Dynamical Models of Decision MakingOptimality, human performance, and

principles of neural information processing

Jay McClelland

Department of Psychology and Center for Mind Brain and Computation

Stanford University

How do we make a decision given a marginal stimulus?

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

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.

Two Problems with the DDM• 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.

Hard -> Easy

RT

Errors

CorrectResponses

Prob

. Cor

rect

Hard

Easy

Two Solutions• 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 Activation of neurons responsive to

Selected vs. non-selectedtarget from Chelazzi et al (1993)

Selected

Usher and McClelland (2001)Leaky Competing Accumulator

Model• Addresses the process of deciding

between two alternatives basedon external input (1 + 2 = 1) with leakage, self-excitation, mutual inhibition, and noisedx1/dt = 1-(x1)+f(x1)–f(x2)+1

dx2/dt = 2-(x2)+f(x2)–f(x1)+2

• 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.

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.

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

Roles of (k = – ) and

Change of Coordinates (Bogacz et al, 2006)

x1

Time-accuracy curves for different |k-|

|k-= 0|k-= .2|k- = .4

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

Leak-dominant Inhibition-dom.

Favored earlyFavored late

S3

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

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.

Neural Basis of Decision Making in Monkeys (Roitman & Shadlen, 2002)

RT task paradigm of R&T.Motion coherence anddirection is varied fromtrial to trial.

Neural Basis of Decision Making in Monkeys: Results

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

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?

Wong & Wang (2006)

7200- and 2-variable Models both account for the behavioral data

… and the physiological data as well!

Some Extensions• 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.

• Future work– Effects of involuntary attention– Combined effects of outcome value and stimulus uncertainty on

choice