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Jay McClelland

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Dynamical Models of Decision MakingOptimality, human performance, andprinciples of neural information processing

Jay McClelland

Department of Psychology and Center for Mind Brain and ComputationStanford University

- 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

- 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.

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

- Ratcliff (1978):
- 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-selectedtarget from Chelazzi et al (1993)

- Addresses the process of decidingbetween two alternatives basedon 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.

- 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.

- 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

x1

|k-b| = 0

|k-b| = .2

|k-b| = .4

- 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’sBut there are clusters bunched together at the end (0, 2 or 4).

Inhibition-dom.

Leak-dominant

Favored early

Favored late

S3

Subjects show both kindsof biases; the less the bias,the higher the accuracy,as predicted.

- 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.

RT task paradigm of R&T.

Motion coherence anddirection is varied fromtrial to trial.

Data are averaged over many different neurons that areassociated with intended eye movements to the locationof target.

- 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?

Wong & Wang (2006)

… and the physiological data as well!

- Usher & McClelland, 2004
- 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.

- Leaky competing accumulator model with
- Future work
- Effects of involuntary attention
- Combined effects of outcome value and stimulus uncertainty on choice