Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November

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Population CodingAlexandre PougetOkinawa
Computational Neuroscience CourseOkinawa, Japan
November 2004
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Outline

• Definition
• The encoding process
• Decoding population codes
• Quantifying information Shannon and Fisher
  information
• Basis functions and optimal computation

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Outline

• Definition
• The encoding process
• Decoding population codes
• Quantifying information Shannon and Fisher
  information
• Basis functions and optimal computation

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Receptive field
s Direction of motion
Response
Stimulus
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Receptive field
s Direction of motion
Trial 1
Trial 2
Trial 3
Trial 4
Stimulus
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(No Transcript)
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Tuning curves and noise

• Example of tuning curves
• Retinal location, orientation, depth, color, eye
  movements, arm movements, numbers etc.

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Population Codes
100
100
s?
80
80
60
60
Activity
Activity
40
40
20
20
0
0
-100
0
100
-100
0
100
Direction (deg)
Preferred Direction (deg)
Tuning Curves
Pattern of activity (r)
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Bayesian approach

• We want to recover P(sr). Using Bayes theorem,
  we have

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Bayesian approach

• Bayes rule

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Bayesian approach

• We want to recover P(sr). Using Bayes theorem,
  we have

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Bayesian approach

• If we are to do any type of computation with
  population codes, we need a probabilistic model
  of how the activity are generated, p(rs), i.e.,
  we need to model the encoding process.

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Activity distribution
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Tuning curves and noise

• The activity ( of spikes per second) of a
  neuron can be written as
• where fi(q) is the mean activity of the neuron
  (the tuning curve) and ni is a noise with zero
  mean. If the noise is gaussian, then

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Probability distributions and activity

• The noise is a random variable which can be
  characterized by a conditional probability
  distribution, P(nis).
• The distributions of the activity P(ris). and
  the noise differ only by their means (Eni0,
  Erifi(s)).

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Examples of activity distributions

• Gaussian noise with fixed variance
• Gaussian noise with variance equal to the mean

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• Poisson distribution
• The variance of a Poisson distribution is equal
  to its mean.

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Comparison of Poisson vs Gaussian noise with
variance equal to the mean
0.09
0.08
0.07
0.06
0.05
Probability
0.04
0.03
0.02
0.01
0
0
20
40
60
80
100
120
140
Activity (spike/sec)
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Population of neurons

• Gaussian noise with fixed variance

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Population of neurons

• Gaussian noise with arbitrary covariance matrix
  S

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Outline

• Definition
• The encoding process
• Decoding population codes
• Quantifying information Shannon and Fisher
  information
• Basis functions and optimal computation

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Population Codes
100
100
s?
80
80
60
60
Activity
Activity
40
40
20
20
0
0
-100
0
100
-100
0
100
Direction (deg)
Preferred Direction (deg)
Tuning Curves
Pattern of activity (r)
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Nature of the problem
In response to a stimulus with unknown value s,
you observe a pattern of activity r. What can you
say about s given r?
Bayesian approach recover p(sr) (the posterior
distribution)
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Estimation Theory
Activity vector r
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Estimation Theory
r
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Estimation theory

• A common measure of decoding performance is the
  mean square error between the estimate and the
  true value
• This error can be decomposed as

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Efficient Estimators

• The smallest achievable variance for an unbiased
  estimator is known as the Cramer-Rao bound, sCR2.
• An efficient estimator is such that
• In general

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Estimation Theory
Activity vector r
Examples of decoders
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Voting Methods

• Optimal Linear Estimator

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Linear Estimators
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Linear Estimators
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Linear Estimators
X and Y must be zero mean
Trust cells that have small variances and large
covariances
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Voting Methods

• Optimal Linear Estimator

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Voting Methods

• Optimal Linear Estimator
• Center of Mass

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Center of Mass/Population Vector

• The center of mass is optimal (unbiased and
  efficient) iff The tuning curves are gaussian
  with a zero baseline, uniformly distributed and
  the noise follows a Poisson distribution
• In general, the center of mass has a large bias
  and a large variance

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Voting Methods

• Optimal Linear Estimator
• Center of Mass
• Population Vector

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Population Vector
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Voting Methods

• Optimal Linear Estimator
• Center of Mass
• Population Vector

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Population Vector
Typically, Population vector is not the optimal
linear estimator.
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Population Vector
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Population Vector

• Population vector is optimal iff The tuning
  curves are cosine, uniformly distributed and the
  noise follows a normal distribution with fixed
  variance
• In most cases, the population vector is biased
  and has a large variance

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Maximum Likelihood

• The maximum likelihood estimate is the value of
  s maximizing the likelihood P(rs). Therefore, we
  seek such that
• is unbiased and efficient.

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Maximum Likelihood
100
80
60
Activity
40
20
0
-100
0
100
Direction (deg)
Tuning Curves
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Maximum Likelihood
Template
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Maximum Likelihood
Activity
0
Template
Preferred Direction (deg)
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ML and template matching

• Maximum likelihood is a template matching
  procedure BUT the metric used is not always the
  Euclidean distance, it depends on the noise
  distribution.

