1. Zen, and the Art of Neural Decoding using an EM Algorithm Parameterized Kalman Filter and Gaussian Spatial Smoothing Michael Prerau, MS

2. Encoding/Decoding Process Generate a smoothed Gaussian white noise stimulus Generate a random kernel, D and convolve with the stimulus to generate a spike rate Drive Poisson spike generator Decode and find K Use K to decode from new stimuli “real time”

3. Encoding/Decoding Process

4. Encoding/Decoding

5. Stimulus

6. Decoded Estimate

7. State-Space Modeling Hidden State: Where sputnik really is Observations: What the towers see State equation: How sputnik ideally moves Observation equation: If we knew where sputnik was, how would that relate to our observations? Parameters:

8. State-Space Modeling Observations State estimate

9. The Kalman Filter Gaussian state The actual stimulus intensity Gaussian observations The filtered estimate State Equation Observation Equation State Estimate

10. State Equation: Random Walk AR Model Observation Equation: Linear Model Parameters The Kalman Filter Application to the Intensity Estimate where

11. Complete Data Likelihood Log-likelihood The Kalman Filter Application to the Intensity Estimate

12. Forward Filter Derivation Most likely hidden state will maximize log-likelihood: Maximize for x k and solve: Arrange Kalman style:

13. For hidden state variance, first take the 2 nd derivative of the log likelihood: Then take the negative of the inverse for the variance of the hidden state: Forward Filter Derivation

14. The EM Algorithm Suppose we don’t know the parameter values? Use the Expectation Maximization (EM) Algorithm (Dempster, Laird, and Rubin, 1977) Iterative maximization E-step: Take the most likely (Expected value) value of the state process given the parameters M-step: Maximize for the most likely parameters given the estimated state values

15. E-Step for Intensity Model Take the expected value of the joint likelihood: We will encounter terms such as: Can be solved with the state-space covariance algorithm (De Jong and MacKinnon, 1988)

16. Example : M-Step for Intensity Model For the M-Step, maximize with respect to each parameter. Set equal to zero and solve

17. M-Step for Intensity Model M-Step Summary:

18. The EM Algorithm

20. Kalman Estimate

21. 2D Gaussian Spatial Smoothing

22. Gaussian Spatially Smoothed Estimate

23. Kalman Filtering the Gaussian Smoothed Estimate

25. Comparison

27. Stimulus S est KalmanGaussian Smoothed Kalman

28. fin