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1 / 41Inference and Computation with Population Codes
13 November 2012
Inference and Computation with Population Codes
Alexandre Pouget, Peter Dayan, and Richard S. Zemel
Annual review of neuroscience 2003
Presenter : Sangwook Hahn, Jisu Kim
2 / 41Inference and Computation with Population Codes
13 November 2012
Outline
1.Introduction
2.The Standard Model ( First Part )
1. Coding and Decoding
2. Computation with Population Codes
3. Discussion of Standard Model
3.Encoding Probability Distributions ( Second Part
)
1. Motivation
2. Psychophysical Evidence
3. Encoding and Decoding Probability Distributions
4. Examples in Neurophysiology
5. Computations Using Probabilistic Population Codes
3 / 41Inference and Computation with Population Codes
13 November 2012
Introduction
Single aspects of the world –(induce)> activity in multiple
neurons
For example
– 1. Air current is occurred by predator of cricket
– 2. Determine the direction of an air current
– 3. Evade with other direction from predicted predator’s move
air cur-
rent
4 / 41Inference and Computation with Population Codes
13 November 2012
Introduction
Analyze the example at the view of neural activity
– 1. Air current is occurred by predator of cricket
– 2. Determine the direction of an air current
( i. population of neurons encode information about single
variable
ii. information decoded from population activity )
– 3. Evade with other direction from predicted predator’s move
air cur-
rent
5 / 41Inference and Computation with Population Codes
13 November 2012
Guiding Questions (At First Part)
Q1:
How do populations of neurons encode information about single
variables?
How this information can be decoded from the population activity?
How do neural populations realize function approximation?
Q2:
How population codes support nonlinear computations
over the information they represent?
6 / 41Inference and Computation with Population Codes
13 November 2012
The Standard Model – Coding
Cricket cercal system has hair cells (a) as primary sensory neurons
Normalized mean firing rates of 4 low-velocity interneurons
s is the direction of an air current (induced by predator)
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13 November 2012
The Standard Model – Encoding Model
Mean activity of cell a depends on s
– : maximum firing rate
– : preferred direction of cell a
Natural way of describing tuning curves
– proportional to the
threshold projection
of v onto
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13 November 2012
The Standard Model – Decoding
3 methods to decode homogeneous population codes
– 1. Population vector approach
– 2. Maximum likelihood decoding
– 3. Bayesian estimator
Population vector approach ( sum )
– : population vector
– : preferred direction
– : actual rates from the mean rates
– : approximation of wind direction (r is noisy rates)
9 / 41Inference and Computation with Population Codes
13 November 2012
The Standard Model – Decoding
Main problem of population vector method
– It is not sensitive to the noise process that generates
– However, it works quite well
– Estimation of wind direction to
within a few degrees is possible
only with 4 noisy neurons
10 / 41Inference and Computation with Population Codes
13 November 2012
The Standard Model – Decoding
Maximum likelihood decoding
– This estimator starts from the full probabilistic encoding
model
by taking into account the noise corrupting neurons activities
– A
– A
– If is high -> those s values are likely to the observed
activities
– If is low -> those s values are unlikely to the observed
activities
rms = root mean
square
deviation
11 / 41Inference and Computation with Population Codes
13 November 2012
The Standard Model – Decoding
Bayesian estimators
– Combine likelihood P[r|s] with any prior information about
stimulus s
to produce a posterior distribution P[s|r] :
– If prior distribution P[s] is flat, there is no specific prior
information of s
and this is renormalization version of likelihood
– Bayesian estimator does a little better
than maximum likelihood
and population vector
12 / 41Inference and Computation with Population Codes
13 November 2012
The Standard Model – Decoding
In homogenous population
– Bayesian & Maximum likelihood decoding >>> population vector
– ‘the greater the number of cells is ,
the greater the accuracy is’
since more cells can provide more information about stimulus
13 / 41Inference and Computation with Population Codes
13 November 2012
Computation with Population Code
Discrimination
– If there are and where is a small angle,
we can use Bayesian poesterior (P[s|r]) in order to discriminate
those
– It is also possible to perform discrimination based directly on
activities by computing a linear :
– : usually 0 for a homogeneous population code
– : Relative weight
14 / 41Inference and Computation with Population Codes
13 November 2012
Computation with Population Code
Noise Removal
– Maximum likelihood estimator is unclear
about its neurobiological relevance.
