##Dynamic control models for strategic interaction **_John Pearson_**, Shariq Iqbal, Caroline Drucker, Michael Platt [pearsonlab.github.io/sfn-talk-2016](https://pearsonlab.github.io/sfn-talk-2016)

Shariq Iqbal

Caroline Drucker

Jean-Francois Gariépy

Michael Platt

Penalty Shot

Complexity tax

  • each trial a different length
  • how to average, align?
  • need to "reduce" dynamics
### Our approach - borrow from control theory, time series - structured black box models (pieces make sense) - neural networks for flexible fitting

Modeling I

Observed positions at each time ($y_t$): $$ y_t = \begin{bmatrix} y_{goalie} & x_{puck} & y_{puck} \end{bmatrix}^\top $$
Control inputs ($u_t$) drive changes in observed positions: $$y_{t + 1} = y_t + v_{max} \sigma(u_t)$$
Goal: predict control inputs from trial history: $$u_t = F(y_{1:t})$$

Modeling II

One (of many) assumptions we could make: $$ u_t = f(y_t, y_{1:t-1}) + \eta_t $$
Control signals are driven by a sum:
  • current state: $y_t$
  • past trajectory: $y_{1:t-1}$
  • nonlinear transformation: $f$
  • noisy "driving" input: $\eta_t$

Modeling III

$$ u_t = f(y_t, y_{1:t-1}) + \eta_t $$
How do we interpret this?
  • $f$: fixed, deterministic, reactive
  • $\eta$: trial-specific, "innovation," residual

Model fitting

Variational Bayes autoencoder
  • Encoding model:
    • $f$: deterministic feedforward neural net
    • $\eta$: $AR(p)$ time series

  • Decoding model:

It fits!

A tougher test

Generated Trials

Innovations matter

###Conclusions - Dynamic control tasks let us leverage motor behavior to study cognitive and social decisions. - Structured black-box models allow us to carve behavior into interpretable pieces. - Control models suggest candidate signals for correlation with neural data.