1. Statistical learning and optimal control: A framework for biological learning and motor control Lecture 2: Models of biological learning and sensory- motor integration Reza Shadmehr Johns Hopkins School of Medicine

2. Various forms of classical conditioning in animal psychology Table from Peter Dayan Not explained by LMS, but predicted by the Kalman filter.

3. Kalman filter as a model of animal learning Suppose that x represents inputs from the environment: a light and a tone. Suppose that y represents rewards, like a food pellet. Animal’s model of the experimental setupAnimal’s expectation on trial n Animal’s learning from trial n

4. Sharing Paradigm Train: {x1,x2} -> 1 Test: x1 -> ?, x2 -> ? Result: x1->0.5, x2->0.5 10203040 0 0.5 1 1.5 y yhat 010203040 0 0.2 0.4 0.6 0.8 w1 w2 10203040 0.2 0.25 0.3 0.35 0.4 0.45 0.5 P11 P22 10203040 0.1 0.15 0.2 0.25 0.3 0.35 0.4 k1 k2 010203040 0 1 x1 x2 y * 10203040 0 0.5 1 1.5 y yhat 010203040 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 w1 w2 Learning with Kalman gainLMS

5. Blocking Kamin (1968) Attention-like processes in classical conditioning. In: Miami symposium on the prediction of behavior: aversive stimulation (ed. MR Jones), pp. 9- 33. Univ. of Miami Press. Kamin trained an animal to continuously press a lever to receive food. He then paired a light (conditioned stimulus) and a mild electric shock to the foot of the rat (unconditioned stimulus). In response to the shock, the animal would reduce the lever-press activity. Soon the animal learned that when the light predicted the shock, and therefore reduced lever pressing in response to the light. He then paired the light with a tone when giving the electric shock. After this second stage of training, he observed than when the tone was given alone, the animal did not reduce its lever pressing. The animal had not learned anything about the tone.

6. Blocking Paradigm Train: x1 -> 1, {x1,x2} -> 1 Test: x2 -> ?, x1 -> ? Result: x2 -> 0, x1 -> 1 010203040 0 1 x1 x2 y * 10203040 0 0.5 1 1.5 y yhat 010203040 -0.25 0 0.25 0.5 0.75 1 1.25 w1 w2 10203040 0.1 0.2 0.3 0.4 0.5 P11 P22 10203040 0 0.1 0.2 0.3 0.4 0.5 0.6 k1 k2 Learning with Kalman gainLMS 010203040 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 w1 w2 10203040 0 0.5 1 1.5 y yhat

7. Backwards Blocking Paradigm Train: {x1,x2} -> 1, x1 -> 1 Test: x2 -> ? Result: x2 -> 0 0102030405060 0 1 x1 x2 y * 0102030405060 0 0.5 1 1.5 y yhat 0102030405060 -0.2 0 0.2 0.4 0.6 0.8 1 w1 w2 0102030405060 0.1 0.2 0.3 0.4 0.5 P11 P22 0102030405060 -0.4 -0.2 0 0.2 0.4 k1 k2 Learning with Kalman gainLMS 0102030405060 0 0.5 1 1.5 y yhat 0102030405060 0 0.2 0.4 0.6 0.8 1 w1 w2

8. Different output models Case 1: the animal assumes an additive model. If each stimulus predicts one reward, then if the two are present together, they predict two rewards. Suppose that x represents inputs from the environment: a light and a tone. Suppose that y represents a reward, like a food pellet. Case 2: the animal assumes a weighted average model. If each stimulus predicts one reward, then if the two are present together, they still predict one reward, but with higher confidence. The weights a1 and a2 should be set to the inverse of the variance (uncertainty) with which each stimulus x1 and x2 predicts the reward.

9. General case of the Kalman filter A priori estimate of mean and variance of the hidden variable before I observe the first data point Update of the estimate of the hidden variable after I observed the data point Forward projection of the estimate to the next trial nx1 mx1

10. DM Wolpert et al. (1995) Science 269:1880 Motor command Sensory measurement State of our body Application of Kalman filter to problems in sensorimotor control

11. When we move our arm in darkness, we may estimate the position of our hand based on three sources of information: proprioceptive feedback. a forward model of how the motor commands have moved our arm. by combining our prediction from the forward model with actual proprioceptive feedback. Experimental procedures: Subject holds a robotic arm in total darkness. The hand is briefly illuminated. An arrow is displayed to left or right, showing which way to move the hand. In some cases, the robot produces a constant force that assists or resists the movement. The subject slowly moves the hand until a tone is sounded. They use the other hand to move a mouse cursor to show where they think their hand is located. DM Wolpert et al. (1995) Science 269:1880

13. Motor command Sensory measurement State of the body A B C The generative model, describing actual dynamics of the limb The model for estimation of sensory state from sensory feedback For whatever reason, the brain has an incorrect model of the arm. It overestimates the effect of motor commands on changes in limb position.

