Quasi-Continuous Decision States in the Leaky Competing Accumulator Model Jay McClelland Stanford University With Joel Lachter, Greg Corrado, and Jim Johnston as well as Juan Gao and Marius Usher

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 + for one alternative, – for the other, with standard deviation . 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 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 and direction is varied from trial to trial.

Neural Basis of Decision Making in Monkeys: Results Data are averaged over many different neurons that are associated with intended eye movements to the location of target.

Hard Prob. Correct 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

Usher and McClelland (2001) Leaky Competing Accumulator Model Addresses the process of deciding between two alternatives based on external input, with leakage, mutual inhibition, and noise: dy 1 /dt = I 1 -y 1 –f(y 2 )+ 1 dy 2 /dt = I 2 -y 2 –f(y 1 )+ 2 f(y) = [y] + Participant chooses the most active accumulator when the go cue occurs This is equivalent to choosing response 1 iff y 1 -y 2 > 0 Let y = (y 1 -y 2 ). While y 1 and y 2 are positive, the model reduces to: dy/dt = I-y+ I=I 1 -I 2 =-=  -   11 22 y1y1 y2y2

Wong & Wang (2006) ~Usher & McClelland (2001)

Time-accuracy curves for different |k-| or || |k-  = 0 |k-  =.2 |k-  =.4

Prob. Correct

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.

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

These results rule out leak dominance X Still viable

The Full Non-Linear LCA i Model y1y1 y2y2 Although the value of the difference variable is not well-captured by the linear approximation, the sign of the difference is approximated very closely.

Three Studies Related to these Issues Integration of reward and payoff information under time controlled conditions –Gao, Tortell & McClelland Investigations of decision making with non- stationary stimulus information –Usher, Tsetsos & McClelland Does the confidence of a final decision state vary continuously with the strength of the evidence? –Lachter, Corrado, Johnston & McClelland

Timeline of the Experiment

Proportion of Choices toward Higher Reward

Sensitivity varies with time

Three Hypotheses 1.Reward acts as an input from reward cue onset til the end of the integration period 2.Reward influences the state of the accumulators before the onset of the stimulus 3.Reward introduces an offset into the decision

Fits Based on Linear Model

Fits based on full LCA i

Relationship between response speed and choice accuracy

Different levels of activation of correct and incorrect responses in Inhibition-dominant LCA Final time slice correct errors

High-Threshold LCA i

Preliminary Simulation Results

Three Studies Related to these Issues Integration of reward and payoff information under time controlled conditions –Gao, Tortell & McClelland Investigations of decision making with non- stationary stimulus information –Usher, Tsetsos & McClelland Does the confidence of a final decision state vary continuously with the strength of the evidence? –Lachter, Corrado, Johnston & McClelland

Continuous Report of Confidence Lachter, Corrado, Johnston & McClelland (in progress) Expt 1: Probability bias included; trial could end at any time Expt 2: No probability bias; observers terminate trial when they have determined their best answer

Trial Duration Protocol and performance as a function of time for participants grouped by performance

Results and Descriptive Model of Data from 1 Participant