information-theoretic stimulus design for neuroscience...

Information-theoretic stimulus design

for neurophysiology & psychophysics

Christopher DiMattina, PhD

Assistant Professor of Psychology

Florida Gulf Coast University

Optimal experimental design

Part 1

2 OCNS 2014 Info Theory Workshop - Quebec City, CA

Consider a simple problem

• Estimate the slope of a line through the origin from noisy

input-output data {(xi, yi)}i = 1:N

yi = a∙xi + (noise)i

x in [-2 , 2]


System identification

a∙x x + y

noise

system

observations inputs


Standard approach

• Choose N inputs xi uniformly from [-2, 2] , observe yi

• Obtain maximum a posteriori (MAP) estimate for slope

parameter a


Accuracy & time tradeoff

• More data confidence intervals get tighter

• Experiment takes longer


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

high accuracy = low posterior entropy

Claude Shannon


Solution

• For linear regression with Gaussian noise, posterior is a Gaussian with µa = y/x, σ2

a = (σn/x) 2

• entropy = C + ln (σa) = C + ln |σn/x|

Posterior entropy is minimized

at the endpoints

singularity at x = 0


Put all stimuli at the endpoints

less entropy

more entropy


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?


Sensory neuroscience

Part 2


Sensory neuroscience

• Psychophysics

• Neurophysiology (single-unit, fMRI)

F(x,θ)

x inputs

y observations

goal: estimate θ

Reviews: Wu, David & Gallant (2006), Sharpee (2014)


Psychometric functions

• F(x, θ) relates stimulus parameter(s) to probability correct

• Model parameters θ are slope and threshold

[from Palamedes website: http://www.palamedestoolbox.org/ ]


http://www.palamedestoolbox.org/

http://www.palamedestoolbox.org/

Tuning curves

[from David Heeger’s website: http://www.cns.nyu.edu/~david/ ]

• F(x, θ) relates stimulus parameter(s) to neural response

• Model parameters θ are peak and tuning width


http://www.cns.nyu.edu/~david/

http://www.cns.nyu.edu/~david/

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)


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)


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)


Firing rate optimization

(Yamane et al., 2008)

Perhaps the most popular application of adaptive

stimulus design is firing rate optimization


Model estimation & comparison

• Active data collection can help to more efficiently estimate

and compare models

(DiMattina, 2009)


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


Old news in Psychophysics

• Staircase method (Cornsweet, 1962)

• PSI Method – Adaptive information-theoretic approach

(Kontsevich & Tyler, 1999) – 230 citations and counting!


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)


Generalized Linear Model

• Estimate receptive fields with fewer trials

Lewi et al. (2009)


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


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)


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)


OED for nonlinear models

Part 3


Work at Johns Hopkins

• Goal was to develop methods for on-line estimation and

comparison of generic nonlinear neural models


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)


Representing the posterior

• Evolving posterior pn(θ) is a Gaussian mixture

• After each observation, we update each peak recursively using Extended Kalman Filter (EKF) equations (Alspach & Sorenson, 1972)


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)


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)



Estimating network structure

• Nonlinear network model (nearly 300 parameters total)

• Want to recover network structure using input-output data



Estimating network structure

• Much more effective at recovering input filters and

network structure than IID (white-noise) stimuli



Multiple models

• The correct nonlinear model is often unknown

• Might want to estimate several models and generate

critical stimuli to compare models



Two phase experiment



Comparison criterion

• Bayes Information Criterion (Swartz, 1978; Bishop, 2006)

ln 𝑃(𝐷) = ln 𝑝 𝐷 𝜃𝑀𝐴𝑃 −𝑀

2ln𝑁

Other good criteria: Minimize model space entropy (Cavagnaro et al. 2010)

rewards good fit penalizes model complexity


Optimal stimuli for model comparison

• 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



Modeling nonlinear neurons

Part 4


Collaborative effort

• Wanted to test this approach in experiments

• Collaborated with Eric Young, William Tam and Eyal Dekel

William Tam Chris DiMattina

Eric Young Kechen Zhang

Eyal Dekel


Test bed

• Inferior colliculus of the awake marmoset monkey


Stimuli

• Wide-band, steady-state acoustic spectra

Yu & Young (2000)

(Wolfe et al., 2012)


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


Experimental set-up

Tam et al. (2011)


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


Could characterize nonlinear neurons

Tam et al. (2011)


More neurons

Tam et al. (2011)


Predicting effective, ineffective stimuli

Tam et al. (2011)


Comparing models

Tam et al. (2011)


Largest and smallest difference

Tam et al. (2011)


Cumulative difference

Tam et al. (2011)


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


High-dimensional

psychophysics

Part 5


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


Breaks down in higher dimensions


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 high-dimensional psychometric models


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


Future goals

Part 6


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)


Psychophysical studies

• How do subjects combine information from multiple neurons

responding to a stimulus to make perceptual decisions?

(DiMattina, Fox & Lewicki 2012)


Software toolbox

• MATLAB toolbox containing various methods for optimal

experimental design for psychophysics and neuroscience


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


Thank You

• OCNS & Information theory workshop

• Alex Dimitrov

• Colleagues at Johns Hopkins

• Kechen Zhang, Eric Young, Eyal Dekel, William Tam

• Florida Gulf Coast University


information-theoretic stimulus design for neuroscience...

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