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Maximum Likelihood

• The maximum likelihood estimate is the value of
  s maximizing the likelihood P(rs). Therefore, we
  seek such that

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Maximum Likelihood

• If the noise is gaussian and independent
• Therefore
• and the estimate is given by

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Maximum Likelihood
Activity
0
Preferred Direction (deg)
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Gaussian noise with variance proportional to the
mean

• If the noise is gaussian with variance
  proportional to the mean, the distance being
  minimized changes to

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Bayesian approach

• We want to recover P(sr). Using Bayes theorem,
  we have

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Bayesian approach

• The prior P(s) correspond to any knowledge we may
  have about s before we get to see any activity.
• Note the Bayesian approach does not reduce to
  the use of a prior

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Bayesian approach
Once we have P(sr), we can proceed in two
different ways. We can keep this distribution for
Bayesian inferences (as we would do in a Bayesian
network) or we can make a decision about s. For
instance, we can estimate s as being the value
that maximizes P(sr), This is known as the
maximum a posteriori estimate (MAP). For flat
prior, ML and MAP are equivalent.
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Bayesian approach
Limitations the Bayesian approach and ML require
a lot of data (estimating P(rs) requires at
least n(n-1)(n-1)/2 parameters)
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Bayesian approach
Limitations the Bayesian approach and ML require
a lot of data (estimating P(rs) requires at
least O(n2) parameters, n100, n210000)
Alternative estimate P(sr) directly using a
nonlinear estimate (if s is a scalar and P(sr)
is gaussian, we only need to estimate two
parameters!).
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Outline

• Definition
• The encoding process
• Decoding population codes
• Quantifying information Shannon and Fisher
  information
• Basis functions and optimal computation

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Fisher Information
Fisher information is defined as and
it is equal to
where P(rs) is the distribution of the neuronal
noise.
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Fisher Information
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Fisher Information

• For one neuron with Poisson noise
• For n independent neurons

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Fisher Information and Tuning Curves

• Fisher information is maximum where the slope is
  maximum
• This is consistent with adaptation experiments
• Fisher information adds up for independent
  neurons (unlike Shannon information!)

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Fisher Information

• In 1D, Fisher information decreases as the width
  of the tuning curves increases
• In 2D, Fisher information does not depend on the
  width of the tuning curve
• In 3D and above, Fisher information increases as
  the width of the tuning curves increases
• WARNING this is true for independent gaussian
  noise.

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Ideal observer

• The discrimination threshold of an ideal
  observer, ds, is proportional to the variance of
  the Cramer-Rao Bound.
• In other words, an efficient estimator is an
  ideal observer.

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• An ideal observer is an observer that can recover
  all the Fisher information in the activity (easy
  link between Fisher information and behavioral
  performance)
• If all distributions are gaussian, Fisher
  information is the same as Shannon information.

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Population Vector and Fisher Information
1/Fisher information
Population vector
CR bound
Population vector should NEVER be used to
estimate information content!!!! The indirect
method is prone to severe problems
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Outline

• Definition
• The encoding process
• Decoding population codes
• Quantifying information Shannon and Fisher
  information
• Basis functions and optimal computation

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• So far we have only talked about decoding from
  the point of view of an experimentalists.
• How is that relevant to neural computation?
  Neurons do not decode, they compute!
• What kind of computation can we perform with
  population codes?

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Computing functions

• If we denote the sensory input as a vector S and
  the motor command as M, a sensorimotor
  transformation is a mapping from S to M
• Mf(S)
• Where f is typically a nonlinear function

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Example

• 2 Joint arm

q2
y
q1
x
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Basis functions

• Most nonlinear functions can be approximated by
  linear combinations of basis functions
• Ex Fourier Transform
• Ex Radial Basis Functions

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Basis Functions
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Basis Functions

• A basis functions decomposition is like a three
  layer network. The intermediate units are the
  basis functions

y
X
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Basis Functions

• Networks with sigmoidal units are also basis
  function networks

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A
B
Z
2
3
Z
X
Y
Y
X
C
D
Z
Y
X
Linear Combination
Z
Basis Function Layer
Y
X
Y
X
Y
X
Y
X
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Basis Functions

• Decompose the computation of Mf(S,P) in two
  stages
• Compute basis functions of S and P
• Combine the basis functions linearly to obtain
  the motor command

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Basis Functions

• Note that M can be a population code, e.g. the
  components of that vector could correspond to
  units with bell-shaped tuning curves.

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Example Computing the head-centered location of
an object from its retinal location
Head position

Gaze
Fixation point
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Basis Functions
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Basis Function Units
Ri
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Basis Function Units
Ri
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Visual receptive fields in VIP are partially
shifting with the eye
Fixation point
Head-centered location
Retinotopic location
Screen
(Duhamel, Bremmer, BenHamed and Graf, 1997)
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Summary

• Definition
• Population codes involve the concerted activity
  of large populations of neurons
• The encoding process
• The activity of the neurons can be formalized as
  being the sum of a tuning curve plus noise

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Summary

• Decoding population codes
• Optimal decoding can be performed with Maximum
  Likelihood estimation (xML) or Bayesian
  inferences (p(sr))
• Quantifying information Fisher information
• Fisher information provides an upper bound on the
  amount of information available in a population
  code

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Summary

• Basis functions and optimal computation
• Population codes can be used to perform arbitrary
  nonlinear transformations because they provide
  basis sets.

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Where do we go from here?

• Computation and Bayesian inferences
• Knill, Koerding, Todorov Experimental evidence
  for Bayesian inferences in humans.
• Shadlen Neural basis of Bayesian inferences
• Latham, Olshausen Bayesian inferences in
  recurrent neural nets

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Where do we go from here?

• Other encoding hypothesis probabilistic
  interpretations
• Zemel, Rao