• 1. finding a single scalar value seems unreasonable
because population codes seem to be used throughout the
brain
• 2. while finding maximum likelihood value is difficult in
general
– Solution : utilizing recurrent connection within population
to make it behave like an autoassociative memory
• Autoassociative memories use nonlinear recurrent
interactions
to find the stored pattern that most closely matches a noisy
input
15 / 41Inference and Computation with Population Codes
13 November 2012
Computation with Population Code
Basis Function Computations
– Function approximation compute the output of functions
for the case of multiple stimulus dimensions.
– For example,
– sh : head-centered direction to a target
sr : eye-centered direction
se : position of eyes in the head
16 / 41Inference and Computation with Population Codes
13 November 2012
Computation with Population Code
Basis Function Computations
17 / 41Inference and Computation with Population Codes
13 November 2012
Computation with Population Code
Basis Function Computations
– linear solution for homogeneous population codes
(mapping from one population code to another, ignoring noise )
18 / 41Inference and Computation with Population Codes
13 November 2012
Guiding Questions (At First Part)
Q1:
How do populations of neurons encode information about single
variables?
-> p.6~7
How this information can be decoded from the population activity?
-> p.8~12
How do neural populations realize function approximation?
-> p.13~14
Q2:
How population codes support nonlinear computations
over the information they represent?
-> p.15~17
19 / 41Inference and Computation with Population Codes
13 November 2012
Encoding Probability Distributions
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13 November 2012
Motivation
The standard model has two main restrictions :
We only consider uncertainty coming from noisy neural activities.
(internal noise)
: Uncertainty is inherent, independent of internal noise.
We do not consider anything other than estimating the single value.
: Utilizing the full information contained in the posterior is crucial.
21 / 41Inference and Computation with Population Codes
13 November 2012
Motivation
“ill-posed problems” : images do not contain enough information.
The aperture problem.
: Images does not unambiguously specify the motion of the object.
Solution - probabilistic approach.
: perception is conceived as statistical inference giving rise to proba-
bility distributions over the values.
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13 November 2012
Motivation
23 / 41Inference and Computation with Population Codes
13 November 2012
Psychophysical Evidence
Perceived speed of a grating increases with contrast.
Nervous system seeks the posterior distribution of velocity given the
image sequence, obtained through Bayes rule:
High contrast -> likelihood function becomes narrow
-> likelihood dominates product
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13 November 2012
Psychophysical Evidence
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13 November 2012
Encoding and Decoding Probability Distribu-tions
Log-likelihood method :
The activity of a neuron tuned to prefer velocity v is viewed as re-
porting the log-likelihood function of the image given the motion
Provides a statistical interpretation, and decoding only involves the
simple operation of exponentiating to find the full likelihood.
Some schemes for computing require that the likelihood only have
one peak.
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13 November 2012
Encoding and Decoding Probability Distribu-tions
Gain encoding for Gaussian distributions :
Using Bayesian approach to decode a population pattern ->
Assuming independent noise in the response of neurons.
-> posterior distribution converges to Gaussian.
Gain of the population activity controls the standard deviation of the
posterior distribution.
Strong limitation : only viably work for simple Gaussians.
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13 November 2012
Encoding and Decoding Probability Distribu-tions
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13 November 2012
Encoding and Decoding Probability Distribu-tions
Convolution encoding :
Can deal with non-Gaussian distributions that cannot be character-
ized by a few parameters, such as their means and variances.
Represent the distribution using a convolution code, obtained by
convolving the distribution with a particular set of kernel functions.
