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The Computing Brain: Focus on Decision-Making
Cogs 1 March 5, 2009 Angela Yu
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Understanding the Brain/Mind? A powerful analogy: the computing brain
Behavior Neurobiology ? Cognitive Neuroscience ≈ A powerful analogy: the computing brain
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An Example: Decision-Making
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An Example: Decision-Making
What are the computations involved? THUR 74/61 2. Uncertainty ? 1. Decision 3. Costs: Health? Coolness?
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Monkey Decision-Making
Properties: info/time slowed down, linear in time, in controlled exp. Random dots coherent motion paradigm
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Random Dot Coherent Motion Paradigm
Properties: info/time slowed down, linear in time, in controlled exp. 30% coherence 5% coherence
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How Do Monkeys Behave? Accuracy vs. Coherence <RT> vs. Coherence
(Roitman & Shadlen, 2002)
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What Are the Neurons Doing? MT: Sustained response
Saccade generation system MT: Sustained response Time Response 99.9% MT tuning function Direction () Preferred Anti-preferred Time Response 25.6% (Britten & Newsome, 1998) (Britten, Shadlen, Newsome, & Movshon, 1993)
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What Are the Neurons Doing?
Saccade generation system LIP: Ramping response (Roitman & Shalden, 2002) (Shadlen & Newsome, 1996)
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A Computational Description
1. Decision: Left or right? Stop now or continue? Properties: info/time slowed down, linear in time, in controlled exp. (t): input(1), …, input(t) {left, right, wait}
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Timed Decision-Making
wait wait L R L R L R
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A Computational Description
2. Uncertainty: Sensory noise (coherence) Properties: info/time slowed down, linear in time, in controlled exp.
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Timed Decision-Making
wait wait L R L R L R
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A Computational Description
3. Costs: Accuracy, speed Properties: info/time slowed down, linear in time, in controlled exp. cost() = Pr(error) + c*(mean RT) Optimal policy * minimizes the cost
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Timed Decision-Making
+: more accurate wait wait -: more time L R L R L R
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Mathematicians Solved the Problem for Us
Wald & Wolfowitz (1948): Optimal policy: accumulate evidence over time: Pr(left) versus Pr(right) stop if total evidence exceeds “left” or “right” boundary “left” boundary “right” boundary Pr(left)
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Model Neurobiology Model LIP Neural Response
Saccade generation system Model LIP Neural Response
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Model Behavior Model RT vs. Coherence Saccade generation system
boundary Coherence
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Putting it All Together
Saccade generation system MT neurons signal motion direction and strength LIP neurons accumulate information over time LIP response reflect behavioral decision (0% coherence) Monkeys behave optimally (maximize accuracy & speed)
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Understanding the Brain/Mind? A powerful analogy: the computing brain
Behavior Neurobiology ? Cognitive Neuroscience ≈ A powerful analogy: the computing brain
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Sequential Probability Ratio Test
Log posterior ratio undergoes a random walk: rt is monotonically related to qt, so we have (a’,b’), a’<0, b’>0. b’ a’ rt In continuous-time, a drift-diffusion process w/ absorbing boundaries.
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Generalize Loss Function
We assumed loss is linear in error and delay: What if it’s non-linear, e.g. maximize reward rate: (Bogacz et al, 2006) Wald also proved a dual statement: Amongst all decision policies satisfying the criterion SPRT (with some thresholds) minimizes the expected sample size <>. This implies that the SPRT is optimal for all loss functions that increase with inaccuracy and (linearly in) delay (proof by contradiction).
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Neural Implementation
Saccade generation system LIP - neural SPRT integrator? (Roitman & Shalden, 2002; Gold & Shadlen, 2004) Time LIP Response & Behavioral RT LIP Response & Coherence
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Caveat: Model Fit Imperfect
(Data from Roitman & Shadlen, 2002; analysis from Ditterich, 2007) Accuracy Mean RT RT Distributions
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Fix 1: Variable Drift Rate
(Data from Roitman & Shadlen, 2002; analysis from Ditterich, 2007) (idea from Ratcliff & Rouder, 1998) Accuracy Mean RT RT Distributions
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Fix 2: Increasing Drift Rate
(Data from Roitman & Shadlen, 2002; analysis from Ditterich, 2007) Accuracy Mean RT RT Distributions
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A Principled Approach Trials aborted w/o response or reward if monkey breaks fixation (Data from Roitman & Shadlen, 2002) (Analysis from Ditterich, 2007) More “urgency” over time as risk of aborting trial increases Increasing gain equivalent to decreasing threshold
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Imposing a Stochastic Deadline
(Frazier & Yu, 2007) Loss function depends on error, delay, and whether deadline exceed Optimal Policy is a sequence of monotonically decaying thresholds Time Probability
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delta, gamma, exponential, normal
Timing Uncertainty (Frazier & Yu, 2007) Timing uncertainty lower thresholds Time Probability Theory applies to a large class of deadline distributions: delta, gamma, exponential, normal
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Some Intuitions for the Proof
(Frazier & Yu, 2007) Continuation Region Shrinking!
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Summary Review of Bayesian DT: optimality depends on loss function
Decision time complicates things: infinite repeated decisions Binary hypothesis testing: SPRT with fixed thresholds is optimal Behavioral & neural data suggestive, but … … imperfect fit. Variable/increasing drift rate both ad hoc Deadline (+ timing uncertainty) decaying thresholds Current/future work: generalize theory, test with experiments
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