Information-theoretic stimulus design for neurophysiology & psychophysics

Transcription

1 Information-theoretic stimulus design for neurophysiology & psychophysics Christopher DiMattina, PhD Assistant Professor of Psychology Florida Gulf Coast University

2 2 Optimal experimental design Part 1

3 3 Consider a simple problem Estimate the slope of a line through the origin from noisy input-output data {(x i, y i )} i = 1:N y i = a x i + (noise) i x in [-2, 2]

4 4 System identification system x a x + y inputs observations noise

5 5 Standard approach Choose N inputs x i uniformly from [-2, 2], observe y i Obtain maximum a posteriori (MAP) estimate for slope parameter a

6 6 Accuracy & time tradeoff More data confidence intervals get tighter Experiment takes longer

7 7 Efficient stimulus selection How can we efficiently choose our inputs x to get the most accurate estimates for a fixed number of observations? This question can be re-phrased using information theory Claude Shannon high accuracy = low posterior entropy

8 8 Solution For linear regression with Gaussian noise, posterior is a Gaussian with µ a = y/x, σ 2 a = (σ n /x) 2 entropy = C + ln (σ a ) = C + ln σ n /x singularity at x = 0 Posterior entropy is minimized at the endpoints

9 9 Put all stimuli at the endpoints less entropy more entropy

10 10 Optimal experimental design This simple example shows how optimal experimental design (OED) can greatly reduce the number of stimuli needed to estimate model parameters How can this be applied in sensory neuroscience?

11 11 Sensory neuroscience Part 2

12 12 Sensory neuroscience Psychophysics Neurophysiology (single-unit, fmri) F(x,θ) x y inputs observations goal: estimate θ Reviews: Wu, David & Gallant (2006), Sharpee (2014)

13 13 Psychometric functions F(x, θ) relates stimulus parameter(s) to probability correct Model parameters θ are slope and threshold [from Palamedes website: ]

14 14 Tuning curves F(x, θ) relates stimulus parameter(s) to neural response Model parameters θ are peak and tuning width [from David Heeger s website: ]

15 15 Neural models F(x, θ) relates stimulus parameters to neural responses Model parameters θ are weights, thresholds, etc (Simoncelli et al., 2004) (Riesenhuber & Poggio, 2000)

16 16 Non-adaptive stimulus generation Traditionally investigators attempt to identify models F(x, θ) using fixed stimulus ensembles This approach is non-adaptive (open-loop) Simoncelli et al. (2004)

17 17 Active data collection Recently, in sensory neuroscience there has been a great interest in closed-loop data collection (DiMattina & Zhang, 2013) Reviews: Benda et al. (2007), Paninski et al. (2007), DiMattina & Zhang (2013)

18 18 Firing rate optimization Perhaps the most popular application of adaptive stimulus design is firing rate optimization (Yamane et al., 2008)

19 19 Model estimation & comparison Active data collection can help to more efficiently estimate and compare models (DiMattina, 2009)

20 20 Old news in Statistics & Machine Learning Lindley (1956) first showed that information theory could be applied to compare experimental designs MacKay (1992) showed that training of neural networks could be speeded up with stimuli maximizing mutual info

21 21 Old news in Psychophysics Staircase method (Cornsweet, 1962) PSI Method Adaptive information-theoretic approach (Kontsevich & Tyler, 1999) 230 citations and counting!

22 22 News in Neuroscience Lewi, Butera & Paninski (2009) developed a fast implementation of information-theoretic stimulus design for the Generalized Linear Model (GLM) Used Laplace approximation of the posterior density Lewi et al. (2009)

23 23 Generalized Linear Model Estimate receptive fields with fewer trials Lewi et al. (2009)

24 24 Limitations of the GLM GLM is essentially a single-layer perceptron Cannot model many nonlinear neurons like those found in the auditory or higher visual systems Frank Rosenblatt

25 25 Nonlinear auditory neurons GLM cannot model non-monotonic rate-level tuning seen in auditory neurons Cannot model complex non-linear properties like harmonic combination sensitivity Kadia & Wang (2003)

26 26 Nonlinear visual neurons Neurons in IT can be modeled as combining inputs from subunits tuned to shape features One does not know the subunit parameters a hidden unit problem Brincat & Connor (2004)

27 27 OED for nonlinear models Part 3

28 28 Work at Johns Hopkins Goal was to develop methods for on-line estimation and comparison of generic nonlinear neural models

29 29 Neural networks A reasonable starting point because of their universal approximation properties and large body of work Method is applicable to arbitrary firing rate models F(x,θ)

