Decision Dynamics and Decision States in the Leaky Competing Accumulator Model

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Decision Dynamics and Decision States in the
Leaky Competing Accumulator Model

• Jay McClellandStanford University

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Is the rectangle longer toward the northwest or
longer toward the northeast?
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Longer toward the Northeast!
1.99
2.00
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A Classical Model of Decision MakingThe 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
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A Problem with the DDM

• 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)

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Usher and McClelland (2001)Leaky Competing
Accumulator Model

• Proposes accumulators of noisy evidence, with
  leakage, and mutual inhibition
• dy1/dt I1-gy1bf(y2)x1
• dy2/dt I2-gy2bf(y1)x2
• f(y) y
• In time controlled tasks, choose response 1 iff
  y1-y2 gt 0
• Let y (y1-y2). While y1 and y2 are positive,
  the model reduces to dy/dt I-lyx
• II1-I2 l g-b xx1-x2

y2
y1
I1
I2
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Time course of stimulus sensitivity in the linear
approximation
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Time-accuracy curves for different k-b or l
k-b 0
k-b .2
k-b .4
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The Full Non-Linear LCAi Model
y1
Although the value of the differencevariable is
not well-captured by thelinear approximation,
the sign of thedifference is approximated very
closely.
y2
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Result of fitting the full model to individual
participant data (Usher McClelland, 2001)
Prob. Correct
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Distinguishing Leak Dominance From Inhibition
Dominance
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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.
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The earlier the pulse, the more it matters(Kiani
et al, 2008)
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These results rule out leak dominance
Still viable
X
The bounded DDM and the full non-linear LCAi are
also still viable.
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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.

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Quasi-Continuous, Quasi-Discrete, Reversible
Decision States in the Non-Linear LCAi
Non-Linear
Linear
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v
Distribution of winners activationswhen
correctalternative wins
Distribution of winners activationswhen
incorrectalternative wins
v
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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.

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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.

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Integration of reward and stimulus
informationGao, Tortell McClelland PLoS One,
2011
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Proportion of Choices toward Higher Reward
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Fits based on full LCAi
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Relationship between response speed and choice
accuracy
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An AccountHigh-Threshold LCAi
Gao McClelland, (in preparation)
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v
Distribution of winners activationswhen
correctalternative wins
Distribution of winners activationswhen
incorrectalternative wins
v
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v
Distributionof activationswhen
correctalternative wins
Distribution of activationswhen
incorrectalternative wins
v
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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.

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Toward Continuous Measures of Decision States
Lachter, Corrado, Johnston McClelland (in
progress)
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• 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

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Results and Descriptive Model of Data from 1
Participant
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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.

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Follow-up

• Log scale ranging from 10001 to 11 to 11000
  (extends and reshapes range)
• Very explicit instructions about contingencies,
  marks on scale.
• Paid for points, length of session depends on
  participants pacing of trials
• Ten sessions per participant

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Two Participants Session 1
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Session 10 for each participant(this and next 6
slides)
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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.

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Decision making with non-stationary stimulus
information(Tsetsos. Usher McClelland in press)
Phase durationdistribution
Evidence Switching Protocol in the Correlation
Condition
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Individual Data from Correlation Condition
Primacy region
Indifference tostarting phase
P(C), A/B at start
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Simulations of Two Correlated TrialsTop A/B
start high Bottom C starts high
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Only LCAi can explain gt50 choice of C even when
A/B phase comes firstDissimilar favored
firstDissimilar favored secondAverageTop low
noiseBottom higher noise
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Individual Data from Correlation Condition and
Model Coverage
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Explaining Individual Differences in theLCA
Balanced,Strong LI
I gt L
Lots ofNoise
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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.

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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.