14. Initial conditions: the subject can see the hand and has no uncertainty regarding its position and velocity Forward model of state change and feedback Actual observation Estimate of state incorporates the prior and the observation Forward model to establish the prior and the uncertainty for the next state

15. 00.20.40.60.811.21.4 0 5 10 15 20 00.20.40.60.811.21.4 Timesec 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Actual and estimated position Kalman gain Bias at end of movement (cm) Variance at end of movement (cm^2) Total movement time (sec) 00.20.40.60.811.21.4 -1.5 -0.5 0 0.5 1 1.5 Motor command u Time of “beep” For movements of various length A single movement Pos (cm)

16. Puzzling results: Savings and memory despite “washout” Gain=eye displacement divided by target displacement Result 1: After changes in gain, monkeys exhibit recall despite behavioral evidence for washout. Kojima et al. (2004) Memory of learning facilitates saccade adaptation in the monkey. J Neurosci 24:7531. 1

17. Result 2: Following changes in gain and a long period of washout, monkeys exhibit no recall. Result 3: Following changes in gain and a period of darkness, monkeys exhibit a “jump” in memory. Puzzling results: Improvements in performance without error feedback Kojima et al. (2004) J Neurosci 24:7531.

18. The learner’s hypothesis about the structure of the world A 1. The world has many hidden states. What I observe is a linear combination of these states. 2.The hidden states change from trial to trial. Some change slowly, others change fast. 3.The states that change fast have larger noise than states that change slow. slow system fast system state transition equation output equation

19. 050100150200250300 -1.5 -0.5 0 0.5 1 1.5 y yhat 050100150200250300 0 1 050100150200250300 -0.4 -0.2 0 0.2 0.4 0.6 w1 w2 Simulations for savings x1 x2 y * The critical assumption is that in the fast system, there is much more noise than in the slow system. This produces larger learning rate in the slow system.

20. 050100150200250300 -0.5 0 0.5 1 x1 x2 y * 050100150200250300 -1.5 -0.5 0 0.5 1 1.5 y yhat 050100150200250300 -0.4 -0.2 0 0.2 0.4 w1 w2 Simulations for spontaneous recovery despite zero error feedback error clamp In the error clamp period, estimates are made yet the weight update equation does not see any error. Therefore, the effect of Kalman gain in the error- clamp period is zero. Nevertheless, weights continue to change because of the state update equations. The fast weights rapidly rebound to zero, while the slow weights slowly decline. The sum of these two changes produces a “spontaneous recovery” after washout.

21. Mean gain at start of recovery = 0.83 Mean gain at start of recovery = 0.86 Mean gain at end of recovery = 0.87 % gain change = 1.2% gain change = 14.4% Mean gain at end of recovery = 0.95 Target extinguished during recoveryTarget visible during recovery Changes in representation without error feedback Seeberger et al. (2002) Brain Research 956:374-379.

22. Massed vs. Spaced training: effect of changing the inter-trial interval Learning reaching in a force field ITI = 8min ITI = 1min Discrimination performance (sec) Rats were trained on an operant conditional discrimination in which an ambiguous stimulus (X) indicated both the occasions on which responding in the presence of a second cue (A) would be reinforced and the occasions on which responding in the presence of a third cue (B) would not be reinforced (X --> A+, A-, X --> B-, B+). Both rats with lesions of the hippocampus and control rats learned this discrimination more rapidly when the training trials were widely spaced (intertrial interval of 8 min) than when they were massed (intertrial interval of 1 min). With spaced practice, lesioned and control rats learned this discrimination equally well. But when the training trials were massed, lesioned rats learned more rapidly than controls. Han, J.S., Gallagher, M. & Holland, P. Hippocampus 8:138-46 (1998)

23. Escape latency (s) Training trial (bin size=4) 4 trials per day for 4 days 16 trials in one day Performance in a water maze Commins, S., Cunningham, L., Harvey, D. & Walsh, D. (2003) Behav Brain Res 139:215-23 Aboukhalil, A., Shelhamer, M. & Clendaniel, R. (2004) Neurosci Lett 369:162-7. Cue-dependent saccade gain adaptation When eyes are looking up, increase saccade gain, when eyes are looking down, decrease gain. (break period: 1 min)