29 / 41Inference and Computation with Population Codes
13 November 2012
Encoding and Decoding Probability Distribu-tions
Motivation : Fourier transform
-periodic, odd function ()
Encoding :
Decoding :
30 / 41Inference and Computation with Population Codes
13 November 2012
Encoding and Decoding Probability Distribu-tions
Use large neuronal population of neurons to encode any function by
devoting each neuron to the encoding of one particular coefficient.
The activity of neuron a is computed by taking the inner product be-
tween a kernel function assigned to that neuron and the function be-
ing encoded.
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13 November 2012
Encoding and Decoding Probability Distribu-tions
Encoding schemes
Kernel – sine function :
Kernel – Gaussian : Gaussian kernel
Kernel – Gaussian, :
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13 November 2012
Encoding and Decoding Probability Distribu-tions
Decoding scheme - Anderson’s approach
Activity if neuron a is considered to be a vote for a particular decod-
ing basis function .
Overall distribution decoded :
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13 November 2012
Encoding and Decoding Probability Distribu-tions
Decoding scheme - Zemel’s approach
Probabilistic approach : recover the most likely distribution over s,
Can be achieved using a nonlinear regression method such as the
Expectation-Maximization algorithm.
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13 November 2012
Examples in Neurophysiology
Uncertainty in 2-AFC (2-alternative forced choice)
: examples offer preliminary evidence that neurons represent proba-
bility distributions, or related quantities, such as log likelihood ratios.
There are also experiments supporting gain encoding, convolution
codes, and DDPC, respectively.
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13 November 2012
Computations Using Probabilistic Population Codes
Experiment by Ernst & Banks (2002) : judge the width of a bar
The optimal strategy : Recovering the posterior distribution over the
width w, given the image V and haptic H
Using Bayes rule :
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13 November 2012
Computations Using Probabilistic Population Codes
If we use convolution code for all distributions
– multiply all the population codes together term by term
– requires neurons that can multiply or sum : achievable neural
operation
If the probability distributions are encoded using the position and
gain of population codes
– Solution : Deneve et al. (2001)
– Some limitations
– Performs a Bayesian inference using noisy population codes
37 / 41Inference and Computation with Population Codes
13 November 2012
Computations Using Probabilistic Population Codes
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13 November 2012
Guiding Questions(At Second Part)
Q3: How may neural populations offer a rich representation of such
things as uncertainty in the aspects of the stimuli they represent?
# 21 ~ # 24
Probabilistic approach : perception is conceived as statistical infer-
ence giving rise to probability distributions over the values.
Hence stimuli of neural populations represents probability distribu-
tions, which gives information of uncertainty.
39 / 41Inference and Computation with Population Codes
13 November 2012
Guiding Questions(At Second Part)
Q4: How can populations of neurons represent probability
distributions? How can they perform Bayesian probabilistic
inference?
#25 ~ #31 (for first), #37 ~ #39 (for second)
Several schemes have been proposed for encoding probability
distributions in populations of neurons : Log-likelihood method, Gain
encoding for Gaussian distributions, Convolution encoding.
Bayesian probabilistic inference can be done by multiply all the
population codes (convolution encoding), or using noisy population
codes (gain encoding)
40 / 41Inference and Computation with Population Codes
13 November 2012
Guiding Questions(At Second Part)
Q5: How multiple aspects of the world are represented in single
populations? What computational advantages (or disadvantages)
such schemes have?
# 25 ~ # 28 (first)
Log-likelihood : likelihood
Gain encoding : mean and standard deviation
Convolution encoding : probability distribution
41 / 41Inference and Computation with Population Codes
13 November 2012
Guiding Questions(At Second Part)
Q5: How multiple aspects of the world are represented in single
populations? What computational advantages (or disadvantages)
such schemes have?
# 25 ~ # 28 (second)
Log-likelihood : decoding is simple, but some distribution limitation
Gain encoding : strong distribution limitation.
Convolution encoding : can work for complicated distribution.
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