, then clicks to terminate trial • Mixed difficulty levels: • Stimuli differ by 1, 2, 4, 8, or 16 dots

31. Results and Descriptive Model of Data from 1 Participant

32. Quasi-Discrete, Quasi-Continuous Decision States • Bi-modality indicates a degree of discreteness, consistent with the bifurcation expected in the model. • The position of each mode should shift as the difference in the number of dots increases, according to the model.

35. Follow-up • Log scale ranging from 1000:1 to 1:1 to 1:1000 (extends and reshapes range) • Very explicit instructions about contingencies, marks on scale. • Paid for points, length of session depends on participant’s pacing of trials • Ten sessions per participant

36. Two Participants: Session 1

37. Session 10 for each participant(this and next 6 slides)

44. Predictions • We should be able to find signs of differences in ‘strength’ of decision states associated with correct and incorrect responses. • We should be able to see signs of bifurcation even when we ask for a continuous response. • We should be able to see evidence of recovery of suppressed alternatives if the input changes.

45. Decision making with non-stationary stimulus information(Tsetsos. Usher & McClelland in press) Phase durationdistribution Evidence Switching Protocol in the Correlation Condition:

46. Individual Data from Correlation Condition Primacy region Indifference tostarting phase P(C), A/B at start

47. Simulations of Two Correlated TrialsTop: A/B start high Bottom: C starts high

48. Only LCAi can explain >50% choice of C even when A/B phase comes firstDissimilar favored firstDissimilar favored secondAverageTop: low noiseBottom: higher noise

49. Individual Data from Correlation Condition and Model Coverage

50. Explaining Individual Differences in theLCA Balanced,Strong L&I I > L Lots ofNoise