1. Real-time optimization of neurophysiology experiments Jeremy Lewi 1, Robert Butera 1, Liam Paninski 2 1 Department of Bioengineering, Georgia Institute of Technology, 2. Department of Statistics, Columbia University

2. Neural Encoding The neural code: what is P(response | stimulus) Main Question: how to estimate P(r|x) from (sparse) experimental data?

3. Curse of dimensionality Both stimuli and responses can be very high-dimensional Stimuli: Images Sounds Time -varying behavior Responses: observations from single or multiple simultaneously recorded point processes

4. All experiments are not equally informative Possible p(r|x) possible p(r|x) after experiment A Goal: Constrain set of possible systems as much as possible How: Maximize mutual information I({experiment};{possible systems}) Possible p(r|x) possible p(r|x) after experiment B

5. Adaptive optimal design of experiments Assume: parametric model p(r|x,θ) of responses r on stimulus x prior distribution p(θ) on finite-dimensional parameter space Goal: estimate θ from data Usual approach: draw stimuli i.i.d. from fixed p(x) Adaptive approach: choose p(x) on each trial to maximize I(θ;{r,x})

6. Theory: info. max is better 1.Info. max. is in general more efficient and never worse than random sampling [Paninski 2005] 2.Gaussian approximations are asymptotically accurate

7. Computational challenges 3.Computations need to be performed quickly: 10ms – 1 sec Speed limits the number of trials 1. Updating the posterior: p(θ|x,r) Difficult to represent/manipulation high dimensional posteriors 2.Maximizing the mutual information I(r;θ|x) High dimensional integration High dimensional optimization

8. Solution Overview 1.Model responses using a 1-d GLM Computationally tractable 2.Approximate posterior as Gaussian easy to work with even in high-d 3.Reduce optimization of mutual information to a 1-d problem

9. Neural Model: GLM We model a neuron using a general linear model whose output is the expected firing rate. The nonlinear stage is the exponential function; also ensures the log likelihood is a concave function of θ.

10. GLM Computationally tractable 1.log likelihood is concave 2.log likelihood is 1-dimensional

11. Updating the Posterior 1. Approximate the posterior, as Gaussian. Posterior is product of log concave functions Posterior distribution is asymptotically Gaussian 2.Use a Laplace approximation to determine the parameters of the Gaussian, μt, Ct. μt = peak of posterior Ct – negative of the inverse hessian evaluated at the peak

12. Updating the Posterior 3.Update is rank 1 4.Find the peak: Newton’s method in 1-d 5.Invert the Hessian: use the Woodbury Lemma: O(d 2 ) time += log priorlog likelihoodlog posterior

13. Choosing the optimal stimulus Maximize the mutual information  Minimize the posterior entropy Posterior is Gaussian: Compute the expected determinant –Simplify using matrix perturbation theory Result: Maximize an expression for the expected fisher information Maximization Strategy –Impose a power constraint on the stimulus –Perform an eigendecomposition –Simplify using lagrange multipliers –Find solution by performing a 1-d numerical optimization Bottleneck: Eigendecomposition – takes O(d 2 ) in practice

14. Running Time 1.Updating the posterior O(d 2 ) d- dimensionality 2.Eigen decomposition O(d 2 ) 3.Choosing the stimulus O(d)

15. Simulation Setup Compare: Random vs. Information maximizing stimuli Objective: learn parameters

16. A Gabor Receptive Field high dimensional Info. Max converges to true receptive field Converges faster than random 25x33

17. Non-stationary parameters Biological systems are non-stationary –Degradation of the preparation –Fatigue –Attentive state Use a Kalman filter type approach Model slow changes using diffusion

18. Non-stationary parameters θ i follow Gaussian curve whose center moves randomly over time

19. Assuming θ is constant overestimates certainty  poor choices for optimal stimuli Non-stationary parameters

20. Conclusions 1.Efficient implementation achievable with: 1.Model based approximations Model is specific but reasonable 2.Gaussian approximation of the posterior Justified by the theory 3.Reduction of the optimization to a 1-d problem 2.Assumptions are weaker than typically required for system identification in high dimensions 3.Efficiency could permit system identification in previously intractable systems

21. References 1.A. Watson, et al., Perception and Psychophysics 33, 113 (1983). 2.M. Berry, et al., J. Neurosci. 18 2200(1998) 3.L. Paninski, Neural Computation 17, 1480 (2005). 4.P. McCullagh, et al., Generalized linear models (Chapman and Hall, London, 1989). 5.L. Paninski, Network: Computation in Neural Systems 15, 243 (2004). 6.E. Simoncelli, et al., The Cognitive Neurosciences, M. Gazzaniga, ed. (MIT Press, 2004), third edn. 7.M. Gu, et al., SIAM Journal on Matrix Analysis and Applications 15, 1266 (1994). 8.E. Chichilnisky, Network: Computation in Neural Systems 12, 199 (2001). 9.F. Theunissen, et al., Network: Computation in Neural Systems 12, 289 (2001). 10.L. Paninski, et al., Journal of Neuroscience 24, 8551 (2004)

22. Acknowledgements This work was supported by the Department of Energy Computational Science Graduate Fellowship Program of the Office of Science and National Nuclear Security Administration in the Department of Energy under contract DE-FG02-97ER25308 and by the NSF IGERT Program in Hybrid Neural Microsystems at Georgia Tech via grant number DGE-0333411.

23. Spike history posterior mean after 500 trials stimulus filterspike history filter

24. Previous Work System Identification 1.Minimize variance of parameter estimate Deciding among a menu of experiments which to conduct [Flaherty 05] 2. Maximize divergence of predicted responses for competing models [Dunlop06] Optimal Encoding 1.Maximize the mutual information input and output [Machens 02] 2.Maximize response hill-climbing to find stimulus to which V1 neurons in monkey respond strongly [Foldiak01] Efficient stimuli for cat auditory cortex [Nelken01] 3.Minimize stimulus reconstruction error [Edin04]

25. Derivation of Choosing the Stimulus I We choose the stimulus by maximizing the conditional mutual information between the response and θ. Neglecting higher order terms, we just need to maximize:

26. Derivation of Choosing the Stimulus II So we just need to minimize Therefore we need to maximize

27. Maximization We maximize the above subject to a power constraint by breaking it up into an inner and outer problem. To maximize this expression, we express everything in terms of the eigenvectors of C t.. are the projection of the mean and stimulus onto the eigenvectors.

28. Maximization II We maximize the inner problem using lagrange multipliers: To find the global maximum we perform a 1-d search over λ 1, for each λ 1 we compute F(y(λ 1 )) and then choose the stimulus which maximizes F(y(λ 1 ))

29. Posterior Update: Math