Real-time optimization of neurophysiology experiments

Jeremy Lewi1, Robert Butera1, Liam Paninski2

1 Department of Bioengineering, Georgia Institute of Technology, 2. Department of Statistics, Columbia University

Neural Encoding

The neural code: what is P(response | stimulus)

Main Question: how to estimate P(r|x) from (sparse) experimental data?

Curse of dimensionalityBoth stimuli and responses can be very high-dimensional

Stimuli:• Images• Sounds• Time -varying behavior

Responses:• observations from single or multiple simultaneously recorded

point processes

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

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})

Theory: info. max is better1. Info. max. is in general more efficient and never worse than

random sampling [Paninski 2005]

2. Gaussian approximations are asymptotically accurate

Computational challenges

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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

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

Neural Model: GLM

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The nonlinear stage is the exponential function; also ensures the log likelihood is a concave function of θ.

GLMComputationally tractable

1. log likelihood is concave

2. log likelihood is 1-dimensional

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Updating the Posterior1. 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

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Updating the Posterior3. Update is rank 1

4. Find the peak: Newton’s method in 1-d

5. Invert the Hessian: use the Woodbury Lemma: O(d2) time

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log prior log likelihood log posterior

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(d2) in practice

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Running Time1. Updating the

posterior O(d2)

d- dimensionality

2. Eigen decomposition

O(d2)

3. Choosing the stimulus

O(d)

Simulation SetupCompare: Random vs.

Information maximizing stimuli

Objective: learn parameters

A Gabor Receptive Field

• high dimensional• Info. Max converges to true receptive field• Converges faster than random• 25x33

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

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Non-stationary parameters

• θi follow Gaussian curve whose center moves randomly over time

• Assuming θ is constant overestimates certainty poor choices for optimal stimuli

Non-stationary parameters

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

References1. 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)

AcknowledgementsThis 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.

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Previous Work

System Identification1. 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 Encoding1. 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]

Derivation of Choosing the Stimulus I

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Neglecting higher order terms, we just need to maximize:

Derivation of Choosing the Stimulus II

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To find the global maximum we perform a 1-d search over λ1 , for each λ1 we compute F(y(λ1)) and then choose the

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Posterior Update: Math

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