Neural Implementations of Bayesian Inference

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Neural Implementations of Bayesian Inference
Alexandre Pouget Department of Brain and
Cognitive Sciences University of Rochester
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Outline

• Encoding probability distributions with spikes
• Bayesian inference with spikes Multisensory
  integration
• Bayesian inference with spikes Decision making
• Alternative schemes
• Maximum likelihood estimation

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Visuo-Tactile Integration
(Ernst and Banks, Nature, 2002)
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Visuo-Tactile Integration
Bimodal p(sVision,Touch)
ap(sVision) p(sTouch)
Probability
p(sVision)
S (Width)
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Main Issues

• How do cortical neurons represent probability
  distributions?
• How do they take products of distributions?
• How do we make optimal decisions? How do neurons
  collapse distributions onto maximum likelihood
  estimates?

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Main Issues

• And how do they do so given the high level of
  variability in neuronal responses in cortex?

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Poisson Variability in Cortex
The variability is Poisson-like p(rs) (rspike
counts) is bell shaped with variance proportional
to the mean (Fano factors within 0.3-1.8, Fano
factor for a Poisson process is 1)
Trial 1
Trial 2
Trial 3
Trial 4
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Probabilistic population code

• As an example, we consider a population of
  neurons with Gaussian tuning curves and
  independent Poisson variability.

rr1,r2,,rn
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Activity
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Activity (Spike count)
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Stimulus
Preferred stimulus
Population pattern of activity on a single trial
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Population codes

• Standard approach estimating

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Population vector
r
80
60
Activity (spike count)
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20
0
-45
0
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Preferred stimulus
Underlying assumption population codes encode
single values.
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Probabilistic population codes

• Alternative compute a posterior distribution,
  p(sr) from (Foldiak, 1993 Sanger 1996).

100
r
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60
Activity (spike count)
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20
0
-45
0
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Preferred stimulus
Variability in neural responses for a constant
stimulus Poisson-like
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Probabilistic population codes

• For independent Poisson noise product of experts
  

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For independent Poisson noise, There
fore, the gain encodes the certainty associated
with the encoded variable.
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Gain and variance

• For independent Poisson noise, we have

This is average of the width of the posterior
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Experimental Evidence

• Contrast

Anderson et al, Nature 2000
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Experimental Evidence

• Contrast
• Motion Coherency
• Retinal Eccentricity

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Outline

• Bayesian inference multisensory integration

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Inferences with probabilistic population codes
100
g1
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Vision
C1
Activity
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40
20
0
-45
0
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Preferred S
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80
C2
Activity
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g2
Touch
40
20
0
-45
0
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Preferred S
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100
g1
80
C1
Activity
60
40
20
100
gg1g2
0
-45
0
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80
Preferred S
Activity
60

40
20
100
0
80
-45
0
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C2
Activity
Preferred S
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g2
40
20
0
-45
0
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Preferred S
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Visuo-Tactile Integration
Bimodal p(sVision,Touch)
ap(sVision) p(sTouch)
Probability
p(sVision)
S (Width)
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Normalization

• Divisive normalization can be used to keep
  neurons within their firing range.

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Assumptions

• Neural noise independent Poisson noise
• Gaussian tuning curves
• Unimodal Gaussian probability distributions over
  the stimulus
• Is this more general?

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Bayesian decoder
100
r1
80
C1
Activity
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40
20
r1r2
100
0
-45
0
45
80
Preferred S
Activity
60

40
20
100
r2
0
80
-45
0
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C2
Activity
Preferred S
60
40
Bayesian decoder
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0
-45
0
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Preferred S
Bayesian decoder
Bayesian inference
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Variability requirements

• Exponential distributions

Covariance matrix of r
Derivative of the tuning curves
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Kernel h(s)
Covariance matrix of r
Derivative of the tuning curves
Covariance between r and s
Local optimal linear estimator!
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Covariance requirements

• This family includes any distribution in which
  the covariance matrix is proportional to the
  mean, regardless of the form of the correlations.
  
• Any exponential distribution with a fixed Fano
  factor works.

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Tuning curve requirements

• The tuning curve f(s) can take any shape.
  However, h(s) has to be the same in all
  populations. What if its not the same?

