Decision Dynamics and Decision States:the Leaky Competing Accumulator Model

Psychology 209March 4, 2013

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

Longer toward the Northeast!

2.00”

1.99”

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.

• Mean of the noisy samples is +m for one alternative, –m for 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, based on value of the relative evidence variable.

The DDM is an optimal model, and it is consistent with some data from

neurophysiology• It achieves the fastest possible decision on average for

a given level of accuracy

• It can be tuned to optimize performance under different kinds of task conditions– Different prior probabilities– Different costs and payoffs– Variation in the time between trials…

• The activity of neurons in a brain area associated with decision making seems to reflect the DD process

Neural Basis of Decision Making in Monkeys (Shadlen & Newsome;

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.

HardProb

. Cor

rect

Easy

A Problem with the DDM

• Accuracy should gradually improve toward ceiling levels as more time is allowed, even for very hard discriminations, but this is not what is observed in human data.

• Two possible fixes:– Trial-to-trial variance in the

direction of drift– Evidence accumulation may

reach a bound and stop, even if more time is available

Goals for a Neurally Inspired Model of Decision Making

• Incorporate principles of neural processing

• Build a bridge between abstract statistically-grounded approaches and details of physiology

• Explain existing data

• Make predictions and see if they are borne out in data

• Offer a new way of thinking about the nature of decision states

Usher and McClelland (2001)Leaky Competing Accumulator Model

• Addresses the process of decidingbetween two alternatives basedon external input, with leakage, mutual inhibition, and noise:

dy1/dt = I1-gy1–bf(y2)+x1

dy2/dt = I2-gy2–bf(y1)+x2f(y) = [y]+

• Participant chooses the most active accumulator when the go cue occurs

• This is equivalent to choosing 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]

I1 I2

y1 y2

Wong & Wang (2006)

~Usher & McClelland (2001)

( ) ( ) ( )d t R t R t

( ) (1 )td t kS e l

The Full Non-Linear LCAi Model

y1

y2

Although the value of the differencevariable is not well-captured by thelinear approximation, the sign of thedifference is approximated very closely.

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

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

Prob

. Cor

rect

Kiani, Hanks and Shadlen 2008Random 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.

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

These results rule out leak dominance

XStill viable

Quasi-Continuous, Quasi-Discrete, Reversible Decision States in the Non-

linear LCAi

Quasi-continuous, quasi-discretedecision states

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.

PredictionsWe 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 if the input changes.

v

v

Distribution of winner’s activationswhen incorrectalternative wins

Distribution of winner’s activationswhen correctalternative wins

Gao, Tortell and McClelland (in press) Experiment on Effect of

Reward on Decision Dynamics

Proportion of Choices toward Higher Reward

Fits based on full LCAi*

*Reward affects the initial state of the accumulators, before the stimulus starts to affect them.

Relationship betweenchoice and RT for eachparticipant and combined

Data from Gao et al (2012)

An Account:Two-Stage LCAi

Response StageModel

PredictionsWe 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 see evidence of recovery of suppressed alternatives if the input changes.

Bifurcation in the LCAi

Bimodality in Decision StatesLachter, Corrado, Johnston & McClelland (in progress)

Results and Descriptive Model of Data from 1 Participant

Experiment 2• Used much finer scale, much more practice & data per participant

• Found evidence that some participants show a bifurcation while others show un-imodal responses

• Mapping to response scale appears to be non-linear in many participants

PredictionsWe 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 see evidence of recovery of suppressed alternatives if the input changes.

Reversability in the LCAi

• If activation of loser cannot go below 0, reversal of decision states can occur

• This leads to a predicted interaction of timing by duration.

Experiment

• Participants viewed random dot motion stimulus presentations of varying durations

• Three types of trials:– Constant evidence

• fixed non-zero coherence throughout trial– Early evidence

• non-zero coherence in first half, 0 in second half– Late evidence

• 0 coherence in first half, non-zero in second half

Interaction of timing by durationin one participant

PredictionsWe 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.

Conclusions• Evidence from several studies is consistent with

the idea of quasi-continuous, quasi discrete, sometimes reversible, decision states, although, in general, data from only some of the participants plays a strong role in selection between models.

• The LCAi model provides a simple yet powerful framework in which such states arise.

• Alternative models considered 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 data might be addressed with other approaches.

Quasi-continuous, quasi-discretedecision states