1. Dynamical Models of Decision Making Optimality, human performance, and principles of neural information processing
Jay McClelland
Department of Psychology and Center for Mind Brain and Computation Stanford University

2. How do we make a decision given a marginal stimulus?

3. An Abstract Statistical Theory (Random Walk or Sequential Probability Ratio Test)
How do you decide if an urn contains more black balls or white balls?
We assume you can only draw balls one at a time and want to stop as soon as you have enough evidence to achieve a desired level of accuracy.
Optimal policy:
Reach in and grab a ball
Keep track of difference between the # of black balls and # of white balls.
Respond when the difference reaches a criterion value C.
Produces fastest decisions for specified level of accuracy

4. The Drift Diffusion Model
Continuous version of the SPRT
At each time step a small random step is taken.
Mean direction of steps is +m for one direction, –m for the other.
When criterion is reached, respond.
Alternatively, in ‘time controlled’ tasks, respond when signal is given.

5. Two Problems with the DDM
Easy
Accuracy should gradually improve toward ceiling levels, even for very hard discriminations, but this is not what is observed in human data.
The model predicts correct and incorrect RT’s will have the same distribution, but incorrect RT’s are generally slower than correct RT’s.
Prob. Correct
Hard
Errors
Correct
Responses RT
Hard -> Easy

6. Two Solutions Selected Ratcliff (1978): Usher & McClelland (2001):
Add between-trial variance in direction of drift.
Usher & McClelland (2001):
Consider effects of leakage and competition between evidence ‘accumulators’.
The idea is based on properties of populations of neurons.
Populations tend to compete
Activity tends to decay away
Selected
Activation of neurons responsive to
Selected vs. non-selected target from Chelazzi et al (1993)

7. Usher and McClelland (2001) Leaky Competing Accumulator Model
Addresses the process of deciding between two alternatives based on external input (r1 + r2 = 1) with leakage, self-excitation, mutual inhibition, and noise
dx1/dt = r1-l(x1)+af(x1)–bf(x2)+x1
dx2/dt = r2-l(x2)+af(x2)–bf(x1)+x2
Captures u-shaped activity profile for loosing alternative seen in experiments.
Matches accuracy data and RT distribution shape as a function of ease of discrimination.
Easily extends to n alternatives, models effects of n or RT.

8. Discussion of assumptions
Units represent populations of neurons, not single neurons – rate corresponds to instantaneous population firing rate.
Activation function is chosen to be non-linear but simple (f = []+). Other choices allow additional properties (Wong and Wang, next lecture).
Decay represents tendency of neurons to return to their resting level.
Self-excitation ~ recurrent excitatory interactions among members of the population.
In general, neurons tend to decay quite quickly; the effective decay is equal to decay - self-excitation
(In reality competition is mediated by interneurons.)
Injected noise is independent but propagates non-linearly.

9. Testing between the models
Quantitative test:
Differences in shapes of ‘time-accuracy curves’
Use of analytic approximation
Many subsequent comparisons by Ratcliff
Qualitative test:
Understanding the dynamics of the model leads to novel predictions

10. Roles of (k = l – a) and b

11. Change of Coordinates (Bogacz et al, 2006)x1

12. Time-accuracy curves for different |k-b|

14. Assessing Integration Dynamics
Participant sees stream of S’s and H’s
Must decide which is predominant
50% of trials last ~500 msec, allow accuracy assessment
50% are ~250 msec, allow assessment of dynamics
Equal # of S’s and H’s But there are clusters bunched together at the end (0, 2 or 4).
Leak-dominant
Inhibition-dom.
Favored early
Favored late

15. Subjects show both kinds of biases; the less the bias, the higher the accuracy, as predicted.

16. Extension to N alternatives
Extension to n alternatives is very natural (just add units to pool).
Model accounts quite well for Hick’s law (RT increases with log n alternatives), assuming that threshold is raised with n to maintain equal accuracy in all conditions.
Use of non-linear activation function increases efficiency in cases where there are only a few alternatives ‘in contention’ given the stimulus.

17. Neural Basis of Decision Making in Monkeys (Roitman & Shadlen, 2002)
RT task paradigm of R&T.
Motion coherence and direction is varied from trial to trial.

18. Neural Basis of Decision Making in Monkeys: Results
Data are averaged over many different neurons that are associated with intended eye movements to the location of target.

19. Neural Activity and Low-Dimensional Models
Human behavior is often characterized by simple regularities at an overt level, yet this simplicity arises from a highly complex underlying neural mechanism.
Can we understand how these simple regularities could arise?

20. Wong & Wang (2006)

21. 7200- and 2-variable Models both account for the behavioral data
… and the physiological data as well!

22. Some Extensions Usher & McClelland, 2004 Future work
Leaky competing accumulator model with
Vacillation of attention between attributes
Loss aversion
Accounts for several violations of rationality in choosing among multiple alternatives differing on multiple dimensions.
Makes predictions for effects of parametric manipulations, some of which have been supported in further experiments.
Future work
Effects of involuntary attention
Combined effects of outcome value and stimulus uncertainty on choice