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Tuning curves Identical Gaussians
Activity
Preferred S
Cue 1
Cue 2
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Tuning Curves Gaussians with different widths
Cue 1
Cue 2
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Tuning curves Gaussians vs Sigmoids
40
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Activity
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10
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-50
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Preferred S
Cue 1
Cue 2
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Tuning curve requirements

• Let say r1 has gaussian tuning curves and r2 uses
  sigmoidal tuning curves. Then, the optimal
  combination is a linear combination.
• The matrix A exists if the tuning curves are
  basis sets.

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Distribution over s

• p(rs) does not have to be a normal distribution
  over s.

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Prior Distributions

• Prior are easily incorporated
• Prediction baseline activity in cortex (e.g.
  before the start of a trial) should encode the
  prior distribution
• There is evidence for this idea in LIP (Glimcher
  and Platt) and the superior Colliculus (Basso and
  Wurtz).

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Summary

• Linear combinations of PPCs are equivalent to
  optimal Bayesian inference when the variability
  follows an exponential distribution. This works
  for
• all covariance matrices that are proportional to
  the mean (Fixed Fano factor)
• any set of tuning curves that forms a basis set
• any probability distribution over s
• any prior distribution over s

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Integrate and fire neurons

• Can we get a similar result with realistic
  networks of spiking neurons, such as integrate
  and fire neurons?

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Integrate and fire neurons

• Output layer
• 1200 conductance-based integrate-and-fire
  neurons, 1000 excitatory, 200 inhibitory
• Lateral connections
• High Fano factors (0.3 to 1)
• Correlated activity
• Linear in rates

100
100
g1
Input near-Poisson correlated spike trains with
different gains and slightly different means
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Activity
Activity
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g2
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20
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0
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0
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Preferred S
Preferred S
Cue 1
Cue 2
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Test cue 1 alone
r1
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Activity
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40
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0
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Preferred S
100
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Activity
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Preferred S
Cue 1
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Test cue 2 alone
r2
100
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Activity
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40
20
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0
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Preferred S
100
80
Activity
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Preferred S
Cue 2
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Test cue1 and cue2 together
r3
100
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Activity
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40
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0
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Preferred S
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Activity
Activity
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Preferred S
Preferred S
Cue 1
Cue 2
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Compare the distributions
r3
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Activity
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How does p(r3s) compare to p(r1s)p(r2s)?
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Preferred S
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100
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Activity
Activity
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60
40
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20
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Preferred S
Preferred S
Cue 1
Cue 2
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p(r3s) versus p(r1s)p(r2s)
Cue 1
Activity
Preferred S
Cue 2
Activity
Preferred S
Identical tuning curves
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p(r3s) versus p(r1s)p(r2s)
Cue 1
Activity
Mean
Variance
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3
95
2.5
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Preferred S
2
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Variance of p(r3s)
Mean of p(r3s)
92
1.5
91
0.5
90
0
89
0
0.5
1
1.5
2
2.5
3
89
90
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Cue 2
Activity
Mean of p(r1s)p(r2s)
Variance of p(r1s)p(r2s)
Preferred S
Different tuning curves and different correlations
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p(r3s) versus p(r1s)p(r2s)
Cue 1
Activity
Preferred S
Cue 2
Activity
Preferred S
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Experimental prediction

• Multisensory neurons should be linear on average

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Experimental prediction

• The main results in the literature are nonlinear
  combinations (superadditivity)!

Wallace, Meredith, and Stein, J Neurophys 1998
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Experimental prediction

• The main results in the literature are nonlinear
  combinations (superadditivity)!
• In fact, nonlinearity is the criteria used to
  define multisensory areas in fMRI
• Are we already proven wrong?

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Experimental prediction
Perrault, Vaughan, Stein, and Wallace. J
Neurophys 2005
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Inference over time

• Can we generalize this approach to inference over
  time, and more generally time varying signals?

