The optimal observer model evaluates the effect of the orientation of the ridges, of the goal direction, and of the reliability of tactile signal on the direction of hand motion. We used the same notation as (26), tailored to the issue of the current study. Refer to Table 1 for the list of symbols used in the model equations. The term G indicates the goal direction, which is either straight ahead in experiments 1 and 2 (goal direction, G = 0°) or toward a virtual target in experiment 3 (G = [ − 15°,0°,15°]). At time t, the internal state of the system, X^t, is the estimate of the motion direction of the hand (1D variable). The ideal observer adjusts his or her direction of motion to compensate for the difference between the state estimate, X^t, and the goal direction, G. To link the (measured) motor behavior and the (latent) observer model, we assumed that the change in the direction of motion in the unitary time interval, Δθ, is equal to the motor command, ut. As illustrated in Fig. 4, a forward model predicts the next motion direction as the sum of the state estimate and the motor commandX^(t+1)=X^t+ut

The output of the forward model is compared with the direction of hand motion as measured by the somatosensory system, θ^(t+1), obtaining the following error termE=θ^(t+1)X^(t+1)

The measured direction is equal to a weighted sum of the two sensory signals from proprioception and touch, P and T, respectivelyθ^(t+1)=wTT(t+1)+wPP(t+1)

We assumed that the two weight terms, wT and wP, are constant within each experimental session. To a first approximation, we assumed that the proprioceptive signal provides an accurate estimate of the actual direction of hand motion, i.e., P¯=θ. Instead, the estimate from touch, T, is always orthogonal to the orientation of the ridges, in accordance with previous literature (16, 17). Last, the state estimate is updated on the basis of the error termX^(t+1)=X^(t+1)+K(t+1)(E)where K(t + 1) is the Kalman gain (0 ≤ K(t + 1) ≤ 1). At time t + 1, the Kalman gain is computed asK(t+1)=σX̂(t+1)2σX̂(t+1)2+σθ̂(t+1)2where σX̂(t+1)2 is the variance of the forward model and σθ̂(t+1)2 is the variance of the sensory measurement. According to the model, a perceived deviation from the goal direction, e.g., to the left, θ^t>0 produces an update in the state estimate, triggering a correction movement to the right and vice versa. Participants do not apply corrections to the motion direction if either E or K are equal to zero.

We simulated the outcome of the model and evaluated whether the response of the ideal observer matched the real data. In each simulated experiment, we simulated 75 trials, including five plate orientations with 15 repetitions each. Each trial consisted of a simulated hand trajectory divided in 100 discrete steps of unitary length. The three free parameters of the model and the model input (motor goal and ridge orientation) were set as explained in Results. In each step, we updated the direction of motion, θt (which is the output of the simulation), by adding the change in direction occurred during the unitary interval, Δθ(t + 1)θ(t+1)=θt+Δθ+ϵ(t+1).with Δθ = ut. In the equation above, ϵ(t + 1) is the sum of the error term related to motor noise, ϵ(ut), and one related to the noise of the state estimate, ϵ(X̂t). The two error terms were sampled from two Gaussian distributions with parameters N(0,σut2) and N(0,σX̂t2), respectively. The variance of the internal estimate, σX̂t2, the variance of the forward model, σX̂(t+1)2, and the Kalman gain, K(t + 1), were computed in each iteration following Kalman filter equations (26). Simulated data were generated in R language (R version 3.4.4).

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