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

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

3. Longer toward the Northeast!
2.00”
1.99”

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. A Problem with the DDM Easy Prob. Correct Hard
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)
Hard
Prob. Correct
Easy

6. Usher and McClelland (2001) Leaky Competing Accumulator Model
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] y1 y2 I1 I2

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

8. Time-accuracy curves for different |k-b| or |l|

9. The Full Non-Linear LCAi Modely1
Although the value of the difference variable is not well-captured by the linear approximation, the sign of the difference 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

17. v
Distribution of winner’s activations when correct alternative wins
Distribution of winner’s activations when incorrect alternative 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 information Gao, 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 activations when correct alternative wins
Distribution of winner’s activations when incorrect alternative wins v

26. v
Distribution of activations when correct alternative wins
Distribution
of activations when incorrect alternative 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. Mixed difficulty levels:
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
Decision making with non-stationary stimulus information (Tsetsos. Usher & McClelland in press)
Phase duration distribution
Evidence Switching Protocol in the Correlation Condition:

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

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

48. Only LCAi can explain >50% choice of C even when A/B phase comes first Dissimilar favored first Dissimilar favored second Average Top: low noise Bottom: higher noise

49. Individual Data from Correlation Condition and Model Coverage

50. Explaining Individual Differences in the LCA
Balanced, Strong L&I
I > L
Lots of Noise

51. 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 see evidence of recovery of suppressed alternatives.

52. Conclusions
Evidence from several studies is consistent with the idea of quasi-continuous, quasi discrete, sometimes reversible, decision states.
The LCAi model provides a simple yet powerful framework in which such states arise.
Alternative models considereed have difficulties addressing aspects of the data.
More work is needed to understand if the LCAi will turn out to be fully adequate, and how the full set of data might be addressed with other approaches.