30 30 Representing the posterior Evolving posterior p n (θ) is a Gaussian mixture After each observation, we update each peak recursively using Extended Kalman Filter (EKF) equations (Alspach & Sorenson, 1972)

31 31 Choosing the next stimulus We chose the peak with the most weight and found the best stimulus for reducing the entropy of that Gaussian Quite often most of probability mass was on only a few bumps, so this approach is reasonable DiMattina & Zhang (2011)

32 32 Not just a good idea For nonlinear models with hidden units, it may not be possible to recover the true model parameters with white noise stimuli (DiMattina & Zhang, 2010, 2011) DiMattina & Zhang (2011)

33 33 Estimating network structure Nonlinear network model (nearly 300 parameters total) Want to recover network structure using input-output data DiMattina & Zhang (2011)

34 34 Estimating network structure Much more effective at recovering input filters and network structure than IID (white-noise) stimuli DiMattina & Zhang (2011)

35 35 Multiple models The correct nonlinear model is often unknown Might want to estimate several models and generate critical stimuli to compare models DiMattina & Zhang (2011)

36 36 Two phase experiment DiMattina & Zhang (2011)

37 37 Comparison criterion Bayes Information Criterion (Swartz, 1978; Bishop, 2006) ln P(D) = ln p D θ MAP M 2 ln N rewards good fit penalizes model complexity Other good criteria: Minimize model space entropy (Cavagnaro et al. 2010)

38 38 Optimal stimuli for model comparison DiMattina & Zhang (2011) Both models fit data about equally well Stimuli optimized for increasing the expected BIC increment did a good job of discriminating the models IID stimuli and stimuli optimized for model estimation did poorly

39 39 Modeling nonlinear neurons Part 4

40 40 Collaborative effort Wanted to test this approach in experiments Collaborated with Eric Young, William Tam and Eyal Dekel Kechen Zhang Eric Young Chris DiMattina William Tam Eyal Dekel

41 41 Test bed Inferior colliculus of the awake marmoset monkey

42 42 Stimuli Wide-band, steady-state acoustic spectra Yu & Young (2000) (Wolfe et al., 2012)

43 43 Underlying circuitry There are theories of the underlying functional circuitry of its main input, the Dorsal Cochlear Nucleus (Young, 1998) Can use a model of this circuitry as a candidate model for the neurons in the IC, which have similar properties auditory nerve inputs

44 44 Experimental set-up Tam et al. (2011)

45 45 Facts Searched over a pre-computed set of stimuli (~ 6000) Auditory nerve model front-end (Bruce et al. 2003) Took only about 300 stimuli (~ 5 minutes) to estimate model parameters

46 46 Could characterize nonlinear neurons Tam et al. (2011)

47 47 More neurons Tam et al. (2011)

48 48 Predicting effective, ineffective stimuli Tam et al. (2011)

49 49 Comparing models Tam et al. (2011)

50 50 Largest and smallest difference Tam et al. (2011)

51 51 Cumulative difference Tam et al. (2011)

52 52 Conclusions Demonstrates that optimally designed stimuli may be effectively used in neurophysiology experiments to estimate models Very helpful for comparing nonlinear models Hope to extend implementation to more complex and generic receptive field models for vision science

53 53 High-dimensional psychophysics Part 5

54 54 Standard PSI Represent posterior density using a 2-D grid of particles, search a 1-D grid of stimuli to minimize expected entropy DiMattina & Zhang (2014), in preparation

55 55 Breaks down in higher dimensions

56 56 High-dimensional questions Many people in psychophysics are interested how observers combine multiple cues (Knill & Saunders 2003) How do we combine multiple cues to detect edges (DiMattina, Fox & Lewicki 2012)? We need methods for efficiently estimating highdimensional psychometric models

57 57 Faster implementation Applied to 2-D and 3-D examples of nonlinear cue combination All three implementations are tractable + give same results as Grid-Psi

58 58 Future goals Part 6

59 59 Future work Higher-dimensional models with multiple subunits For instance, complex cells integrate inputs from many Gabor-like subunits (Chen et al. 2007) (Chen et al. 2007)

60 60 Psychophysical studies How do subjects combine information from multiple neurons responding to a stimulus to make perceptual decisions? (DiMattina, Fox & Lewicki 2012)

61 61 Software toolbox MATLAB toolbox containing various methods for optimal experimental design for psychophysics and neuroscience

62 62 Book As adaptive stimulus generation methods are becoming more prevalent in brain and cognitive sciences, it may be time for a multi-method, multi-disciplinary edited volume

63 63 Thank You OCNS & Information theory workshop Alex Dimitrov Colleagues at Johns Hopkins Kechen Zhang, Eric Young, Eyal Dekel, William Tam Florida Gulf Coast University