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![Page 1: 1 / 41 Inference and Computation with Population Codes 13 November 2012 Inference and Computation with Population Codes Alexandre Pouget, Peter Dayan,](https://reader031.fdocuments.in/reader031/viewer/2022032708/56649e8b5503460f94b90bdf/html5/thumbnails/1.jpg)
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
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2 / 41Inference and Computation with Population Codes
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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
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3 / 41Inference and Computation with Population Codes
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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
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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
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5 / 41Inference and Computation with Population Codes
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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?
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6 / 41Inference and Computation with Population Codes
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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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7 / 41Inference and Computation with Population Codes
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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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8 / 41Inference and Computation with Population Codes
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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)
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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
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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
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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
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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
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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
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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
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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
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Computation with Population Code
Basis Function Computations
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Computation with Population Code
Basis Function Computations
– linear solution for homogeneous population codes
(mapping from one population code to another, ignoring noise )
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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
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Encoding Probability Distributions
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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.
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21 / 41Inference and Computation with Population Codes
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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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Motivation
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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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Psychophysical Evidence
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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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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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Encoding and Decoding Probability Distribu-tions
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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.
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Encoding and Decoding Probability Distribu-tions
Motivation : Fourier transform
-periodic, odd function ()
Encoding :
Decoding :
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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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Encoding and Decoding Probability Distribu-tions
Encoding schemes
Kernel – sine function :
Kernel – Gaussian : Gaussian kernel
Kernel – Gaussian, :
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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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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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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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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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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
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Computations Using Probabilistic Population Codes
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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.
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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)
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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
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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.