real-time optimization of neurophysiology experiments jeremy lewi 1, robert butera 1, liam paninski...
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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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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
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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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.
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)
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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• 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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We choose the stimulus by maximizing the conditional mutual information between the response and θ.
Neglecting higher order terms, we just need to maximize:
Derivation of Choosing the Stimulus II
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Therefore we need to maximize
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Maximization
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To maximize this expression, we express everything in terms of the eigenvectors of Ct..
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Maximization II
We maximize the inner problem using lagrange multipliers:
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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
stimulus which maximizes F(y(λ1))