Temporal Influence on Subject Reaching Strategies

Matt BalcarrasIrene TamagnoneLeonie Oostwoud WijdenesAndrew BrennanDeborah BaranyYashar Zeighami

CoSMo 2012

“Kalman, maybe?”

Outline

• How do reaching strategies depend on recent experience and current sensory information?

• Analyzed Körding & Wolpert (2004) data from the ★DREAM★ database

• Original paper did not consider recent trial effects on performance

Outline

• Three approaches

1. Regression analysis

2. Prior evolution

3. Kalman Filter

Temporal Structure

Regression Analysis

Kalman Filter

PriorEvolution

Körding & Wolpert (2004)

End pointhand position

Deviation from midpointand endpoint positions

End pointcursor position

Regression Analysis

• Predictors

1. Mid-point hand position

2. Mid-point cursor position

3. Five-trial running mean

4. Cumulative mean

Regression Coefficients

Estimating the Prior

€

μposterior = r⋅ μ likelihood + 1 − r( )⋅ μ prior

r =σ prior

2

σ prior2 +σ likelihood

2

MSE =σ prior

2 ⋅σ likelihood2

σ prior2 +σ likelihood

2 +σ motor2

€

μposterior = −(end point position)

μ likelihood = lateral shift (based on feedback at midpoint)

μ prior = estimated lateral shift

r = estimated weight

σ motor2 = motor noise

Berniker, Voss, & Körding, 2010

Prior Estimate Over Time

Trial Bin

Pri

or (

cm)

Likelihood Variance

€

σ likelihood2 =

MSE −σ motor2

r

€

μposterior = r⋅ μ likelihood + 1− r( )⋅ μ prior

€

σ likelihood2 : 0.29 0.43 0.60 ∞

σ0 σM σL σ∞

Implementation of Kalman Filter

€

At =α 1 −α1

t −11 −

1

t −1

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥,C = 1 0[ ]

μ =0

0

⎡

⎣ ⎢

⎤

⎦ ⎥,∑ =

1/4 0

0 1/100

⎡

⎣ ⎢

⎤

⎦ ⎥

ε(σ s) =0.36 if cond(σ

s) = 1 ; 0.67 if cond(σ

s) = 2

0.8 if cond(σs

) =3 ; 100 if cond(σs

) =4

⎧ ⎨ ⎪

⎩ ⎪

• Our state vector : Current perturbation Pt and perturbation mean μt computed on all the previous trials

• Our observer: Midpoint Cursor Position

)(

,1

1

smpt

tt

t

tt

t

t

hP

Cy

NP

AP

σμ

μμμ

where:

• Parameter α • Weights the contributions of the previous perturbation and

of the whole history of perturbations in the current prediction

• Is optimized on the training data (first 1000 trials)

Kalman Filter Fit

KalmanFilter

SubjectData

Trial

End

Han

d Po

sitio

n (c

m)

α (across subjects)=0.35

R2 = 0.18

Typical Subject

Conclusions• Recent trial history and online cursor

feedback play a significant role in predicting end position

• The prior is learned quickly and is stable over time

• Likelihood variance increases with uncertainty

• The Kalman filter reveals that subjects trust the estimation of the previous trial

• Under uncertain conditions, temporal factors influence subject strategy suboptimally

M I L A D Y