24. A A AAA The learner’s hypothesis about the structure of the world 1. The world has many hidden states. What I observe is a linear combination of these states. 2.The hidden states change from trial to trial. Some change slowly, others change fast. 3.The states that change fast have larger noise than states that change slow. 4.The state changes can occur more frequently than I can make observations. Inter-trial interval

25. 050100150200250300 0 0.5 1 1.5 y yhat 050010001500200025003000 0 0.5 1 1.5 y yhat ITI=2 ITI=20

26. When there is an observation, the uncertainty for each hidden variable decreases proportional to its Kalman gain. When there are no observations, the uncertainty decreases in proportion to A squared, but increases in proportion to state noise Q. 100010201040106010801100 0.106 0.108 0.11 0.112 0.114 0.116 0.118 P22 100010201040106010801100 0.0126 0.0128 0.013 0.0132 0.0134 0.0136 P11 Uncertainty for the slow stateUncertainty for the fast state ITI=20 Beyond a minimum ITI, increased ITI continues to increase the uncertainty of the slow state but has little effect on the fast state uncertainty. The longer ITI increases the total learning by increasing the slow state’s sensitivity to error.

27. 020406080100120140 0 0.5 1 1.5 y yhatspaced yhatmassed 020406080100120140 -0.2 0 0.2 0.4 0.6 0.8 w1massed w1spaced w2massed w2spaced 020406080100120140 0.2 0.4 0.6 0.8 k1massed k1spaced k2massed k2spaced Observation number Performance in spaced training depends largely on the slow state. Therefore, spaced training produces memories that decay little with passage of time.

28. ITI=14 ITI=2 ITI=98 Performance during training Test at 1 week ITI=14 ITI=2 ITI=98 Testing at 1 day or 1 week (averaged together) Pavlik, P. I. and Anderson, J. R. ( 2005). Practice and forgetting effects on vocabulary memory: An activation-based model of the spacing effect. Cognitive Science, 29, 559-586. Spaced training results in better retention in learning a second language On Day 1, subjects learned to translate written Japanese words into English. They were given a Japanese word (written phonetically), and then given the English translation. This “study trial” was repeated twice. Afterwards, the were given the Japanese word and had to write the translation. If their translation was incorrect, the correct translation was given. The ITI between word repetition was either 2, 14, or 98 trials. Performance during training was better when the ITI was short. However, retention was much better for words that were observed with longer ITI. (The retention test involved two groups; one at 1 day and other at 7 days. Performance was slightly better for the 1 day group but the results were averaged in this figure.)

29. Data fusion Suppose that we have two sensors that independently measure something. We would like to combine their measures to form a better estimate. What should the weights be? Suppose that we know that sensor 1 gives us measurement y1 and has Gaussian noise with variance: And similarly, sensor 2 has gives us measurement y2 and has Gaussian noise with variance: A good idea is to weight each sensor inversely proportional to its noise:

30. To see why this makes sense, let’s put forth a generative model that describes our hypothesis about how the data that we are observing is generated: Observed variables Hidden variable Data fusion via Kalman filter

31. See homework for this priors our first observation variance of our posterior estimate

32. Notice that after we make our first observation, the variance of our posterior is better than the variance of either sensor. What our sensors tell us The real world

33. -20246810 0 0.1 0.2 0.3 0.4 0.5 -2.502.557.51012.515 0 0.1 0.2 0.3 0.4 Sensor 1Sensor 2 Combined Sensor 1 Sensor 2 Combined Combining equally noisy sensorsCombining sensors with unequal noise Mean of the posterior, and its variance probability

34. musclesMotor commands force Body part State change Sensory system Proprioception Vision Audition Measured sensory consequences Forward model Predicted sensory consequences Integration Belief What we sense depends on what we predicted

35. Duhamel et al. Science 255, 90-92 (1992) The brain predicts the sensory consequences of motor commands

36. Vaziri, Diedrichsen, Shadmehr (2006) Journal of Neuroscience Combining sensory predictions with sensory measurements should produce a better spatial estimate of the visual world

38. Vaziri et al. (2006) J Neurosci

40. How to set the initial var-cov matrix In homework, we will show that in general: Now if we have absolutely no prior information on w, then before we see the first data point P(1|0) is infinity, and therefore its inverse in zero. After we see the first data point, we will be using the above equation to update our estimate. The updated estimate will become: A reasonable and conservative estimate of the initial value of P would be to set it to the above value. That is, set: