Causal inference in motor adaptation Kunlin Wei & Konrad Koerding Rehabilitation Institute of Chicago During motor adaptation the nervous system constantly uses error information to improve future movements. Many models assume that the nervous system uses errors in a linear fashion (Kawato, 1987; Wolpert and Kawato, 1998; Thoroughman and Shadmehr, 2000; Scheidt et al., 2001). However, allow us to illustrate an important problem with this linear error compensation strategy by a simple example. If we reach for a cup and our hand does not move far enough by a few millimeters, it makes sense to adapt and use larger motor commands in the future. If, on the other hand, we miss it by more than the length of our arm, it is very likely that this error was not caused by our body (for example someone playing a practical joke on us) and we should not adapt linearly in response to this error. We hypothesize that whenever the visual system detects an error, the central nervous system (CNS) needs to consider two scenarios. The error may be induced by extrinsic factors, unrelated to the motor plant such as the displaced cup in the above example. Alternatively, the error may be induced by intrinsic factors within the motor plant, such as muscle fatigue. The size of the perceived visual error could be (and according to our considerations should be) used by the nervous system to calculate how likely each of the two scenarios is. As a result of this estimated likelihood the nervous system should strongly adapt to small errors (more likely to be caused by the body) and only weakly adapt to very large errors (less likely to be caused by the body). Here we present a systematic Bayesian treatment of such problems. The model calculates how likely an error is caused by the environment or by the body and derives an adaptation strategy based on the inferred cause of the error. The model predicts that adaptation should be a sub-linear function of the size of errors. We test this prediction in a reaching experiment where the size of errors is systematically varied. Our experiment requires subjects to make straight reaching movements to a target 15 cm away in a virtual reality setting where the visual feedback of the hand is represented as a cursor. The cursor is only shown at the end of reaching movements and its position is perturbed by one out of 9 possible values: 0, ±1, ±2, ±4 and ±8 cm. This manipulation essentially specifies the size of the visual error for each reach. Proprioception and vision are two perceptual cues available for estimating the actual error and the estimated error will in turn guide future movements. Our results (Figure 1) demonstrate that, indeed, adaptation linearly depends on the size of visual errors when the error size is small. It becomes sub-linear when the visual error increases further. This nonlinear behavior is well predicted by our Bayesian causal inference model. Large errors have less relative influence on adaptation. Moreover, the model infers that large errors lead to very small probabilities of the cursor’s position being caused by the hand position (Figure 2A). The influence of previous errors decreases over time in a fashion that is roughly fit by an exponential function, which is predicted by numerous state space models and also Kalman filter learning models (Figure 2B). We proceed to fit the model with two other published studies on force field learning (Wei et al., 2005) and saccadic adaptation in monkeys (Robinson et al., 2003). The model provides satisfactory explanations for the nonlinear relationship between error size and adaptation (Figure 3). Causal inference, the estimation of the probability of causal hypotheses, has been extensively studied in high-level cognitive activities (e.g. Tenenbaum et al., 2006) and more recently in sensory integration (e.g. Kording et al., in press). Our study shows that it is also performed continuously and effortlessly in motor adaptation. Our results suggest that there exists an optimal error size for adaptation/learning and this is relevant for rehabilitation procedures where error feedback in virtual reality is often provided.

Figure 1: The average deviation from the target is plotted as a function of the size of the visual disturbance and Δt (the trial lag between the disturbance and the current trial). The average deviation is the mean of endpoint errors of all the trials following one type of disturbance. Adaptation, as expressed by the deviations, is a sub-linear function of the disturbance. The larger Δt , the smaller the influence of a disturbance. Figure 2: (A) The normalized probability of visual errors being inferred as relevant feedback as a function of the size of visual disturbances. The smaller the visual disturbance, the more likely the visual error is inferred as relevant. (B) The influence of the visual disturbance on future trials is estimated from the data and is plotted as a function of trial lag Δt . The influence of the visual disturbance decreases exponentially with time. Figure 3: Data and model predictions for other published experiments. Model predictions are shown in red. (A) The adaptation rate in a visuo-motor adaptation task (Wei et al. 2005) is plotted as a function of visual error gain. (B) The adaptation gain in saccades is plotted as a function of the visual error size (Robinson et al. 2003). References Kawato M, Furukawa, K., and Suzuki, R. (1987) A hierarchical neural-network model for control and learning of voluntary movement. Biological Cybernetics 57:169-185. Kording KP, Beierholm U, Ma WJ, Quartz S, Tenenbaum JB, Shams L (in press) Causal inference in Cue combination. PLOSOne. Robinson FR, Noto CT, Bevans SE (2003) Effect of visual error size on saccade adaptation in monkey. J Neurophysiol 90:1235-1244. Scheidt RA, Dingwell JB, Mussa-Ivaldi FA (2001) Learning to move amid uncertainty. J Neurophysiol 86:971-985. Tenenbaum JB, Griffiths TL, Kemp C (2006) Theory-based Bayesian models of inductive learning and reasoning. Trends Cogn Sci 10:309-318. Thoroughman KA, Shadmehr R (2000) Learning of action through adaptive combination of motor primitives. Nature 407:742-747. Wei Y, Bajaj P, Scheidt R, JL P (2005) A Real-Time Haptic/Graphic Demonstration of how Error Augmentation can Enhance Learning. In: IEEE- International Conference on Robotics and Automation. Barcelona, Spain. Wolpert D, Kawato M (1998) Multiple paired forward and inverse models for motor control. NeurNetw 11:13171329.