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Outline

• Bayesian inference decision making

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Binary Decision Making
Shadlen et al.
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Binary Decision Making

• The Bayesian strategy involves computing the
  posterior distribution given all activity
  patterns from MT up to the current time,
• Therefore, all we need to do is add the activity
  patterns over time.
• This predicts that decision neurons act like
  integrators

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Bayesian decoder
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Activity
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Preferred S
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LIP
Roitman Shadlen, 2002 J. Neurosci.
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Outline

• Alternative schemes

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Alternative schemes
Log likelihood ratio (Shadlen et al Deneve)
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Log Likelihood

• Race models and Bayesian approach

Temporal sum
over
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Differences between Log odds and PPCs

• With PPCs, LIP neurons do not compute the
  activity difference between MT neurons with
  opposite direction preferences
• PPCs and log odds turn products into sums but for
  log odds, sums are products regardless of the
  noise distribution. Not so for PPCs
• At the end of the integration, LIP encodes the
  posterior distribution over direction, i.e, LIP
  knows how much it can trust its choice

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Alternative schemes
Log likelihood ratio (Shadlen et al Deneve)

• Log probability
• (Barlow Rao Jazayeri, Movshon)

• Probability
• (Anastasio et al Simoncelli Hoyer and
  Hyvarynen Rao Koechlin et al)

• Convolution codes
• (Anderson Zemel, Dayan, Pouget)

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Alternative schemes
Si
Log likelihood ratio (Shadlen et al Deneve)
100
80
60

• Log probability
• (Barlow Rao Jazayeri, Movshon)

Activity
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20
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-90
0
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Stimulus

• Probability
• (Anastasio et al Simoncelli Hoyer and
  Hyvarynen Rao Koechlin et al)

• Convolution codes
• (Anderson Zemel, Dayan, Pouget)

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Alternative schemes
Si?
Si?
Log likelihood ratio (Shadlen et al Deneve)
100
80
60

• Log probability
• (Barlow Rao Jazayeri, Movshon)

Activity
40
20
0
-90
0
90
Stimulus

• Probability
• (Anastasio et al Simoncelli Hoyer and
  Hyvarynen Rao Koechlin et al)

• Convolution codes
• (Anderson Zemel, Dayan, Pouget)

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Alternative schemes
Si?
Log likelihood ratio (Shadlen et al Deneve)
100
80
60

• Log probability
• (Barlow Rao Jazayeri, Movshon)

Activity
40
20
0
-90
0
90
Stimulus

• Probability
• (Anastasio et al Simoncelli Hoyer and
  Hyvarynen Rao Koechlin et al)

• Convolution codes
• (Anderson Zemel, Dayan, Pouget)

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Alternative schemes
The convolution codes and the log likelihood fail
to account for contrast invariance
0.04
Log P(sr)
0.02
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Orientation (deg)
Contrast invariance
Prediction for Convolution codes
Prediction for Log Likelihood
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Outline

• Maximum likelihood estimation

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Decision Making
Superior Colliculus
LIP
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Maximum Likelihood
Activity
0
Preferred Direction (deg)
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Neural implementation

• Attractor networks

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Optimal decision making
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Superior Colliculus
Preferred saccade direction
LIP
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Preferred saccade direction
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Nonlinear Networks

• Networks in which the activity at time t1 is a
  nonlinear function of the activity at the
  previous time step.

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Line Attractor Networks

• Attractor network with population code
• Periodic variable
• Translation invariant weights

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Line Attractor Networks

• Fixed point when

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Line Attractor Networks

• Computing the weights

Desired profile
Desired profile over u
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Line Attractor Networks

• The problem with the previous approach is that
  the weights tend to oscillate. Instead, we
  minimize
• The solution is

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Weight Pattern
5
4
Amplitude
2
0
-2
0
Difference in preferred orientation
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Optimal decision making
Superior Colliculus
LIP
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-100
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Preferred saccade direction
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Optimal decision making
A maximum likelihood estimate minimizes this
variance
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Superior Colliculus
Preferred saccade direction
LIP
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Preferred saccade direction
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Is the network an ML estimator?
Variances
Above Maximum Likelihood
Pop Vector
Network
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Optimality constaint
Eigenvector with eigenvalue equal to 0
Covariance matrix of r
Derivative of the tuning curves
Covariance between r and s
Local optimal linear estimator!
This network is effectively projecting its input
on the LOLE
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General Results

• Line attractor networks (stable smooth hills) are
  equivalent to maximum likelihood estimators.
• This result holds regardless of the exact form
  of the nonlinear activation function.

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Performance Over Time
6
5
4
3
Standard Deviation (deg)
2
1
0
-1 0 5 10
15
Time ( of iterations)
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Optimal decision making
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Sensorimotor transformation lecture
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f(S)
S
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S
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Kalman and Particle Filters